Physics of Semiconductors (1)Shingo Katsumoto
Institute for Solid State Physics, University of Tokyo
April 12, 2021
In the first half of this fiscal year (FY2021), I am assigned to the lecture on ”Physics of Semiconductors.”
It’s been a long time since I gave a lecture in one semester last time (8 years), and several new themes, which I
want to introduce, have appeared. I am not very good at giving lectures like a machine-gun and want to take acourse with a comparatively small amount of content. Though, maybe it’s not enough for motivated students
who want to learn a lot. Therefore, the notes follow the lecture and cover the advanced content that would
open the eyes for more expansive fields. I would also like to introduce references for those who want to expand
their studies.

Chapter 1 General Properties of Semiconductors
1

What characterizes semiconductors?

Semiconductors refer to a form of solid that is usually classified by electrical conduction. Metal have large densities of states around the Fermi levels (that is, the Fermi surfaces exist) and are good conductors while insulators
have their Fermi levels deep inside the wide energy gaps and interrupt electric currents. Semiconductors standsomewhere between them. They usually have comparatively narrow energy band-gaps, low but finite electric
conductance at high temperatures, become insulating with lowering the temperautres.
However, such a viewpoint is not always usuful nowadays. For example, a fine insulator with a large bandgapof 5.5 eV at room temperature, such as diamond, is also called a semiconductor, and devices are being madefrom it while materials with zero bandgap, such as graphene, are also important targets of researches in thesemicondutor field. Rather, it seems to me that the property of “structure sensitive” fits better into the recent
usage of the term “semiconductors.” This is a long-used expression, which indicates the property that theelectric conduction is sensitive to the ultra-small amount of impurities. Although, after the appearances ofheterojunctions, MOS structures, superlattices, nanostructures, etc., I think the same expression is applicableto the sensitiveness of the transport properties on such real space structures.
In most cases, the object of such structure sensitive transportation is the electric charge, but recently thespin current in which the magnetic moment is transported by spin has also become an important research
object. Research on spintronic devices is also active, and there is a possibility that some will eventually become
practical.
For the spin current, which is the flow of magnetic momentum, the non-magnetic metals whose spins arecanceled by the time-reversal symmetry, are like an empty space. It can be seen as a system similar to asemiconductor, which is a charge neutral space due to the charge cancellation of the nuclei and electrons. In
fact, the inside of the metal is almost equipotential under normal experimental conditions, but a spin Hall spincurrent may exist. In spintronics, these systems are also “structurally sensitive” and look like semiconductors
from the eye of semiconductor researchers. However, this is rather a unique view of myself, and usually,
semiconductors are defined as those that are structurally sensitive in electric conduction.

2

Crystal Structures

2.1

LatticeA solid classified into crystal commonly has a spatially periodic structure of basis, which is also a certain
structure of atoms. We represent such a state of matter as a lattice. “Spatially periodic structure” can beE1-1

represented as follows. An arbitrary point in a crystal with spatial coorinate r has an infinite number of
equivallent points r ′ , which is represented, in the case of three-dimensional lattice, with three constant vectors
ai (i = 1, 2, 3) and three integers li (i = 1, 2, 3) as∑r′ = r +li ai = r + R.
(1.1)
i=1,2,3Then the unit of the period is a certain set of atoms around r. We take an arbitrary point in the unit. Suchpoints form the lattice. We call such points as the lattice point.
The basis is the unit of the periodicity in the crystal and should be taken as to have the minimum number ofatoms. ai in eq.(1.1) are called primitive vectors, while R is called a lattice vector. The parallelpiped withai as the edges contains a single basis is called primitive cell, with which we can fill up the space without gap.
Primitive vectors often can be taken in multiple ways and usually taken as to make the symmetry of the latticehighest. A primitive cell is defined as a polygon which contains single basis and fills up the entire space without
gap. Then there are infinite ways to define a primitive cell other than the above mentioned paralledpiped.
When a block with multiple primitive cells is taken as the unit of period and the periodic structure has a higher
symmetry, the block is more convenient for the unit. We then consider a unit cell, which may consist of single
or multiple primitive cells.
As mentioned above, a crystal is composed of a unit structure and a lattice. The example of diamondstructure, which often appears in group-IV semiconductors, is illustrated in Fig.1.1. In Fig.1.1(a) the atomic
positions are indicated by middle-sized spheres, of which colors (black and white) indicate two diﬀerent atomic
sites in the crystal. The basis is composed of a black and a white atoms and a primitive cell can be taken as tocontain these two sites. A point in the primitive cell, e.g. the position of black atom, can be taken as the lattice
point. The consequent lattice is, as shown in (b), face centered cubic (fcc). Let ex,y,z be the unit vectors of
the Cartesian coodinate system, then the primitive vectors can be taken as
a1 =

a0
a0
a0
(ex + ey ), a2 = (ey + ez ), a3 = (ez + ex ).
2
2
2

(1.2)

The primitive cell of the parallelpiped spanned by the vectors in eq.(1.2) is drawn with solid lines in Fig.1.1(a).
On the other hand, the cubic drawn in the figure is often taken as a unit cell.
The lattices are classified by seven crystal system and additonal lattice point (no point, face-centered,body-centered, base-centered) into 14 species of Baravais lattice.

2.2

Bravais lattice

The number of crystal structures is huge, maybe infinite, if we count, e.g. diﬀerences in molecular structuresof organic crystals. On the other hand, the number of independent lattice structures is as small as 14 as shownin Fig.1.2. These 14 lattices are called three dimensional Bravais lattice.
The definition of the Brave lattice classification is based on the discussion of spatial symmetry. The spatial
symmetry of a manifold is defined by whether the manifold is invariant for the symmetry operations, such as
rotation, reflection, translation, etc. For detailed discussion see, e.g. Ref.[1, 2]. Here we briefly summarize how
we reach the 14 Bravais lattice.
We first classify the lattices with the relative lengths of primitive translatinal vectors a1 ，a2 ，a3 and theangles defined by two edges θ12 ，θ23 ，θ31 (see the right-down inset of Fig.1.2). And for the rotational symmetry

Figure 1.1: (a) Diamond structure.

a 0 The circles corespond to atomic poa2
z y
x
(a)

a3
a1
a0

(b)
E1-2

sitions, while the thin cylinders correspond to covalent bonds. There isa single atom species though the two
positions identified with colors, arediﬀerent. The sold lines indicate thea 0 primitive cell. (b) Face-centered cubic lattice of the diamond crystal of(a). a1−3 are the primitive vectors.

cubictetragonal

orthorhombic

monoclinic

triclinic

hexagonal

rhombohedral

primitive

bodycentereda3
facecentered

q 31

q 23
a2

q 12

a1

basecentered
Figure 1.2: Bravais lattices of three-dimension. The parameters for the classification are illustrated in theright-down space.
around a primitive translational vector, the angle π/2 is a spacial value and whether the angles θij are π/2 or
not is the other condition. These conditions leads us to the classification in Tab.1. This classification is called
crystal systems. In three dimension, possible crystal systems are seven species in in Tab.1 and on the firstcolumn in Fig.1.2.
In the classification of crystal system, the focus is on the symmetry in the positions of neighboring lattice
points. There are, however, some cases, in which we need to consider the relation between the next neighboring
lattice points. For example, we take two (simple) cubic lattices and put them so as to lattice points of one ofthem are placed to the center of cubic in the other lattice. In this case if we look at the neighboring relation,
the cubic symmetry seems to be lost but the second next ones are originally in cubic symmetry and the lattice
is still classified into cubic crystal system but contains an additonal lattice point at the center of cubic. The
lattice, which contains an additional lattice point at the center of simple cubic, is called body-centered cubic(bcc).
In this way, the additional lattice points to the simple crystal system is another condition for the classification.
The positions of such additional points are face-centerd, body-centered and base-centered. As a consequence,
we get 14 Bravais lattice shown in Fig.1.2.
Because we have ambiguity in taking “lattice”, the classification by Bravais lattice also has ambiguities. Fora simplest example, in an fcc lattice crystal, if we take the unit as a single face-centered cubic, then the lattice

cubic
tetragonal
orthorhombic
monoclinic
triclinic
hexagonal
rhombohedral(trigonal)

θ12
π/2
π/2
π/2
π/2
π/2
π/2
θ0

θ23
π/2
π/2

θ31
π/2

a1 ，a2 ，a3
a1 = a2 = a3
a1 = a2 ̸= a3
a1 ̸= a2 ̸= a3

π/2
2π/3
θ0

θ0 ̸= π/2

a1 = a2
a1 = a2 = a3

Table 1: Conditions for Bravais lattice classification in three dimension. The difinitions of the parameters areshown in the right-down panel in Fig.1.2.
E1-3

(a)

(b)

(c)

S
C

C
TCNQ

(d)

(e)

N

BEDT-TTF

Figure 1.3: Examples of crystal structures. The centers of spheres indicate the positions of nuclei. (a) bccstructure (Li, Na, Ba etc.), (b) fcc-structure (Al, Ni, Ag, Au etc.), (c) hcp-structure (Mg, Co, Cd etc.). These
three often appear in crystals of metals. We get (a) and (b) with simply putting atoms at the lattice points of bcc
and fcc lattices in Fig.1.2. (c) is classified into simple hexagonal lattice with taking two atomic positions whichhave a little darker color, as the basis. (d) is zinc-blende, which often appears in compound semiconductors. (e)is the structure of BEDT-TTF-TCNQ, which is one of organic semiconductors. It is diﬃcult to see the structurein the BEDT-TTF molecults in the main panel. The inset at the left-down shows the molecule structure in theview from the direction vertical to the molecular plane.
is simple cubic. Also, the rhombohedral lattice can viewed as a composite of three regular hexagonal prisms
with 120◦ rotations to each other.
In Bravais lattices, the abbrebiations fcc for face-centered cubic and bcc for body-centered cubic are frequently used. These lattices often appear in metal crystals and the primitive cells often consist of single atoms.
Then for the crystal structures, fcc and bcc are also often used. The hexagonal close-packed structure shown in
Fig.1.3(c) is also used as a crystal structure that appears approximately well in metal crystals, and the abbreviation hcp is used. The abbreviations for crystal structure are bcc, fcc, and hcp, but there is no hcp “lattice”
in the sense defined here. That is, in Fig.1.3, the three atomic positions in the middle of the structure are notequivalent to the peripheral atomic positions, and one atom cannot be taken as a unit structure 1 . The basic
structure can be taken as a combination of one atomic position in the upper and lower surfaces of Fig.1.3(b)and one atomic position in the middle, and the Brave lattice is a hexagonal lattice.
Bravais lattice is a classification that focuses on the symmetry of the lattice and is important in the discussionof symmetry, but the fact that the symmetry of the lattice and the symmetry of the crystal are not the samemeans. It is clear from the fact that the unit cell is regarded as a lattice “point” in the lattice and the details
in the unit cell are discarded. Let us take the diamond structure in Fig.1.1 again as an example. The positionof the lower left apex of the regular tetrahedron, which is a part of the ptimitive cell, is the basic cell position,and the spatial arrangement is fcc in Fig.1.2. The gray colored parallelpiped is a primitive cell containing two
atoms. In Fig.1.1, the diﬀerence of the two atomic positions is indicated by shades of colors. In the diamondstructure, the atom species is the same for the two. In the case these are alternatively occupied by diﬀerent
species of atoms, e.g. Ga and As, the crystal structure is caled zinc-blende(Fig.1.3(d)). That is, zinc-blende
structure also belongs to fcc Bravais lattice. On the other hand, two atoms in the primitive cell are of the same
1 On the web, many “hexagonal close-packed lattices” are searched, but in these explanations, the term “lattice structure” wasused for ”crystal structure”, and this combination was created. Also, “closest packing” mathematically means packing the spheres
most densely. Crystals that have a mathematically perfect hcp structure are not known in real atoms because of their anisotropy.

E1-4

kz
b2

b3

k z [001]
ky

X

L

kx

K

(a)

X

S K

G

S

W

kx
[100]
k-G

k
G/2

L

D

b1

G

(b)

(c)

Figure 1.4: (a) Reciprocal lattice of
fcc-lattice. The structure is classified
k y to bcc. (b) A way to “cut” the recipX [010]rocal space to obtain Brillouin zone.
Namely, cut at G/2 with the plane
perpendicular to G, where G is a reciprocal lattice vector. (c) The first
Brillouin zone obtained with the cuttings described in (b). The points indicated as Γ，X，L, · · · are the points
with high symmetries.

species in the diamond structure, though of the diﬀerent species in the zinc-blende structure. The former issymmetric for the inversion operation at the midpoint along the axis connecting the two positions while thelatter is asymmetric.
Another expample is in Fig.1.3(e), which shows an organic molecular crystal called (BEDT-TTF)2 TCNQ.
The atomic positions take a complecated form though the basis is single molecule and the lattice is triclinic.
It is easy to understand the basis has strong anisotropy due to the atomic structure of the molecule and thesymmetry of the crystal and that of the lattice is diﬀerent. The symmetries of crystals are classified by thesymmetry operations to 230 space groups.

2.3

Reciprocal lattice, Brillouin zone

Because the lattices of crystals have discrete translational symmetry, a potential U (r)(r is spatial coordinate)
caused by the lattice can be expanded in the Fourier seriese as∑
U (r) =
UG eiGr .
(1.3)
GFrom the periodicity in (1.1), U (r + R) = U (r). Then we obtain the condition for G as
G · R = 2πn(n : integer), ∵ eiG·R = 1.

(1.4)

The vectors G which fulfill the condition (1.4) are called reciprocal lattice vector. Just like the real-spacelattice, if we define primitive reciprocal lattice vectors as
ai · bj = 2πδij(i, j = 1, 2, 3),(1.5)

then we can write down bj (j = 1, 2, 3) with |A| ≡ a1 · (a2 × a3 ) as
2πa3 × a1
2πa1 × a2, b3 =
.
(1.6)
|A|
|A|
∑
A reciprocal lattice vector can be represented as G = i=1,2,3 hi bi (hi : integer). Generally a function withthe periodicity of a lattice can be Fourier expanded with the reciprocal lattice. It is legitimate to say a lattice
and the corresponding reciprocal lattice are in the relation of Fourier transformation 2 .
As we considered primitive cells in spatial lattices, we can define the units of periodic repetition in reciprocal
spaces. That is the Brillouin zone. A general way to obtain Brillouin zones is described in Fig.1.4(b). Let
us see how to obtain the first Brillouin zone around the origin. The procesure is simply to cut the reciprocal
space with planes containing the points G/2 and perpendicular to G, where G are the reciprocal lattice vectors
starting from the origin. The minimum space (polyhedron) around the origin surrounded by such surfaces isb1 =2πa2 × a3
,
|A|

b2 =

2 As an analytic expression, it is enough to remember that the Fourier transform of a regular series of δ-functions is, again, aregular series of δ-functions. This is a very general principle. An example is the optical frequency comb.

E1-5

the first Brillouin zone. This way of “cutting at G/2” will have meaning in considering the band structure,
which will be discussed in the next chapter.
The example of fcc-lattice is shown in Fig.1.4. First, the primitive reciprocal lattice vectors are obtained
from eq.(1.6). Then we find the reciprocal lattice for the fcc-lattice is bcc-lattice as shown in Fig.1.4(a). Next
we apply the method in (b). The reciprocal lattice vectors pointing the nearest neighbor reciprocal lattice
points are the primitive reciprocal lattice vectors ±b1 ，±b2 ，±b3 . There are equivalent eight planes which
cut the vectors vertically at the midpoints. The polyhedron covered with these planes is a regular octahedron.
However, the planes which cut the vectors pointing the next nearest neighbor reciprocal lattice points at the
midpoints, also cut the octahedron around the vertices. The procesure thus results in the first Brillouin zone
shown in Fig.1.4(c), where Γ，X，L, · · · indicate the points with high symmetry. The points are often used in
the display of band structure.

2.4

Crystals often used as semiconductors

Which materials shold be called “semiconductors” is a diﬃcult
II
III
IV
V
VI
problem, and some scholars propose the classification of “every
4 Be
5B
6C
7N
8O
material that is not metal”. In fact, diamond, which was a typical2s2
2s2 2p 2s2 2p2 2s2 2p3 2s2 2p4
insulator a while ago, has recently completely established itself
12 Mg
13 Al
14 Si
15 P
16 S
as a semiconductor. Here, let’s have a quick look at the simple
3s2
3s2 3p 3s2 3p2 3s2 3p3 3s2 3p4
and clear “crystals” of the spatial periodic structure, and those
30 Zn
31 Ga
32 Ge
33 As
34 Se
that are often used as semiconductors in the industry.
3d10
3d10
3d10
3d10
3d10
2
2
2
2
2
3
As specific examples, we take materials consist of compara4s
4s 4p 4s 4p 4s 4p 4s2 4p4
tively small numbers of elements from Group II to Group VI in
48 Cd
49 In
50 Sn
51 Sb
52 Te
the periodic table. In the right table, we show the part of the
4d10
4d10
4d10
4d10
4d10
2
2
2
2
2
3
periodic table under consideration with the electronic orbital oc5s
5s 5p 5s 5p 5s 5p 5s2 5p4
cupation. We can guess from the table that the semiconductors
composed of these elements takes similar lattice structures. Here we mainly introduce crystal structures.
2.4.1

Group IV semiconductors

Elementary semiconductors of C, Si, Ge take diamond structure (Bravais lattice is fcc). The bonds inthese crystals are dominated by covalent binding of sp3 hybrid orbitals. Silicon (Si) is of course the king ofsemiconductors in the industry. Tin (Sn) are metals in many phases but the form called α-Sn (gray tin) is asemiconductor with diamond crystal structure.
Creation of low dimensional electron system is a big charm of semiconductor physics, and it is also very
important in the semiconductor industry. In the case of Si, a metal-oxide-semiconductor (MOS) structure haslong been used to create two-dimensional electron systems. Since the oxide layer generally takes an amorphous
structure, the interfacial scattering probability of two-dimensional electrons is high, and it is diﬃcult to obtain
an electron system with high mobility. On the other hand, a two-dimensional electron system with high mobility
has been realized by a method of forming a bf heterojunction using mixed crystals of Si-Ge.
semiconductorCGe
Si

lattice constant Å
3.566835.646135.43102energy gap (RT eV)
5.47
0.66
1.12

E1-6

electron mass m0
0.25
1.64, 0.082
0.98, 0.19

hole mass
0.2
0.04, 0.28
0.16, 0.49

2.4.2

III-V compound semiconductors

Semiconductors made by combining group III elements and group
V elements on a one-to-one basis, group III and group IV atoms
occupy the lattice points of the diamond structure alternately, thatis, zinc blende structure. Al, Ga, In are often used as group III, andAs, P, Sb, etc. are often used as group V. Many kinds of compound
semiconductors are formed by these combinations, and more kinds ofsemiconductors can be synthesized by further mixing diﬀerent kindsof elements to form bf mixed crystal. Strictly speaking, these are no [001]Ga[010]
longer crystals because they have lost the spatially regular structure,but most of the concepts in the crystals work well by consideringAs
some blunting due to lattice disorder.
[100]
In reality, these compound semiconductors and mixed crystal
semiconductors are often synthesized by bf epitaxial growth, andfor this purpose, the crystal form and lattice constant should be similar between the heterogeneous semiconductors to be joined. This will be described later.
Many of III-V semiconductors have a direct gap at the origin of the reciprocal lattice space, Γ point, and are
therefore often used for optical devices. In addition, there are many combinations that can form high-quality
heterojunctions by epitaxial growth, and they are frequently used for devices for high-speed operation.
2.4.3

III-N compound semiconductors

In nitride semiconductors, whose applications have expanded rapidly forblue light emitting diodes, high-frequency, high-power devices, GaN, InN,
and AlN are currently the main research targets. They take hexagonal
Wurtzite crystal forms. They are usually grown by use of epitaxial growth
and annealed at high temperature to improve their quality. The figure in
the right shows the crystal structure of Wurzeit (for the case of ZnO).
2.4.4

O

Zn

II-VI compound semiconductors

Group II-VI semiconductors take various crystal forms such as zinc-blende,
wurtzite, and chalcopyrite. There are various compounds such as ZnO and
CdTe, and before GaN became the leading player in blue-color optical devices, the II-VI system was mainly studied as a candidate. ZnO is still
considered to be a material that threatens the GaN system if the device
characteristics and manufacturing method are improved because the material is easily available. The ZnO system tends to have a small structuresuch as nanotubes, and it is not easy to form it into a thin film device. On the other hand, its application asa nanostructure device is also attracting attentions. The Hg system is said to have a “negative bandgap” andhas become well known for its use in constructing topological materials.

2.5

Organic semiconductor materials

Organic thin films as semiconductors are attracting attention because they are lightweight, flexible, and inexpensive. Most organic solids are molecular solids in which intermolecular bonds are formed by van der Waals
forces. The qualities of organic semiconductors have been imporoved and the various concepts of semiconductor
physics are now applicable to organic ones. However, it is often more realistic to consider the electronic state in
the molecule and the solid state as an aggregate of them separately, reflecting the fact that they are molecularsolids. Especially in the case of macromolecules, on one hand Bloch electrons and bands in the molecule aregood approximation, on the other hand, the electrical conduction of the whole solid should be analyzed withthe theory for amorphous solids developed in the 1970s and 1980s.

3

Crystal growth

In order to utilise the structural sensitivity of semiconductors as functions to explore condensed matter physics,
to setup them as laboratories of quantum and many-body eﬀects and to use them as devices, we need to obtain,

E1-7

as the starting point, obtain crystals with very low concentrations of defects and impurities. For that, theoriginal materials with ultra high purity, higher than those in reagents by orders, should be prepared withcheap cost, huge amount, in very short time, and with very low environmental load. The crystal growth ishence a high field in the semiconductor industrial science. The physics, the main issue of this lecture, is notdirectly connected to that field but I would like to introduce some in a very short time.
Crystal growth methods of inorganic semiconductors can be classified to one for three-dimensional bulk
growth and another for two-dimensional growth on wafers of crystals cut from three dimensional ingot. The
latter is called epitaxial growth.

3.1

Growth of bulk crystals

Mining and refinement of source materials are very important processes before the crystal growth, and weneed to choose the best degree of material quality and refinement method considering the cost and the final
product. In the case of crystalline silicon, it is said that astonishing purity of 11N(99.999999999%) is required
for substrates of MOS-LSI 3 , which is called “semiconductor grade”.
On the other hand, a solar cell device generally has an area 10 orders of magnitude wider than that of
MOS-LSI, and the tolerance for leakage current per area also diﬀers by a few orders of magnitude. Hencefor them, the purity of 6N∼7N is enough under reduction of impurities that form non-radiative recombination
centers or pn characteristics degrading deep levels. Such wafers are called in “solar grade”.
In the latter, usually low quality Si called “metal grade” is used as a starting material. There have beenlong term seekings for purification method with low power consumption and some new progress has been madethough the world market is now dominated by companies which provides cheap wafers produced with traditional
method in 2013. Such situation is largely aﬀected by international aﬀairs or economic atmosphere. I am sorry
but must say that “basic researches” are aﬀected by such political situation in reality.
Bulk single crystals of inorganic semiconductors are usually obtained from gradual solidification by coolingfrom high temperature melts. This is comparatively easy for single element semiconductor Si or Ge. In thegrowths of compound semiconductors, mixed melts of multiple elements should be prepared and the diﬀerence
in melting point, vapour pressure and mutual solubility are the possible problems.
3.1.1

Czochralski process

In Czochralski (CZ) process, as illustrated in Fig.1.5 a thin seed crystal is put down to the surface of a meltfrom source materials, and a thick cylindrical crystal is pulled up. The grown crystal is formed in a cylindricalshape because the seed is rotated during the pulling up growth process. This is a representative method to
SeedSingle Si Crystal
Quartz Crucible
Water cooled chamber
Heat shield
Carbon heater
Graphite Crucible
Crucible support
Spill Tray
ElectrodeFigure 1.5: Schematic drawing of Czochralski process. Left: Three-dimensional schematic illustration. Right:
Cross-sectional illustration.
From http://people.seas.harvard.edu/ jones/es154/lectures/lecture 2/materials/materials.html

3 Here they are using a special definition of “purity”. I have experienced that such an ultra-pure Si ingot contains a considerable

amount of oxygen measured from low temperature magnetic susceptibility measurement with a SQUID magnetometer. 11N ishence the value on the ignorance of these impurities. Oxygen has little eﬀect on logic LSIs but is a problem in the application for
power devices.

E1-8obtain a dislocation free crystal of Si. The thin disk form popular for LSI wafer appears after slicing thecylindrical columnar shape.
Such a primitive CZ method cannot generally be applied
heaterto compound semiconductors due to large diﬀerence in theevaporation pressure. Actually CZ method is often adopted
in growth of III-V semiconductors GaAs, InP, GaP, etc. but
GaAs crystal
As
GaAs melt
not in the primitive form because the group V materials
seed crystal
have much higher vapour pressures than those of the groupquartz boat
quartz tube
III, resulting in the rapid escape of group V materials fromthe melts. Instead, Liquid Encapsulated Czochralski
jacket with heater movements(LEC) process, in which the melt of the sources is encapT
1250 º C
sulated with an inert liquid like B2 O3 .
610 º C

3.1.2

Boat methodx

Another popular method for bulk-growth of compound
Figure 1.6: Schematic illustration of a boat semiconductors is the one called “boat method”. The boat
method is further classified into horizontal Bridgeman (HB)
method (horizontal Bridgeman process).
method and temperature gradient freeze method. In theformer, a furnace with two temperature regions is moved
along a boat, in which the source materials are melt, andfrom one end the melt is freezed into a single crystal.
Figure 1.6 shows a schematic illustration of HB method for the case of GaAs. Initially a metallic solid As
is set to one end of a quartz tube, which also contains a quartz made boat. In the beginning Ga melt and aseed crystal are in the boat. The side of the As metal is heated to 610 ◦ C while the other side to 1250 ◦ C. As
sublimates severely above 600 ◦ C and gets into Ga melt forming GaAs melt. At 1250 ◦ C, GaAs is in melting
phase and at 610 ◦ C in solid phase. As the furnace moves to the right in the figure, a GaAs single crystal issolidified from the end of the seed crystal to the right.
3.1.3

Zone melting method

As mentioned in the footnote in pageE1-8, “ultrahighpurity” Si actually often contains high concentration
RF heating coilof oxygen, which mainly comes from the crucibles during the growth. For power MOS FET or other devices
GaAs substrate
in which such oxygen causes troubles, single crystals
are grown by floating zone melting (FZ) method. In
susceptor
the initial stage, a rod of polycrystal with a high purity is prepared in standing manner and a seed crystal is put on top of the rod. At first a zone of thepolycrystal rod from the top is melted e.g., by con- H 2 N 2
centration of infrared beam with confocal method orby rf loss heating. The melt in contact with the seed
TMG
AsH 3
crystal changes into single crystal and the melted zone
slowly goes down to form a single crystal rod. During
the process the melt does not touch any other materials and the high purity of polycrystal is kept. On the Figure 1.7: Extremely simplified schematic illustration
other hand, such big radiuses of grown rods as those of MOVPE (MOCVD) apparatus of GaAs deposition.
The “suscepter” absorbs the power of RF and gets heat.
in CZ method cannot be obtained.

3.2

Epitaxial growth of thin films

Epitaxial growth, in which thin crystal films are grown with deposition of materials onto crystal substrates,
is classified into liquid phase epitaxy (LPE), vapor phase epitaxy (VPE), and epitaxy in vacuum or invery low pressure gas. Here I will pick up metal organic vapor phase epitaxy (MOVPE) and molecular beam
epitaxy (MBE) from the number of epitaxial growth methods.

E1-9

3.2.1

Metal-organic vapor phase epitaxy

Also often called as Metal organic chemical vapour deposition (MOCVD). Often used for the growth of compound
semiconductors. Let us see the case of GaAs.
In general in epitaxial growth of a thin film crystal, component materials are carried onto the substratewith some carriers or with some other method, and react with the substrate surface to form single crystal films.
Therefore the keys for the growth are the surface states of the substrate, the way of carrying the materials, thedynamics of deposited atoms, etc. In the case of MOVPE, the sources are carried by hydrogen and nitrogengases. Ga is put on tri-methyl gallium ((CH)3 Ga, TMG), and As on arsine (AsH3 ). They are carried onto
the substrate and decomposed into atoms on the surface by heating. Then they are chemically bonded to thesurface atoms to form GaAs crystal films. Omitting all the intermediate chemical reactions and the initial andthe final states can be written as(CH3 )3 Ga+AsH3 −→ GaAs+3CH4 .
TMG and arsine have low vapor pressures and as shown in Fig.1.7, liquids of them are bubbled with hydrogento be vaporized. Hydrogen gas is deoxidizing atmosphere for GaAs surface. Thus flat and high quality films canbe grown though the actual chemical reaction is not so simple. Doping of impurities, growth of mixed crystals
are possible with preparation of materials. All of metal organic gases of group II or III, arsine or phosphine ofgroup V are explosive and at the same time nerve gases. They are extremely dangerous and should be treatedwith highest care and rigid safeguards.
3.2.2

Molecular beam epitaxy

Molecular beam epitaxy (MBE) is a representative growth method of ultra-thin semiconductor films. Characteristic features are: (1) deposition in ultra-high vacuum; (2) single crystalline substrates and various methods
for surface cleaning; (3) heating of substrates to activate the motions of deposited atoms; (4) stoichiometricdeposition of materials is not always required; (5) in situ characterization of grown films in various ways ispossible because the growth front is always on the surface to the vacuum.
Figure1.8(a) is a schematic
illustrationof an MBE magate valve
control panelload lock chamber
growth chamberchine, (b) shows a photographoutlet of liquid nitrogenload lockof a real machine. In orderchamberquadrupole
for keeping ultra-high vacuum
mass spectrometergate valve
in the growth chamber, (a)“pre-evacuation chamber(s)” isbeam flux monitor
substrate rotation
used for loading and unloadsubstrate holder
RHEED
ing of substrates. Molecular
RHEED screen
screenoutlet of liquid nitrogenmolecular beam cellelectron gun
growthbeam cells (Knudsen cells, Kchamber
shutter rotation
shutters
cells; Langmuir cells, L-cells),
which have sources of evaporation in them, are kept at
molecular beam cells
intermediate temperatures for(a)(b)
them to avoid adsorption of gas
Figure 1.8: (a) Schematic illustration of an MBE machine. (b) Photograph of molecules. While the growth,a real machine (RIBER S32).
the shrouds covering substrate,heating system and molecularbeam cells are cooled down with continuous flow of liquid nitrogen to adsorb outgassed molecules. When evaporations are going on, the source molecules are inside the chamber and the total gas pressure increases, whichmakes the quality of vacuum obscure. To monitor the quality, we need a partial pressure gauge for gas species.
That requirement, not 100 % is fulfilled by a mass spectrometer, with which partial pressure can be measured
as a function of the ratio of the molecular mass to the charge.
Substrates for growths are introduced into the preparation (pre-evacuation) chamber after surface cleaning
with chemical etching and protection of the cleaned surface with oxidation. The oxide film at the surface isevaporated simply by heating the substrate in ultra-high vacuum.
To confirm the evaporation and to see the growth mode during the growth, we need some in-situ monitor
of the surface state. For that purpose a conventional method is refractive high energy electron diﬀraction,
RHEED. In the RHEED technique, as illustrated in Fig.1.8(a), electron beam with 15∼30keV acceleration is
injected onto the surface with very shallow angle and the diﬀraction pattern of reflected beam is imaged on theillumination screen. The image reflects the atomic structure of the surface, that is, it is the pattern of reciprocal
E1-10

lattice. Because the incident beam goes onto the surface with very shallow angle, when the surface is a mirror,
the diﬀraction is close to two-dimensional, that is, the diﬀraction pattern is a set of vertical reciprocal lattice“rods”.The image on the screen is a slice of the reciprocal rods with a plane almost parallel to the rods. Actual
diﬀraction patterns of rods have some widths due to various reasons and in such a two-dimensional growth,
images like upper-left of Fig.1.9 are obtained.
The image in Fig.1.9 has a strong diﬀraction spot at upper-center. This is due to the simple mirror-likereflection from the surface (mirror spot) and the more flat is the surface, the higher the intensity is. After opening
the shutters molecular beams reach the substrate and the growth starts. Molecules or atoms migrate on thesurface of the substrate with thermal activation after adsorbation and hit the lattice points at last, formingstrong bonding to substrate crystals. This is one of the possible mechanisms for crystal growth and such a“state of growth” is called “growht mode”. The growth mode mentioned above is called layer-by-layer mode.
Brightness (arb.)

In the initial stage of layer-by-layermode, one atomic monolayer growth conGaAs RHEED intensity oscillation
tains a cycle from a flat surface through a
1.5
rough surface to a new flat surface. Sucha single cycle causes one period in inten1
sity oscillation of mirror spot. The oscillation hence makes it possible for us to
shutter open
monitor the growth of each atomic layer.
0.5
0
2
4
6
8
10
The oscillation damps in proceeding of theTime (s)growth due to some incoherency though inmany cases growth interruption recovers
the flatness for lowering the surface energyof roughness. These properties opens up a
Figure 1.9: Upper-left: RHEED image for two-dimensional MBE
way to “flat surface growth” of “migration
growth. The intense spot at upper-center is the mirror spot. The
enhanced epitaxy”, in which the intensity
right shows the oscillation of the intensity at the mirror spot as theof mirror spot is monitored and the shutfilm grows. The cartoons illustrate the surface states at indicatedters are controlled to keep the highest inpoints in the intensity oscillation of the mirror spot.
tensity in the oscillation.
With increasing substrate temperatures, generally the dominant growth mechanism changes into “step flow mode”, in which migrating atoms
on the surface attach to the edges of surface steps causing widening of terraces, that is, flow of steps. In thismode no oscillation occurs.

References[1] N. W. Ashcroft and N. D. Mermin, “Solid State Physics” Chapter 4 (Brooks/Cole Publishing, 1976).
[2] D F Johnston, Rep. Prog. Phys., 23, 66 (1960).
[3] D. K. Ferry, “Semiconductors: Bonds and bands” (IOP Publishing, 2013).

E1-11

Chapter 2 Band structure, eﬀective mass approximation
In solid-state physics, the term “band structure” refers to the energy dispersion relations of the crystal eigenstates in the reciprocal lattice space introduced in the previous chapter. The theme of this chapter is introduction of the concept and how to calculate it theoretically. In addition, we will introduce the eﬀective mass
approximation, which is indispensable for handling band electrons in a simple and clear view.

1Band electrons

In free space, the kinetic energy of an electron takes a continuous value from zero. On the other hand, thekinetic energy takes a discrete value in the bound state in the localized potential of the nucleus. There are twoviews on the energy eigenstates in the periodic potential. One is the perturbation to the state in free space,which creates a section (energy gap) where the eigenvalues do not exist, and the energy eigenvalues are cut tobands. The other is that the discrete level due to the localized potential spreads in a band due to the tunnel
between the adjacent sites. The former is called nealy free electron approximation (NFEA), and the latteris called tight-binding approximation (TBA).

1.1

Bloch theorem

It is needless to proove the Bloch theorem, which is very basics of the solid state physics. For the reference, thetheorem is described as follows. Energy eigenstates in a periodic potential are expressed in the real space (r)expression as
ψnk (r) = unk (r) exp(ik · r),(1.1)
where n is the band index, unk is a function with the lattice periodicity, i.e.
∀R ∈ {(lattice vector)}, unk (r) = unk (r + R).

(1.2)

Here k is the wavenumber.

1.2

Nearly free electron approximation (NFEA)

We write the equation for the eigenstates in a lattice potential as][ℏ2 2∇ + V (r) ψ(r) = Eψ(r),H ψ(r) = −2m0
where V (r) is the lattice potential.
From the periodicity of V (r), uk (r), they can be Fourier expanded as∑∑
V (r) =
VG eiG·r , uk (r) =CG eiG·r ,
G

(1.3)

(1.4)

G

where G are the reciprocal lattice vectors. With substituting (1.1) and (1.4) into the Schrödinger equation(1.3), we obtain][}∑∑ { ℏ22
VG−G′ CG′ ei(k+G)·r = 0.
(k + G) − E) CG +
2m0
′
G

G

Because each term in the sum of G should be zero, the following simultaneous equations for {CG } are obtained.
}]∑ [{ ℏ2(k + G)2 − E δGG′ + VG−G′ CG′ = 0.
(1.5)
2m0
′
G

The condition for eq.(1.5) to have non-trivial solutions is
[{ 2
}
]
ℏ
2
= 0.
(k + G) − E δGG′ + VG−G′
2m0
GG′

(1.6)

In NFEA, we consider the pertubation δVG−G′ to (V (r) = 0)
ψ(r) = eik·r , C0 = 1, CG = 0 (G ̸= 0), E =
E1-12

ℏ2 k2
.
2m0

(1.7)

As a result of pertubation, δCG is caused. In (1.5), the terms δV δC, δEδC are in the higher order to be ignored.
Then
ℏ2
2m0
−VG
[(k + G)2 − k2 ]δCG + VG = 0 ∴ δCG = 2.
2m0ℏ (k + G)2 − k2
However the approximation collapses at(k + G)2 − k2 = 0.

(1.8)

Therefore around the point (1.8), we approximate that only C0 and CG are non-zero. The we can write down
(1.6) as
ℏ2 2
k −E
V−G
2m0
= 0,
(1.9)
2
ℏ
2
VG
(k + G) − E
2m0
which gives the energy eigenstates as
E=

1 (0)
1
[E (k) + E (0) (k + G)] ±2
2√[E (0) (k) − E (0) (k + G)]2 + 4|VG |2 ,(1.10)

where E (0) (k) ≡ ℏ2 k2 /2m0 . The result indicates the appearance of the energy separation of ±VG (bandgap
or forbidden band). For a system with the lattice constant a, the condition (1.8) is 2a cos θ = nλ(n is an
integer, λ is the wavelength of electron). This is nothing but the Bragg condition for diﬀraction of waves. Thus
the result can be interpreted as the electron wave get a Bragg reflection from the lattice and the interference
between the waves creates a standing wave, which results in the bandgap.

1.3

Reduced zone expression

A Bloch function can be written as fowllows with G a reciprocal lattice vector asψnk (r) = unk (r)eik·r = unk (r)e−iG·r ei(k+G)·r .
Function v(r) ≡ unk (r)e−iG·r also has the periodicity v(r) = v(r + R), where R is lattice vectors. ψnk canthus be expressed as
ψnk (r) = ξn′ k+G (r)(1.11)
with another Bloch function ξn′ k . Namely, the expression in (1.1) has the arbitrariness of reciprocal lattice
vectors. In other words, when a wavefunction has a spatial modulation of lattice period, there is ambiguity
whether the modulation is included in the lattice periodic part u(r) or in the plane wave part eik·r .
On the other hand, the system represented by Schrödinger equation (1.3)has the time-reversal symmetry andE(k) = E(−k). The above two relations on E(k) leads to E(G + k) = E(G − k). That is, E(k) is symmetricto the zone boudaries.
The arbitrarity in (1.11) leads to the arbitrarity in the representation of E(k). As Fig.1.2(a), in extended
zone representation, E(k) is represented as a single-valued function of k while as in Fig.1.2(b), in reducedzone representation, representation of E(k) is folded into the first Brillouin zone.

E(k)
2V 0

E(k)
k
2V 0
p/a

k

(a)

(b)

p/a

E1-13

Figure 1.1: (a) In NFEA, a bandgapof (1.10) appears at k = G/2. (b)Blowup of the region around the
bandgap in figure (a). The cartoons
explain why the standing waves get
the energy gap

E(k)

-3p/a

-2p/a

-p/a

0

E(k)

p/a

2p/a

(a)

3p/a

k

-p/a

0

p/a

k

(b)

Figure 1.2: Two ways of expression for the energy bands of NFEA. (a) Extended zone expression. (b) Reducedzone expression.
distance : reciprocal lattice
number of points
0:(0,0,0)1√
3: (1,1,1)，(1,1,-1)，(1,-1,1)，· · ·82:(2,0,0)，(0,2,0)，(0,0,2)，(-2,0,0),·
·
·
6
√
8
:
(2,2,0)，(2,0,2)，(0,2,2)，(-2,2,0)
，
·
·
·
12
√
11 : (3,1,1)，(1,3,1)，(1,1,3)，(-3,1,1)，· · ·24......................................................................
Table 2: Classification of reciprocal lattice points with the distance from Γ-point (unit G0 ≡ 2π/a).
In Fig.1.2(b), the second and the third bands is obtained from cutting for the first Brillouin zone [−π/2, π/2]of the overlap of the extended zone representations from the two neighboring reciprocal lattice points k = ±2π/a.
This is a natural consequence that the reduced zone representation is possible from the arbitrariness of thereciprocal lattice vectors as in (1.11).

1.4

Empty lattice approximation

In NFEA, in the limit of V0 → 0, this is nothing but a free space and the energygap disappears, the dispersion
relation is simply paraboric. However, the free space is not a system in which the spatial periodicity of thelattice is lost, and the continuous translational symmetry also includes the periodic translational symmetry of
the lattice. Rather, it can be considered that the lattice of the empty primitive cell remains. In the empty
lattice approximation, we hence consider the reduced zone representation of the parabolic energy dispersion.
′
In Bloch function representation, the plane wave function eik ·r is separated into the lattice periodic part unk (r)ik·rand the crystal wavenumber part e
and apply reduced zone representaiton.
Let’s take an example with a three-dimensional crystal. Consider the reciprocal lattice and Brillouin zone
in Fig.1.4 in the case of fcc. First, to obtain the reduced zone representation, since the principle of reducedzone representation is indefiniteness of the reciprocal lattice vector as in (1.11), consider the reciprocal lattice offcc and the bcc lattice of Fig.1.4 (a), and draw parabola with the origin at each reciprocal lattice point. Then
we cut the diagrams with the first Brillouin zone shown in Fig.1.4(c). The reciprocal lattice points we need toconsider in this drawing are summarized in Tab.2. The farer from the origin, the higher the energy branch of
the parabora from the reciprocal lattice point.
The problem with a three-dimensional band structure expression is how to display it. It is not possible todraw multiple parabolas in a three-dimensional space. Usually we only draw the energy dispersion on some
representative lines in the reciprocal space. Fig.1.3 shows a way of drawing often used to display the band
structure. The energy dispersions on the lines which connect points with high symmetries are drawn. As in

E1-14

the figure the line goes along L→ Γ →X→K→ Γ. (a) shows the empty lattice approximation while (b) shows
the realistic band dispersion in Si calculated with empirical pseudo-potential approximation (we will see in thenext week). There is no bandgap in the empty lattice approximation naturally. On the other hand, we see clear
resemblance between them. When we go into realistic calculations with gaps, the diagram is usuful to seew
which branch corresponds to which reciprocal point. Furthermore, when a level repulsion causes energygap, we
need to consider symmetry of the lattice and the empty lattice approximation is also usuful for seeing that.
[-1,-1,-1][1,-1,-1][0,-2,0]

4[0,2,0][2,0,0][1,-1,-1]

3[1,1,-1][1,1,1][1,-1,1][2,0,0][1,1,1]

1

L

(a)

-2

G 8+
G7

+

X5

-4

5
4

-6

3

-8

Si

2
[0,0,0][0,0,0]

0

0

Z
10
9
8
7
6

2

[1,1,1]X5

2

E ( k ) (eV)[-1,1,1][2,0,0]E ( h 2 G 0 2 /2 m )4[-2,0,0]

1
-12

(3)
(1)

G

(2)
k

X K

(4)

X5

-10

[0,0,0]

G

L

(b)

G 6+
G

k

X K

G

Figure 1.3: (a) Empty lattice approximation of fcc-lattice. Three numbers in [· · · ] indicate corresponding origins
of parabolas. (b) Realistic band structure of Si calculated by empirical pseudo-potential method.

1.5

Tight binding approximation

In the next week we begin with tight-binding approximation.

E1-15

Physics of Semiconductors (2)Lecture note on
18th April (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

2.1.5 Tight-binding approximation
In the nearly free electron approximation (NFEA), the lattice potential causes interference of electron waves in freespace and opens “slits", namely energy gaps in the original continuous energy spectrum. On the other hand in the view oftight-binding approximation (TBA), the energy bands with finite widths are formed from originally discrete energy levelsin spatially localized potential at each site by electron tunneling between the different sites.

u( r+2R )u( r+R )u( r )u( r - R )V( r )f( r+2R )f( r+R )f( r )

f(r-R)

Fig. 2.4 Schematic diagram of TBA. The periodic potential (blue lines)
is the sum of atomic potentials represented by broken lines. In TBA, the linear combination of localized orbitals drawn by redlines should be in the formof Bloch function.

The concept is illustrated in Fig. 2.4. The Hamiltonian for single-site is written as
Ha = T̂ + u,

(2.12)

where T̂ , u are the kinetic energy and the site-localized potential respectively. Let Ri be the potision of site i, then thereal-space representation of Ha can be written as Ha (Ri ) = T̂ + u(r − Ri ). If we write the normalized orthogonal
eigenstates of Ha (0) as {ϕn }, thenHa (Ri )ϕn (r − Ri ) = ϵn ϕn (r − Ri ),(2.13)

where n is the level index of the discrete states. We write the periodic potential obtained by overlapping of u as V (r),the total Hamiltonian for such system illustrated in Fig. 2.4 is
H = T̂ + V (r).

(2.14)Let us consider the solution of the eigenequation of the above Hamiltonian. For simplicity we ignore the direct overlapp
integral of wavefuntions ϕn between the neighboring sites(⟨ϕn (r − Ri )|ϕn (r − Rj )⟩ = δij ). We also ignore the matrix
elements of H between the states of different n. Then the total wavefunction can be written in the linear combinationof ϕn Because all of the lattice sites are equivalent, each coefficient of the linear combination should be in the form of√c/ N where |c| = 1, c is constant for r. Furthermore, the linear combination should be written in the form of the Blochfunction. From the above requirements the eigenstate should be written in the following form.
[]
1 ∑ ik·Ri
eik·r ∑ −ik·(r−Ri )ψnk (r) = √eϕn (r − Ri ) = √eϕn (r − Ri ) .
N iNiSince the last part of [· · · ] has the periodicity of lattice, the form of (2.15) is in the Bloch form.

2-1(2.15)

E h( k )
2t

0

k

-p/a

p/a

-2t

Fig. 2.5 Band dispertion relation in onedimensional tight-binding approximation.
Inthe model the hopping integral is finite only for nearest neighbor and t, in which situation cosine-shaped
band structure is obtained. The insets illustrate thestanding wave amplitude, phase at the band bottom
and the top.

The expectation value of H (energy expectation value) with ψnk is⟨ψnk |H |ψnk ⟩ = N −1∑

eik·(Rj −Ri ) ⟨ϕn (r − Ri )|[T̂r + V (r)]|ϕn (r − Rj )⟩

i,j

=N

−1

∑

eik·(Rj −Ri ) ⟨ϕn (r − Ri )|[T̂r + u(r − Ri ) + V (r) − u(r − Ri )]|ϕn (r − Rj )⟩i,j= ϵn + N −1
= ϵn +∑∑

eik(Rj −Ri ) ⟨ϕn (r − Ri )|[V (r) − u(r − Ri ))]|ϕn (r − Rj )⟩i,jeik·Rj ⟨ϕn (r)|[V (r) − u(r))]|ϕn (r − Rj )⟩.

(2.16)j

From the second line to the third line, we have used the fact that T̂r + u(r − Ri ) is the local Hamiltonian of the site i andignored the overlapp integral. Then to the fourth line, because of the infinite integration over r is taken, we have shhifted
the origin to each Ri for the integration, which results in simply N -times of the result and the normalization is considered.
So if we write the difference between the lattice potential V (r) and the localized potential u(r) as v(r) ≡ V (r) − u(r),En (k) = ϵn + ⟨ϕn (r)|v(r)|ϕn (r)⟩ −∑eik·Rj tn (Rj ),(2.17)

Rj ̸=0where the hopping integral tn (Rj ) is defined as
tn (Rj ) ≡ −⟨ϕn (r)|v(r)|ϕn (r − Rj )⟩.

(2.18)Let us restrict ourselves to one-dimensional systems. For simplicity, we assume the hopping integral has a finite value
t only for the nearest neighbor sites. Then the sum in (2.17) is only for Rj = ±a, where a is the lattice constant. The
result gives

En (k) = ϵn − αn − t(eika + e−ika ) = ϵn − αn − 2t cos ka.

(2.19)

The cosine band in (2.19) is common for one-dimensional systems. αn ≡ −⟨ϕn (r)|v(r)|ϕn (r)⟩ is the on-site energy
shift due to the change from the localized potential to the crystal potential and called crystal field contribution.
Figure 2.5 shows the cosine band thus obtained. In the case of NFEA, unperturbed is the band bottom k = 0 and theperturbation is stronger with coming close to the edge of Brillouin zone. In the case of TBA, the starting point is the band
center. When t = 0 the band is flat, and with increasing t it broadens to ±2t.
At the band bottom, as we know from (2.19), the eigenstate is in the mode of standing wave. As illustrated in one of
the insets, the localized wavefunctions on the sites are synchronously summed up wih the same phase at the bottom. On
the other hand, they are summed up with opposite phase for neighboring site at the band top to be standing wave withhigher energy. These features are the same as those in a double well potenntial with a tunneling matrix element t between
them. The bottom corresponds to the bonding orbital while the top to the anti-bonding orbital.

2-2

2.2 Band structure measurement, calculation
The importance of the energy band structure, that is, the energy dispersion relation in the first Brillouin zone, to materials science and engineering cannot be overstated. The views from the two extremes in the previous section are usefulfor understanding the band concept and for obtaining a general physical perspective on the phenomena that occur there.
For quantitative analysis of experimental data or for selection of materials to design devices with desired characteristics,
precise band parameters are required.
In the band structure, to analyze the optical response and hot electron characteristics, the dispersion in a wide rangeof crystal wavenumbers is important. Even in the optical response, it is necessary to know the dispersion of the bandedge precisely for the light emission from the band edge, problems such as excitons, and electrical conduction to a low
electric field. Let’s call the former “global band structure" and the latter “band edge structure". Here are some typicalexperimental and theoretical methods to obtain for each. Appendix 2A introduces one of the ab-initio calculation methods
that calculate without directly relying on the measured values of the experiment.

2.2.1 Measurement of global band structure: angle-resolved photoemission spectroscopy
In recent years, the resolution of photoemission spectroscopy has
improved remarkably, and angle-resolved photoemission spectroscopy(ARPES) is now a means for directly obtaining a global band structure.
The right figure shows the concept of photoemission spectroscopy. Letν

the binding energy of electrons measured from the Fermi level EF , hν theenergy of incoming light, ϕ the work function of the specimen. Then in
the light absorption process, the energy conservation law
E ν = hν − ϕ − EB

Electron energy

E be the energy of photo-electron measured from the vacuume level, EB(2.20)

hnvacuum level

Emission
intensityEFholds. In the experiments, Ekin = ϕ + E ν is measured and the relation is
simply written as
EB = hν − Ekin ,(2.21)

Electron density of states

which gives the binding energy. In the above ordinary photoemission spectroscopy, because the photo-electrons arecollected independent of the angles, the total density of states is measured.
In the emission of photo-electron, the sum of the parallel component of crystal wavenumber k∥ and the recprocal lattice
vector G is conserved. Let ki be the wavenumber of the initial state, kf the wavenumber of electrons emitted into thevacuum, then(ki + G)∥ = kf∥ .
The energy conservation reads
Ei (ki ) + hν = Ef (ki ) =ℏ2 kf2+ ϕ.
2m0(2.22)(2.23)

If we know the workfunction ϕ, the dispersion relation of the final states in the crystal Ef (k), then we can obtain thedispersion relation Ei (k). Since the energy of the final state is high, the effect of lattice potential on it is small, hence

2-3

0
h 2 kv2
2m0

E

-2
-4

E(k)

V0
hn
0

-6
-8

ce
EF

-10

f
-12

kc

-14

kv

L

(a)

G

K

X

G

(b)

Fig. 2.6 (a) Illustration o the procesure to determine the dispersion relation in ARPES measurement. (b) Band
structure obtained for GaAs with the vertical emission method.

Ef (k) is often replaced with that of free electron. Care should be taken that the wavenumber is still a crystal wavenumber
and the energy conservation law (2.23) becomesEi (ki ) + hν =

ℏ2 kf2ℏ2 (ki + G)2+ V0 =
+ ϕ.
2m0
2m0

(2.24)

Here, still the potentail V0 for the zero-kinetic energy “free electron in the crystal" is not known. But V0 is also determinedfor various interpretations of experiments to be consistent.
As a simplest method among ARPES, a possible way to obtain the dispersion relation is to measure photo-electrons
emitted veratically to the surface. In this case, as k∥ = 0, the energy conservation gives
Ei (ki ) + hν =

ℏ2 |(ki + G)⊥ |2ℏ2 kf2+ V0 =
+ ϕ,
2m0
2m0
√

and from|(ki + G)⊥ | =

2m
ℏ2

( 2 2
)
ℏ kf
+ ϕ − V0 ,2m0(2.25)(2.26)

ki and then the dispersion can be obtained.

2.2.2 Global band structure calculation: empirical pseudo potential method
Time-independent part Schrödinger equaiton in a periodic potential V (r) = V (r + R)(R is an arbitrary lattice vector)(

is written as
H ψ(r) =)ℏ2 2−∇ + V (r) ψ(r) = Eψ(r).
2m(2.27)

And we write a solution of the above in the Bloch form with dropping the band index as
ψ(r) = eik·r uk (r).

(2.28)The lattice periodic part V (r), uk (r) can be written in the Fourier expansion with the reciprocal lattice vector G as
V (r) =∑

VG eiG·r , uk (r) =

G

∑
G

2-4

CG eiG·r .

(2.29)

Substituting (2.28) and (2.29) into (2.27), we obtain[]}∑∑ { ℏ22(k + G) − E) CG +VG−G′ CG′ ei(k+G)·r = 0.
2m
′
G

G

Because each term in the summation of G should be zero, we obtain simultaneous equations for {CG } as}]
∑ [{ ℏ2(k + G)2 − E δGG′ + VG−G′ CG′ = 0,2m′(2.30)

Gand the condition for the existence of non-trivial solution is
[{ 2
}
]
ℏ
2
(k + G) − E δGG′ + VG−G′
= 0.
2m
GG′

(2.31)

If we can perform perfect expansion of (2.30) in an actual crystal, the solution of the secular equation (2.31) gives theaccurate band structure E(k). Equation (2.31) tells that requirements for this calculation are the coefficient VG of Fourier
expansion of periodic lattice potential.
In pseudo potential method we calculate “effective" VG under the following concepts.
(1) Structures of valence band and conduction band below and above the Fermi level determine the properties of semiconductors. The outermost electrons of consisting atoms are forming these bands. The inner electrons are strongly boundaround nuclei and can be included into the periodic crystal potential. Hence we apply the above secular equation only forthe outermost electrons.
(2) (Characteristic for pseudo potential method) In the vicinity of nuclei, V (r) can be approximated with r being the¯
distance from the nucleons and Z being the atomic number as
V (r) = Ze/r,and the outermost electron wavefunction should have stronger space modulation in the amplitude. On the other hand,
far from the nucleus, the inner electrons (let the number be Zc ) screens the potential and the effective atomic number
decreases to Z ′ = Z − Zc . Furthermore if we take into account the electron-electron mutual interaction in, e.g., mean
field approximation, the screening gives faster decay of potential than r−1 resulting in much weaker spatial modulationof the wavefunction. As we have seen in the tight-binding approximation (Sec.2.4), the band structure is dominated withthe overlapping of wavefunctions in neighboring sites, that is, this weak potential part determines the band structure inpractice.
If we perform the Fourier expansion of V (r) itself, strong spatial modulation around the nuclei introduces lots of highfrequency components, which are nothing to do with the band structure. Such high frequency coefficients VG not only
introduce useless calculations but also make it difficult to solve the secular equation eq.(2.31).
The above consideration brings about the central concept of pseudo potential method. We look for a “pseudo potential"
which simplifies the wave function around the nucleus but reproduces the tailing part of the wavefunction. Then obtain
VG for this pseudo potential and solve the secular equation (2.31).
The simplest example can be as in the left figure

W p( r )0

rc

Wp (r) = 0 (r < rc ), Wp (r) = Z ′ e/r (r ≥ rc ).

r(2.32)Taking rc to an appropriate value, we can approximately reproduce

Z 'er

the tail of wavefunctions, keeping the eigen energies. Because thepotential around the ion core is flat, a pseudo potential with small

wavenumber expansion can be constituted. Summing up Wp on the positions of unit cells Rj , we obtain a pseudopotential for the crystal potential as
Vp (r) =∑Wpα (r − Rj − τα ),

j,α

2-5

(2.33)

Fig. 2.7 (a) Unit cell of a zinc blende crystal (GaAs).
Alternate occupation of lattice points in diamond structure with group III (Ga) atoms and group IV (As) atoms.
In the tetrahedron indicated by broken lines, apices areoccupied with As and the center with Ga. (b) A primitive
cell contains a Ga an an As atoms. Let the edge length of
the unit cell in (a) a, and take the lattice point at the center between Ga and As then their coordinates are fromthe lattice point (a/8)(1, 1, 1) and (−a/8)(1, 1, 1) respectively.

t1
t2
[001]

R

Ga

[010]
As
[100]

(a)

(b)

where α is the index of atomic positions in the unit cell and τα are relative positions of constituting atoms from a certain
point in the unit cell.
Because potential (2.33) has the lattice periodicity, it can be Fourier expanded with wavenumber of reciprocal lattice
points K.
vp (K) =∫ ∑Wpα (r − Rj − τα )e−iK·r

j,α

dr
V

′

Let r ≡ r − Rj − τα , N : number of unit cells, Ω : unit cell volume, From e−iK·Rj = 1
∫
∫
∑
′
′
1 ∑ −iK·Rj ∑ −iK·τα 11eeWpα (r ′ )e−iK·r dr =Wpα (r ′ )e−iK·r dr ′ ,=e−iK·τα
N j
Ω Ω
Ω Ω
α
α
∑
=
(2.34)
e−iK·τα wpα (K).
αwpα (K) is the Fourier transform of (2.32), and depends on the atomic species α, that is the strength and functional formof nuclear potential, not on the crystal structure, and called form factor. On the other hand e−iK·τα depends only on the
crystal structure and called structure factor. This separation of factors make it possible to estimate band structure basedon the analogies between the crystals.
In the case of zinc blende structure, from Fig. 2.7(b), we can write τ1 = −a(1/8, 1/8, 1/8) = −τ2 ≡ τ . Then (2.34)
is written as
vp (K) = eiK·τ1 vp1 (K) + e−iK·τ1 vp2 (K) = (vp1 + vp2 ) cos K·τ + (vp1 − vp2 ) sin K·τ= vps (K) cos K·τ + vpa (K) sin K·τ .

(2.35)

Here vps and vpa are symmetric and anti-symmetric part of the form factor vp , sin and cosine functional part is the structure
factor. In the case of diamond structure (like Si or Ge), from the symmetry vp1 = vp2 and vpa = 0.
To obtain the value of form factor we need detailed functional form of the pseudo potential. Here comes the idea of“empirical" pseudo potential method, in which we rather determine the form factors from experiments as fitting parametersthan to calculate them deductively from the specific form of pseudo potential.
2.2.2.1 Global band structures of semiconductors with diamond and zinc blende structures
In the case of diamond structure vpa (K) = 0, and because of the original concept of pseudo potential, we only need√to calculate vps (K) for reciprocal lattice points with small |K|. Here we restrict |K| ≤ 11, then aK/2π = (000)，(111)，(200)，(220)，(310) and their reversed points, 51 in total as in page ?? (for a while we drop the commas between
the vector components. Accordingly the size of the matrix in the left hand side in (2.30) is 51×51.
The potential for |K| = 0 just shifts the energy and is set to zero. And from the above approximation, the terms for√|K| > 11 are also dropped. The atomic potential is as in (2.32), supposed to have spherical symmetry. The form factors,
which are the Fourier transform fo the potential, should also be a function of the absolute value of wavenumber. We thus
2-6

vps (111)vps (220)vps (311)

vpa (111)

vpa (200)

vpa (311)Si−2.856

0.544

1.088

0

0

0

Ge

−3.128

0.136

0.816

0

0

0

GaAs

−3.128

0.136

0.816

0.952

0.68

0.136

CdTe

−2.72

0

0.544

2.04

1.224

0.544

Tab. 2.3 Empirically obtained form factors of pseudo potential from optical reflection coefficients for representative
fcc semiconductors in unit of eV．The values are taken from M L. Cohen and T. K. Bergstresser, Phys. Rev. 141, 789(1966).

4

4

D1

2

2

G 25 ’

0

X1
D5

-2

E ( k ) (eV)E ( k ) (eV)

0

D 2’

-4

-6

-8

-2

-4

-6

-8

Si

-10

-10

-12

-12

L

G

k

X K

G

(a) Si

L

GaAs

G

k

X K

G

(b) GaAs

Fig. 2.8 Band structures (solid lines )of (a) Si and (b) GaAs, obtained from the form factors in Tab.2.3. Brokenlines show the results of linear muffin-tin orbital (LMTO) method, which is one of the simplest “first principles
calculations". The two results are shifted to have the same value at the top of valence band. Energy zero E(k) = 0 is
taken to the Fermi energy in the pseudo potential calculations.

only need to know the form factors for (111), (200), (220), (311) below the distance√

11. Even among them for (200) the

structure factor cos K·τ is zero and we do not need to know the value or can put it as zero. And the last three are required.
Then form factors are determined to fit to experimentally measured quantities. A way for further decreasing of the numberof parameters is to determine rc in (2.32) to obtain vps (K) and perform the iteration to explain the experiments. Table
2.3 shows three parameters vps (K) chosen as to fit to the optical reflection coefficients in experiments for representative
diamond and zinc blende structure semiconductors.
Procedures of pseudo potential calculation for zinc blende semiconductors are similar to the above though vpa (K) is
now finite. In the case of GaAs in Tab.2.3, since Ga and As locate in the both sides of Ge in the periodic table, thevalue of Ge is also adopted for vps (K). From (2.35), the anti-symmetric term is proportional to sin K ·τ , there is thus
no contribution from (220), and those from (110)，(200)，(311) should be considered. Table 2.3 shows the results to
reproduce optical measurements. Similarly in the case of a II-VI semiconductor CdTe, the value of Sn (gray tin) forvps (K) is adopted and others are obtained from experiments.
Having the values of vp (K), we substitute them into (2.31) and solve the eigenvalue problem of the 51×51 matrix andobtain E(k). Global band structures thus obtained are shown in Fig. 2.8.
Because the above calculations do not take care of the spin-orbit interaction, which actually has important contributions
2-7

Fig. 2.9 Global band structure of Ge calculated with empiricalpseudo potential method. The spin-orbit interaction is taken into
account and the top of the valence band shows spin-orbit splitting. From ref.[2].

to the band structure, in particular at the top of valence band, the three bands are degenerated. In pseudo potential
calculations which take into account the spin-orbit contribution, one of the three branches goes down. In the results forGe shown in Fig. 2.9, clear spin-orbit splitting at the top of the valence band is observed.
Above obtained band structures of Si, GaAs, Ge (representative diamond, zinc blende type semiconductors) are shownin Fig. 2.8 and Fig. 2.9. The bottoms of conduction band of Si are close to X-points though a bit inside the first Brillouinzone, while it is at Γ-point in GaAs and they are at L-points in Ge. Schematic drawing of equipotential surfaces are thus
shown in Fig. 2.10, which are probably familiar to the readers. From Fig. 2.8 and Fig. 2.9, we see that the expressions in
Fig. 2.10 are a bit exaggerated.
As can be seen above, Si has six equivalent bottoms in the first Brillouin zone, which are called valleys. In metallic
doped n-type samples, the number of Fermi surfaces is that of valleys, to which we should pay attention in performing,e.g., some integration over the Fermi surfaces. GaAs has a single valley at Gamma-point and the effective mass is almost
isotropic. Ge has valleys at L-points and there are 8 equivalent L-points just at the Brillouin zone boundaries. Henceeach valley is divided by the neighboring zones and the effective valley number is 4.

2.2.3 Band structure at band edges: Effective mass
Whan a Bloch type electron wavefunction ψnk (r) has a dispersion relation En (k), the group velocity is written asvn (k) = ℏ−1 ∇k En (k).

(2.36)dvn
dk∇k ∑ ∂En (k)=· ∇k (∇k En (k)) = 2
Fj .
dt
ℏdt
ℏ j=x,y,z ∂kj(2.37)

Hence, the acceleration is given as

Here, F = dp/dt = ℏdk/dt is a vector of a “force". Now we define the inverse effective mass tensor 1/m∗ , which is
the inverse matrix of the effective mass tensor with)(1 ∂ 2 E(k)
1
≡ 2
.
∗
m ij
ℏ ∂ki ∂kj
Then (2.37) can be re-written as

dvi (k) ∑
=
dt
j

(

2-8

1
m∗

(2.38))
Fj ,ij(2.39)

kz

kz

GaAs

Si

ky

ky

kx

kx

(a)

(b)
kz

Ge

ky

kx

(c)
Fig. 2.10 Schematic drawings of equipotential surfaces for (a) Si, (b) GaAs, (c) Ge conduction band valleys fromthe information obtained in the pseudo potential calculation of Fig. 2.9. In the case of Ge, which has the valleys
at L-points, because the region is limited to the first Brillouin zone, the boundaries cut the centers of the spheroidal
valleys.

which is equivalent to
Fi =

∑

m∗ij

j

dvj (k).
dt(2.40)For simplicity, we consider an energy band with an isotropic energy dispersion E(k) = ak 2 . m∗ , in general is a tensor,
becomes a scalar ℏ2 /(∂ 2 E(k)/∂k 2 ) = ℏ2 /2a. Let it be more specific. Consider the case of eq. (1.9), where a gap opens
up in the nearly free electron approximation (NFEA). Around ∆k ∼ 0,
[
√
(
)2 ]
ϵz
∆k
2m0 V0E± ≈ ϵz ± V0 1 +,kg ≡,2V0 kgℏ(2.41)

which reads to the effective mass ofm∗ = ±2V0ϵgℏ2 2V0 2m0=±
m0 = ± m0 .
2 ϵz ℏ2ϵz
ϵz(2.42)

Here ϵz is the band width，ϵg the band gap. In this naive approximation, the ratio between the band width and the band
gap determines the effective mass, namely, the wider the energy gap in comparison with the band width, the heavier theeffective mass. This is a kind of “toy model" but can predict at least some trend in the effective masses in the same typeof energy bands, i.e., withe the same symmetry at the same point in the reciprocal lattices. For example we can see suchtendency in the effective masses at Γ point of conduction band in GaAs, InP, InAs.

2-9

2.2.4 Measurement of band-edge structure: cyclotron resonance
The information given from ARPES introduced in Sec. 2.2.1 is limited to the region below EF as we can see from theprinciple. And the precision is, at present, not enough for quatitative discussion on electric conduction or optical actions.
Cyclotron resonance has long been used as a means of experimentally obtaining information on the band-edge of the
conduction band and valence band. This method is based on the fact that the motion of the particle with charge q andmass m in the magnetic field with flux density B projected to the plane perpendicular to the magnetic field is a circularmotion with cyclotron frequency
ωc =

qB
.
m

(2.43)

Let’s consider in classical approximation. The motion of equation for the particle with charge q and effective mass
→ istensor ←m←→←→ dv + m v = q(E + v × B).
m(2.44)dt
τThe electric field of microwave oscillates as Ee−iωt and the velocity as ve−iωt then()
1 ←→v = q(E + v × B).
m
−iω +
τ

(2.45)

Once we assume the oscillatio in v as above, the resonance condition does not depend on E and we put E = 0 for
simplicity. For simpler expression we write ω ′ = ω + i/τ , B = B(α, β, γ), and put
 −1
m1
←
→−1 =  0
m
0

0
m−1
2
0


m−1
3

.

(2.46)

When the lattice system is cubic or rhombic this is justified. Then the equation of motion is written asiω ′ m1 vx + q(vy Bz − vz By ) = 0,iω ′ m2 vy + q(vz Bx − vx Bz ) = 0,iω ′ m3 vz + q(vx By − vy Bx ) = 0.
For the above to have non-trivial solution,iω ′ m1−qBγqBβ
From the condition ωc τ ≫ 1,−qBβqBα = 0.
iω ′ m3(2.47)

m1 α2 + m2 β 3 + m3 γ 2.
m1 m2 m3

(2.48)qBγiω ′ m2
−qBα
√

qB
ωc =
= qB
mc

mc is obtained from the experiment with eq.(2.43) and called cyclotron mass.
In experiment, the condition ωc τ ≫ 1 should hold and for that the impurities in the specimen should be very small. At
low temperatures the carrier concentration discussed later is very low, and carriers are excited by light illumination. In
Fig. 2.11(a), we show an example of thus obtained cyclotron resonance in the absorption of 24 GHz microwave. Several
peaks are observed and assigned to electron (excitation in conduction band) and to hole (excitation in valence band).
While the electrons have negative charge, the holes have positive. Hence if we use circular polarized microwave, theabsorption intensity changes according to the direction of rotation. The assignment was done as above.
In the practical analysis, the symmetry of the crystals tells that the constant energy surfaces should be spheroids. We
thus take the spatial coordinate to the main axes of a spheroid. Then the effective mass is represented by a tensor with

2-10

(a)
Fig. 2.11 (a) Example of microwave
absorption signal in cyclotron resonance. (b) Magnetic field angular dependence of cyclotron mass in Ge conduction band. (c) The same for Si.
From [1].

(b)

(c)

diagonal terms:ml along the main axis and two mt along the other two axes. Let θ be the angle between the magnetic
field and the main axis, (2.48) gives

(

1
mc

)2
=

sin2 θ
cos2 θ
+
.
m2t
mt ml

(2.49)The magnetic field angular dependences of cyclotron mass are displayed for the conduction bands in Ge and in Si in Fig.
2.11(b) and (c) respectively. From the analysis, we can get the directions of spheroids (and the number from the crystal
symmetry) and the values of mt and ml .

2.2.5 Band-edge structure calculation: k·p perturbation
k·p perturbation is an adequate method to obtain highly accurate band structures around band edges. Though in
empirical pseudo potential method we can reproduce band structure from very few parameters with comparatively simple
calculation, in k·p perturbation we need to increase the number of bands included in the calculation, which increases thedimension of matrices and large scale calculation is required.
The basics of k·p perturbation is eq.(1.4) in the first hour of this lecture. Substituting Bloch function eikr unk (r) into
the original Schrödinger equation, we obtain the equation for the lattice periodic part unk (r). In three dimensional space(1.4) can be written as
[ 2 2
]
ℏ ∇
ℏ2 k 2ℏ2−+ V (r) +−ik · ∇ unk (r) = Enk unk (r).
2m0
2m0
m0

(2.50)

Here the Bloch wavenumber k is a parameter (c-number) and not an operator.
Now we redefine the unperturbed Hamiltonian H0 and k-dependent eigen energy as
H0 ≡ −ℏ2 ∇2+ V (r),2m0

En′ (k) = Enk −

ℏ2 k2
.
2m0

Then the perturbation term can be written as
H ′ (k) = −i

ℏ2
ℏ
k·∇=
k · p̂,m0
m0
2-11(2.51)

from which we can easily guess the source of the naming “k· p".
(2.51) is zero for k = 0. Hence we set the unperturbed state as k = 0. Assume that we obtain the exact eigenstates{uj0 (r)} for k = 0, then they form a complete set and an eigenstate for finite k can be expanded as
unk (r) =∞∑cnj (k)uj0 (r).

j=0These cnj (k) can be obtained from the perturbation of H ′ (k). This is the concept of k·p perturbation.
In the above k = 0 is taken to the unperturbed point assuming some avoided level crossing due to the high symmetry.
Such avoided level crossing results in ∂E/∂k = 0 and the approximation is practically good around the point because
H ′ (k) is small around it. Similar may occur in other points with high symmetry and k·p expansions around such points
are also available. Also as we have seen so far, physical properties of semiconductors are determined with band structures
around such symmetric points. We cannot perform, of course, the infinite summation hence cut the summation aroundthe band n which is under consideration. The accuracy of the k·p perturbation usually determined by the number of bands
taken into account.
(a) the case of non-degenerate ui0 (r)uik (r) = ui0 (r) +∑ ⟨j|H ′ |i⟩j̸=i

Ei − Ej

ui0 (r), Ei (k) = Ei (0) + ⟨i|H ′ |i⟩ +∑ |⟨i|H ′ |j⟩|2
j̸=iEi − Ej(2.52)

are obtained as the first order perturbation. Here we have used abbreviation |i⟩ for |ui0 (r)⟩. From ⟨i|j⟩ = δij and⟨i|∇|i⟩ = 0,
Ei (k) = Ei (0) +ℏ2 k 2ℏ4 ∑ ⟨i|k · ∇|j⟩⟨j|k · ∇|i⟩
− 2
.
2m0
m0
Ei − Ej(2.53)j̸=i(b) the case ui0 (r) has degeneracy
When u00 (r) has n-fold degeneracy, we take an orthogonal basis {uj00 (r)} (j = 1, · · · , n) and write the functionsn∑Aij (k)|0j⟩.
in short form as |0j⟩. Perturbed wavefunction is approximated with the linear combination |ui0k ⟩ =j=1

Substituting this into (2.50) gives [H0 + H ′ − E0 (k)]u0k = 0. With taking inner product with |0i⟩, equationn∑Aij (k)[⟨0i|H0 |0j⟩ + ⟨0i|H0′ |0j⟩ − ⟨0i|E0 (k)|0j⟩]

j=1

=

n
∑

Aij (k)[⟨0i|H ′ |0j⟩ + (E0 − E0 (k))δij ] = 0(2.54)

j=1

is obtained. The secular equation for this simultaneous equation to have non-trivial solution is|⟨0i|H ′ |0j⟩ + (E0 − E0 (k))δij | = 0,(2.55)

which gives the dispersion relation E0 (k). From the solution Aij (k), we obtain approximate set of eigenfunctions
corresponding to k.

2.2.6 Spin-orbit interaction
For rigorous derivation of spin-orbit interaction we should go back to Dirac equation, for which we do not haveenough time unfortunately. Here without any derivation, we adopt the Hamiltonian for spin-orbit interaction as
Hso = −ℏ
σ · p × (∇V ).
4m20 c22-12(2.56)

And just discuss the effect on the band structure. σ = (σx , σy , σz ) is a vector which has Pauli matrices:
(
σx =

)
(
0 1
0
, σy =
1 0
i

)
(
−i
1
, σz =
0
0

)
0
,
−1

as its elements. (2.56) is added to (2.50) and with (1.5), we obtain][ 2pℏ2 k 2ℏℏ
p·σ×∇V |nk⟩ = En (k)|nk⟩,
+V +
+
k·π+
2m0
2m0
m0
4m20 c2ℏπ ≡p+
σ × ∇V.
4mc2(2.57)(2.58)We expand the solution, again with the basis of the band bottom |ν0⟩. This time we need to take care of spin freedom and∑write |ν, σ⟩ ≡ |ν0⟩ ⊗ |σ⟩ , and expand as |nk⟩ = ν ′ ,σ′ cn,νσ |ν ′ , σ ′ ⟩. With taking inner product with ⟨ν, σ|, we obtain
the eigen equation as
]
}
∑ {[
ℏ
ℏ2 k 2νν ′νν ′δνν ′ δσσ′ + k · Pσσ′ + ∆σσ′ cnν ′ σ′ = En (k)cnνσ .
Eν ′ (0) +
2m
m
′
′

(2.59)

ν ,σ

where
′

′

νν
′ ′
Pσσ
∆νν
′ ≡ ⟨νσ|π|ν σ ⟩,σσ ′ ≡ℏ2⟨νσ|[p · σ × (∇V )]|ν ′ σ ′ ⟩.
4m2 c2(2.60)The dispersion relation again is obtained with solving the eigen value problem. In this stage, it is often a good approxi′ννmation to drop the spin-orbit part. In such cases it can be written as π = p，Pσσ′ = δσσ ′ Pνν ′ .

2.2.7 Wavefunctions at Γ-point in fcc semiconductors
In empirical k·p method, without detailed knowledge of wavefunctions, the parameters required for the band calculation
are extracted theoretically and the values are obtained from experiments. Many of such parameters are zero around highly
symmetrical points making the calculation easier. Hence the knowledges of spatial symmetries in crystal and in atomic
orbitals are important. Though the theory of space group gives systematic discussion to this problem, again due to thetime limitation, we restrict ourselves to the discussion around Γ-point in fcc semiconductors.
Bravais lattice is fcc for group IV semiconductors with diamond structure and group III-V semiconductors with zinc
blende structure. Here we name them “DZB" semiconductors. As is guessed from the structure in Fig.5.10(b), in chemicalbond theory the crystal formation can be understood along covalent bonding between neighboring sp3 hybrid orbitals.
In the group III-V semiconductors, for each atom to form sp3 hybrid, it needs to be ionized. Hence the crystals are also
formed with the ionic bonding. The most effective atomic orbitals on the band structures in these semiconductors are s
and p. In DZB structure, there are two atoms per a single lattice point in the simplest fcc structure (Fig.5.10). In Fig.5.9,√
substituting 2 × 4 = 8 into Z, we see that the energy gap opens around the degeneracy points in the distance around 3
from Γ point.
We consider a function |S⟩, which has the lattice translational symmetry though also has the same angular symmetry
as s orbital in the vicinities of nuclei. For that, we first take a linear combination of atomic orbitals (LCAO) of s orbital|s⟩|us ⟩ =∑

aiβ |siβ ⟩,i,βwhere i is the index of unit cells, β is the relative index of atoms in a unit cell (as is in the pseudo potential calculation).
Though the above function satisfies the crystal translational symmetry, it is not a solution for the Schrödinger equation
with lattice potential. Hence we assume that we can modify the form of |s⟩ to make the linear combined function a

2-13

solution for the Schrödinger equation with keeping the rotational symmetry in |s⟩ characteristic to the s-orbital. We write
thus obtained LCAO wavefunction as |S⟩, which must satisfy
[ 2 2
]
ℏ ∇
H0 |S⟩ = −+ V (r) |S⟩ = Ec |S⟩.
2m0

(2.61)In the same way we define |X⟩, |Y ⟩, |Z⟩, which have angular symmetries of px , py , pz respectively around nuclei,translational symmetry at the same time.
At Γ-point, the bottom of conduction band is mostly made from s orbitals while the top p orbitals. Hence, though theapproximation is rough, we assume the above defined functions satisfy the unperturbed (k = 0) equation
[ 2 2
]
ℏ ∇
H0 |ζ⟩ = −+ V (r) |ζ⟩ = Eb |ζ⟩,2m0

(2.62)

where ζ ∈ {S, X, Y, Z}, Eb is Ec for ζ = S and Ev for others. It may be a problem whether such functions as |S⟩, |X⟩,
· · · actually exist. The space group theory says we can adopt lattice periodic functions with the same angular symmetries
as s or pα orbitals around the point at which parabola with bottoms at (±1, ±1, ±1) degenerate in the empty lattice
approximation. *1 ．
For the convenience to take into account the spin-orbit interaction, we transform basis from |X⟩, |Y ⟩, |Z⟩ to√√|+⟩ ≡ (|X⟩ + i|Y ⟩)/ 2, |0⟩ ≡ |Z⟩, |−⟩ ≡ (|X⟩ − i|Y ⟩)/ 2,
which correspond to eigen functions of angular momentum |p+1 ⟩, |p0 ⟩, |p−1 ⟩ respectively. With the direct product ofthese four basis functions for the orbital part and two for the spin part (↑, ↓), eight basis functions in total, roughest
k · p perturbation calculation, in which the orbital degeneracy and the spin-orbit interaction are taken into account can be
performed.
The perturbation Hamiltonian to |nk⟩ is taken as
H ′ + HSO = −iℏ2ℏσ · (p × ∇V ),k·∇−
m0
4m20 c2(2.63)

in which we have dropped higher order terms from (2.58) and put π = p. The matrix elements between |S⟩, |X⟩, · · · areℏℏℏ⟨S|px |X⟩ =⟨S|py |Y ⟩ =⟨S|pz |Z⟩,
m0
m0
m0

(2.64)

3iℏ⟨X|[∇V × p]y |Z⟩ = (cyclic replacement of xyz)
4m20 c2(2.65)

P ≡∆≡−and their conjugate elements. Others are zero due to the symmetries around nuclei.
Hence non-zero matrix elements of H ′ are∓P⟨Sα|H ′ |0α′ ⟩ = P kz δαα′ , ⟨Sα|H ′ | ± α′ ⟩ = √ (kx ± iky )δαα′ ,2(2.66)and their conjugate elements Hji′ = (Hij′ )∗ , where α, α′ are spin coordinates. As for HSO ,⟨± ↑ |HSO |± ↑⟩ = −⟨± ↓ |HSO |± ↓⟩ = ±∆/3,√⟨±α|HSO |0α′ ⟩ = (1 − δαα′ ) 2∆/3,(2.67)and others are zero. From (2.62), unperturbed Hamiltonian H0 has⟨Sα|H0 |Sα′ ⟩ = δαα′ Ec , ⟨{+, 0, −}α|H0 |{+, 0, −}α′ ⟩ = δαα′ Ev .

*1 e.g. see Inui, Tanabe, Onodera, “Applied group theory" (Shokabo, 1976) Chapter 11 (in Japanese).

2-14(2.68)

E
conductionEc
heavy hole E v + D /3
light holespinsplit off

Fig. 2.12 Band structure for diamond and zinc blende semiconductors calculated from the lowest order k·p perturbation withadopting only S and P orbitals. Spin-orbit splitting exists though
the heavy hole mass is the same as that of the vacuum electron,
that is, the hole mass is negative in this calculation.

E v - 2 D /3
k

From the above we obtain the secular equation and thus the energy eigen values En (k). H ′ is an 8×8 matrix in thepresent basis though if we fix the wavenumber vector to z direction, i.e., k = (0, 0, k), it becomes
[
]
Hd 0
,
0 Hd
thus is broken down to 4×4 matrices and


Ec
 0
Hd = 
kP ∗
0

0
E√
v − ∆/3
2∆/3
0

√ kP
2∆/3
Ev
0


0

0

.
0
Ev + ∆/3(2.69)

From this the secular equation to obtain the eigenvalue λ is obtained as∆
λ = Ev + ,
(3
)(
)
(
)
2∆
∆
∆
2 2
(λ − Ec ) λ − Ev +λ − Ev −− |P | k λ − Ev +
= 0.
3
3
3
In the second equation we approximate that the term of |P |2 k 2 is small then obtain the energies for the conduction band
Ec (k), and the valence band Evj (k) as[]ℏ2 k 2|P |2 k 2 2
1
Ec (k) = Ec +
+
+
,
2m
3
Eg
Eg + ∆∆ ℏ2 k 2+,3
2m0∆ ℏ2 k 2
2|P |2 k 2
Ev2 (k) = Ev +
+
−
,
3
2m0
3Eg
Ev1 (k) = Ev +Ev3 (k) = Ev −2∆ ℏ2 k 2
|P |2 k 2
+
−
.
3
2m0
3(Eg + ∆)(2.70)(2.71)(2.72)(2.73)The band structure around k = 0 thus far obtained is displayed in Fig. 2.12. Small mass of the conduction band, twodifferent masses at the top of the valence band, and lowered energy of spin split-off band due to the spin-orbit couplingin the valence band, which properties are well known from optical measurements, cyclotron measurements, etc, arereproduced qualitatively though in particular, the heavier valence band mass is that of the vacuum electron, that is, the2-15

hole effective mass is predicted to be negative apparently different from the real band structure. This is, of course, due
to the coarse approximation, which is to the first order perturbation based on the degenerate four bands. The accuracy
is enhanced by enhancing the order of perturbation to second, and by taking the surrounding bands into account. At
present front of calculation, due to algorithm developments, and enhancement in computational performance have madeit possible to perform calculations including over 20 bands and results with high accuracy which can be even used atcomparatively high k[4].
Another way to utilize the result of k·p “empirically" is, as is in the pseudo potential method, to represent the resultsof second order k·p perturbation with a small number of parameters (e.g. Luttinger parameters) and to determine themfitting to the experiments. In the case of valence band in diamond and zinc blende semiconductors, the energies can be
expressed as√∆+ Ak 2 ± B 2 k 4 + C 2 (kx2 ky2 + ky2 kz2 + kz2 + kx2 ),3
2∆
Evsp (k) = Ev −+ Ak 2 ,3
Ev (k) = Ev +(2.74)(2.75)and A, B, C are obtained from, e.g. cyclotron resonance.

2.3 Band structure of graphene
One of the ways to form a two dimensional electron system is to utilize two-dimensional crystals (two-dimensional materials). Graphene is the representative two-dimensional material. Graphene provides a good expample for the applicationof tight-binding calculation and we would like to see how the things go in a practical (though simplest) example.
The crystal structure of single-layer graphene is show in Fig. 2.13(a), which is a simple honeycomb structure of carbon
atoms. The diamond drawn in the figure is the unit cell and the primitive lattice vectors and the primitive reciprocal lattice
vectors are written as
(√
a1 =

)
( )
(
(
√ )
√ )
0
3a/2
4π/ 3a−2π/ 3a, a2 =, b1 =, b2 =
.
a
a/2
0
2π/a

(2.76)

Henceforce we calculate the electronic states of graphene under simplest approximation. Because the approximationis rough, quantitative comparison with experiments is difficult. However, the results help understanding properties ofgraphene, e.g. the Dirac points appear at the Fermi level in pure graphene. Carbon belongs to group-IV and the outmost
electrons exist in the orbitals 2s, 2px , 2py , 2pz . It is easy to see that these orbitals form sp2 -hybrids and the electronic
states separate to σ-electrons (sp2 ) and π-electrons (pz ). σ-electrons form the honeycome through covalent bonding andthe energy bands lie at low energy region. Then the electronic states placed around the Fermi level are π-electrons. Hence
we consider Schrödinger equation on π-electrons on the honeycomb lattice.
We write the equation asψ = H ψ,(2.77)

and as Fig. 2.13(a), we separate the lattice sites to A-sites and B-sites on different sub-lattices. We consider a kind oftight-binding approximation between the two-sites. That isψ = ζA ψA + ζB ψB ,∑ψA =exp(ikrj )ϕ(r − rj ),(2.78)(2.79a)

j∈A

ψB =

∑

exp(ikrj )ϕ(r − rj ),

j∈B

2-16

(2.79b)

y

ky

A

a2

b2
K
a1

B

x

M
K

b1

G

(a)

kx

(b)

Fig. 2.13 (a) Two dimensional cryatal structure of graphene. Carbon atoms form a honeycomb lattice. It can be also
viewed as an overlap of two face-centered square lattices placed at A and B positions. (b) Reciprocal lattice of (a).
b1 , b2 are the primitive reciprocal lattice vector corresponding to a1 , a2 . The centtral point of the first Brillouin zoneis Γ-point and as other points with high symmetries, K-point and M-point are indicated in the figure.

where ϕ(r) is atomic wavefunction of π-electrons, rj are the lattice points. Here we write the matrix elements of the
Hamiltonian between the each sub-lattice wavefunctions as∗
HAA = ⟨ψA |H |ψA ⟩, HBB = ⟨ψB |H |ψB ⟩, HAB = HBA= ⟨ψA |H |ψB ⟩.

(2.80)

And the number of atoms in the system is 2N , that is⟨ψA |ψA ⟩ = ⟨ψB |ψB ⟩ = N.

(2.81)

Let ⟨ψA |ψB ⟩ be zero. We substitute (2.78) to (2.77). The condition of have non-trivial (ζA , ζB ) givies the cecular equation
HAA − N E
HBA
LastlyHAB= 0.
HBB − N E()√
E = (2N )−1 HAA + HBB ± (HAA − HBB )2 + 4|HAB |2 ≡ hAA ± |hAB |,(2.82)(2.83)

where we have used HAA = HBB , which comes from the symmetry, and we use lower cases for the quantities per atomwith being devided by (2N )−1 .

y

a
a ö
æ
, ÷
çè 2 3 2 ø

d2
d1

a
a ö
æ
,ç÷
2 ø
2 3
è

x
æ
ç
è

a
ö
,0 ÷
3
ø

d3
Fig. 2.14 Vectors indicating three directional couplings between nearest
neighbor carbon atoms.

2-17

E

kx
ky
Fig. 2.15 Left: surface plot of eq.(2.87). The figure shows the appearance of Dirac points, where the vertices of energy corns crash at
the K-point. Upper: Schematical drawing of a Dirac point.

HAB =

∑

exp [ik(rj − rl )] ⟨ϕ(r − rl )|H |ϕ(r − rj )⟩r .

(2.84)

l∈A,j∈B

We further approximate that the off-diagonal matrix elements of H just exist between the nearest neighbor sites. For the
calculation we take the atom indicated as A in Fig. 2.13(a) as the center atom. The vectors from A to the nearest neighboratoms 1, 2, 3 are di (i = 1, 2, 3) respectively. As is apparent from the figure,
)
)
(
(
kx a
kx
ky
kx
ky
k · d1 = √ , k · d2 = − √ +a, k · d3 = − √ −
a,
2
2
3
2 3
2 3

(2.85)

where a = |a1 | = |a2 |. The terms ⟨ϕ(r − rl )|H |ϕ(r − rj )⟩r should be equal due to the symmetry and we write it as ξ.
Consequently the residual resonant integral from the crystal structure is the repetition of the above and3∑hAB = exp(ik · dj ) ξ.

(2.86)

j=1

Substituting eqs.(2.85), (2.86) into eq.(2.83), we get the following expressio for the energy.
√
√
3kx a
ky a
ky a
E = hAA ± ξ 1 + 4 cos
cos+ 4 cos2.
222(2.87)The second term is the perturbation from the nearest neighbor resonant integral, which vanishes at K-point in the reciprocal space((kx , ky ) =

4π
0, ±
3a

)

(
,

2π
2π
√ ,±
3a
3a

)

(
,

2π
2π
−√ , ±
3a3a).

(2.88)

We write ky = 4π/3a and around kx = 0(one of the K-points), eq. (2.87) can be approximated as(

4π
E kx ,3a√)≈ hAA +

3ξa|kx |.
2(2.89)

Namely, at the K-point the upper band has a lower pointed shape. Because the same for the lower band and as a result,at the K-point, as shown in Fig. 2.15, the band structure called Dirac point, which has no energy gap, no effective mass,
appears.
2-18

Equation (2.87) is for a very simplified model. Just like a cosine band appeared in the tight-binding model in onedimension, the model itself does not have realistic meaning. However the model tells that the reason why we have theDirac points at K-points is that the exsittence of three equivalent resonant integrals in eq. (2.86). The inference holds forthe band calculation with any level precision since it is based on the symmetry. That meas the K-points in real graphene
are really Dirac points.

Appendix 2A: Band structure calculation based on density function theory
There are many electrons in an actual substance, and the wave function that expresses that state has fermion symmetry
called antisymmetry with respect to the particle exchange operation. This causes an electron correlation effect. In
addition, a Coulomb repulsive force acts between the electrons. In the semi-empirical band calculation, the effect of theseelectron-electron interactions is taken into account when the parameters of the one-electron band picture are obtained fromthe fit to the experimental values, but in the so-called first principle (ab initio) calculation, direct treatment of the electronelectron interaction is required. Calculations that incorporate electron-electron interactions require enormous amountsof calculation for high accuracy, computer resource savings are hence required. The density functional theory (DFT) isvery advantageous in that point, thus used in many ab-intio calculations. Nowadays, calculation packages sometimes
give us answers even without knowledge of calculations inside. Here, hoewever, we will have a brief look at very basics
of ab-initio band calculations[6].
It has become clear that the ab-intio calculations with various approximations reproduce the qualitative features ofthe semiconductor band structure, but on the other hand, even the band gap, which is the most basic quantity cannot be
reproduce without taking into account the quantum correlation effect though this is not an easy task. Incorporating thecorrelation effect properly is not easy even with DFT, and various theoretical modifications are made that can be said to be
ad hoc, which is far from the “first principle" in some cases. It is necessary to pay attention to what kind of approximation
is used for the calculation and how accurate it is.

2A.1 Kohn-Sham equation
Hohenberg and Kohn showed that the energy of interacting electron gas with electron density distribution ρ(r) in theexternal potential v(r) can be expressed with a universal functional of density F {ρ(r)} as∫
E{ρ(r)} = F {ρ(r)} + v(r)n(r)dr,(2A.1)

and also showed that E{ρ(r)} takes minimum for the true electron density ρ(r). The proof is for the case of ground state
without degeneracy but the limitation was removed by Levy. Herer we skip the proof.
Let the Hamiltonian without v(r) beNNℏ2 ∑ 2 ∑Hf = −∇ +V (ri − rj ),2m i=1 i i>j(2A.2)and the density functional F {ρ} is generally written as
F {ρ} = ⟨ΨN |Hf |ΨN ⟩.

(2A.3)

ΨN is the wavefunction to give ρ(r). To obtain the form of F {ρ}, we write it in the form(0)
F {ρ} = T {ρ} + U {ρ} + Exc
{ρ}.

2-19

(2A.4)

T {ρ} is the kinetic energy, U {ρ} is the mean field expression of the Coulomb interaction between the electrons as
U {ρ} =

e2
8πϵ0

∫

ρ(r)ρ(r ′ )drdr ′ ,|r − r ′ |(2A.5)and is called Hartree term. The residucal Eex {ρ}(0) is called exchange-correlation energy.
Even the kinetic energy T {ρ} is difficult to be expressed explicitly with ρ and we make modification as follows. We
consider an imaginary electron system without interaction in an effective potential veﬀ (r).
[
]
ℏ2 2
−
∇ + veﬀ (r) ψi = ϵi ψi (r).
2m

(2A.6)We also assume the electron density of this system coinsides with that of the interacting electron system.
ρ(r) =∑|ψi (r)|2 .

(2A.7)

ψi∗ (r)∇2 ψi (r)dr.

(2A.8)i

The kinetic energy of the imaginary system isℏ2 ∑2m i=1N

TS {ρ} = −∫We now renormalize Eex to include the difference between TS {ρ} and T {ρ} as(0)Eex {ρ} = Eex+ T {ρ} − TS {ρ}.

(2A.9)

Then the total energy can be written as∫
E{ρ} = TS {ρ} + U {ρ} + Eex {ρ} +v(r)n(r)dr.

(2A.10)We minimize E{ρ} with the variation of ψi (r)∗ . For that the normalization condition ⟨φi |φi ⟩ = 1 leads to theintroduction of Lagrange multipliers −ϵi . Then the minimum condition is written as
[
]
∫
δE{ρ}
ℏ2 2
e2
ρ(r ′ )′
= −∇ + v(r) +dr + µex (r) − ϵi ψi (r) = 0,
δψi∗
2m
4πϵ0
|r − r ′ |where µex (r), defined asµex (r) =δEex {ρ}δρ(r)(2A.11)(2A.12)is the quantity called exchange-correlation potential. The condition can be written in an eingenvalue equation as
]
[
∫
e2
ℏ2 2
ρ(r ′ )′
∇ + v(r) +dr + µex (r) ψi (r) = ϵi ψi (r).
(2A.13)−2m
4πϵ0|r − r ′ |Eq. (2A.13) is called Kohn-Sham equation.
In summary, the many-body effects are put into Eex {ρ} in the above. The next step is how to calculate this term. In
most frequently used method is local density approximation (LDA),∫
Eex {ρ} = ϵex {ρ(r)}ρ(r)dr,(2A.14)then for ϵex {ρ(r)}, the exchange-correlation energy for uniform electron gas with the density ρ. There are many othermethods for the calculation.

2-20

References[1] G. Dresselhaus, A. F. Kip, and C. Kittel, Phys. Rev. 98, 368 (1955).
[2] M. L. Cohen and J. R. Chelikowsky, “Electronics Structure and Optical Properties of Semiconductors" (Springer,
1988).
[3] C. Hamaguchi, “Basic Semiconductor Physics" 3rd ed. (Springer, 2017).
[4] L. C. Lew Van Voon, M. Willatzen, “The k·p method" (Springer, 2009).
[5] P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864 (1964).
[6] D. Sholl and J. A. Steckei, “Density Functional Theory: A Practical Introduction" (Wiley, 2009).

2-21

Lecture note on
18th April (2021)

Physics of Semiconductors (3)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

2.3.1 Effective mass approximation
Let us consider the effect of spatially non-uniform perturbation. For that we add the perturbation potential U (r) to the
Schrödinger equation in crystals to obtain
[ 2 2
]
ℏ ∇
−
+ V (r) + U (r) ζ(r) = [Ĥ0 + U (r)]ζ(r) = Eζ(r).
2m(2.90)

ζ(r) can be expanded with the Bloch functions ψnk , which are the eigenstates of Ĥ0 as
ζ(r) =∑∑

f (n, k)ψnk (r) =n,kf (n, k)unk (r)eik·r .

(2.91)

n,kWith taking the inner product with ψn′ k′ after substitution of (2.91) to (2.90),[E0 (n′ , k′ ) − E]f (n′ , k′ ) +∑⟨n′ , k′ |U |n, k⟩f (n, k) = 0,(2.92)

n,k

where ψnk is written as |n, k⟩. The second term, the transition mediated by U (we write it as Un′ k′ ,nk ), represents thescattering from |n, k⟩ to |n′ , k′ ⟩. U and u∗n′ k′ unk are Fourier transformed into∫∑
U (r) = dqUq e−iq·r , un′ k′ (r)unk (r) =bn′ k′ nk (G)eiG·r .
GThe transformation of u∗n′ k′ unk is a Fourier series on the reciprocal lattice because the term has the lattice periodicity.
The coefficients bn′ k′ nk are written with the unit cell space Ω0 , the unit cell volume v0 as∫
dr −iG·r ∗bn′ k′ nk (G) =eun′ k′ (r)unk (r).
Ω0 v0
∫
∫
∑
′
∴ Un′ k′ ,nk = dqUq
bn′ k′ nk (G) drei(k−k +q+G)·r .
G

The last integral can be performed to be (2π)3 δ(k − k′ + q + G), and the integration over q gives
Un′ k′ ,nk = (2π)3∑Uk′ −k−G bn′ k′ nk (G).

(2.93)

G

U (r) is assumed to have much slower spatial variation than the lattice potential. Then as Uq , it is enough to restrict
ourseleved to |q| ≪ π/a, i.e. much smaller values than that of the Brillouin zone edge. The approximation corresponds
to k′ − k ∼ G. We further assume that U does not cause strong scattering that drives the state to the zone edge, thenG takes only ⃗0. Also |U | is smaller than the band gap, then there is no interband scattering via U , i.e. there is no matrix
element for n ̸= n′ . Then we can approximate
Un′ k′ ,nk ≈ Uk′ −k δn′ n .
(2.92) is written as[E0 (k′ ) − E]f (n, k′ ) +

∑
k

E3-1

Uk′ −k f (n, k) = 0.

(2.94)(2.95)

F( r )z( r )U( r )E

V ( r )+ U ( r )rFig. 2.16 A perturbation pontetial U (r)is superposed on the crystal potential
V (r). The figure illustrates the potential and the wavefunction ζ(r), the envelope function F (r) for the system withthe crystal potential V (r) the slowly varing perturbation potential U (r).

Nest we consider the expansion in (2.91). In the present approximation, only the region k ∼ 0 is considered for unk ,and u is almost constant for k(≈ un0 ). Then we take it out from the sum over k.
∑
ζn (r) = un0
f (n, k)eik·r = un0 Fn (r),(2.96)kwhere the index n is attached to ζ with ignoring the intermixing of the bands. Here, Fn (r) defined asEnvelope function
Fn (r) ≡∑f (n, k)eik·r(2.97)kis the inverse Fourier transformation of f (n, k), and called envelope function. Fn (r) should be a slowly varing function
over the scale of lattice constant(Fig. 2.16).
For the unperturbed dispersion relation, we apply that of a particle with the effective mass, and for simplicity theeffective mass m∗ is assumed to be isotropic. Then E0 (k) = ℏ2 k2 /2m∗ is substituted to (2.95) to give∑
ℏ2 kk′2
f (k) +
Uk′ −k f (k) = Ef (k′ ).
∗2m(2.98)k

Here we omit writing n. The above can be inverse-Fourier transfomed with the care that the second term becomesconvolution because it already has the summation over k to giveEffective mass equation
]
[ 2 2
ℏ ∇
+ U (r) F (r) = EF (r).
2m∗


(2.99)
∗The equation takes the form of the Schrödinger equation of the paticle with the mass m and the potential U (r). That is,for the envelop function, the problem is now a particle with the effective mass in the perturbation potential. This way ofhandling the problems on the level of envelope function is called effective mass approximation, and eq.(2.99) is called
effective mass equation. In this sense, the Bloch states can be viewed as plane waves in the effective mass approximation.
The viewpoint is very usuful in designing various quantum systems in solids with semiconductor technologies. We
test the approximation for the shallow impurity states in the next chapter. We also use it in many places in this lecture.
We should be careful, however, that the envelope function is not the wavefunction itself. The difference becomes clear,particularly when the perturbation potential has a sharp spatial variation.

E3-2

Chaper 3 Carrier statistics and impurity doping

In this chapter we consider the energy distribution of carriers in semiconductors. We introduce the concept of carrier
doping with very little amount of impurities, which brings drastic changes in the electric conduction.

3.1 Carrier statistics in intrinsic semiconductors
We call a pure semiconductor without any impurity as an intrinsic semiconductor. Of course this is just an idea,
but e.g. non-doped Si for LSIs’ can be considered as an intrinsic semiconductor. And under some conditions othersemiconductors can also be treated as intrinsic semiconductors.

3.1.1 Density of states
We consider a simple lattice system which has a state per a unit cell with an edge length of a. We take the system size
as L = N a in one dimension. For an n-dimensional system, the volume (2π/L)n contains a single state in k-space(Fig.
3.1(a)). Given the kinetic energy as E(k) = ℏ2 k 2 /2m, the number of states per volume between E and E + dE (Fig.
3.1(b)) devided by dE is
1
D(E) = d
L

(

L
2π

)d

dVd (k)1 dVd (k) dk
1 m0 dVd (k)
=
=
,
d
dE
(2π)
dk dE
(2π)d ℏ2 kdk(3.1)

where Vd (k) is the volume of d-dimensional sphere with the radius of k. This D(E) is called energy density of state.
Because V1 = 2k ，V2 = πk 2 ，V3 = 4πk 3 /3 (Fig. 3.2),

√
2m0
m0
1
(0)
(0)
, Dd=2 =,Dd=1 =
πℏ
E
πℏ2


√
2m3 √
(0)
Dd=3 = 2 30 E,
π ℏ


(3.2)

where the factor 2 comes from the freedom of spin.
In the case of electrons in crystals, the above expressions for density of states are applicable with replacing the mass
with the effective mass where non-parabolicity of the band is ignorable, e.g., around tops and bottoms of the bands. When
we cannot apply the parabolic approximation, we need to go back to the definition of the density of states. For a three

kz

kz

E+dE
ky

E

ky

kx
2p/L
(a)

kx
(b)

E3-3

Fig. 3.1 (a) The red dots represent possible
wavenumber in k-space in 3d empty lattice
approximation for simple cubic. (b) Counts
the number of dots in the spherical shell fromE to E + dE.

Fig. 3.2 Schematic diagrams ofdensity of states for 1, 2, 3 dimentions in (3.2).

dimensional system it is given from∫
D(E) =dSk
2.
3 ∇ E(k)(2π)kE(k)=E(3.3)

The integral is over the equi-energy surface E(k) = E in k-space.

3.1.2 Concept of holes
The total current Jv.b. carried by a full valence band is zero by can∑celing of counter-going electrons (Jv.b. = v.b. (−e)vk = 0). When astate with crystal momentum k is empty as seen in the left, the current is∑Jv.b. (k) =(−e)vk′ − (−e)vk = evk ,
(3.4)
v.b.

as if there is a particel with the charge +e and the velocity vk . Such a
many-body state in valence band is called hole.
We write the wavenumber of hole as kh , then it should be the variationof the total wavenumber (momentum) due to the creation of the hole.
∑kh =ke′ − ke = −ke .
(3.5)
v.b.

When an electric field E is applied, the valence electrons are accelerated
and move in the k-space. The “hole" follows the movement, that is, theequation of motion for holes is the same as that for electrons. However,
we define a hole has the charge +e, then the acceleration by the electricfield should be opposite to that for electrons and for consistency the signof the effective mass should be opposite.
m∗

dv
dv
= (−e)E → (−m∗ )
= eE.
dt
dt

The kinetic energy decreases with the creation of a hole and if we takethe origin of energy at the top of valence band, we get
)
(
)
(
1
1
=−
, Eh (kh ) = Eh (−ke ) = −Ee (ke ). (3.6)
m∗h ijm∗e ijForm (3.6), m∗h is positive around the valence band top and the dispersion is obtained with 180◦ rotation of the electron
dispersion. The density of states Dh (E) is the same as De (E). The above definitions make it possible to treat the holesas positive charge carriers. The hole band drawn in the lower left is used to be consistent with the electron dispersion andthe hole picture. We need to be careful that the frequently-used whilte hole picture (actually in the left) which is really anelectron dispersion and the “white hole" is placed at ke which is kh .
E3-4

3.1.3 Carrier distribution in thermal equilibrium
Let us see how electrons and holes distribute in energy space at a finite temperature obeying the Fermi distributionfunction. For a while we treat general properties, which hold also for doped semiconductors. The effect of doping can be
included in the position of the Fermi level EF . The numbers of electrons and holes which exist in E ∼ E + dE are
ge (E)dE = De (E)f (E)dE,(3.7a)

gh (E)dE = Dh (E)[1 − f (E)]dE ≡ Dh (E)fh (E)dE.

(3.7b)

Here we introduced the hole distribution function as (Fig. 3.3(c)),
1.
1 + exp(EF − E)/kB T )fh (E) = 1 − f (E) =(3.8)

For the density of states, we use those of particles with the effective masses. From (3.2),
√
√
2m∗3
e
De (E) =E − Ec (conduction band),
2 3
√π ℏ
√
2m∗3
h
Dh (E) =Ev − E (valence band).
23
π ℏ(3.9a)(3.9b)

Here Ec , Ev are the bottom of conduction band and the top of valence band respectively as in Fig. 3.3(a).
Hence the distributions of electrons and holes at a finite temperature should be as in Fig. 3.3(b), giving the electron
concentration in the conduction band n, the hole concentration p in the valence band as
√
√
∫ ∞
∫ ∞
2m∗3
E − Ec dEen=ge (E)dE =
,
2
3
π ℏ
Ec
Ec 1 + exp(E − EF )/kB T
√
√
∫ Ev
∫ Ev
2m∗3
Ev − EdEh.
p=gh (E)dE =
2 ℏ3
π
1
+
exp(E
F − E)/kB T−∞−∞(3.10a)(3.10b)In the case of fF (E) ≪ 1(E ≥ Ec )，fh (E) ≪ 1(E ≤ Ev ), the distribution can be approximated by Maxwellian asfF (E) ∼ exp(EF − E)/kB T,
We apply the identity∫ ∞√fh (E) ∼ exp(E − EF )/kB T.

xe−x dx =

0

E

E(k)

re(E)

(3.11)√π2E

f h( E )Ec

electronkFig. 3.3 (a) Schematic diagram of

hole

Ev

EF
f(E)

(a)

(b)

(c)
E3-5

1

energy bands. (b) Density of states
and the distributions of electronsn(E)(blue-gray)，p(E)(white)．
(c) Electron distribution function
f (E)(red solid line), and hole distribution function fh (E)(blue brokenline).

with x = (E − EF )/kB T to obtain()
EF − Ec
,
kB T
( ∗
)3/2
(
)
(
)
mh kB TEv − EFEv − EF
p=2
exp≡ Nv exp
.
2πℏ
kB T
kB T

n=2

m∗e kB T
2πℏ

)3/2

(

exp

EF − Ec
kB T)(≡ Nc exp(3.12a)(3.12b)Nc and Nv are the coefficients which give n, p respectively in the situation the energy states are concentrated at Ec and
Ev . They are called effective density of states. From (3.10a) and (3.10b) we obtain
Law of mass action
)
(
)
(
Eg
Ev − Ec= Nc Nv exp −= n2i
np = Nc Nv exp
kB T
kB T



(3.13)


Here the width of forbidden band Eg ≡ Ec − Ev is called energy gap, and ni は真性半導体の場合のキャリア濃度である．Equation (3.13) does not depend on the position of EF , which varies, e.g. with doping. In other words, the productnp in the thermal equilibrium is determined only by the temperature and the species of semiconductor.
In intrinsic semicondutors, there is no space charge and the charge neutral condition leads to n = p hence is written as
ni in the above law of mass action. The relation n = p in intrinsic semiconductors leads toEF =Ec + Ev
kB T
Nv
3kB T
mh
Ec + Ev
+
ln
+
ln
=
,
2
2
Nc
2
4
me

(3.14)

which gives the position of EF . At low temperatures the second term gets small and EF comes close to the middle of theband gap.

3.2 Impurity doping
In semiconductors, very small amount of impurities give drastic change in the material properties. Such addition of
impurities is called doping *1 .

3.2.1 Donors and acceptors
As a typical example, the case of Si is shown schematically in Fig. 3.4. In Si pure crystal, as in (a), a Si atom hasfour nearest neighbor atoms, which have 4 covalent electrons. As a result each atom has eight electrons in the outmost
shell, which fill up 3s and 3p orbits forming the closed shell geometry. When the center atom is replaced with an Sb(group-V) atom, there is an excess electron for the closed shell structure as in (b). On the other hand, the positive charge
in the nucleous excesses the negative one of surrounding electrons by +e, which forms a Coulomb potential around theSb atom. The excess electron is excited to the conduction band or loosely trapped in the bound state in the Coulomb
potential. An impurity that emits electrons to the conduction band or the shallow levels is called donor.
When the center atom is replaced with a B atom (group-III), which is just the opposite of Sb, there are not enoughelectrons to form a closed shell structure. Therefore, holes are created in the valence band to supplement the electrons,
but as a result, the electron charge becomes extra around the B atom, and a Coulomb potential of only −e is generated.
Impurities that emit holes into the valence band and shallow levels in this way are called acceptors(acceptor).

*1 In so called studies in strongly correlated systems, which began with the studies of high-Tc superconductors, addtion of atoms wtih concentra-

tions even far above 1% is called “doping" as far as the crystal structure is unchanged. Such regions are called “alloying" in the semiconductor
fields. Furthermore, enhancement of carrier concentrations with application of strong electric field is sometimes called “electric field doping",
and the addition of impurities is called “chemical doping." Here, however, I follow tranditional epxression in the field of semiconductors.

E3-6

Fig. 3.4 (a) Illustration of electronic structure in the outmost shell in a Si atom in a Si crystal. (b) In the case of
replacement with an Sb atom. There is an excess electron for Sb to form the closed shell structure. The excess electron
should go out from the covalent system. (c) In the case of replacement with a B atom, which opens a “hole" in thevalence band.

The situation is a little more complicated in the case of compound semiconductors than in Group IV elemental semiconductors. For example, when a group III-V semiconductor is doped with a group IV element, replacing the group III
site becomes a donor, and replacing the group V site becomes an acceptor. Elements whose donor / acceptor changes
depending on the doping method are called amphoteric.

3.2.2 Effective mass approximation for shallow hydrogen-like levels
Regarding the Coulomb potential formed by donors and acceptors, semiconductors generally have a relatively large
permittivity due to the polarization of valence electrons, and this impurity attraction potential is considerably weaker thanin vacuum. Therefore, the binding energy of the impurity bound state is smaller than that of the hydrogen atom, and inmany cases, it spreads over several unit cells, and the effective mass approximation can be applied.
For the case of isotropic effective mass, taking the origin at the impurity potision we write the effective mass equationfrom U (r) = −e2 /4πϵ0 ϵr as

[ 2 2
]
ℏ ∇
e2
−
F (r) = EF (r),−2m∗
4πϵ0 ϵr(3.15)

which as the same form as that of hydrogen atom other than the effective mass m∗ and the relative permittivity ϵ. Hence
we can readily apply the results for hydrogen atom. Here we write the effective Rydberg constant and the effective
Bohr radius as
Ry ∗ =

e2 m∗
m∗ 1
=
Ry,
2(4πϵϵ0 )2 ℏ2
m ϵ2

a∗B =

4πϵϵ0 ℏ2m
= ∗ ϵaB∗2m em(3.16)

respectively. The eigenenergy is then represented as
En = Ec −Ry ∗n2(n = 1, 2, · · · )(3.17)()
1r
exp − ∗ .
πa∗3
aB
B

(3.18)and the wavefunction corresponding to 1s state is√ψ1s (r) =An expample of semiconductor with such an isotropic effective mass is GaAs. At the conduction band minimum
placed at Γ-point, ϵ ≈ 11.5, m∗ ≈ 0.067m. a∗B = 172aB = 91 Åis sufficiently longer than the lattice constant 5.65 Åandguarantees the legitimacy of the effective mass approximation. From Ry ∗ = 5.07 × 10−4 Ry = 5.57 × 103 m−1 thebinding energy of 1s state is as small as 6.9 meV.

E3-7

Semiconductor

Calculated binding

Experimental binding

energy (meV)energy (meV)
SiGa (5.84); GeGa (5.88)

GaAs

5.72

InP

7.14

7.14

InSb

0.6

TeSb (0.6)

CdTe

11.6

InCd (14); AlCd (14)

ZnSe

25.7

SAs (5.87);

Tab.
3.1 Effective mass approximation for hydrogen-like impurities and the measured value ofbinding energies.

AlZn (26.3); GaZn (27.9)
FSe (29.3); ClSe (26.9)

Tab. 3.1 shows the comparison of the value given by (3.17) and experimentally measured values for isotropic effective
mass condition. The agreement is satisfactory.
Then what if the effective mass is anisotropic and there are six conduction band valleys, as in Si? We consider theeffective mass approximation for the valley along (0,0,1). The equation for the spheroidal surface is[]ℏ2 kx2 + ky2(kz − k0 )2
E1 (k) =
+
.
2
mt
ml
Then the effective mass equation is
]
( 2
)
[
∂2
e2
∂
ℏ2 ∂ 2ℏ2+−
F (r) = EF (r).
−−2mt ∂x2∂y 2
2ml ∂z 24πϵ0 ϵr(3.19)(3.20)The eigenfunction can be approximated by the variational method assuming an anisotropic exponential function. As atrial function we take a and b as the paramters and write down as
( √
)
√
1
x2 + y 2
z2
F1s (r) =
exp −
+ 2 .
πa2 b
a2
b

(3.21)The stationary condition gives numerical solutions as a = 2.5 nm, b = 1.42 nm, E = 29 meV. However the experiments
give 33 meV for Li (the shallowest case), 45 meV for P manifesting that the approximation is not appropriate. Further
discussion will be given in Appendix 3B.

3.3 Carrier statistics in doped semiconductors
Let us consider the case we dope donors uniformly with the density ND . At absolute zero all the electrons emitted
from the donors are bound to the donors. *2 At finite temperatures some of them are excited to the conduction band andcan carry electric charges. We call them “carriers" or “electrons". Let n be the density of such electrons and nD be thedensity of electrons bounded at the donors. From the charge neutrality condition we get n + nD = ND .
Now we estimate Helmholtz free energy F = U −T S by considering the number of cases W for assigning nD electrons
to ND states. From S = kB ln W ,[F = ED nD − kB T ln 2nD

]
ND !
.
nD !(ND − nD )!

ED is the position of the bound state measured from the bottom of the conduction band and 2nD is due to the spin
degeneracy. We assume that the Coulomb repulsion prevents double occupation of a localized state with two electrons.

*2 In so called degenerate semiconductors the following discussion does not hold.

E3-8According to Starling approximation ln N ! ∼ N ln N − N , the chemical potential (Fermi energy) is given as[]∂F
2(ND − nD )µ = EF == ED − kB T ln.
∂nDnD
And from this

[
(
)]−1
1
ED − EFnD = ND 1 + exp
2
kB T(3.22)(3.23)

is obtained. The factor 1/2 on the exponential function is due to the spin degeneracy.
Similarly, for uniform doping of acceptors with density NA , the density of electrons bounded to the acceptors nA is)]−1[(
EA − EF.
nA = NA 1 + 2 exp
kB T(3.24)

Here we have a factor 2 instead of 1/2 but the density of holes bounded to the acceptors is pA = NA −nA and symmetricalwith nD having a factor 1/2.
From (3.22), if we dope only “shallow" donors, for which the effective mass approximation holds, EF comes to ED
at T → 0. ED should be much smaller than Eg . Accordingly from (3.23), the electron concentration n becomes muchhigher than that of the intrinsic semiconductor at finite temperatures. This type of semiconductors are called n-type.
Similarly doping of acceptors enhances the hole concentration p. We call them p-type.
When donors and acceptors co-exist, the semiconductor becomes n-type for ND ≫ NA and p-type for ND ≪ NA . In
the former, some of the electrons emitted from donors are captured to acceptors and almost all the acceptors are ionized.
In the latter, the other way around. In both cases we say such semiconductors are compensated.
Remember the semiconductor equation (3.13) then the product np does not depend on the doping. If one of n , p
increases with doping, then the other decreases. In the case of n-type semiconductor under ND ≫ NA , n is much higher
than p by many orders, and we call the electrons majority carriers and the holes minority carriers. The other way
around in the case of p-type semiconductors.
Even in the presence of donors and acceptors eq.(3.12) hold and simultaneous satisfaction of them gives n, p and EF .
To obtain EF with knowledge of n, p approximate expressions
[ (
)
)]
(
n
n
EF ≈ EC + kB T ln
+ 2−3/2
,
NC
NC
[ (
)
(
)]
p
p
EF ≈ EV − kB T ln
+ 2−3/2
NVNV(3.25a)(3.25b)are convenient. In the region where (3.23), (3.24) hold the last term can be omitted.
In an n-type semiconductor with compensation, p,nA can be ignored and the electrically neutral condition isn + N A = ND − n D .
Substitution of eq.(3.23) givesn + NA1= exp
ND − NA − n2(

ED − EFkF T(3.26)).

(3.27)

Equation (2.22) holds for the case of doped semiconductors with shifts of EF , multiplication of each side of the equation
results in()
n(n + NA )1∆ED
= Nc exp −, ∆ED ≡ Ec − ED .
ND − NA − n2
kB T(3.28)The temperature dependence of carrier concentration n described by eq.(3.28) has the following four characteristic
regions:

E3-9

IV

log n

III
exhaustionII

I

impurity II

E g /2
DE D /2

impurity I

intrinsic

Fig. 3.5 Characteristic four temperature regions of an n-type semiconductor with compensation. Schematical temperature dependence of carrier concentration n is plotted versus 1/T in semi-log scale.

DE D
1/k B T

I. Impurity (Freeze-out) region I: At very low temperatures and the case of n ≪ NA ≪ ND ,
(
)
ND Nc
∆ED
n≈
exp −
,
2NA
kB T

(3.29)

where n decreases with lowering the temperature in an Arrhenius type with an activation energy of ∆ED .
II. Impurity (Freeze-out) region II: In middle temperature range, in the case of NA ≪ n ≪ ND ,
(
n≈

Nc ND
2

)1/2

(
)
∆ED
exp −
,
2kB T

(3.30)

where the temperature dependence shows again an Arrhenius type but with a different activation energy, which isa half of that in the impurity region I.
III. Exhaustion (Saturation) region: Temperature is higher than ∆ED (kB T > ∆ED ). The exponential function ineq.(3.28) is now almost a constant (∼ 1) andn ≈ ND − NA .

(3.31)

Electrons once captured in donors are “exhaustively" excited to the conduction band and work as carriers.
IV. Intrinsic region: At higher temperatures where direct thermal excitation for the valence band to conduction bandcannot be ignored in comparison with ND , the temperature dependence of the carrier concentration asymptoticallyapproaches to that in an intrinsic semiconductor described as eq.(3.10b), (3.14).

We show the behavior in Fig. 3.5 schematically. For semiconductor devices, the exhaustion region III is mostly used.

Appendix 2B: Wannier functions and the effective mass approximation
There is a way to derive the effective mass approximation by using expansion with the Wannier function. Thouhg it isessentially the same as in Sec.2.3.1, Wannier functions have several convenient points and we may use them afterwards.
In that case, I will introduce it again, but let’s take a quick look at what it is as an appendix.

E3-10

2B.1 Wannier function
The Wannier function is defined as Fourier transform of Bloch function as follows.
1 ∑wn (r − Rj ) = √exp(−ik · Rj )ψnk (r).
N k

(2B.1)The Bloch function is usually given in the coordinate representation but here the spatial coordinate is a parameter and it
is now taken as a function of wavenumber k. The summation over k is inside the Brillouin zone. The Bloch function
is a product of a lattice periodic function and a plane wave and sperad over the space. On the other hand, the Wannierfunction has tendency to localized to the lattice point Rj . This is understood by making the lattice-periodic function in
the Bloch function a constant, which makes the Wanner function completely localized on Rj . Equation (2B.1) can beseen as the expantion of the Wannier function with the Bloch function. Conversely the Bloch function can be expandedby the Wannier function as

1 ∑ψnk (r) = √exp(ik · Rj )wn (r − Rj ).
N k(2B.2)

An advantage in the Wannier function is the orthgonality, that is⟨wn∗ ′ (r − Rj ′ )|wn (r − Rj )⟩ = δjj ′ δnn′ .

(2B.3)

We skip the proof but straightforwardly performed with using the summatoins on the lattice and the reciprocal lattice.
The Wannier function is normalized if it is defined as (2B.1) and also forms a complete set.

2B.2 Derivation of effective mass approximation
The problem is the addition of perturbation potential U (r) to the crystal Hamiltonian H0 , that is[H0 + U (r)]ϕ(r) = Eϕ(r).

(2B.4)The derivation goes almost parallely as that with the Bloch funtion. First we expand the wavefunction with the Wannierfunctions as
ϕ(r) =∑Fn (Rj ′ )wn (r − Rj ′ ).

(2B.5)n,j ′We assume that H1 does not have the elements for interband transition (the amplitude is too small) and we drop n
henceforth. Tne Wannier function w(r − Rj ) is simply written as |j⟩. Equation (2B.5) is substituted into eq. (2B.4) andwith takeing the inner product with ⟨j|, the orthogonality (2B.3) leads to∑∑⟨j|H0 |j ′ ⟩F (Rj ′ ) +⟨j|U (r)|j ′ ⟩F (Rj ′ ) = EF (Rj ).
j′(2B.6)j′As seen above |j⟩ is localized to Rj and because U (r) is slowly varying function in the scale of lattice constant, we
can approximate as∑j′⟨j|U (r)|j ′ ⟩ ≈∑U (Rj ′ )⟨j|j ′ ⟩ = U (Rj ).

(2B.7)

j′

The term of crystal Hamiltonian can be written with shifting the spatial origin as⟨j|H0 |j ′ ⟩ = ⟨w(r)|H0 |w(r − (−Rj ′ + Rj ))⟩ ≡ h0 (Rj − Rj ′ ).

(2B.8)The Bloch funcition ψk (r) is the eigenstate of H0 and the application of the effective mass approximation to the eigenenergy gives⟨ψk (r)|H0 |ψk (r)⟩ = E0 (k) =E3-11ℏ2 k 2.
2m∗(2B.9)

ψk (r) can be expanded as (2B.2) and leads to1 ∑1 ∑exp[−ik · (Rj − Rj ′ )]⟨j|H0 |j ′ ⟩ =exp[−ik · (Rj − Rj ′ )]h0 (Rj − Rj ′ )
N ′
N ′
j,j
j,j
∑
=
exp(−ik · Rj )h0 (Rj ).

E0 (k) =(2B.10)j

The inverse transformation gives
h0 (Rj ) =1 ∑E0 (k) exp(ik · Rj ).
N(2B.11)kThough F (Rj ) is just defined on the lattice point as in (2B.5), since the spatial variation in U (r) is slow, the differences
between the values of F for the neighboring lattice point are small and smooth interpolation is possible. The operator of
spatial shift by a is exp(−a · ∇)*3 , then we can write
F (r − Rj ) = exp(−Rj · ∇)F (r)to obtain∑

h0 (Rj ′ )F (r − Rj ′ ) =j′∑h0 (Rj ′ ) exp(−Rj ′ · ∇)F (r).

(2B.12)j′

On the other hand for (2B.10),
E0 (k)F (r) =∑h0 (Rj ′ ) exp(−ik · Rj ′ )F (r).

(2B.13)j′We formally inverse Fourier transform (2B.12) and (2B.13). Then these equations have the common right hand side if
we make replacement of k → −∇. Therefore we can write∑

h0 (Rj ′ )F (r − Rj ′ ) = E0 (−i∇)F (r).

(2B.14)j′

All the above results are restored into (2B.6) and we replace Rj with a continuous variable r. Then we obtain][ℏ2 2− ∗ ∇ + U (r) F (r) = EF (r).
2m

(2B.15)Here I have introduce the proof in the textbook [1]. It has, though, a small jump in the logic from (2B.13)→(2B.14). Also
the description of Rj → r is rather vague. The derivation in [2] is more strict but it needs the width of paper. For thetransformation Rj → r, clearly written as “to be strict, this should be done in variations."

Appendix 3A: Methods for impurity doping
Various methods have been developed for impurity doping. A part of it is introduced in the following. Impurities to be
doped are called dopants, and base crystal is called host.

3A.1 Mixing of impurities to the raw material
We have introduced the semiconductor crystal growth method, but especially in the method of growing bulk crystals
from a material melt, if impurities are mixed in the raw material in advance, doping may be performed with relatively
good uniformity. In many cases, for example, even in the Czochralski method, a concentration gradient is generated in
the crystal growth direction by segregation. For this reason, various growth measures are taken, such as adding a dopant*3 This can be confirmed, for example by the Taylor expansion.

E3-12

to the melt crucible in order to obtain a uniform doping concentration. In addition, in the case of amphoteric impurities,
there is a possibility that some impurities will be compensated depending on the growth conditions.
In epitaxial thin film growth, by controlling the dopant and irradiating it on the growth surface, it is possible to starta stepwise distribution and create various concentration distributions with high non-equilibrium while maintaining crystallinity. This bf modulation doping method plays a major role in occupying an extremely important position in thesemiconductor industry for semiconductor thin films.

3A.2 Thermal diffusion method
A method in which the host is kept at a high temperature in a state where the dopant is present in a high concentrationon the surface of the host, and is mixed inside by heat diffusion. Methods for increasing the concentration of the surface
include contacting with the vapor of the dopant and pre-depositing a thin film of the dopant on the host surface. In the
figure below, the dopant and host wafer are simultaneously enclosed in a quartz tube and heated together so that the vaporof the dopant flows to the surface of the high-temperature wafer. In some cases, the whole is put into the furnace without
creating a flow. In addition, if some of the constituent elements of the host have a high vapor pressure, it is necessary to
suppress the separation from the surface by mixing the vapor of this element.
In the thermal diffusion method, the concentration is high near the surface and low as it goes inside. It is usually used
to form a device near the surface. It is used for integrated circuit formation because the doping region can be patternedby masking the wafer when the vapor is applied.

Fig. 3A.1 Schematic diagram of thermal diffution doping.

3A.3 Ion implantation
The ion implantation method is used not only for doping but also for cutting and oxidation of the inner layer. As in Fig.
3A.2(a), ions entering from the source are bent by a magnetic field, passed through a diaphragm for mass spectrometry,
then narrowed down by a lens, irradiated on a wafer, and scanned by an XY voltage.
As in the imaginary figure in (b), Since the ions that reach the surface have high kinetic energy, they invade the crystal
in a non-equilibrium manner and stop at a depth corresponding to the average kinetic energy. Since the crystallinity of thepassing region decreases due to the collision of ions and the dopant is not always in the stable position, it is often annealed
after implantation. The distribution of dopants is generally represented by the Gaussian distribution after annealing.
In addition to doping, it was once actively used as one of the Silicon on Insulator (SOI) techniques that form an oxide
film inside by implanting oxygen ions and annealing as described above. At present, the seemingly primitive method ofbonding after surface oxidation is mainly used.

E3-13

(a)

(b)
Fig. 3A.2 (a) Illustration of ion implantation doping. Ions coming out of the source are subjected to mass spectrometry using a magnetic field to sort them, and then the focused beam is scanned onto the wafer. (b) Imaginary
illustration of the host surface during the ion implantation.

3A.4 Irradiation with neutrons
Currently, it is rarely used, but there is an interesting doping method that uses neutrons. For that the following nuclear
reaction is used.
[ n + 30 Si ] → 31 Si → [31 P + β ]With reactor neutrons, extremely uniform doping can be performed without compromising crystallinity. However, due toproblems such as throughput, this method remains at the research level.

Appendix 3B: Shallow donors in Si
For the improvement of the effective mass approximation for the shallow donor levels in Si, we consider the effectof multiple (6) valleys. Since the impurity potential also has a significant magnitude and steepness near the center, it ispossible that it has matrix elements between the eigenfunctions attached to the degenerate valleys. We consider the donor
wavefunction χ(r) = F (r)ψ(r), where F (r) is the envelope function and ψ(r), for each valley and obtain the donor
function as a linear combination of them as
ϕ(i) (r) =

6
∑

(i)

αj χj ,

(3B.1)

j=1

where j is the index of valley and i is the index of symmetry reflecting that of surrounding atoms. Here we assume there
is no mixing between the eigenstates with different quantum number in the trap potential.
While the isotropic potential approximation does not hold in the vicinity of impurities, the spatial symmetry of thecrystal field created by the atoms around the impurity ions must be taken into consideration when taking a linear bond.
It is convenient to use point group theory for the discussion, and I will use it here as well. Regarding the point group,
if I have time, I would like to supplement the minimum knowledge in the appendix of lecture notes etc.[?, 3]. In theE3-14

case of Si, the nearest neighbor atoms are at the apexes of a regular tetrahedron containing impurity ions. The point
group corresponding to the symmetry is expressed as the symbol Td . The elements of point group have correspondences
to the representations, which express symmetry operations. The independent elements have one-to-one correspondence
to the reduced representations. The reduced representations in Td group are A1 , E and T1 , which have single, double,
triple degeneracy respectively and there are six elements. The index i in eq.(3B.1) correspons to these six elements. The
coefficients for these elements are as in the following table.
j
(1)
αj
(2)
αj
(3)
αj
(4)
αj
(5)
αj
(6)
αj

normalization const.
√
1/ 6

1

2

3

4

5

6

expression

1

1

1

1

1

1

A1

1/2

1

1

−1

−1

0

0

E

1/2
√
1/ 2
√
1/ 2
√
1/ 2

1

1

0

0

−1

−1

E

1

−1

0

0

0

0

T1

0

0

1

−1

0

0

T1

0

0

0

0

1

−1

T1

Tab. 3.2 Linear combination coefficients for the donor states in Si.

Then we index the donor states with the quantum number of valley wavefunction chij and the above reduced representation. Here the qunantum number (qn) of χj is determined by main qn n, directional qn l, magnetic qn m but withanisotropy. Hence the indices are like 1s(A1 ), 1s(E) etc.
From Tab. 3.2, we see that in the elements other than A1 , the wavefunctions are superposed in the inverse phase andthe amplitude at the origin is small. In A1 , all the wavefunctions are superposed in phase and the amplitude at the origin
is large. Hence the state of 1s(A1 ) largely deviates from the effective mass approximation and that causes large decreasein the eigenenergy. The larger the positive charge in the nuclear the larger the binding energy. This eneryg splitting iscalled valley-orbit splitting.
Effective mass theory

Li

P

As

Sb

Bi

32

32.5

45

53.7

43

70.6

Tab. 3.3 Donor binding energy in Si (meV)

As in Tab. 3.3, such tendency really appears in 1s(A1 ). Pantelides and Sah gave theoretical calculation of the valleyorbit splitting, which reproduces the experiments well[4]．

References[1] C. Hamaguchi, “Basic Semiconductor Physics" 3rd ed. (Springer, 2017).
[2] P. Yu and M. Cardona, “Fundamentals of Semiconductors", (4th ed. Springer, 2010).
[3] M. Tinkham, “Group Theory and Quantum Mechanics" (Dover, 2003).
[4] S. Pantelides and C. T. Sah, Phys. Rev. B 10, 621-637 (1974); ibid. 638-658 (1974).

E3-15

Lecture note on
25th April (2021)

Physics of Semiconductors (4)Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

3.3.1 Degenerate semiconductors
So far we treat the impurity states in semiconductors as isolated. In such cases, as shown in Fig.3.2, charge carriers
disappear at low temperatures, the conductivity is lost, and the system is insulating. Now we consider doping to higher
impurity densities, where the average distance between the dopants is similar to or less than the spatial size of the
wavefunctions. Then the overlapping of wavefunctions enables tunneling between the impurity sites. Such tunnleings
may form a kind of conducting network in the crystal and with further increasing the impurity concentration finallyspreads the network over the whole crystal, which now has a finite conductance at the lowest temperature, thus is a metal.
This problem – metal-insulator transitions, MIT – has been one of the most important problems in condensed matterphysics, and huge amount of efforts have been devoted for years. The field of MIT extends over various phenomena incondensed matter physics, far beyond the doped semiconductors. We have not reached the final answer through greatamount of knowledges have been accumulated. There are so many textbooks, very few of which are listed in references([1]∼[5]).
In the above we have defined the MIT as the spatial size of the wavefunctions at the Fermi level. The phenomenon isobserved in the energy space as follows. With overlapping of neighboring wavefunctions, as we have seen in the tightbinding model (regular, coherent case), the energy levels broadened and a band is formed, which we call an impurity
band. Even under the formation of impurity band, in which the density of states is continuous, the wavefunction at the
Fermi level is not necessarily spread over the entire crystal. It was first pointed out by Anderson that the electrons in a
potential with a certain degree of disorder are spatially localized. This is called Anderson localization. Hence, somelower part of the impurity band is usually localized and the boundary is called a mobility edge.
It is well known that as shown in Fig. 3.6, in many matrix crystals and species of dopants, an empirical relation,∗
n1/3
c aB = 0.26

(3.32)100
InSb

n c 1/3a B *=0.26

a B * (nm)

10
Ge:Sb
Ge:P

Ge:As
CdS:ClCdS:In
Si:Sb
Si:As
Si:P

1

GaP:ZnCH 3 OH :Li
WSe 2 :TaWO 3 :Na

0.1

56
logn c 1/3

Ar:Na
Xe:Hg

7

E4-1

Fig. 3.6 Experimental values for the critical concentrations of the MIT and the effective Bohrradiuses for various matrix semiconductors anddopants (the element symbols put after the colons).
The data are plotted in log scale. The unit for nc is
cm−3 . The line indicates the empirical relation ineq.(3.32). The data are taken from P. Edwards and
M. Sienko, Phys. Rev. B 17, 2575 (1978).

holds between the critical impurity concentration nc for the MIT and the effective Bohr radius a∗B . This criterion isnatural from the view of impurity band formation and there are many trials to derive it from more rigorous theoreticalbackground.
The largest difficulty in solving this problem lies in the treatment of disorder, which makes it impossible to utilize thecoherence of the scattering from the crystal lattice. In the band theory, the coherence brings about great simplicity. In thecase of MIT in disordered systems, one should directly treat the disorder itself.
Though the final answer has not been found, or it is not even known whether there is such one or not, many new
physical idea have been developed, which have greatly expanded our knowledges on random systems. The concepts havebeen applied various fields such as organic semiconductors. We do not go into this problem further due to the space-time
limitations.
The MIT in ordered systems is also an important and difficult problem, particularly in so called strongly correlated
systems. Here I just list a review paper[6], which was published a quarter century ago. I will not go into this problem inthis lecture.
There are a number of devices which utilize such degenerate semiconductors. A representative is the Esaki diode(tunnel diode), which first attracted attensions as a device for fast switching and actually was used in counter circuits inthe experiments of high energy physics. Recently, the Esaki diode is also used in the interfacial connection of multiple
junction solar cells. In many of ordinary solar cells, highly-doped degenerate semiconductors are used as the upper
layers of the junctions. The high level doping is also used in the IMPATT diodes for high-frequency use, and p-i-n typephoto-diode, etc.

3.3.2 Excitons
Here we introduce the concept of exciton, which is a bit tail subject as “carrier statistics” but we can view
it as an application of the effective mass approximation.

Exciton has long been the central theme of opticalproperties[7], but even more extensive research is still underway, such as the BEC of exciton polaritons. In solids,
the bound states formed by Coulomb force between quasiparticles of positive and negative charges are called exciton.
When the quasiparticles are spread over several lattice constants,
the exciton is called “Wanner type.” When the charge polarization
occurs within a molecule or over very few lattice points, it is called“Frenkel type.” The latter often found in organic semiconductors,
in which the molecules at the lattice points are comparatively well
separated. Here we concentrate ourselves on the Wannier type.
Let us consider an exciton state with an electron excited to the
conduction band, and a hole excited to the valence band. These
spread over several lattice points or more, and effective mass approximation can be applied. Based on the free state of both electrons
and holes, even if they create a bound state, the degree of freedomof the movement of the center of mass remains, and the ”wave number” and kinetic energy due to this remain. This wavenumber should
be derived from the overall wavenumber conservation since the concept of holes is also introduced by considering the conservation of
the total wavenumber. The “mass” of the exciton also would be introduced simply by taking the sum of effective masses of electrons
and holes, as me + mh . I have used the expression “would” because the Coulomb force, which is most natural candidate

E4-2

for the force in the equation of motion for electrons and holes, works in the opposite directions, the accelerations for
two kinds of particles with different effectve masses is a bit complicated for the treatment. Anyway we assume that theeffective mass approximation holds. Then the creatio energy of an exciton from the state without the electron and thehole, hence including the electron-hole pair creation energy, is written as(n)Eex(k) = Eg +e4 m∗r 1ℏ2 k 2−,2(me + mh )2ℏ2 n2n = 1, 2, · · · ,(3.33)

where the third term is the binding energy of the electron and the hole. We assume the system is isotropic and the excitonis hydrogen-like. mr is the reduced mass defined as111= ∗ + ∗.
∗
mr
me
mh

(3.34)The second term in (3.33) is the kinetic energy of the parallel motion. The dispersion described in eq.(3.33) is illustrated
in the figure in the previous page. The existence of such bound states can be confirmed by checking, e.g. the opticalabsorption spectra, which we will see in the next chapter.
However, for example in optical absorption experiments, there often appear many absorption peak spectra which cannot
be interpreted simply with eq.(3.33). The candidates for the interpretation of those observations are, the excitons trappedby impurity potentials (bound excitons), the exciton molecules made of more than two excitons, or such complicatedexcited states.

In the above illustration, the cases for the number of charged excitation including zero or single donor is from three tofour (corresponding to hydrogen molecule or its charged state) are listed. Such excited states are called exciton comlexes.
In this chapter, we introduced electrons excited in the conduction band, holes excited in the valence band, and excitons,
which are bound states of these excitations.
All of them are many-body states of electrons, but they can be treated as if “particles” are freely moving in the space of acrystal, which is different from the vacuum. Such free particle-like pictures, in which many-body effects are renormalized
are called quasi-particle.

E4-3

Chapter 4 Optical response of bulk semiconductors

Many of the substances called “semiconductors” have bandgaps around the energy region of electromagnetic wave
called “light”, and have characteristic optical responses. The optical response is one of the most important subject as wellas the carrier transport. In the optical devices such as detectors or emitters, semiconductors are mostly used as active
materials. In this chapter, as the first look, we see basic optical properties of semiconductor bulk materials.

4.1 Optical response of two-level systems
In order to consider the optical response of semiconductor bulk, we should investigate the relationship between lightand the transition between the extended electronic states of the valence band and conduction band that we have seen
so far. But here, we begin with the optical response of a much simpler “two-level system.” The reason why we devote
our pages to such basic matters here is that we want to confirm the zero-point oscillation of the electromagnetic field
and the state of photons in particular. The following two sections are for the lecture to be just self-contained. For more
complete description, see the textbooks listed in [8]. If the reader already has such knowledge, the skip to Sec.4.1.3 isrecommended. In addition, if he/she is already used to the two-level systems, a further skip to Sec.4.2 is also OK.

4.1.1 Quantizationo of electromagnetic field
We have a very short look at the quantization of electromagnetic field to consider the states of photons[8]. As thebasics we start with the one-dimensional harmonic oscillator, which subject appears in the beginning part of elementary
quantum mechanics. The problem is discribed as the Schrödinger equation;
[
]
ℏ2 d2
mωh2 x2−
ϕ = Eϕ.
+2m dx22(4.1)

The second term in the parenthis in the left hand side represents the potential characteristic for the harmonic oscillator.
We define a dimensionless variable q with√x=

and rewrite (4.1) as

We introduce down and up operators
(
)
1
d
a= √
+q ,
2 dq

ℏωh
2

(
−

(4.2)

)
d2
2
+
q
ϕ = Eϕ.
dq 2

1
a =√
2
†

ℏ
q,
ωh m

(
)
d
− +q ,
dq[a, a† ] = 1, (others) = 0,

where the commutation relation is readily derived from the definition. Then eq.(4.3) is furthre rewritten as()
1†ℏωh a a +ϕ(≡ Ĥϕ) = Eϕ.
2
If we define the number operator asn̂ ≡ a† a.

E4-4

(4.3)

(4.4)

(4.5)

(4.6)

Because n̂ and Ĥ commute with each other ([n̂, Ĥ] = [Ĥ, n̂] = 0), they have common eigenfunctions. Here we assume|w⟩ is a eigen function common for Ĥ and n̂ with eigen[values ϵ, γ respectively. From the commutation relation eq.(4.4),
we seeĤ(a† |w⟩) = (γ + ℏωh )(a† |w⟩),

Ĥ(a|w⟩) = (γ − ℏωh )(a|w⟩),(4.7)

that is, a† |w⟩, a|w⟩ are also such simultaneous eigenfunctions with the energy eigenvalues of up and down shifts by ℏωhrespectively. Let |0⟩ be the simultaneous eigenstate with the lowest energy eigenstate ϵ0 . Since there is no eigenstate withenergy eigenvalue of ϵ0 − ℏωh , the above equation leads to a|0⟩ = 0. Furthermore, ϵ0 = ℏωh /2 is concluded.
On the other hand, the eigenstates with higher energy eigenvalues than ϵ0 can be obtained by sequential application of†a to |0⟩. The eigenvalues are()
1
En = ℏωh n +2(n = 0, 1, 2, · · · ).

(4.8)

And the commutation relation tells that the operatore an (a† )n works as multiplication of n!. Then the normalized
eigenfunction for the eigenvalue En can be obtained from the normalized |0⟩ is written as(a† )n|n⟩ = √ |0⟩.
n!

(4.9)

Also from a|0⟩ = 0, a solution of |0⟩ = φ0 (q) is straightforwardly obtained as
( 2)
dφ0
1
q
+ q 2 φ0 = 0 ∴ φ0 = 1/4 exp −
.
dq
2
π

(4.10)

Based on the above knowledge, we go to the electromagnetic field. Our starting point here is the fact that the electromagnetic field is mathematically equivalent to a set of harmonic oscillators *1 . We take Coulomb gage (divA = ⃗0), andexpand the vector potential A with the plane waves as follows.
∑A(r, t) =(Akλ ei(k·r−ωkλ t) + A∗kλ e−i(k·r−ωkλ t) ),(4.11)k,λ(ωk = c|k|,A∗kλ = A−kλ ).

Here λ represents the freedom of polarization. From the selection of Coulomb gauge, the electromagnetic wave shouldbe transverse and λ represents two-dimensional freedom. From the Maxwell equaiton E = ∂A/∂t, B = rotA, theenergy of electromagnetic field E in volume V is written as∫E=2[ϵ0 E 2 (r, t) + µ−1
0 B (r, t)]

V

∑
d3 r
= 2ϵ0 Vωkλ (Akλ · A∗kλ ),2(4.12)k,λbecause the terms with exp(±2ik · r) vanish with spatial integration.
Then we introduce variables(vectors) Qkλ , Pkλ as√√
Qkλ = ϵ0 V (Akλ e−iωkλ t + A∗kλ eiωkλ t ), Pkλ = dQkλ /dt = iωkλ ϵ0 V (−Akλ e−iωkλ t + A∗kλ eiωkλ t ).
E is expressed as
E=

1∑ 2
2
(Pkλ + ωkλ
Q2kλ ),
2(4.13)(4.14)

kλwhich tells the electromagnetic field is described as a set of harmonic oscillators in Q space. Then the field can bequantized with changing P , Q to operators and require the following commutation relations.
[Q̂k′ λ′ , P̂kλ ] = iℏδkk′ δδδ′ ,(others) = 0.

(4.15)*1 According to the literature[10], this is called “Jeans theorem.” Actually, in ref.[9], that “theorem” is proven. The discussion then leads to theRayleight-Jeans law. However, there is more famous “Jeans theorem”, which is on the distribution of particles with gravitational interactions[9,
10]

E4-5

The Hamiltonian is in the same form with (4.3).
Ĥ =

1∑ 2
(P̂kλ + ωk2 Q̂2kλ ).
2(4.16)

kλcreation/annihilation operators, which corresponds to up/down operators, area†kλ = √1(ωk Q̂kλ − iP̂kλ ),2ℏωkakλ = √

1
(ωk Q̂kλ + iP̂kλ ).
2ℏωk(4.17)

From (4.15), the commutation relations[akλ , a†k′ λ′ ] = δkk′ δλλ′ ,(others) = 0(4.18)are derived. Finally (4.16) can be quantized in the Hamiltonian form as
(
)
∑
1
Ĥ =
ℏωk a†kλ akλ +.
2(4.19)

kλThe vector potential, for example, can also be written in the form of operator as
√
[
]
∑
ℏ
Â(r, t) =ekλ akλ ei(k·r−ωk t) + a†kλ e−i(k·r−ωk t) .
2ϵ0 ωk V(4.20)kλ4.1.2 States of photons
Corresponding to eq.(4.6), the operator†n̂kλ ≡ akλ
akλ(4.21)

represents the revel of excitation in the mode (k, λ) from the ground state |0⟩kλ . Single step of the excitation corresponds
to the energy of ℏωk , which is also the energy of single photon in the Einstein relation of photon quantum. Hence n̂kλ in(4.21) can be interpreted as photon number operatorin mode (k, λ). As in (4.9), there are eigenfunctions of (4.19) inwhich the number of photons in mode (k, λ) is nkλ . We use the expression that the symbol {· · ·α } represents the set ofelements with α as the index. Then the state is represented as |{nkλ }⟩.
From (4.9), we call the state described in the form
Number state]∏ (a† )nkλ
kλ√
|0⟩|{nkλ }⟩ =
nkλ !
kλ
[

(4.22)



as number state .
The expectation value of the energy of the number state is⟨{nkλ }|Ĥ|{nkλ }⟩ =

∑
kλ

(
ℏωk

1
nkλ +
2

)
.

(4.23)

A state with multiple mode is a superpositon of eigenstates with different eigenvalues and is not an eigenstate of the total
Hamiltonian. On the other hand, the expectation value of the electric field is from (4.20),⟨{nkλ }|Ê|{nkλ }⟩ = −⟨{nkλ }|(∂ Â/∂t)|{nkλ }⟩ = 0.

(4.24)

That is the expectation value of the electric field is zero. This does not mean the time-average makes it zero. Even for themeasurements in very short time, the average over many measurements is zero. Just the same for the magnetic field. On
the other hand the quantum fluctuation in the electric field is)∑ ℏωk (
1
1
2⟨{nkλ }|Ê |{nkλ }⟩ =nkλ +=⟨{nkλ }|H|{nkλ }⟩,
ϵ0 V
2
ϵ0 V
kλ

E4-6

(4.25)

which is non-zero. Furthermore, even for the photon number zero state, each mode has the fluctuation of ℏωk /(2ϵ0 V ),which is called zero-point motionof electromagnetic field. The zero-point fluctuation corresponds to 1/2 in the energy
expression of (4.23). This is very important property for the spontaneous emission of photon. The reason of using spacefor free electromagnetic field is to describe this clearly.
The properties of number state described above indicate that it is difficult to coherently superimpose the oscillating
electromagnetic field of multiple photons to obtain the oscillating electromagnetic field as in the classical picture in theenergy eigenstate where the number of photons is fixed. On the other hand, by superimposing several states, it is possible
to create a state with a finite expected value of the electromagnetic field. For example, the number states for a single
mode (hence for a while we omit writing the mode index as |n⟩) can be summed up with Gaussian weight to get
Coherent state
(
)
(
) ∞
|α|2
|α|2 ∑ αn√ |n⟩,|α⟩ = exp −exp(αa† )|0⟩ = exp −
2
2
n!
n=0



(4.26)



where α is a complex parameter. The state expressed as (4.26) is called coherent state. When the annihilation operator√
is applied, from a|n⟩ = n|n − 1⟩,a|α⟩ = α|α⟩,(4.27)
that is the coherent state is the eigenstate of the annihilation operator with the eigenvalue of α. This means that thecoherent state is a superposition of an infinite number of states, and even if quantum mechanical ”measurement” isperformed on the single photon in it, the whole state remains unchanged. If we measure the photon number in this state,
the probability of detecting n-photons ise−|α| |α|2n,n!
2

P (n) = ⟨n|α⟩ =(4.28)

which is a Poissonian distribution. We write the complex parameter α in the amplitude and the phase as α = |α|eiϕ .
Then the expectation values of the electric field and the magnetic field are√2ℏωk⟨α|Ê(r, t)|α⟩ = −|α|ekλ sin(k · r − ωk t + ϕ),
ϵ V
√ 0
2ℏ
⟨α|B̂(r, t)|α⟩ = −|α|k × ekλ sin(k · r − ωk t + ϕ).
ϵ0 ω k V(4.29a)(4.29b)This means classical electromagnetic wave is reproduced in the coherent state.

4.1.3 Basic optical processes in two-level systems
The two-level system composed of two qunatum states is also called qubit in the field of quantum information and is
the most basic quantum system. As in Fig. 4.1, we consider a two-level electronics system of (|a⟩, |b⟩) with the energy
eigenvalues (Ea , Eb ). We take these

H0 |a⟩ = Ea |a⟩,

H0 |b⟩ = Eb |b⟩(4.30)

as the basis and the general state can be written as
ψ(t) = ca (t)e−Ea t/ℏ |a⟩ + cb (t)e−Eb t/ℏ |b⟩.

(4.31)In Fig. 4.1, three basic optical processes in the two-level system are illustrated. (a) is the optical absorption, in which
the electron absorbs the photon energy and makes transition of |a⟩ → |b⟩. (b) is the spontaneous emission of photon
associated with the transition |b⟩ → |a⟩ of the electron initially excited to |b⟩. (c) is the slimulated emission, in which
the first photon comes to the excited state |b⟩ to sitimulate the emission of the second photon coherent to the first one.
E4-7

Fig. 4.1 Three basic optical processes in two-level systems (a) optical absorption, (b) spontaneous emission of aphoton, (c) stimulated emission of a photon

We write the Hamiltonian of the system with electromagnetic field in non-relativistic approximation as
Hop =(p + eA)2+ V (r),2m(4.32)

where A is the vector potential and we treat it as perturbation. We then drop the term of A2 and the perturbation
Hamiltonian H ′ can be defined as

Hop ≈ H0 +e
A · p ≡ H0 + H ′ .
m(4.33)For simplicity, we assume H ′ does not have the diagnal terms.
⟨a|H ′ |a⟩ = ⟨b|H ′ |b⟩ = 0.

(4.34)

We consider the case that a plane electromagnetic wave is applied to the two-level system, which wave is described in
Coulomb gauge (divA = 0) as
A = A0 ep cos(kp · r − ωt).

(4.35)

As we saw in the previous section, this means a coherent state comes to the two-level system. The perturbation Hamiltonian isH′=

eA0
ep · p̂ cos(kp · r − ωt).
m

(4.36)

This approximation is called dipole approximation from the following reason. The matrix element of H ′ for |a⟩ → |b⟩is with writing A0 cos(kp · r − ωt) as A,
eA
eA
m
iA
ep · ⟨b|p̂|a⟩ =⟨b|ep · [r̂, H0 ]|a⟩ =(Eb − Ea )ep · ⟨b|(−e)r̂|a⟩.
m
m
iℏ
ℏ

(4.37)The last term is the transition element of the electric dipole momen operator and the transition by the Hamiltonian (4.36)
is called dipole transition.
Substituting (4.30) to the Schrödinger equation iℏ∂ψ/∂t = (H0 + H ′ )ψ, we obtain
[
]
dca
dcb
iℏ
|a⟩e−iEa t/ℏ +|b⟩e−iEb t/ℏ = ca H ′ |a⟩e−iEa t/ℏ + cb H ′ |b⟩e−iEb t/ℏ .
dt
dt(4.38)Taking inner products with ⟨a| and ⟨b| leads to the following simultaneous differential equations for (ca , cb ).

dca
i


= − cb ⟨a|H ′ |b⟩e−iω0 t ,dtℏ dcb = − i c ⟨b|H ′ |a⟩eiω0 t ,
a
dt
ℏ

E4-8

ω0 ≡

Eb − Ea.
ℏ(4.39)

4.1.4 Optical absorption, emission(1)

Frist we consider the optical absorption in (a). We take ca (0) = 1, cb (0) = 0 as the initial condition and ca (t) = 1 as
the starting point of the sequential substitution method to get the approximate solution.
c(1)a (t) = 1,
(1)

cb (t) = −

i
ℏ

c(2)
a (t) = 1 −

∫ t
0

1
ℏ2

′

⟨b|H ′ |a⟩(t′ )eiω0 t dt′ ,∫ t(4.40a)′dt′ ⟨a|H ′ |b⟩(t′ )e−iω0 t

[∫ ′
t

0

]
dt′′ ⟨b|H ′ |a⟩(t′′ )eiω0 t′′.

(4.40b)0

The expression ⟨b|H ′ |a⟩(t) is to clarify that ⟨b|H ′ |a⟩ is a function of t.
To consider ⟨b|H ′ |a⟩(t), we drop the term of photon wavenumsin 2 [( w-w 0 ) t /2]/( w-w 0 ) 2 (arb.)

ber kp from (4.36) because of the following reason. We consider-2p

the process in which the electron wavenumber varies as 0 → ke tot =2enhance the kinetic energy by ℏ2 ke2 /2m0 . If we consider a pho-t =1.5ton with the energy that matches to this energy enhancement, the√photon wavenumber kp is obtained from ke =2kec kp . Heret =1

kec = m0 c/ℏ is the product of electron rest mass m0 and the speedof light c, corresponding to a very large momentum. Then ke ≫ kp

t =0.5and we can ignore the contribution of photon momentum.
Now we write

-p

0
w-w 0

p

Vba ≡ ⟨b|2p

eA0
ep · p̂|a⟩m(4.41)

and from (4.40a),
[
]
∫ t
′
i
Vba ei(ω0 +ω)t − 1 ei(ω0 −ω)t − 1
cb (t) ≃ − Vba+
dt′ cos ωt′ eiω0 t = −ℏ2ℏω0 + ω
ω0 − ω
0
Vba sin[(ω0 − ω)t/2] i(ω0 −ω)t/2
≃ −i
e
.
ℏ
ω0 − ω

(4.42)

Then if we apply the oscillating electromagnetic field from t = 0, the probability amplitude of |b⟩ at time t isPb (t) = |cb (t)|2 ≃|Vba |2 t sin2 [(ω0 − ω)t/2]ℏ2 2 (ω0 − ω)2 (t/2)(4.43)

As is well known the last factor goes to a delta function in the limit of t → ∞ (limt→∞ sin2 [(ω0 − ω)t/2]/(ω0 −ω)2 (t/2) = πδ(ω − ω0 )). Actually in the plot of sin2 [(ω0 − ω)t/2]/(ω0 − ω)2 , the peak at ω − ω0 = 0 grows high andsharp with the increase of t. Hence the factor represents the energy conservation.
When we take the initial condition as cb (0) = 1, ca (0) = 0, that is, the excited state |b⟩ with photon field of ω, thetransition |b⟩ → |a⟩, which is the reversed process of the optical absorption, occurs with the emission of a photon of ω0 .
This emitted photon is coherent to the existing coherent photon state[11]. This is the stimulated emission in (c).
Then how we treat spontaneous emission in (b) without photon field outside? As we saw in Sec. 4.1.1, even in thevacuum with zero number of photons, the zero-point quantum fluctuation of electromagnetic field exists for each mode.
In the spontaneous emission process, the photon emission is “stimulated” by these zero-point fluctuations. Most of thelight emission from semiconductor devices (other than lasers) is by this spontaneous emission. In that sense, we arelooking at zero-point quantum fluctuation when we are facing electric displays.

E4-9

4.1.5 Rabi oscillation
In the perturbation term Vba cos(ωt), the cosine part can be expressed as (eiωt + e−iωt )/2, which means the decomposition into two terms rotation on the complex plane with the angular frequency of ±ω. Among them, the component
important for the transition is with ω ∼ ω0 and the term of rotation with −ω has the frequency very far from ω0 and canbe ignored as⟨a|H ′ |b⟩ =Vab iωte .
2(4.44)

This kind of approximation is called rotation wave approximation, in which the simultaneous differential equations are

i
dca


= − cb Vab e−i(ω0 −ω)t ,dt2ℏ dcb = − i c V ei(ω0 −ω)t .
a ba
dt
2ℏ

(4.45)

This can be written into a differential equation for a single variable as
d2 cb
dca
|Vab |2
+
i(ω
−
ω
)
+
= 0.
0
dt2
dt
(2ℏ)2

(4.46)The solution is obtained straightforwardly as
cb (t) = c+ eiλ+ t + c− eλ− t ,√1(δ ± δ 2 + |Vab |2 /ℏ2 ),2

λ± ≡Under the initial condition |ca (0)| = 1, cb (0) = 0,i|Vab | iδt/2 cb (t) =esin(ωR t/2),

ωR ℏ
[
(
)
)]
(
ωR t
δ
ωR t

iδt/2


c
(t)
=
e
cos
−
i
sin
 a
2
ωR
2
are obtained whereωR ≡√δ 2 + |Vab |2 /ℏ2δ ≡ ω0 − ω.

(4.47)(4.48a)(4.48b)(4.49)

is called Rabi frequency.
The oscillation between the two-levels caused by the electromagnetic wave (photon) with energy close to the energy
difference between the two levels is called Rabi oscillation. When the photon energy is tuned to the energy difference(δ = 0), the Rabi frequency is proportional to the magnitude of the electromagnetic irradiation. The Rabi oscillation
is widely used in various resonance phenomena utilized to get information inside materials, or quantum information
processing, etc.

4.1.6 Oscillator strength, selection rule
For the one-dimensional harmonic oscillator, which we consider in the beginning of this chapter, from (4.2)，(4.4), we√

can writex̂ =ℏ(a + a† ),2mωh(4.50)

which leads to the dipole transition elements of (4.37), corresponding to |0⟩ → |1⟩ is√⟨1|(−e)x̂|0⟩ = −e

E4-10

ℏ
≡ µ10 .
2mωh(4.51)

The probability of the dipole transition |a⟩ → |b⟩ is indicated by the transition dipole moment;µba ≡ ⟨b|(−e)r̂|a⟩.

(4.52)

Then for the “unit” of the strength, we take the transition dipole moment µ10 for the one-dimensional harmonic oscillator
with the characteristic frequency ωh = (Eb − Ea )/ℏ. For the probability we need to take the square of the absolute value,
we define oscillator strength asfba =|µba |22mωba=|µba |2 .
2|µ10 |
e2 ℏ(4.53)

The character f is commonly used and we also call it as “f -value.”
When there are multiple possible final states |b⟩, we use b as the index of all such states. Then the f -values sutisfy thefollowing sum rule.

∑fba = 1.

(4.54)b

For the system with N -electrons, the right hand side is N .
When the system has multiple directional oscillators with random directions, the effective transition dipole mement⟨µeﬀ ⟩ is given by taking the avarage as µba /3. Then the oscillator strength is expressed as
′
fba
=

2mωba
|µba |2 .
3e2 ℏ(4.55)

When the system has spatial inversion symmetry, the eigenstates of the Hamiltonian should have the parity for the
spatial inversion operation. That is, for an eigenstate ϕn (r), the following should hold.
ϕn (−r) = ±ϕn (r).

(4.56)+, − correspond to even and odd parity respectively. In the expression (4.52), r has the odd parity then if |a⟩, |b⟩ havethe same parity, the integration gives zero for µba and the dipole transition is forbidden. As above, the rule that dominates
the possibility of a transition along with symmetry, quantum number etc. is called selection rule.

4.2 Interband transition and optical response
So far we have seen very basic knowledges on the optical response of two-level systems. We now expand the concepts
and the discussions to the electronic states in solids, in which both ground states and excited states are extended over the
crystals.

4.2.1 Absorption of light with interband transition
Materials absorb electromagnetic wave in various ways. Free carrier absorption, impurity absorption, absorption bylattice vibration, etc. though the main absorption used in the optical devices is the absorption due to the interband
transition of electrons. Thus in this sub-section, we will see the very basics of the interband transition absorption.
For simplicity, we write a plane electromagnetic wave with a linear polarization propagating along z axis with vector
potential A as
A = A0 e exp[i(kp · r − ωt)].

(4.57)

The wavenumber kp is (0, 0, kp )，e is the polarizaiton vector and we put ex = (1, 0, 0). The electric field E = −∂A/∂t,
the magnetic field H = µ−1 rotA (µ is the permeability of the medium), then the energy flow density (Poynting vector)isI = ⟨E × H⟩ =ϵ0 cn̄ω 2 A20
ez .
2

E4-11

(4.58)

√n̄ is the refractive index (light speed in the medium is c′ = 1/ ϵ1 ϵ0 µ1 µ0 (ϵ1 , µ1 are the ratio of dielectric constant and√that of magnetic permeability to those of vacuum) n̄ = c/c′ = ϵ1 µ1 ), ez = (0, 0, 1).
The absorption of light causes the exponential damping of the intensity |I| as I(z) = I0 exp(−αz). The damping
constant α is the absorption coefficient. From this definition α = −dI/Idz = −dI/Ic′ dt. Thus if we assign the
averaged number of photons absorbed in the unit time and the unit volume as W , then the decreasing rate of I is written
as ℏωW giving
α=

ℏωW
2ℏωW
=
.
I
ϵ0 cn̄ω 2 A20(4.59)

Among the various absorption mechanisms, that caused by a valence electron absorbing a photon and being excited tothe conduction band, is called fundamental absorption. The fundamental absorption begins just above the band gap.
The absorption just at the band gap is called “band edge absorption”.
We write the Hamiltonian of the system as H = (p + eA)2 /2m0 + V (r) and treat A as a perturbation. With ignoring
A2 , H = H0 + (e/m0 )A · p. Bloch functions in conduction band and valence band are written as |ck⟩ = uck eikr ,|vk⟩ = uvk eikr respectively and the perturbation term causes the transition from the valence band to the conduction
band with the probability Wvc per unit volume in the Fermi’s golden rule approximation as
Wvc =

2πe
πe2 2
A |M |2 δ(Ec (k) − Ev (k′ ) − ℏω),|⟨ck|A · p|vk′ ⟩|2 δ(Ec (k) − Ev (k′ ) − ℏω) =ℏm0
2ℏm20 0(4.60)∑ i(kp +k′ −k)·Rl ∫ed3 r i(kp +k′ −k)·r ∗′M=euck (r)e · (p + ℏk )uvk′ (r) = l
d3 ru∗ck (r)e · (p + ℏk′ )uvk′ (r)
V
V V
Ω
∫
N
d3 ru∗ck (r)e · (p + ℏk′ )uvk′ (r). (4.61)= δkp +k′ −k,KVΩ∫

Here, l is the label of lattice points, V , Ω are the volumes of the system and the unit cell respectively. K is a reciprocallattice vector, kp a photon wavenumber, N the total number of the lattice points, N Ω = V .
In eq.(4.61) we implicitly consider a direct excitation of an electron by the electromagnetic field of a photon. Sucha transition is called a direct transition. The necessary condition for the momentum conservation in a fundamental
absorption is kp + k′ − k = K, though practically from common values of band gaps, effective masses and lattice
constants, it turns to be K = 0. Also within the dipole transition approximation, kp can be ignored and we can put

E(k)hn direct

k
(a)

hn g

hn g

hn

hn g

q

k
(b)

hn

k
(c)

Fig. 4.2 Illustrations of optical response due to the interband transition of an electron. (a) Optical absorption withdirect interband transition. (b) Optical absorption with indirect interband transition. (c) Photoluminescence, in which
optically excited electron-hole pair recombine for the light emission.

E4-12k = k′ . uck (r), uvk (r) belong to different eigenvalues hence the term of ℏk′ in (4.61) vanishes giving∫M=

d3 r ∗uck (r)e · puvk (r).
Ω Ω

(4.62)

From (4.59) and (4.62), we assume k-dependence of M is weak and obtain the expression for the absorption coefficientfor direct transition as∑πe2
δ(Ec (k) − Ev (k) − ℏω).
|M |22n̄ϵ0 ωcm0

αda =(4.63)k

The summation part on k is called joint density of states. Let us write it as Jcv (ℏω) and Ec (k) − Ev (k) as Ecv (k), andturn the summation on k in an integral form to get
Jcv (ℏω) =∑∫δ(Ecv (k) − ℏω) = 2kd3 k
δ(Ecv (k) − ℏω).
(2π)3(4.64)

We transform the integral in k-space into that on the infinitesimal area dS on an equi-energy surface and on the energy
Ecv . By writing the k-component perpendicular to the equi-energy surface as k⊥ , the integration can be transformed intodk⊥
dEcv = dS|∇k Ecv |−1 dEcv ,
dEcv
∫
2
dS
∴ Jcv (ℏω) =.
3(2π)|∇k Ecv (k)|Ecv =ℏω

d3 k = dSdk⊥ = dS

(4.65)

From the above we see that we have absorption anomalies around the points where the integrand of (4.65) vanishes.
Let us consider the case of a direct gap semiconductor as illustrated in Fig. ??(a), and assume Ecv = Eg , ∇k Ecv = 0 at
k = k0 . In the expansion of Ecv around k0 , the first order term is zero and taking the second order term we get
Ecv (k) = Eg +

∑ ℏ2
i

2ξi

(ki − ki0 )2 .

(4.66)For simplicity let ξi > 0(i = 1, 2, 3). With variable translation (ℏ/(2ξi )1/2 )(ki − ki0 ) = si ,Ecv = Eg +∑√s2i ≡ Eg + s2 , d3 k =i

8ξ1 ξ2 ξ3ds1 ds2 ds3 ,ℏ3We also consider the integration in s-space with that on equi-energy surfaces and on the energy. Because |∇s Ecv | = 2s,
2
Jcv =
(2π)3

√

8ξ1 ξ2 ξ3
ℏ3

∫

dS
1
=
2s
2π 2

√

√
3/2
8ξ1 ξ2 ξ3 √2 mr √ℏω−E=ℏω − Eg .
gℏ3
π 2 ℏ3(4.67)

The calculation in the last line is for a direct gap semiconductor as illustrated in Fig. ??(a), based on the assumption of∗−1+ m∗−1
isotropic effective mass at the band edge, i.e., ∀i ξi = mr . From m−1
this is the reduced mass for anr = mehelectron-hole pair. After all, (4.67) is the density of states in a three dimensional k space, that is just a re-calculation of
the density of states in a three dimensional system (2.14). In this case, from the expression of the absorption coefficientin a direct transition (4.63),α(ℏω) =e2 (2mr )3/2 |M |2 √ℏω − Eg ,2πϵ0 m20 n̄ωcℏ3(4.68)

is obtained. The factor in the left hand side other than the joint density of states
fvc =2|M |2,m0 ℏω

(4.69)

in which |M |2 /ω is representing the strength of the transition. The dimensionless quantity fvc is called oscillator
strength.

E4-13

Appendix 4A: Rate of stimulated emission, spontaneous emission
Here we consider many identical two-level systems with states (|a⟩, |b⟩). They are placed in the electromagnetic fieldwith the energy density spectrum U (ω), where ω is the angular frequency. There is no direction interaction between
the two-level systems while they are in equilibrium with the electromagnetic field, which is in thermal equilibrium, i.e.
has the energy distribution of Planck law of radiation, and the momentum distributes isotopically. The rate of opticalabsorption (frequency per unit time) for ω ∼ ω0 is obtained from (4.37) as|⟨b|H ′ |a⟩| = |E0 ep · ⟨b|(−e)r̂|a⟩| = |E0 ep · µba |,(4A.1)

where E0 = ωA0 is the amplitude of the oscillation in electric field. From(4.43), the absorption probability is proportional
to the square of the above, hence to E02 ∝ U . In the form of equation the absorption rate Wba is written as
Wba = Bba U (ω),(4A.2)

with Bba a coefficiant. We write the emission rate as the sum of the rate for spontaneous emission, which is independentof U and the rate for the stimulated emission, which is proportional to U .
Wab = A + Bab U ((ω).

(4A.3)As seen in Sec.4.1.4, the optical absorption and the stimulated emission are in the relation of reversed process,
Bba = Bab ≡ B.

(4A.4)

We write Eb − Ea = ℏω, and let the numbers of the two-level systems in the states a, b as Na , Nb respectively. Then()ℏω
Nb = Na exp −.
(4A.5)kB T
Because the system is in equilibrium, the event frequencies of emissions and absorption should be the same, i.e.
BU Na = (A + BU )Nb .

(4A.6)

These leads to the following expression for U (ω).
U (ω) =A
1.
B exp(ℏω/kB T ) − 1(4A.7)We request this to be equivalent to the Planck law of radiation
U (ω) =1ℏω 3,
π 2 c3 exp(ℏω/kB T ) − 1

and obtainAℏω 3= 2 3.
B
π c(4A.8)(4A.9)

These coefficient A, B are called Einstein A coefficient, B coefficient respectively.
In the discussion of transition probability (4.43), we have considered photons with single energy ℏω. Now we consider
a finite width δω of the energy distribution around ω0 with the photon density (4A.8). We write the electric field amplitudefor ω0 as E0 , then the energy density is ϵ0 E02 /2 *2 .
ϵ0

E02
=
2

∫ ω0 +δω/2
U (ω)dω.

(4A.10)ω0 −δω/2*2 From (??), the energy of oscillating electromagnetic field is ⟨(ϵ0 E 2 + B 2 /µ0 )/2⟩ = ϵ0 ⟨(E 2 )⟩, and then the time average gives ϵ0 E 2 /2.
0

E4-14

And taking the directional average, we obtain
⟨

⟩ ⟨
⟩ µ2
|µab · ep |2 = µ212 cos2 θ = 12 .
3

(4A.11)

The the transition probability (4.43) can be approximated as|µab |2 1|cb (t)| ≃3ℏ2 ϵ0∫ ω0 +δω/2

2

U (ω)
ω0 −δω/2

sin2 [(ω − ω0 )t/2]
πµ2ab
dω
≈
U (ω0 )t.
(ω − ω0 )2
3ϵ0 ℏ2(4A.12)

We replace the integral over the period δω with the infinite integration. And we applied the identity limλ→∞ sin2 λx/λx2 =πδ(x). The transition probability is obtained as |cb (t)|2 /t. Then the discussion leads to the expression of B coefficient as
B=

πµ2ab
πe2
fba .
=
2
3ϵ0 ℏ
6ϵ0 mℏω0(4A.13)

If we use the frequency spectrum ρ(ν) (2πν = ω) instead of the angular frequency spectrum U (ω), the expression needs
correction of 2π, of course.

References[1] N. F. Mott, “Metal-Insulator Transisions” (CRC Press, 1990); 和訳 「金属と非金属の物理」小野嘉之，大槻東巳(丸善, 1996).
[2] 小野嘉之 「金属絶縁体転移」(朝倉書店，2002).
[3] 大槻東巳 「不規則電子系の金属-絶縁体転移」(現代物理最前線 (2) 共立出版，2000).
[4] 米沢富美子 「金属-非金属転移の物理」(朝倉書店，2012).
[5] D. Stauffer and A. Aharony, “Introduction to Percolation Theory” (2nd ed., Taylor & Francis, 2018); 和訳 「パーコレーションの基本原理」小田垣 孝 (吉岡書店, 2001).

[6] M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70, 1039 (1998).
[7] D. C. Reynolds, “Excitons: Their Properties and Uses” (Academic Press, 1981).
[8] 本格的に学ぶには例えば，R. Loudon, “The Quanutm Theory of Light” (3rd ed., Oxford, 2000); P. Meystre and M.
Sargent III, “Elements of Quantum Optics” (Springer, 1990); 松岡正浩 「量子光学」 (裳華房, 2000) など．
[9] J. H. Jeans, Phil. Mag. 10, 91 (1905).
[10] 太田浩一 「マクスウェル理論の基礎」(東京大学出版会，2002)．
[11] 霜田光一「レーザー物理入門」

E4-15

Lecture note on

Physics of Semiconductors (5)12th May (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

4.2.2 Luminescence with interband transition
There are numerous types of semiconductor light emission. A typical example is light emission due to the recombination(pair annihilation) of electron-hole pairs. Minority carriers excited by various methods, including the above lightabsorption, emit their energy as photons by radiative recombination with the majority carriers. When the electronhole pair does not emit a photon and the energy is dissipated to other freedoms, the process is called non-radiative
recombination. Such emission of photons by radiative recombination is called luminescence. Amon them the ones
with comparatively short lifetime are called fluorescence while those with very long lifetime is called phosphorescence.
Luminescence is also classified with the origin of the electron-hole pair creation. The photon-absorption originated emission is called photoluminescence, the electrically stimulated emission (electric field activation of recombination center,
injection of minority carriers, etc.) is called electroluminescence. By some reason minority carriers are trapped in
impurities and some heat pulses cause release of them and lead to luminescence, which phenomenon is called thermoluminescence*1 .
As we saw in the previous section, there are two kinds of photon-emission, stimulated emission and spontaneousemission. In the former the emission probability is proportional to the photon density in the surrounding space while in
the latter the probability is independent of that. If we include the zero-point fluctuation into the photon density, there is nodifference in these two. In practice, however because the former is significant under limited conditions, causing peculiar
phenomena like laser light emission etc., we usually discuss these separately. As this indicates, the density of photons is
a very important factor in the treatment of light emission.
The Planck’s law of radiation gives the density of photons

D (E)with energy E in a material with reflactive index n̄ (we assume
v.b.

a real number ignoring the absorption) as

c.b.

hole

P (E) =

electronEvEc(4.70)

When minority carriers are generated by photoexcitationEf(E)f v( E )8πn̄3 E 31.
h3 c3 exp(E/kB T ) − 1

under light irradiation, the carrier distribution deviates fromthe thermal equilibrium, which is described by single chem-f c( E )ical potential and temperature. Even in such a case, if thesystem is steady in balance, we consider the energy distribution function fc of electrons in the conduction band, that

E Fv

E Fc

E

of electrons in the valence band fv . Because in most cases,
relaxation of distribution by carrier-to-carrier interaction and

Fig. 4.3 Illustration of the concept of pseudo-Fermi levels.

relaxation by intra-band carrier-lattice interaction are much

faster processes than the inter-band carrier recombination, we adopt the approximation described in the following. In
a semiconductor under steady light irradiation, the electrons in the conduction band and those in the valence band are
in quasi-thermal equilibrium state described by the Fermi distribution functions with the same temperature but with the*1 There are many other excitation factors around us, such as electron beams, sound, friction, and chemical reactions, etc.

E5-1

different chemical potentials called quasi-Fermi levels. The difference is caused by the exciation by the light and theslow inter-band transition. Then we write
[
(
)
]−1
E − EFc
fc (E) = exp
+1
,
kB T

[
(
)
]−1
E − EFv
fv (E) = exp
+1
.
kB T

(4.71)Let us consider the process of the photon absorption (energy ℏω) and the excitation of an electron from the valenceband (energy E1 ) to the conduction band (energy E2 ). The frequency of such transition is written as
R(1 → 2) = B12 fv (1 − fc )P (ℏω),(4.72)

where B12 is the transition probability of 1 → 2. Conversely, the frequency of spontaneous emission with the electron
relaxation from E2 to E1 isR(sp, 2 → 1) = A21 fc (E2 )(1 − fv (E1 )),(4.73)

independently of the photon density. The frequency of the stimulated emission is proportional to the photon density as
R(st, 2 → 1) = B21 fc (E2 )(1 − fv (E1 ))P (ℏω).

(4.74)

They should fulfill the balance equation
R(1 → 2) = R(sp, 2 → 1) + R(st, 2 → 1).

(4.75)

Substituting equations (4.70)−(4.74) to the above and the comparison of LHS and RHS gives the following Einstein
relations.

3
8πn̄3 E21
B21 ,
3
3
h c

B12 = B21 .

A21 =(4.76a)(4.76b)

These are identical with eq.(4A.9). Equation (4A.9) is for the angular frequency spectrum and there is the difference inthe conversion factor ℏ.

4.3 Phenomenological treatment of electromagnetic field in materials
In the above we have considered the optical response caused by the photon absorption by interband transition ofelectrons based on the knowledge of two-level systems. This is very important of course, but there are many other opticalprocesses in real crystals. It is also important to look at the optical phenomena from macroscopic perspectives. For
example, the reflactive index can be viewed as a parameter that modifies the speed of light. We have a brief look at sucha classical macroscopic approachs.
Let us begin with the Maxwell equations:
divD = ρ,divB = 0,∂D∂B, rotH = j +,∂t∂t
D = ϵ0 E + P , B = µ 0 H + M .

rotE =(4.77a)(4.77b)(4.77c)Here we assume a non-magnetic insulating material and drop the magnetization M = ⃗0, and the current |j| ≪|∂D/∂t|*2 . These simplifications leads to the following wave equation.
∆E − ϵ0 µ0

∂2P
∂2E
=
µ
,
0
∂t2
∂t2

(4.78)*2 When these are finite, various interesting phenomena are expected even in this macroscopic level. On the microscopic level, we can find numerous

subjects. These are called magneto-optical effects. They are the targets of researches as well as the sources of many usuful experimental
techniques. [2, 3] are recommended for advanced study.

E5-2which is the same as that for the vacuum when P = ⃗0. This means that the polarization P represents the effect of dielectric material in this macroscopic model. In the linear response approximation, P is written with the electric susceptibility
tensor χ as
P = ϵ0 χE.

(4.79)

Equation (4.77c) leads to D = ϵ0 (1 + χr )E and relative dielectric functionor relative permittivity is defined as follows*3 .
D = ϵ0 ϵr E,

ϵr = 1 + χ.

(4.80)

Below for simplicity, we consider isotropic materials and the tensor ϵr can be treated as a scalar ϵr . From eq.(4.79) and(4.80), eq.(4.78) becomes∆E − ϵ0 µ0∂2E∂2E= ϵ0 µ0 (ϵr − 1) 2 .
2∂t∂t

Then we obtain∆E −ϵr ∂ 2 E= 0.
c2 ∂t2(4.81)In the above simplest approximation, the effect of polarization in the material can be taken into account with changing√
the light speed c with c′ = c/ ϵr . Hence the dispersion relation in the vacuum ω = ck is modified as
c2 k2 = ω 2 ϵr (ω, k).

(4.82)



Here ϵr depends on ωr , k, reflecting the properties of materials.
As above, the association of the polarization with the electromagnetic wave inside materials can be taken into account√phenomenologically by considering the relative dielectric function ϵr (ω, k) or the refractive index ñ = ϵr . Above that,the absorption we saw in Sec.4.2.1 can be phenomenologically taken into account with adding the imaginary part to therefractive index.Then the complex refractive index is defined asñ(ω, k) = n(ω, k) + iκ(ω, k).

(4.83)

Then from eq.(4.59), or from the definition I(z) = I0 exp(−αz), the absorption coefficient α is expressed asα=2ωκ(ω, k).
c(4.84)Let us go into a bit “model” of materials. In the Lorentz model, the electromagnetic field in the materials is a setof harmonic oscillators. In the model the mass, the charge, and the spring constant is common as (m, e, ξ) and theelectromagnetic wave interacts with the oscillators through the Coulomb interaction with the charges. The frequencyof the electromagnetic wave is ω and the wavelength is much longer than the distance between the oscillators and theelectromagnetic wave can be approximated by uniform time-dependent electric field, which is written as eE0 e−iωt . The
equation of motion for each oscillator is written as
m

d2 x
dx
+ Γm
+ ξx = eE0 exp(−iωt),
2dt
dt(4.85)

where Γ m is the coefficient representing the energy dissipation (friction in a classical model).
√The eigenfrequency of each oscillator is ωh = ξ/m. In order to find the long-term stable solution of (4.85), we
substitute x(t) = xp exp(−iωt). Thenxp (ω) =eE01
m ωh2 − ω 2 − iωΓ(4.86)*3 Various expressions are use for the dielectric funtion. Here we put the expression “relative” to clarify the unit is taken as the vacuum dielectric

constant ϵ0 . The units in electromagnetism often cause confusions. Textbooks [5, 6, 7] are recommended for those who are intrested in the
problem.

E5-3

is obtained. Let N be the spatial density of the oscillators and we get
P = N (exp (ω)) =

N e2
1
E0 .
2
m ωh − ω 2 − iωΓ(4.87)

The coefficient of E0 in r.h.s. corresponds to χ in (4.79). Then the definition in (4.80) leads to the relative dielectric
functionϵr (ω) = 1 +

1
N e2
.
2
ϵ0 m ωh − ω 2 − iωΓ(4.88)In the above we consider the case of single mode oscillator. If the oscillator has multiple mode and we write fj as the
portion of the mode indicated by index j, (4.88) はϵr (ω) = 1 +N e2 ∑
fj.
ϵ0 m j ωh2 − ω 2 − iωΓj(4.89)

This fj is the oscillator strength we’ve already seen, but with this treatment we understand the wording of “oscillator
strength.”

4.4 Optical response of excitons
The excitons introduced at the end of the last chapter have discrete energy levels below the band gap. In many cases they
appear as prominent peak structures in the absorption/emission spectrum. In bulk semiconductors, they appear mostly at
low temperatures but the situation changes in quantum structures discussed later in this lecture. We do not have time to
go into but the Frenkel-type excitons are now the main origin of the electroluminescence in organic semiconductors. Let
us begin with the excitons in bulk semiconductors.

4.4.1 Absorption/emission by excitons
As we saw in Sec.3.3.2, the kinetic freedoms in excitions can be specified by the electon-hole relative spatial coordinate
r and coordinate of the parallel motion R. Then the wavefunction can be written in the effectve mass approximation as1
ΦnK (r, R) = √ exp(iK · R)ϕn (r).
V

(4.90)The Fourier transform of the above is∫1
d3 re d3 rh e−ike ·re e−ikh ·rh ΦnK (r, R)
V
∫
∗
1
√
d3 rd3 Re−iR·(ke +kh −K) ϕn (r)e−ik ·r
=
V
∫
∗
1
mh ke − me kh=√
d3 re−ik ·r ϕn (r)δK,ke +kh , k∗ ≡.
me + mhV

FnK (ke , kh ) =(4.91)The total wavenumber of the excition K is thus turned out to be
K = ke + kh .

(4.92)For the treatment of optical absorption, we take the initial state before the electron-hole excitation as the ground stateΦ0 = ϕcke ϕvke and calculate the transition probability wif to the state represented as eq.(4.90) with taking kp = ⃗0，ke = −kh and along with the line shown for the case of two-level systems.
wif =

1 ∑
2π e2
|A0 |2|⟨ΦλK | exp(ikp · r)e · p|Φ0 ⟩|2 δ(Eg + Eλ − ℏω)
2
ℏ m
V
λ

2

=

1 ∑

2π e
|A0 |2ℏ m2
V|FλK (ke , −ke )⟨ϕcke |e · p|ϕvke ⟩|2 δ(Eg + Eλ − ℏω).

ke λ

E5-4

(4.93)

(a)(b)

Fig. 4.4 (a) Optical absorption peaks by excitons at lower side in the energy than the fundamental absorption edge
in GaAs[9]. (b) Absorption anomaly by excitons around the fundamental edge in GaAs[10].

From ke = −kh ,FnK (ke , −kh ) =1V∫
d3 re d3 rh exp[−ike · (re − rh )]ΦλK (re , rh ).

(4.94)In (4.93), the summation over ke results in re = rh . FnK takes large values only in a narrow region of ke around ke ≈ ⃗0.
In that region, ⟨ϕcke |e · p|ϕvke ⟩ is almost constant and is M in (4.62). We then obtain
wif =

∑
2π e2
|A0 |2|M |2 |ϕλ (0)|2 δ(Eg + Eλ − ℏω).
2
ℏ m

(4.95)λAgain for simplicity we consider an isotropic system. Because ϕλ (0) is not zero only for s-state,|ϕn (0)|2 =1,πa3ex n3

En = −

Eex.
n2(4.96)The imaginary part of the relative dielectric function ϵr2 (ω) = 2in(ω)κ(ω) is
)
∑ 1 (
πe2
Eex
2 1
ϵr2 (ω) =
|M
|
δ
E
−
−
ℏω
.
g
ϵ0 m2 ω 2
πa3ex n n3
n2

(4.97)In the above the spin degree of freedom 2 is not considered and the result should be multiplied by two.
We do not go into calculation details (see e.g.
[8]), but the twice of (4.97) agrees with (4.68) atthe boundary ℏω = Eg between discrete states
and continuum. Hence we can confirm how goodis the approximation by the comparison of thespectra in experiments.
In eq.(4.97), the part other than δ-functions iscommon and we write it as the constant C.
)(
Eex(4.98)
ϵr2 = Cδ Eg − 2 − ℏω .
nMathematical identitylim

1

Γ →+0 x0 − x − iΓFig. 4.5 Emission spectra of bound excitons in Cu2 O[11].

E5-5

=P

1
+ iπδ(x0 − x)x0 − x(4.99)





tells
ϵr2 = Im





C/π
.
E

 Eg − ex − (ℏω + iδ) n2(4.100)Here we write Γ → +0 as δ. And the Kramers-Kronig relation (4B.2) leads to
ϵr =

C/π
,
Eex
Eg − 2 − (ℏω + iΓ )n(4.101)

with which we can try fitting the data in, e.g. Fig. 4.4(b).
The emission is the reversal process of the absorption and just as the absorption, discrete emission peaks appear at lower
energies than the funcamental emission edge. Figure 4.5 shows an example of photoluminescence spectra of Cu2 O.

4.4.2 Exciton-polariton
Well known as “polaritons” are the quasiparticle created by the combination of optical phonons and photons. Here we
consider, however the combination of photons and excitons. The concept of exciton-polariton is illustrated in Fig. 4.6.
As mentioned in the previous section, an absorption and an emission of photon with an exciton are reversal process toeach other. In an exciton-polariton these processes form a continuous chain. The cycle period of the processes is as short
as a few fs and both the exciton and the photon keep their quantum coherence and the resultant quasiparticle propagatesinside the crystal as a coherent state.
hn

e

e

hn

h

e

hn

h

hn

h

Fig. 4.6 Illustration of the concept of exciton-polariton. A photon creates an exciton and the recombination of theelectron-hole pair recreates a photon. These processes occur in series.
2

We consider the ground state of n = 1 in eq.(4.101), define ω0 as
Eg −Eex ≡ ℏω0 , and the contribution to the dielectrin function otherthan the excitons as ϵs . Then with γ = Γ/ℏ, the relative dielectric
function is written as∆exω0 − ω − iγ).

(4.102)w/w0(
ϵr (ω) = ϵs 1 +1

For the transverse wave with k · E = 0, the angular frequency ωt =
ω0 , the polariton equation (4.82) holds. On the other hand, for thelongitudinal wave ϵr (ω) = 0, the angular frequency ωl is given asωl = ω0 + ∆ex = ωt + ∆ex .

(4.103)0∆ex is called longitudinal-transverse splitting.
Now we consider the wavenumber k = k1 +ik2 , then from (4.82),(4.102) we get

12

k 1 c /w 0 n s
Fig. 4.7 Schematic drawing of the dispersion


relation of exciton-polariton.
(
)

∆ex
ω 2 ϵs2
1+= k1 − k22 , , 2cω0 − ω

ω 2 ϵsπδ(ω − ω0 ) 02 = 2k1 k2 .
c
E5-6

(4.104a)(4.104b)

Equation (4.104b) represents the resonance at ω = ω0 , then we ignore k2 in (4.104a) to get√ω − ω− ∆ex
ek1
ω
=√ ,
ω − ω0ϵs(4.105)which gives the dispersion relation of exciton-polariton.

Appendix 4B: Kramers-Kronig relation
Here we just show well-known Kramers-Kronig relation. Let us consider a complex function with a complex argument
ω asχ(ω) = χ1 (ω) + iχ2 (ω),

χ1 , χ2 ∈ R.

(4B.1)

χ(ω) is analytic in the upper half of ω-plane and diminish faster than 1/|ω| for large |ω|. Then there hold relations
between χ1 and χ2 asχ1 (ω) =

1
P
π

∫ ∞

χ2 (ω ′ ) ′dω ,′−∞ ω − ω

1χ2 (ω) = − Pπ∫ ∞χ1 (ω ′ ) ′dω .
′−∞ ω − ω(4B.2)

Here P represents the Cauthey’s principal value. The above are the Kramers-Kronig relation.

Appendix 4C: Lattice vibration in semiconductors
Lattice vibration is a phenomenon in which an atom vibrates around it with kinetic energy while being localized at
an equilibrium position as a time average position. This is an important subject in semiconductor physics, should be
discussed using at least one whole chapter, but that is impossible due to the lecture time. Here we take a minimum lookat very basics.

4C.1 Lattice vibration in one-dimensional system
Just as in the electron system, we introduce basic concepts in one-dimensional systems.
We consider a one-dimensional latteic with the unit

u 2,i-1

u 1,i

u 2,i

u 1, i+1

cell of length a, which has two atoms with masses M1
and M2 . The shifts of the atoms from the equilibrium positions are written as u1j , u2j (j: integer). The

M1

M2

force working on the atoms is assumed to be harmonicaoscillator-like, that is, the force proportional to the shiftof the distance between neighboring atoms from theFig. 4C.1 Schematic diagram of one-dimensional lattice
vibration.

equilibrium value a/2. Let α be the coefficient for theforce then we get the equation of motion as

d2 u1,j= −α(u1,j − u2,j−1 ) + α(u2,j + u2,j−1 ) = α[−2u1,j + (u2,j − u2,j−1 )],dt2
d2 u2,j= α[−2u2,j + (u1,j − u1,j−1 )].
M2
dt2

M1

E5-7

(4C.1a)(4C.1b)

w
o.p.

a.p.
-p/a

0

Fig. 4C.2 Dispersion relation of one dimensional lattice vibration ω(q) obtained from eq.(4C.5) is plotted for the case
M1 = 2M2 .

p/aq

The equation (4C.1) remains unchanged with the parallel shift operation j → j + n (n is an integer) and the solution
can be written in the form of Bloch function. Let us take x coordinate along the lattice direction and the equations for thewavenumber q are{

u1,j (xj ) = eiqxj u1,q ,
u2,j (xj + a/2) = eiq(xj +a/2) u2,q .

Substituting the above into (4C.1) we obtain
d2 u1,qja M1= 2α(−u1,q + cos u2,q ),
dt2
2
2

d
u
ja
2,q

 M2
= 2α(−u2,q + cos u1,q ).
dt2
2(4C.2)(4C.3)In order to find the solution we assume u(1,2),q ∝ exp(iωt) to write down
2α − M1 ω 2qa−2α cos2qa  
u1,j
u1,q
2 
 ≡ A
 = ⃗0.
2α − M2 ω 2
u2,q
u2,q
−2α cos(4C.4)For the equations to have non-trivial solution {ui,q } other than ⃗0, |A| = 0 leads to
2
ω±
=
α

(

1
1
+
M1
M2

√(

)
±

1
1
+
M1
M2

)2
−4

sin2 (qa/2).
M1 M 2(4C.5)

We consider non-negative ω, and eq.(4C.5) has two modes, the dispersion relations of which are shown in Fig. 4C.2.
The following description of wording does not depend on the dimension.
The modes with linear dispersion around q ≈ 0 are called acoustic modes, those with finite ω and dω/dq = 0 for
q = 0 are called optical modes. The naming acoustic mode comes from the property that the group velocity does notdepend on the frequency just like sound in the air or electromagnetic wave in the vacuum. The naming optical mode
comes from the interaction with photons as the small wavenumber and the large energy. The quantized particles of them
are called acoustic phonon and optical phonon respectively.

4C.2 Lattice vibration in zinc-blende crystals
We consider zinc-blende (ZB) crystals as an example of three-dimensional crystal which has two species of atoms in
the unit cell. The Bravais lattice is fcc but the ZB crystalline structure can be considered as an overlapp of two “fcccrystals”, in which one atom is placed at the lattice point of fcc lattice.

E5-8(to be continued in the next lecture note.)

References[1] 櫛田孝司「光物性物理学」(朝倉書店，2009).
[2] 佐藤勝昭「光と磁気」(朝倉書店，1988).
[3] S. Sugano ed., “Magneto-optics” (Springer, 1999).
[4] A. K. Zvezdin and V. A. Kotov, “Modern Magnetooptics and Magnetooptical Materials” (IOPP, 1997).
[5] 青野 修「電磁気学の単位系」(丸善，1990).
[6] 高梨弘毅「磁気工学入門」(共立出版，2008)[7] 2018 年の SI 単位改定については，沢山解説が出ているが，改定自身というより単位系全般の話を非常に易しく解説したものとして，和田純夫, 大上雅史, 根本和昭「単位がわかると物理がわかる」(ベレ出版, 2002).

[8] 浜口智尋「半導体物理」 (朝倉書店, 2001)．英語版は C. Hamaguchi, “Basic Semiconductor Physics” (Springer,2017).
[9] G.W. Fehrenbach, W. Schafer, and R. G. Ulbrich, J. Luminescence, 30, 154 (1985).
[10] M. D. Sturge, Phys. Rev. 127, 768 (1962).
[11] J. I. Jang, Y. Sun, B. Watkins, and J. B. Ketterson, Phys. Rev. B 74, 235204 (2006).

E5-9

Lecture note on

Physics of Semiconductors (6)19th May (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Chapter 5 Semi-classical treatment of electrical transport

The electric tranport is a response to external perturbations as important as the optical response. Treatment of nonequilibrity to some extent is inevitable for the discusstion of transport. A big difference between the electric transport andthe optical response is, however, in the former the characteristic energy scale is much smaller than that in the latter (∼ Eg ).
In this chapter, we have a brief look at very basic part of the transport in the linear response regime, in semi-classicaltreatment. We will go into the quantum transport in the later chapters.

5.1 Classical transport phenomena
Among transport phenomena interests of physicist mainly lies in quantum transport such as the quantum Hall effect.
In earth-flooding semiconductor devices, however, dominant is the classical transport *1 .
The reason that the classical theories are applicable to transport in semiconductors at room temperatures mainly lies in
the low density of carriers. In bulk transport, for example, the Fermi level EF lies in band-gap, that is, there is no densityof states around EF . When we are looking at the energy distribution of electrons, what we actually see is the tail of the
Fermi distribution function, which can be approximated with Maxwellian.
Heavy doping changes semiconductors into disordered metals, or spatial modulation of materials which shift the positions of Fermi levels above the conduction band edges provide low-dimensional metallic systems. Even in many of such
systems, classical approximations hold around room temperatures. The Fermi degeneration temperature for a systemwith density n and particle mass m is
TF =ℏ2
(3π 2 n)2/3 :3D,2mkB

ℏ2n : 2D.
16πmkB(5.1)

Substitution of typical values for semiconductors give, e.g. for a two-dimensional electron system, which has comparatively large Fermi energy, about 70K for TF . That is, the distribution of kinetic energy is still described by a Maxwellian.
Furthermore, the width of distribution is as large as the Fermi energy making the quantum mechanical interference effectobscure. In this chapter we thus concentrate on the phenomena, which can be described within classical theories forelectron kinematics in solids.

5.1.1 Transport phenomena and transport coefficient“Transport” here means transportation of some physical quantity in real space. In the treatment of such a problem,
we often map the problem onto a set of particles and the transport is transfer of the particles in the model. For example,
consider a stretched string and some local shift from the stretched line. The shift is transmitted on the string as a wave but
we can also treat the shift as a particle, which brings some potential energy. We may consider, then, the transport of the*1 In many devices quantum confinement is working and low-dimensional systems are realized though the transport can be understood withinclassical theories.

E6-1

shift. In solids, we actually have various elementary excitations such as phonons, spin-waves (magnons), etc. In electric
conduction, which is a representative transport in semiconductors, the physical quantity is charge and a particle bringing
it is called a “carrier”. Examples are, of course, electrons and holes, many-body states of band electrons actually.
Needless to say, we first need to exclude trivial motion of the center of mass due to arbitrary selection of inertial system.
We thus assume that the center of mass for the system under consideration sits still in equilibrium without perturbation.
Transport is a response flow of some physical quantity to an external perturbation. *2 . In the case of linear response, likeelectric current for voltage in Ohm law, the coefficient is called transport coefficient.
We often have strongly non-linear response in electric transport in semiconductors though we begin with linear response. As a typical example, in the electric current response to the field, thre linear response between the current density
j and the field E is written as
j = σE,E = ρj = σ −1 j,(5.2)

where σ is the conductivity tensor，ρ is the resistivity tensor. These two are inverse tensor to each other.

5.1.2 Boltzmann equationpf ( r , p , t )d r d p
drdp( v d t, F d t )rFig. 5.1 Illustration of time evolution for particles in an infinitesimal volume drdp in an infinitesimal time dt with a scattering.

Let us consider a distribution function f (r, p, t) in a six-dimensional space of spatial coordinate r and momentum p,i.e., a phase space. The meaning of f is that the ratio of particles in the volume drdp around the point (r, p) in the wholesystem is f (r, p, t)drdp.
In the absence of scattering, the classical equation of motion is described as
dr/dt = v = p/m∗ ,dp/dt = F ,(5.3)

with F as the force working on the particle. Kinematic states of particles in drdp are the same in the first order and so
are the time evolution during dt, giving
f (r + vdt, p + F dt, t + dt) = f (r, p, t).
Some scatterings bring shifts in f as illustrated in Fig. 5.1. We write the coefficient in the shifts as (∂f /∂t)c , that is,
f (r + (p/m∗ )dt, p + F dt, t + dt) + (∂f /∂t)c dt = f (r, p, t).
Expanding f in the left hand side to the first order, we getBoltzmann equation



∂f
p ∂f
∂f
+ ∗·
+F ·
=−
∂t
m ∂r
∂p


(

∂f
∂t

)
(5.4)
c



Equation (5.4) is called Boltzmann equation, and the right hand side is called the collision term.
*2 This definition cannot include supercurrent or diamagnetic current at edge states of quantum Hall effect.

transport phenomena taking the reference of coordinate to crystal lattices.

E6-2

But we usually include them into

The collision term depends on the scattering mechanism and the nature of scattering centers and is generally difficult
for us to calculate. The simplest approximation of this term is the constant relaxation time approximation, in which we
consider a relaxation time τ independent of energy and putConstant relaxation time approximation
(
−

∂f
∂t





)
=−
c

f − f0
,
τ

(5.5)


where f0 is the equilibrium distribution function for F = 0, τ , the relaxation time, is the time for recovery from nonequilibrium states. In spatially uniform systems, ∂f /∂r = 0, and the approximation (5.5) can be generalized to the one
with energy or momentum dependence in τ .
Below, to avoid trivial failure in pure classical pictures, we use some quantum mechanical relation like p = ℏk or
quantum statistics.

5.1.3 Drift current, diffusion current
As currents we here consider electric currents. Net particle flow appears when the distribution function f gets some
anisotropy in p space. Hence we need to consider perturbations in (5.4) other than anisotropy or non-uniformity in p.
The candidates are then F (= −eE), and ∂/∂r. The former perturbation, e.g. acceleration by external electric field,
brings about non-uniformity of distribution function f (r, ℏk, t) in k-space resulting in the flow of carriers in the real
space. That kind of flow is called drift current. The latter is non-uniformity of the distribution in the real space and also
causes carrier transport, which is called diffusion current.
First let us consider a steady uniform electron system under uniform electric field E. From this assumption, ∂f /∂t = 0(steady) and ∂f /∂r = 0 (uniform). We further assume τ only depends on p. Then eq.(5.4) becomes
−eE ·

∂f
f − f0=−∂p
τ (p)∴ f (p) = f0 (p) + eτ (p)E ·

∂f
.
∂p

In the next step of approximation, we take E as a small perturbation. Hence, the 1st order expansion is obtained withreplacing f in the right hand side with f0 as
f (p) ≃ f0 (p) + eτ (p)E · (∂f0 /∂p).

(5.6)

Higher order terms can be obtained by successive replacements. Now eq.(5.6) can be viewed as the first order expansionof f (p) ≃ f0 (p + eτ (p)E with E, which means this f (p) is the one shifted by −eτ (p)E in p space form f0 (p). If τ isconstant for p, the shift is uniform as illustrated in Fig. 5.2.
f( k )

-k F

ky

0

kF

k

0
-k F

kF

kx

Fig. 5.2 Schematic view for constant shift of Fermi sphere ofelectrons under acceleration by external electric field in the spaceof wavenumber. The distribution f (k) shifts from the equilibrium position (indicated by broken line) by a wavevector indicated by small arrows. The upper shows the shift in the distribution and the lower shows the shift of Fermi sphere in twodimensional systems. In realistic systems, the shifts are muchsmaller than that illustrated here.

E6-3

We need to integrate v(k)f (k) in k space to obtain the current. Without loosing generality we take E = (Ex , 0, 0) anderasing integrals of odd functions we obtain
(
) ∫
∫
∫
d3 k
∂f0
d3 k ℏkx∂f0eExℏ2 kx2 ∂f0
v(k)
f
+
eτ
E
·
=
eτ
E
=
D(E)τ
(E)
dE,
0
x
(2π)3
ℏ∂k
(2π)3 m
ℏ∂kx
m
m ∂E

(5.7)

where we assume τ depends only on energy. ℏ2 kx2 /2m, the kinetic energy along x-direction is E/3 from the equipartition
condition．
For a metallic Fermi-degenerated system, ∂f0 /∂E can be approximated as −δ(E − EF ) in (5.7). For a three√dimensional system the density of states is D(E) = A E with a coefficient A, then (5.7) is⟨vx ⟩ = −AeEx 2τ (EF ) 3/2
EF ,
m
3

while the particle density is calculated as
∫ EF
n=
0

2 3/2
D(E)dE = A EF .
3

Putting together the above expressions we obtain the expression for σ = j/E = −e⟨vx ⟩/Ex as
Drude conductivityσ=

e2 nτ (EF )
m


(5.8)


which is well known Drude conductivity．
When the temperature is high, or the particle density is low and the Maxwellian approximation holds, from f0 ≈A exp(−E/kB T ),

[
]
∂f0
A
E
f0
f0
−
=−
exp −
=−
=−
∂E
kB T
kB T
kB T
(2⟨E⟩/3n)

is obtained, in the last equation of which we have used averaged kinetic energy kB T /2 for single kinetic degree of
freedom. The electric conductivity is again given in the Drude form as∫
σ = e2

τ (E)D(E)2E 3nf0
ne2 ⟨τ ⟩EdE =.
3m 2⟨E⟩m(5.9)

Here, ⟨E⟩E represents the average with weight E 3/2 :⟨τ E⟩=⟨τ ⟩E =⟨E⟩∫ ∞
τ (E)E

3/2

/∫ ∞
f0 dE

0

E 3/2 f0 dE.

(5.10)

0

We then proceed to the diffusion current caused by non-uniformity of f in real space. In Boltzmann equation (5.4), F
is set to zero and constant relaxation time approximation (5.5) is applied to the space distribution of f = f0 + f1 asv · ∇f = −f1 /τ,take to the first oder of f1 f1 = −τ v · ∇f0 .

(5.11)

When a constant diffusion current J is flowing through a spatial volume V , it is written as∫
J = (−e)τ v(v · ∇f0 )dr.
V

The direction of ∇f0 is assumed to be constant and along x-axis then the components in v other than vx vanish withintegration since they are odd functions. ⟨vx2 ⟩ = ⟨v 2 ⟩/3 and we further assume that the temperature is uniform andconstant, no spatial variation in ⟨v 2 ⟩, then the current density is∫∂f0
dr = −eτ vx2jx (current density) = −eunit volume

E6-4

∂x

⟨

τ v2
3

⟩

∂n
.
∂x

That is,D = ⟨τ v 2 /3⟩.

j = (−e)D∇n,(5.12)

Here D is diffusion constantand within constant relaxation time approximation,
Einstein relation
D=




τ 2
τ kB Tµ⟨v ⟩ == kB T∗3me(5.13)


Equation (5.13) is called Einstein relation. µ in the right end is the mobility, defined in (5.19), which appears later.

5.1.4 Hall effect
The drift current under magnetic field (flux density B) can be calculated with substituting Lorentz force into F in
(5.4). The straightforward but a bit long calculation is summarized in Appendix A. Here we consider the situation shownin Fig. 5.3, that is, the sample has a finite length along y-axis and infinitely elongated along x-axis and the electric field
E = (Ex , 0, 0) is applied. jy brings the carriers and accumulates them to the edges. The charges at the edges form electricfield Eint = (0, Ey , 0) and in the ultimate steady state jy = 0.
This phenomenon, which generates an electric field vertical both to the current and the magnetic field is the Hall effect.
The linear response coefficient
RH =Ey
J x Bz(5.14)

is called Hall coefficient. Hall field Ey is obtained as follows. From jy = 0,
Ey = −(At /Al )Ex .

(5.15)

Substituting the above and (5A.11b) into (5.14), we obtain the conductivity tensor defined in j = σ̂E as
⟨
⟩
⟨
⟩
ne2
ne2
τ
ne2
ωc τ 2σxx = ∗ Al = ∗
,
σ
=
,
xy
m
m
1 + (ωc τ )2 Em∗ 1 + (ωc τ )2 E
RH = −1At.
ne ωc (A2l + A2t )(5.16)(5.17)

In weak fields, from ωc τ ≪ 1,
RH = −

1 ⟨τ 2 ⟩E
1 Γ(2s + 5/2)Γ(5/2)
rH
=
=
2
2
ne ⟨τ ⟩En(−e) (Γ(s + 5/2))
n(−e)

(
=

1
n(−e)

)
.

(5.18)

Knowing s, we obtain the carrier concentration as well as the sign of charge from the Hall measurement (for holes −eis replaced with e). rH , which is called Hall factor, takes in many cases values around 1 depending on the scattering
mechanism at high temperatures (see Tab. 5.1). Within constant relaxation time approximation (s = 0) or when thesystem is Fermi-degenerated, rH = 1. When s = 0, as eq.(5A.9) tells, eq.(5.18) holds giving the expression shown in thelast parentheses.

BzEy+ + + + + + + +JyJxy- - - - - - - - - - - - - - x

Fig. 5.3 Magnetic field is applied along z-axis. Current along x-axis generates y-component Jy through theLorentz force. The y-component in current results incharge accumulation at the sample edges, which createsHall electric field along y-axis. In steady state, Jy is canceled by the Hall field and the total current is along x-axis.

E6-5

Let v be the average velocity gained by the electrons from the electric field E, the mobilityis defined as v/|E|, and in
the relaxation time approximation, written as
µ=

nev
j
σ
eτ
v
=
=
=
= σ|RH | = ∗ .
|E|
ne|E|
ne|E|
ne
m

(5.19)

E exponent

T exponent

Hall factor

Acoustic phonon

−1/2

−3/2

1.18

Ionized impurity (weak screening)

+3/2

+3/2

1.93

Ionized impurity (strong screening)

+1/2

+1/2

1.18

Neutral impurity

0

1.00

+1/2

1.10

Scattering mechanism

Piezoelectric phonon

Tab. 5.1 Hall factors for various scattering mechanism. E ，T -exponents are for scattering time. See e.g. [1].

5.1.5 Various scatterings
We have considered the Boltzmann equation by relaxation time approximation, but various mechanisms such as scattering with phonons and other degrees of freedom in solids contribute to relaxation. We consider relaxation time for eachrelaxation mechanism, and index each relaxation time τα with index α. Then the frequencies of the relaxations (∝ τα−1 )is summed up to give the total relaxation. This gives the Matthiessen’s rule∑τ −1 =
τα−1 .

(5.20)αIn the relaxation time approximation of classical transport, the carrier scatterings are taken into account through theaveraged scattering time and the Matthiessen’s rule(5.20) into the total relaxation time. Therefore we can infer the
scattering mechanism dominating the present transport by tuning, for example, a parameter which gives different effects
on different scattering times. Scattering of band electrons (holes) have many origins as shown in Fig. 5.4. In this section,
representative scatterings and their characteristics are listed.
Phonon scattering: Quantization of lattice vibration gives phonons. The phonons are classified into acoustic phonons,
which have the dispersion E(kp ) → 0 for wavenumber kp → 0, and optical phonons, which have finite E(kp → 0). Ina plain expression, the difference comes from whether the oscillations of neighboring atoms are in phase or out of phase.
For the band electrons the lattice vibration is distortion in the lattice potential and causes scattering. The scattering ofelectrons causes rebounding of nuclei resulting in the phonon scattering. Such phonon scattering is, from the electronside, inelastic associated with the energy gain/loss.

Fig. 5.4 Classification of scatterings mostly by the origins.

E6-6

The relaxation time due to the acoustic phonon has energy dependence as τ (E) = aph E −1/2 . The averaged scattering
time with energy-weight ⟨τph ⟩E is−1/2⟨τph ⟩E = aph (kB T )√Γ(2)8 πaph= √.
Γ(5/2)
3 kB T(5.21)

In high temperature approximation, the energy distribution of phonons gives aph ∝ (kB T )−1 , then the mobility limitedby the acoustic phonons µph has the temperature dependence asµph ∝ ⟨τ ⟩E ∝ (kB T )−3/2 .

(5.22)On the other hand, optical phonons have large energies around k ≈ 0 and do not affect the transport for weak electricfield. In hot electron transport, in which the electrons are very far from equilibrium by the effect of strong electric field,
the optical phonon scatterings are very important.
Ionized impurity scattering：Impurity atoms in solids
often emit electrons to become positive or trapped negative ions, forming a Coulomb potential for band electrons and causing scattering. In most cases, such potentials are screened by surrounding charge carriers andhave the Yukawa-type (e−r/LD /r) distance (r) dependence rather than the Coulomb-type 1/r. When the ionized impurities have magnetic moments due to the electron spins, they also cause magnetic impurity scattering
the the internal freedom causes peculiar effects like the
Kondo effect. If there is no internal freedom the scattering is simple potential scattering and elastic.
Scattering by Yukawa potential of carriers with
Maxwell distribution, contributes to the scattering time
Fig. 5.5 Hall mobility in GaAs(experiments, points) andfitting by putting various scattering mechanisms withtemperature dependence into (5.20). Red broken lines
indicate temperature dependences of various scattering
mechanisms[2].

asτion ∝ T 3/2 , µion ∝T 3/2
24m∗ λkB T
x , x≡ln(1 + x) − 1+xℏ2(5.23)

for weak screening, i.e. a long LD . For strong screening
with a very short LD , the scattering is δ-function like and the contribution isτion ∝ T 1/2 , µion ∝ T 1/2 .

(5.24)The mobility in GaAs obtained from Hall and conductivity measurements, and the result of fitting by consideringvarious types of scatterings included in eq.(5.20) are shown in Fig. 5.5. The broken lines show temperature dependencesof the scatterings. We see all of these limit the mobility.

5.2 Thermal transport and electric transport
In the Boltzmann equation (5.4), the second and the third term in left hand side representing non-uniformity in thephase space, correspond to drift current and diffusion current respectively. In this subsection we treat the thermoelectriceffect, in which coexistence of the both types of currents should be considered. A temperature gradient in solids causes aheat current (or thermal flux). Here we consider heat transport by charge carriers, i.e. electrons and holes though lattice
vibrations (phonons) also carry heat in solids. Below, we do not consider Joule heating for a while.
E6-7

5.2.1 Thermal conductivity
Thermal flux density along x-direction with carrier concentration n is defined as∫ ∞jqx = ⟨nvx (E − µ)⟩ =vx (E − µ)f (E)D(E)dE.

(5.25)

0

Then thermal conductivity κn under temperature gradient ∇T is defined as
κn = −

jqx.
∂T /∂x(5.26)

In vector format jq = −κ̂∇T ．

5.2.2 Thermoelectric effects
The heat flux in (5.26) should also lead to someT2, x2
T1, x1electric effect. Such complex effects of tempera-

B

B
A

A

ture gradient and electric response are called ther-Bmoelectric effects.

B
(a)

B

Let the temperatures at edges of a conductor A

J

T1 and T2 respectively. Two conductors of an-(b)other material B with the same lengths are con-V

nected to the edges. Other ends of conductors(material B) are connected to a voltmeter with infinite input impedance ((a) in the left figure). In the steady state, there is no net current and the electric current driven byheat flow should be compensated by the voltage VAB measured at the voltmeter. This is called Seebeck effect, and theratio of the voltage to the temperature difference (∆T = T1 − T2 )
SAB =

VAB
∆T

(5.27)

is called Seebeck coefficient. On the other hand as in (b), when there is a junction of A and B set at a uniform temperature,a current J causes heat fluxes QA and QB . In a steady state there is no charge accumulation and J is uniform, that means
QA and QB are different reflecting difference in the thermal transport coefficients. The difference results in heating at
the interface. This is called Peltier effect and the ratio of the heating speed to J,
ΠAB =

QAB
J

(5.28)

is called Peltier coefficient. If we apply a current J to a BAB type junction as in (a), the same current flows with inverted
directions through the two interfaces. Hence if a heating occurs at one interface, a cooling of the same amount of heatoccurs at the other end.
In a uniform conductor with a current J and a temperature gradient (assume along x-direction) ∂T /∂x, cooling or
heating occurs. Heat creation per unit length ∂Q/∂x is proportional to the product of J and ∂T /∂x. This is Thomson
effect and the coefficient
τ=

∂Q/∂x
J(∂T /∂x)(5.29)

is called Thomson coefficient.
Among the above three kinds of coefficient, Kelvin (Thomson) relations
ΠAB = SAB T,

τA − τB = T
E6-8

dSAB
dT

(5.30)

hold (Appendix B). From the relations we can define material specific (combination free) Seebeck coefficient as∫ T
SA (T ) ≡0

τA (T ′ ) ′dT .
T′(5.31)The relation with the coefficient in (5.27) is
SAB = SA − SB .

(5.32)In the measurement of Seebeck effect, we need to connect the sample and the voltmeter with leads, which also have
Seebeck coefficient. Hence the measured voltage is the difference between the Seebeck effects of the sample and the leads.
Equation (5.32) indicates the fact. Thermocouple works as a sensor for temperature difference ∆T with knowledge ofSeebeck coefficients for the two components.

5.2.3 Boltzmann equation and thermoelectric coefficients
Let us look for the relation between the thermoelectric coefficients and the distribution function with Boltzmann equation under relaxation time approximation (5.4), (5.5). In a steady state ∂f /∂t = 0 we rewrite the equation asv · ∇f +F
f − f0∇v f = −.
m
τ (E)(5.33)

We take the approximation that the shifts from equilibrium are small and replace f in the left hand side with f0 .
∇f0 due to temperature gradient ∇T is written as∇f0 = ∇T∂f0.
∂T

In f0 , E and T always appear in the expression −(E − EF )/kB T , which we write a here for short description. Then∂f0
∂f0 ∂E ∂a∂f0
E − EF∂f0 EF − E==(−kB T )=,2∂T∂E ∂a ∂T
∂E
kB T
∂E
T
EF − E ∂f0
.
T
∂E
∂f0
∂f0
And ∇v f0 = ∇v E
= mv
.
∂E
∂E

therefore∇f0 = ∇T

When the electric field E and the temperature gradient ∇T coexist, (5.33) can be written with (5.34) as[]
EF − E∂f0
f = f0 − τ (E)v · −eE +
∇T
.
T
∂E

(5.34a)(5.34b)(5.35)

We take E = (Ex , 0, 0) and the current along x-direction isjx = −e⟨nvx ⟩ = −e

∫ ∞

∫ ∞
vx f (E)D(E)dE = e

0

0

[
]
EF − E ∂T ∂f0
vx2 τ −eEx +D(E)dE.
T∂x ∂E

The Seebeck coefficient is obtained with putting jx = 0 as
Ex=S=∂T /∂x∫ ∞0EF − E ∂f0
vx2 τD(E)dE
eT

∂E

/∫ ∞

∂f0
vx2 τ
D(E)dE
∂E
[0
/∫ ∞
]
∫ ∞
∂f0
1
2 ∂f0
τE
=
EF −
τE
D(E)dE
D(E)dE . (5.36)
eT
∂E
∂E
0
0

Here vx2 is replaced with 2E/3m.

E6-9

Fig. 5.6 Left panel: Schematicof Peltier device. p-type semiconductors and n-type semiconductors
are placed staggered (in the figure
just a single pair) along the current
path. While electric current meanders heat flows one way. Rightpanel: Photo of a Peltier device.
From Akizuki-denshi web site.

Cooling+n-type+-p-type

+
+

J

-

+

heatingheating

In eq.(5.35) inside the parentheses at right hand side, the first term represents the drift current while the second the
diffusion current caused by the temperature distribution. The canceling of these term results in the Seebeck effect,
therefore the Seebeck effect is the result of diffusion current which causes charge non-uniformity inside the sample. The
non-uniformity creates electric field, of which the drift current cancels the diffusion current.
In Maxwellian approximation, ∂f0 /∂E = −f0 /kB T , and we further assume the energy dependence of the relaxation
time as τ ∝ E s , then

1
S=−
eT

[

]
[(
)
]
⟨τ E⟩E
1
5− EF = −+ s kB T − EF .
⟨τ ⟩E
eT
2

(5.37)

This equation tells that if we can measure the temperature dependence of S, we obtain EF and s. The above calculation
is for electrons and for holes −e is replaced with +e, hence measurement of S also gives the sign of carriers. This resultfor Maxwellian approximation does not depend on the carrier concentration, which can be understood as follows. The
Einstein relation (5.13) connects the diffusion constant and the mobility, which are material constants for diffusion anddrift currents respectively. Hence these constants disappear from the balancing equation leaving the temperature. The
carrier concentration also included as the first order in both currents and dropped. In the case of Hall coefficient, the drift
current by external field comes into one side and the carrier concentration remains in the expression.

5.2.4 Peltier device
Peltier and Thomson coefficients can also be obtained from the Kelvin relations. Peltier coefficient also changes its signwith that of carriers. In a material with junctions to n and p-type semiconductors, a current flow through this structure
thus causes heating at one junction and cooling at the other resulting in a heat flow. Such a device is called Peltier device.
Peltier devices once were frequently used in combination with cooling fans for cooling CPUs in PCs. They have long
been used where we need cooling without noises such as refrigerators in bedrooms.

Appendex 4C: Lattice vibration in semiconductors (continued)
Continuing from the last time, let us briefly look at the lattice vibration of sphalerite-type crystals as an example ofthree-dimensional crystal lattice vibration.

4C.2 Lattice vibration in zinc-blende crystals
We consider zinc-blende (ZB) crystals as an example of three-dimensional crystal which has two species of atoms in
the unit cell. The Bravais lattice is fcc but the ZB crystalline structure can be considered as an overlapp of two “fcccrystals”, in which one atom is placed at the lattice point of fcc lattice. We consider two such fcc-crystals with different
atoms with common lattice constant a. We obtain a ZB crystal by placing these two with the shift of a(1/4, 1/4, 1/4).

E6-10

LO
TO
w

LA
TA

0

Fig. 4C.3 Schematic diagram of lattice vibration modes (dispersion relation) in a zinc-blende crystal.

p/aq

Let uα,R be the atomic shift vectors. Here α is the index of the two sublattice, R is the lattice point. The lattice kinetic
energy can be written as
EK =

∑1
α,R

2

Mα u̇2α,R .

(4C.6)On the other hand, with expanding the ptential V (r) to the second order, the potential energy is written as
∑

EP =

∂2V

′

ujα,R ujα′ ,R′

αα′ ,RR′ ,jj ′′∂ujα,R ∂ujα′ ,R′,j = x, y, z.

(4C.7)The equation of motion can be obtained by defining the Lagrangian L ≡ EK − EP , and general coordinate qk ≡ ujα,Rfrom the Lagrange equation;
d
dt

(

∂L
∂ q̇k

)
−

∂L
= 0.
∂qk

In the present case from (4C.6), (4C.7) we get
Mα üjα,R = −

∑

∂2V

′

j′

j
α′ ,R′ ,j ′ ∂uα,R ∂uα′ ,R′ujα′ ,R′ ≡ −∑Cαα′ ,RR′ uα′ ,R′ .

(4C.8)

α′ ,R′

Tensor C corresponds to “force constant” and just depends on the combination αα′ , and on the relative position of unit
cell R′′ = R′ − R. Then with Cαα′ ,RR = Cαα′ (R′′ ) we can write
Mα üjα,R = −∑Cαα′ (R′′ )uα′ ,R+R′′ .

(4C.9)

α,R′′This equation is invariant againg the shifts among the lattice points R → R′ . Then we can write the solution in the formof Bloch function
uα,R (t) = uα (q, t) exp(iq · Rα ).

(4C.10)As the time dependence, we consider the oscillation with angular frequency of ω, and assume
uα,r (t) = √1uα (q, ω) exp[i(q · Rα − ωt)].
MαSubstituting this to (4C.9) results in][∑∑ jj ′∑1√
ω 2 uα (q, ω) =Cαα′ (R) exp(iq · R) uα′ (q, ω) ≡Dαα′ (q)uα′ (q, ω).
M α Mα ′ R
α′
α′
E6-11

(4C.11)(4C.12)

For (4C.12) to have solutions other than the trivial ⃗0,
′

jj
2
|Dαα
′ (q) − ω δαα′ δjj ′ | = 0.

(4C.13)The dispersion relations can be obtained by solving this numerically. The 6th order equation gives 6 modes, in which 3
acoustic modes and 3 optical modes exist. The each 3 are separated into 2 transverse modes and 1 longitudinal mode.
The namings are, then, TA, LA, TO, LO.

Appendix 5A: Galvanomagnetic effectWe consider the response of drift current to magnetic flux B. In the Boltzmann equation (5.4), F is taken as F =−e(E + v × B) and the relaxation approximation (5.5) is applied. With f1 ≡ f − f0 ,
−

e
∂f
f1
(E + v × B) ·
=−
ℏ
∂k
τ

(p = ℏk).

(5A.1)In the first term of the left hand side, f in ∂f /∂k is replaced with f0 . Form dE = v · dp, the second term is ∂f0 /∂k =ℏ(∂f0 /∂E)v and the term of f0 is orthogonal with v × B and vanishes (magnetic field driven force is orthogonal with v
and does not give work). In the second term we take terms to f1 and obtain− ev · E

∂f0
e
∂f1
f1
− (v × B) ·
=− .
∂E
ℏ
∂k
τ

(5A.2)

Here we introduce a vector Ea with the physical dimension of electric field satisfying
f1 = eτ (v · Ea )

∂f0
.
∂E

(5A.3)

This is from the concept that the Lorentz force shifts the Fermi sphere as in Fig. 5.2 and the origin of the shift isrepresented as an electric field. Then the equation is− v · E = −v · Ea +eτeτ(v × B) · Ea , ∴ E = Ea − ∗ B × Ea .
mm

(5A.4)The solution to eq.(5A.4) is given as follows.
Ea =

[
]
( eτ )2
eτ
1
E
+
B
×
E
+
(B
·
E)B
,
1 + ωc2 τ 2
m∗
m∗
e|B|
ωc =
.
m∗

ωc is the cyclotron frequency. Then f1 is given as follows.
[]( eτ )2
eτ
eτ E
∂f0
·
v
+
v
×
B
+
f1 =
(B
·
v)B
.
1 + ωc2 τ 2
m∗
m∗
∂E

(5A.5)(5A.6)(5A.7)

We take the case B = (0, 0, Bz ), E = (Ex , Ey , 0). From vz = 0 and eq.(5A.7), f1 is calculated as
f1 = e

)
(
)]
[ (
∂f0
ωc τ 2ωc τ 2
τ
τ
E
−
E
+
v
E
+
E
.
vx
x
y
y
x
y
∂E
1 + (ωc τ )2
1 + (ωc τ )2
1 + (ωc τ )2
1 + (ωc τ )2

(5A.8)For example, to obtain jx = −en⟨vx ⟩ from this equation take the expectation value of vx with f = f0 + f1 . The
expectation value for f0 is zero and odd components in v is dropped from the integration over k. Then∫jx = 2(−e)vx f (k)

dk
e2
=− 3
3
(2π)
4π

∫

E6-12

∂f0
τ vx2
(Ex − (ωc τ )Ey )dk.
1 + (ωc τ )2
∂E

(5A.9)

The integrand in (5A.9) is the same as that in equilibrium other than vx2 and is a function of kinetic energy E. For ageneral function ξ(E), the principle of energy equipartition gives∫∫22vx ξ(E)dk =Eξ(E)dk.
3m∗(5A.10)With the Maxwellian approximation f0 = A exp(−E/kB T ), and density of states D(E) = AD E 1/2 , eq.(5A.10) leads
to

∂f0
f0
=
, n = AD
∂E
−kB T

∫ ∞

f0 E 1/2 dE =

0

2AD
3kB T

∫ ∞

E 3/2 f0 dE.

0These being substituted into (5A.9) and we obtain
[⟨
⟩
⟨
⟩
]
ne2
τ
ωc τ 2
jx = ∗Ex −Ey ,m
1 + (ωc τ )2 E
1 + (ωc τ )2 E

(5A.11a)≡ (ne2 /m∗ )(Al Ex − At Ey ) (definitions of Al and At ).

(5A.11b)⟨· · · ⟩E is defined in eq.(5.10). jy is obtained in the same way and the conductivity tensor in xy-plane is expressed as
j=

ne2
m∗

(

)
−At
E.
Al

Al
At

(5A.12)

Appendix 5B: Kelvin relations
We consider two species of metals A, B and a junction

Tm

BAB as shown in the right figure. The temperature at thetwo edges is kept to Tm and a unit charge moves from

A

B

Tm

T +D TT

B

one edge to the other quasi-statically. As indicated in
the figure, temperatures at the two junctions are T and
T + ∆T . The voltage between the two edges is VAB .
From the requirement of quasi-static assumption, we apply the first and the second laws of thermodynamics to obtainthe conditions,

VBA + ΠBA (T ) − ΠBA (T + ∆T ) + (τB − τA )∆T = 0
ΠBA (T ) ΠBA (T + ∆T ) τB − τA−+
∆T = 0.
T
T + ∆TT

In the differential formula with ∆T → 0,
dVBA
dΠBA
−
+ τB − τA = 0,dT
dT
From the second equation
τB − τA = T
and we reach∴ SAB =

d
dT

(

ΠBA
T

ΠAB
,
T

d
dT
)
=

(

ΠBA
T

)
=

dΠBA
ΠBA
−
,
dT
T

dSAB
τA − τB=
dT
T

with exchange of A and B.

E6-13

τB − τA.
T(5B.1)

References[1] M. Lundstrom “Fundamentals of carrier transport” 2nd ed. (Cambridge, 2000).
[2] K. Fletcher, P.N. Butcher, J. Phys. C 5, 212- 224 (1972).

E6-14

Lecture note on

Physics of Semiconductors (7)26th May (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Chapter 6 Homo·hetero junctions
So far we have seen the bulk properties of uniform semiconductors. Henceforth we go into the rich physical phenomenon in spatially structures semiconductors, the actions as defices.

6.1 Electrical and optical characteristics of homo pn junctions
The pn junction is one of the first semiconductor devices for electric circuits. For the detailed history of the device, seee.g. [1] (though in Japanese, out-of-print).

6.1.1 Thermal equilibrium
A pn junction, as it expresses, is a junction of a p-type semi-

be diffusion pressures which drive electrons to the p-layer and holes
to the n-layer for increase of entropy S. On the other hand, such

+
+

+
+
+

+

+

+
+

the n-layer and the hole density in the p-layer. Hence there should- - - - - - - -

n ND

+
+

p- and n-layers. In such a junction, the electron density is high in

wn

+
+

junctions, in which the same species of semiconductor is used for

-w p

+

p NA

conductor and an n-type semiconductor. Here we consider homo-Exdiffusions charge up the p-layer to negative and the n-layer to positive creating charge double layer at the junction (charge depletion
layer). This electro-magnetically enhances the internal energy U .
In thermal equilibrium, the double layer width is determined from-Ec

the condition for free energy (U − T S) minimum.

N A wp + N D wn
ee 0

eV bi

We take a simple model of an abrupt junction (Fig. 6.1), andp ∼ n ∼ ni in the depletion layer. We write the built-in voltage due to the pn structure at the interface across the depletion layer

EvEF

Vbi . In the process that an electron moves from the n-layer to thep-layer, the energy increases by eVbi . In the n-layer the electron
density nn ∼ ND , and in the p-layer the semiconductor equation
tells np ∼ n2i /NA . We consider a general case that N1 and N2
electrons are respectively distributed in two boxes with site number

Fig. 6.1 (a) Schematic of an abrupt pnjunction. (b) Electric field E(x) in depletion
layer. x-direction is taken as positive for thefield. (c) Band diagram.

N . The number of cases is W = N CN1 N CN2 . Here only particle
exchanges are considered hence dN1 = −dN2 . Under assumption N ≫ N1,2 , d(ln W ) ≈ ln(N2 /N1 )dN1 (mixing entropy of gases). Applying this to the pn-junction with dN1 = −1, N1 = nn , N2 = np , condition d(U − T S)/dnn = 0

E7-1

E Fn

108

n

p

106

eV

Forward bias

log p ,
logn

0

pp

| J / J 0|

E Fp
x
nn

104

Reverse bias

102

Shockley theory

np

100

pn

0

(a)

10

(b)

e | V |/ k B T

20

Fig. 6.2 (a) Upper panel: quasi-Fermi levels in a pn-junction under external forward voltage V . Lower panel:
Spatial variation of carrier densities. (b) Broken line: I-V characteristics of Shockley theory (eq.(6.11)). Normalizedwith J0 , which is the coefficient in eq.(6.11). Solid line: Realistic I-V characteristics, in which series resistance,
recombination inside the depletion layer, tunneling through localized states are taken into account. The inset is alinear plot of (6.11).

gives
eVbi = kB T ln

Nc Nv
nn
ND NA
= Eg − kB T ln
∼ kB T ln
.
np
n2i
ND NA

(6.1)

(nn ∼ ND , pp ∼ NA ).
In equilibrium it is also required that the chemical potential (Fermi energy) is constant through the junction, independent of the spatial coordinate. Far inside p, n-layers apart from the junction, the band structure should recover thebulk states. Hence the band diagram in Fig. 6.1(c) is drawn. Let the depletion layer widths in p and n-layers wp , wn
respectively, then E(x) is given as− ϵϵ0 E(x) = NA (2x + wp ) + ND wn (x < 0),NA wp + ND (wn − 2x) (x ≥ 0),

where ϵ is the dielectric constant. Then Vbi is calculated as
∫ wn
e
e
ND 2
Vbi =
(ND + NA )wn wp =(ND + NA )w(−E(x))dx =
ϵϵ0
ϵϵ0
NA n
−wp

∵ wn ND = wp NA .

(6.2)

(6.3)

From eqs.(6.1) and (6.3), we obtain the relation between the doping concentrations and the depletion layer width.

6.1.2 Current-Voltage characteristics
In equilibrium of a pn-junction, the net current is zero as a result of balance between the entropy and the internalenergy. An externally applied voltage pushes off the balance and a current flows as a result. When the energy cost islowered by the voltage, the diffusion current causes injection of minority carriers. Minority carrier injection is an actionthat increases density of minority carrier dynamically. The minority carrier injection breaks the semiconductor equation
np = n2i locally. Even in such circumstances, by introducing quasi-Fermi level, we can treat electrons and holes as inquasi-equilibriums and apply the Boltzmann equation to obtain carrier fluxes. The semiconductor equation (law of massE7-2

action) can also be recovered in a bit modified manner. The goal here is to give the net current as a function of external
voltage.
We model the effect of external voltage V as follows. All the voltage drops outside the depletion layer are ignored and
V is applied inside it. Far from the junction, the current is carried by majority carriers, which have high concentrationand the gradient in the chemical potential in such regions is ignorable. Around the depletion layer, imbalance between theinternal energy cost and the increase of entropy causes a flow of carriers. V is applied against Vbi lowering the barrier for
diffusion currents, then the majority carriers flows into the counter layers increasing the minority carrier densities at the
depletion layer edges. The injected minority carriers diffuse into the bulk, recombine with majority carriers and disappear.
The diffusion-annihilation process forms a exponential decay in the steady minority carrier density distribution.
In the above model, we assume that local thermal equilibrium is attained in each thin layer parallel to yz plane throughthe carrier-carrier interaction and the particles can be exchanged between neighboring layers. Quasi-Fermi levels, which
depends on x-coordinate, for electrons (µe (x)) and holes (µh (x)) are introduced as follows,n(x) = Nc exp[−(Ec (x) − µe (x))/kB T ],i.e., µe (x) = Ec (x) + kB T lnp(x) = Nv exp[−(µh (x) − Ev (x))/kB T ],

n(x)
,
Nc

µh (x) = Ev (x) − kB T ln

(6.4a)

p(x).
Nv(6.4b)The diffusion of minority carriers (densities np , pn ) is described by the following diffusion equations.
De

d2 np
np − np0=− G(x),dx2τeDhpn − pn0
d2 p n=− G(x),
dx2
τh

(6.5)

where G(x) represents minority carrier creation e.g. by light illumination and in the dark G(x) = 0. np0 , pn0 areminority carrier concentrations in the bulk regions, De,h , τe,h are the diffusion constant and the lifetime respectively (efor electrons, h for holes). Then minority carrier diffusion lengths for electrons and holes are√√
Le = De τe , Lh = Dh τh .

(6.6)

The solution for (6.5) (pn for x > wn , np for x < −wp ) which satisfies the boundary condition np → np0 (x → −∞)and pn → pn0 (x → ∞), is obtained as()
x + wpnp (x) = δn0 exp+ np0 ,
Le()
x − wnpn (x) = δp0 exp −+ pn0 ,
Lh(6.7)

where δn0 , δp0 are concentrations of injected minority carriers at the edges of the depletion layer. From the definition(6.4b), in the region of diffusion and with ignoring np0 , pn0 in (6.7), the quasi-Fermi levels linearly depend on the
distances as[µe (x) = Ec + kB T]
x + wp
δn0
+ ln
,
Le
Nc

(p),(n)These should join the bulk values EF[µh (x) = Ev − kB T(p),(n)at x → ±∞ respectively and EF]
x − wn
δp0
+ ln
.
Lh
Nv
(p)

(6.8)
(n)

differ by eV , i.e., EF − EF

= eV .

Therefore they are schematically drawn as in Fig. 6.2(a).
We ignore electron-hale recombination inside the depletion layer and assume the currents are limited by the diffusionof minority carriers. Then the net current density is the sum of minority carrier diffusion currents at the two edges of thedepletion layer. From eq.(6.7) and eq.Fig. 6.2(a),δn0 + np0 = n(−wp ) = np0 expeVeV, δp0 + pn0 = p(wn ) = pn0 exp
.
kB T
kB T

(6.9)

The electron diffusion current density at x = −wp in the process (6.5) is thus[]eDe δn0
dnp
eDe
eV
=
je = eDe=
np0 exp−1 .
dx −wp
Le
Le
kB T
E7-3

(6.10)

pp

J

ln p , ln n

nn
V oc

pn
np
Gt e
n p0

V

Gt h
p n0
x

J sc

(a)

(b)

Fig. 6.3 (a) Carrier density distribution around a pn-junction under photo-generation of minority carriers G. Depletion layer edges are indicated by perpendicular broken lines. Bias is taken as shortage V = 0. (b) Schematic I-Vcharacteristics in the dark and under illumination.

The hole current can be calculated in the same way and the net current is given as
[
][
]
[
][
]
De
Dh
eV
De
Dh
eV
j=e
np0 +
pn0 exp− 1 ≈ en2i
+
exp
−1 .
Le
Lh
kB T
Le NA
Lh ND
kB T

(6.11)

Equation (6.11) is the very basics of the Schottky theory of pn-junction. Though the model grabs the essence, real pnjunctions are much more complicated. Important modifications are series resistance, recombination in depletion layer andtunneling conductance through localized level inside energy gap (parallel Ohmic resistance). With these modifications, arealistic characteristics shown in Fig. 6.2(b) differs considerably from the Shockley theory.

6.1.3 Photo-response of pn-junctions
Let us take the simplest model for a pn-junction under illumination assuming majority carrier generation G(x) does
not depend on x (a constant G) in the diffusion equation (6.5). Just as before, the solution for np (x) and pn (x) which
satisfies the boundary condition np → nn0 + Gτe for x → −∞, and pn → pn0 + Gτh for x → ∞ is
[
(
(
)
)
]
(
)
eV
x + wpnp (x) = np0 + Gτe + np0 exp− 1 − Gτe exp
,
kB T
Le
[
(
(
)
)
]
(
)
eV
x − wnpn (x) = pn0 + Gτh + pn0 exp− 1 − Gτh exp −
.
kB T
Lh

(6.12a)(6.12b)The solution for V = 0 is schematically drawn in Fig. 6.3(a).
From the solution, the net current density is given as[]
eV
j = j0 exp− 1 − eG(Le + Lh ),kB T(6.13)

where j0 is the coefficient in front of the parentheses in (6.11). Equation (6.13) is a simple negative shift of (6.11) by
jsc ≡ G(τe + τh ). Figure 6.3(b) shows the characteristics. Real solar cells are more complicated but the common is
the negative shift of the current characteristics with illumination. The parameters which characterize each device are thenegative shift at short-circuit condition |JSC | (short circuit current) and the voltage at open-circuit condition VOC (opencircuit voltage). These depend, of course, on the strength and the spectrum of illumination.
In the characteristics shown in Fig. 6.3(b), the cell pumps out an electric energy under the bias condition in the fourth
quadrant. Current J and voltage V give power W = |JV |. In the forth quadrant |J| ≤ |JSC |, |V | ≤ |VOC | thenW ≤ |JSC VOC |. Then Jmax , Vmax which give the maximum power is determined andFF ≡Jmax Vmax≤1
JSC VOC
E7-4(6.14)

is called filling factor (FF). The better the squareness of the I-V characteristics, the higher the FF. JSC , VOC , and FF areuseful parameters for discussing phenomenology of solar cells, modeling equivalent circuits. In the ideal characteristics(6.13),[]kB T
eG(τe + τh )|JSC | = eG(Le + Lh ), VOC =
ln
+1 .
e
j0

(6.15)

The above is the basics of photoelectric conversion and applied to e.g. solar cells. For the solar cells see the article by
the present author [2] (in Japanese).

6.2 pn-junction transistors
Today, we see two kinds of semiconductor devices invented by agenius named William Shockley. The style of research and development which he began, as well as his devices, has been changing thehuman life. The above expression is not exaggeration, I believe. I
have read a short commentary, which tells “the researchers in Bell
Labs. were doing basic research on the surface states of Ge withputting tips on the surfaces and accidentally found the transistor action”. But this is far from real situation. Walter Brattain and John
Bardeen, who were the direct finders, were doing research aiming atconstruction of “solid state amplifier” under the team leader Shockley. They did not expect such an easy finding probably but theyrealized the amplification certainly because they were doing such
From left, John Bardeen, William Shockley, Walter
Brattain. At AT&T Bell Laboratories, in 1948.

objective research.
The experiment was done a little before the Christmas of 1947
(said to be 12/16. The application for patent was 12/23) Shockley
was out of the labs for a journey. He was thus not so glad hearing
the success. Also the transistor (the term is a combination of transfer

and resistor) which Brattain and Bardeen accidentally found was called “point contact type”, unstable, had low reproducibility. It should have serious obstacles for commercial viability. Their finding might have stimulated Shockley’s fightas an inventor, he was absorbed in thought as a theorist aiming at realization of “reproducible device for amplification”
and finally got the brilliant inspiration of junction transistor, on the new year’s eve allegedly. The theory for the junction
transistor established 1/23 in the next year. The experimental realization was a year later. The event was the glorious dawnof the semiconductor physics, in which artificial structures in solids utilize the structural sensitivity of semiconductors
and create new functions, new stages of physics[3]．

6.2.1 Junction transistor: structure
Figure 6.4 shows basic structure of junction transistor (Bipolar Junction Transistor, BJT, at times just “bipolar transistor”), in which two pn-junctions are placed close to each other. npn and pnp are possible types of junctions. An ohmic
contact to the central layer is required for the device to have three terminals. The terminals at the two ends are called
Collector (C), Emitter (E) respectively and the central one is called Base (B). In the very beginning, the structure wasfabricated with alloying metals which work as dopants to both sides of the base material. The naming “Base” came fromthe fact though lithography and thermal diffusion, ion implantation and epitaxy soon became the dominant methods. As
we will see for the transistor action, the base should be very thin. Thinner than the minority carrier diffusion length.

E7-5Collector (C)

C
C

p
n

Base (B)

JC

C

JE

E

n
B

B

p

p
n

E

B
JB

E

Emitter (E)

(a)

(b)

Fig. 6.4 (a) Schematic structure of pnp transistor. Circuit symbol and the names of terminals. (b) Schematic
structure of npn transistor and the circuit symbol.

Circuit symbols of transistors are shown in Fig. 6.4, which represent connections of two electrodes to the base graphically. Circles are often omitted. pnp and npn are distinguished with the direction of arrow, which indicates direction ofelectric current when minority carriers are injected into the base electrode. Below we consider npn-type and define thedirections of the current as in the figure.

6.2.2 Current-amplification of junction transistors
In the first experiment a constant voltage source is connected to B-C and collector current JC is measured. Inside thestructure B-C is nothing but a pn diode and the result is a well known rectification characteristics (JE = 0 in Fig. 6.5(a)).
Now we connect a constant current source between E and B, and apply finite currents through E. Because B-E is also apn junction, the forward bias is positive for B. As shown in Fig. 6.5(a) VBC − JC curve shifts parallely to negative. The
amount of shift is almost JE .
It should be noted that the characteristics is close to that of a solar cell shown in Fig.2.3(b). The similarity is not acoincidence, rather, the physical situation is almost the same. While In a solar cell, the minority carriers are directly
created by photon irradiation, in a transistor, the minority carriers are injected through the pn junction between E and B
to the other junction between B and C.
The phenomenon occurring in the junctions are summarised as follows. Here we only describe the phenomenon in
conduction band while that in valence band can be discussed in parallel. In an npn junction, a reverse bias voltage to
B(p)-C(n) suppresses the diffusion current from the n-layer to the p-layer. The (reverse) diffusion of electrons from thep-layer to the n-layer is not enhanced by the reverse bias because all the electrons reach from the p-layer to the junction
are swung to the n-layer and it is already saturated at zero-bias. Under the reverse (or zero) bias condition of VBC , let
the other pn-junction (E-B) be under a forward bias condition. This is possible because an Ohmic contact is attachedto the base electrode, hence VEB and VBC can be controlled independently. The forward bias lowers the barrier by thebuilt-in potential in E-B junction and the electrons (majority in the n-layer) diffuse into the base layer and the minority
carrier concentration increases in B. This is the phenomenon called minority carrier injection, which decays over theminority carrier diffusion length through the recombination with majority carriers (holes). Note that the continuity incurrent is hold. The flow by injected electrons is not driven by the electric field but by the density gradient. So the flux
is perpendicular to the junction plane almost ignoring the base Ohmic electrode (the recombination current goes to theelectrode). When the B-layer is much thinner than the minority carrier diffusion length, most of the injected carriers reachthe other junction enhancing the reverse current. In Fig. 6.6(a), this appears as the enhancement of the reverse current,
the amount of which is determined that of injected minority carriers. Hence the current does not depend on VBC as long
E7-6

4

J B = - 20 m A

2N222A

2N222a3

JE = 00

J C (mA)-J C (mA)

2

1mA
2mA

-2

-16mA
-12mA

2

-8mA

3mA
1

4mA

-4mA

-4

5mA
-0.5

0

0

0

0.5

0

V BC (V)

0.5

V BC
B

JE
E

JE

B

-J B

p

n

electron

C

-J C

E

n

electron

n

p

n

1

V CE (V)

JC

C

(a)

(b)

V CE

Fig. 6.5 (a) JC (upside down for convenience) as a function of VBC with JE as a parameter in the circuit shown in
the lower panel. With increasing JE i.e., injecting electrons from E to B, the characteristics resembles to that of anilluminated solar cell. (b) Application of collector-emitter voltage VCE with floating B, almost no current flows dueto the reverse bias in C-B. The biasing B with some currents JC appears according to JB showing saturation for VCE .

as there is no forward current.
Now an amplification circuit can be composed as follows. Let the electrodes C-E be voltage biased as in Fig. 6.6(c).
The amount of minority carrier injection into B-layer is determined by VBE . Hence in this circuit JC strongly depends on
VBE as shown in Fig. 6.6(a). However, the relation is too non-linear for the use of the device in a voltage-input circuit.
Some of the injected minority carriers recombine with majority carriers and some portion flows out to B-electrode. Thebase current JB depends on VBE in the same functional form only but the coefficient as JC because the pn-junction is thesame. JC is thus proportional JB , that is,
JC = hFE JB .

(6.16)The good linearity is confirmed in the measurement as shown in Fig. 6.6(b). hFE is called current amplification
factor. And it is often said that “a bipolar transistor works as a current amplification device” from this face. This is inpractice, true as long as we use it as a black box device in electric circuits. However in physical mechanism, as discussed
above, there is no such causality that a small current drives a larger current. The following expression may be closer toreality: a small current here is just a monitor for voltage to control a large current.
In the usage of a BJT in a circuit, care should be taken that (because it is a “current amplification device”) the input
voltage bias should be set to a low differential resistance region. Particularly in high frequency circuits, the impedance
matching should be taken to the characteristic impedance of the transmission line. One simple “rule” for transistor circuits
is that when a transistor is working as an amplifier, the base-emitter bias voltage should be around the quasi-thresholdvoltage (though as we saw there is no threshold voltage in pn-junctions, in ordinary circuit scale, the I-V curve seems as
if it has).

E7-7

4

2N222AV CE = 6V

2N222AV CE = 6V

10-6

3

2

10-3

J C (A)

10-4

J C (mA)J C (A)

10-2

10-8

1
10-10

J B (A)
10-5

0.01

0

0.1

50

100

150

10-4
200

250

300

J B ( m A)

V BE (V)
(a)

(b)

V BE
B

JC

electronEnFig. 6.6 Characteristics of a transistor measured in a experimental circuits shown in (c). (a) C-E is biased with 6 V
and the voltage between B-E (VBE ) is varied. JC strongly
depends on VBE . (b) JC is re-plotted as a function of JB
and (because outputs of the same diode is observed) very
good linearity is obtained. The inset shows a log-log plot
with a broken line indicating slope 1.

JB

p

n

C

6V

(c)

6.3 Field effect transistors I
Field Effect Transistors (FETs) are now used much more widely in circuits than BJTs. And the idea of FET was born
even long before that of BJT *1 , but for the realization of FET requires technologies even higher than those for BJT andthe realization was later than that for BJT. In these 20 years, Metal-Oxide-Semiconductor (MOS) type FETs are mainly
used but the first FET was realized for Junction FET (JFET), which utilize pn junctions.

6.3.1 pn-junction and depletion layer
For understanding the device action of JFET, the relation between the reverse bias voltage and the depletion layer isimportant. We consider a pn-junction shown in Fig. 6.7, with x-dependent potential ϕ(x). The Poisson equation is givenasd2 ϕ= −aq(x)dx2(a ≡ (ϵϵ0 )−1 ).

(6.17)In the space-charge region (depletion layer) we assume abrupt concentration distribution of dopants and sharp cutting of
the end of depletion layer. Then{q = −eNAq = eND(−wp ≤ x ≤ 0),(0 ≤ x ≤ wn ).

(6.18)*1 Shockley wrote a patent on FET before BJT though many similar ideas had existed before that. We cannot say the patent is as unique as that of

BJT.

E7-8

e ( V bi + V )

p

-

-

+ +

-

-

+ +

-

-

+ +

-w p

n
Fig. 6.7 Simple model of a pn junction

wnLet’s take the asymptotic condition as ϕ(−∞) = 0. When there is external reverse bias voltage V , the boundary conditionat the edges of depletion layer isdϕ= 0,dx −wpϕ(−wp ) = 0,ϕ(wn ) = V + Vbi ,(6.19)

dϕ
= 0.
dx wn

Integration of the above gives{
ϕ(x) =(aeNA /2)(x + wp )2
V + Vbi − (aeND /2)(x − wn )2(−wp ≤ x ≤ 0),(0 ≤ x ≤ wn ).

(6.20)

From the condition for the connection at x = 0
lim ϕ = lim ϕ,

x→+0

x→−0

lim (dϕ/dx) = lim (dϕ/dx),x→+0(6.21)

x→−0

the widths of depletion layer wp , wn are given as follows.
[

]1/2
[
]1/2
2ϵ0 ϵ(V + Vbi )ND
2ϵ0 ϵ(V + Vbi )
NA
wp =
·
, wn =·eNA
ND + NAeND
ND + NA[]1/2
2ϵ0 ϵ(V + Vbi ) NA + ND
wd = wp + wn =
·
.
e
NA ND

(6.22)(6.23)

The charge accumulated in the depletion layer on n-side is Q = eND wd per

1/C 2

unit area giving the effective capacitance (differential capacitance) as
√
√
1
dQ
2ϵϵ0
ϵϵ0 eND√= eND=(V + Vbi )−1/2 .
dV
eND 2 V + Vbi
2(6.24)In a p+ n-structure, that is, NA ≫ ND ,
[
wd ≈

-V
V bi

2ϵϵ0 (V + Vbi )
eND

]1/2
≈ wn .

(6.25)

This means the depletion layer expands in proportional to the square root of thereverse bias voltage plus the built-in potential.
This relationship is frequently used for characterization of pn-junctions. For

example, differential capacitance C(V ) can be measured with applying high frequency voltage source with a small
amplitude and through the phase shift. We plot the data as shown in the left figure (for the convenience, the horizontal
axis is taken to −V ), 1/C 2 versus −V . If ND is spatially uniform, the data points should be aligned on a line. (6.24) isE7-9

valid only for V > 0 and C → ∞ cannot be realized. But with extrapolation from V > 0 the point 1/C 2 = 0 can be
specified and we obtain Vbi from this.
When ND is not uniform spatially or some deep level traps exist, we obtain information of the spatial distribution fromdifferentiating the plot. Application of pulses in V and analysis of transient response under light illumination or related
techniques can bring much of the information inside the semiconductor[4]．

6.3.2 Junction Field Effect Transistors
Figure 6.8 shows a schematic drawing of the JFET structure in a cross sectional view. It is for an n-channel, which has
two electrodes on the both edges. They are called Source (S) and Drain (D) respectively. The channel is sandwiched byp+ layers called Gates (G).
The principle of device action is very simple as can be seen in Fig. 6.8. Applying reverse bias to the gates causes
expansion of white-colored depletion layer according to eq.(6.23). This makes the conduction channel narrower andenhances the channel resistance up to infinity for pinch-off. Thus the current through the device is controlled by the gate
voltage. This is apparently a voltage-controlled device and the input impedance is typically resistance of pn-junction inreverse bias condition. So it is classified into high input impedance device.
A characteristic feature here is that a large source-drain current causes a significant voltage drop across the device,
resulting in gradient of effective reverse bias voltage for the channel-controlling depletion layer. Let us see a simple
model. As before in the model for pn-junctions, we assume the boundaries between depletion layers and conduction
channel are abrupt. Let the gate length L, the thickness of JFET 2wt . We take the channel direction along y-axis. The
depletion layer with wd is√
wd (y) =2ϵϵ0 V (y),eND(6.26)

where V (y) is local voltage at position y between the channel and the gate. V (y) can be obtained by subtracting voltage
along the channel Vch due to the source-drain current from the sum of the built-in potential Vbi and the reverse bias gate
voltage Vg .

V (y) = Vg + Vbi − Vch (y).

We have no injection of minority carrier and only consider the drift current of majority carriers. The electric field alongy-direction is dV /dy. Let the channel depth W and the drift current through the channel is
Jch = eND µndV· 2(wt − wd )W.
dy

G (gate)p
S(source)pD(drain)

n

-3V
D
G

n

p

(6.27)

p

-3V

D
G

S
S
n-channel p-channel

G (gate)L0w d( y )y2w t

Fig. 6.8 Schematic structure of JFET (n-channel) (upper left panel). A
cross-sectional view. The easiest way to form the p+ -layer is alloying themetal, which can work as acceptor inside the semiconductor. The picturein the center shows the way the depletion layer expand (white region) withapplication of reverse bias to the gates. The upper left shows circuit symbols. The (lower) left shows the dimensions of the model adopted in the
text.

E7-10

In steady state there is no charging up and Jch is uniform through the channel thus integration over the channel shouldbe Jch L.
∫ L
Jch L =0∫ L

dV
Jch dy = 2eND µn W
(wt − wd )dy = 2wt eND µn W
dy0

∫ VL (
V0

wd
1−
wt

)
dV.

(6.28)

Let the critical voltage Vc at which the channel is pinched (wd = wt ) and Jch = 0 then Vc = eND wt2 /2ϵϵ0 . Hence from√wd /wt = V /Vc , Jch in this model is obtained as[]
2ND eµn W wt
2
Jch =VL − V0 + √ (V (V0 )3/2 − V (VL )3/2 ) .
(6.29)L
3 Vc
In eq.(6.29), at small voltages, the first linear term in VL is dominant and Jch increases linearly. With increasing the
3/2

voltage, the last VL

term grows and at last the current begins decreasing, which means negative differential resistance.

In actual device, this does not occur and Jch simply saturates with increasing V . The model contains various shortages,e.g., the equipotential lines are straight and along x-axis. Improved models can reproduce the saturation but they are
inevitably complicated. There are also empirical analytical formulas well fit to the experiments but they have no physicalreasoning.

Appendix 6A: Analysis of pn junction transistor
Let us have a brief look at the simplest analysis of charrier statistics in bipolar transistors.

6A.1 Current-voltage characteristics
Figure 6A.1 illustrates the bias conditions and the carrier concentrations in an npn-type transistor. We take the x-axis
along the device current direction, and the depletion layer edge at the emitter side of the base is set to x = 0. The electron(minority carrier) concentration at x = 0 is
np (0) = np0 exp

eVBE
.
kB T

(6A.1)

They diffuse the base region and reach the depletion edge at the other side x = WB . From there the electrons are
immediately swept out to the collector by the electric field in the depletion layer. Hence the electron concentration in thevicinity of WB should be very small.
np (WB ) = np0 exp

−eVBC
≈ 0.
kB T

(6A.2)

Providing that WB is much shorter than the minority carrier diffusion length, we can ignore the carrier recombination andthe diffusion current in the base is constant. Equation (5.12) tells the current is proportional to dnp /dx. Hence np varies
emitter
emitter base collectorn 0E

log n , p

basecollector

linear n , plog n , p

n 0C

p p( x )npnn p( x )

p 0E
(a)

V BE

V BC

(b)

depletion0

p 0C
W B depletionx

Fig. 6A.1 (a) Biasing condition of the npn transistor under consideration. (b) Schematic diagram of carrier concentrations in a npn type transistor. In the base the ordinate is in linear scale while logarithmic in other regions.

E7-11

Fig. 6A.2 Characteristics of transistor 2SC1815 for small signal amplification (from the datasheet). Left panel:Collector-emitter voltage VCE dependence of collector current JC for various base current JB .

linearly against x as illustrated in the figure (the concentration is in the linear scale only in the base region). Hence from
(6A.2) the diffusion current density in the base isjDe = −Denp (0)
dnp
≈ eDe
.
dx
WB

(6A.3)

This is the major part of the collector current and the collector current is with A as the cross setion of the device,
JC = eADenp (0).
WB(6A.4)

From the semiconductor equation np0 ≈ n2i /NA ,JC ≈eADe np0
eADe n2i
eVBE
eVBE
eVBE
≈
≡ JS exp
.
exp
exp
WB
kB T
W B NA
kB T
kB T(6A.5)

JS = eADe n2i /WB NA is the coefficient which is inversely proportional to WB NA .
On the other hand, the base → emitter is in forward biasing condiction while base → collector is reverse biased.
Hence most of the base current flows to the emitter, which is determined by the hole diffution current in the emitter. The
calcution is along the same line as the above collector current and the diffusion part of the base current is
JBh =eADh n2i
eADh
eADh
eVBE
eVBE
=
.
pnE (0) =pnE0 exp
exp
Lh
Lh
kB T
Lh ND
kB T

(6A.6)In the base, minority carrier concentration is enhanced and the recombination current may give some contribution. The
total charge of the minority carriers is Qe = −enp (0)WB A/2. Let τb be the minority carrier life time and the recombination current is
JBr =

Qe
enp (0)AWB
eVBE
=
exp
.
τb
2τb
kB T

Therefore the base current is written as the sum of the above as)(
np0 WBeVBE
Dh n2i+exp.
JB = eA
Lh ND
2τb
kB T

(6A.7)(6A.8)

Then from (6A.5) and (6A.8), the current gain is obtained as(hFE =Dh WB NAWB2+
De Lh ND
2τb De
E7-12

)−1
.

(6A.9)

6A.2 Effect of depletion layer width
Figure 6A.2 shows the characteristics of a transistor numbered 2SC1815 (Toshiba, Co. Ltd.). The right panel showshFE as a function of JC . hFE is almost constant in the low JC region indicating good linearity. On the other hand, the left
panel shows JC as a functionof VCE with JB as a parameter. In this panel, in the region VCE ≈ 0, the base-collector isforward biased and not in the region of current amplification. Even in the current amplificatio region, JC increases with
VCE . This is called the Early effect caused by the widening of the depletion layer thus by the thinning of the base width
WB .
Let ∆W be the variation in the width of base width and the collector current is given as()()
np (0)
np (0)∆W∆W≈ eADe1+≡ JC0 1 +.
JC = eADe
WB − ∆W
WB
W
W(6A.10)∆W grows rapidly with VCE as in (6.23) when VCE is small while the rate lowers with VCE . In Fig. 6A.2, such tendency
is apparent. In Fig. 6.5(b), the Early effect is small and the increase in JC can be approximated to be linear in VCE .

付録 6B：Deep level transient spectroscopy (DLTS)
Here I would like to give qualitative explanation on the basic principles of Deep Level Transient Spectroscopy (DLTS).
For details, see e.g. ref. [4]. We consider modification to effective capacitance (6.24), which depends on the reverse bias
voltage V . Let ND be the shallow donor concentration, NP the one for a deep donor. In the region where this deep donor
responds to change in the bias voltage, the voltage-differential capacitance is expressed as a function of reverse voltage
V as
[
]1/2
2ϵϵ0 (V + Vbi )wd (V ) ≈≈ wn ,e(ND + NP )√ϵϵ0 e(ND + NP )C(V ) =(V + Vbi )−1/2 .
2(6B.1)(6B.2)For simplicity, we consider the situation that the reverse bias Vp is applied and kept for sufficiently long time for electrons
to escape from the depletion layer including the deep levels *2 . Now V is abruptly lowered to V0 < Vp and the carriers arecaptured by the donor levels within w(V0 ) < x ≤ w(Vp ). Shallow donors have high capture rate and can respond within
ms without delay, deep levels, on the other hand, the capture rate strongly depends on temperature and with decreasing
temperature, the average time for capture often elongates from ms to s, min, hour and sometimes day. Then if we open up
a fixed time window and observe the time evolution of C, the time dependence is observed in the time window at sometemperature range and in low or high temperature regions the effect of deep levels does not observed.
Such a process is illustrated in Fig. 6B.1(a). We take t = 0 at the time the reverse bias is changed:Vp → V0 andmeasure the difference in the differential capacitances at t1 and t2 : ∆C = |C(t1 ) − C(t2 )| as a function of temperatureT.
We now assume existence of two species of deep donors, which have temperature dependent capture cross sections
shown in the upper panel of Fig. 6B.1(b). ∆C should show two peaks in the temperature dependence. Analysis of thedata gives the concentration and capture cross section of each deep level, and combination with photo-response, in some

*2 At low temperatures the capture/emission rates of deep levels become very small and it is not rare that we need days for the emission. So thiscondition is, in general, hard to be fulfilled. But the consideration of this does not give significant change and thus we adopt the assumption.

E7-13

s( T )w n( V 0)w n( V p)

T
Deep level 2
C ( V 0)C ( V p)DC

Deep level 1

T
0

t1

t2

t

(a)

(b)

Fig. 6B.1 (a) Upper panel: Illustration that the change in the reverse bias Vp → V0 makes shallow levels and a partof deep levels ready for catching carriers. Lower panel: With progress in capture of carriers, differential capacitanceC(V ) shows transient response. (b) Upper panel: two deep levels exist and assumed temperature dependences of
the capture cross section σ are illustrated. Lower panel: shows how the DLTS signal appears from the temperaturedependence σ(T ).

cases identification of deep levels or at least energy positions can be measured[4]. With variation of V0 and Vp , depth
profile of deep levels can be obtained also.

References[1] 菊池誠「半導体の理論と応用 (中)」(裳華房, 1963).
[2] 勝本信吾 物性研究 電子版 Vol.3, No.3, 033209 (2014 年 8 月号)
http://mercury.yukawa.kyoto-u.ac.jp/˜bussei.kenkyu/pdf/03/3/9999-033209.pdf[3] Jon Gertner, “The Idea Factory: Bell Labs and the Great Age of American Innovation”, (Penguin Press, 2012).
[4] 国府田隆夫, 柊元宏 「光物性測定技術」(東大出版会, 1983).
[5] M. Jaros, “Deep levels in Semiconductors” (CRC Press, 1982).

E7-14

Lecture note on

Physics of Semiconductors (8)2nd June (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Next we see FETs without pn-junction. For transistor action, they utilize phenomena on the surfaces or interfaces. In
homo-type pn-junctions the uniformity of space is broken by impurity doping. They do not use interfaces or surfaces.
This was important for Shockley and co-workers to realize “stable and reproducible” devices because for the semiconductor technologies in those days control of surfaces or interfaces was too difficult for commercial production. Even
the high quality crystalline growth and the accurate doping technique, which are indispensable for the realization of pnjunctions, were surprisingly high technique. However the great strides in semiconductor technologies caught the control
techniques of surfaces and interfaces in incredibly short time. Naturally there were movement to utilize them for device
actions and they overwhelmed bulk shortly. We have a look for these representative modern devices here. But the limitof miniaturization inevitably requires three dimensionality nowadays and we do not know what happens next.

6.3.3 Schottky barrier (junction)
Here we consider junctions between semiconductors and metals. Simple guiding principles are
1. Rigid band approximation,
2. Recovery of bulk states away from the junction,
3. In equilibrium EF (µ) is constant over the space.
On semiconductor surfaces, there usually are surface states with high density of states. Metal-semiconductor junctions
are strongly affected by those states. Here, however, we first look what Anderson’s rule tells about the interface[?]. The
baseline of rigid bands can be taken to an edge of “band”, in which electrons can freely travel between the metal and thesemiconductor. It is usually impossible to find such an energy band inside insulators and semiconductors, which have very
different energy bands. Then such a “band” can be found as the vacuum levels. Then the excitation energy required is socalled work function. Let the work functions in the semiconductor and the metal eϕS and eϕM respectively. GenerallyeϕM ̸= eϕS . On the other hand, from the guiding principle 2., the bulk EF ’s in the metal and in the semiconductor awayfrom the junction should be the same. And EF should be constant throughout the system.
The following procedure, of course, is not real physical process but just a virtual process inside human brain, forconstruction of consistent band alignment. The final result, however, may be realized in the model of junctions though
there still remain many idealizations and reality should be much more complex.
We assume eϕM is larger than eϕS , the semiconductor is doped to n-type and the donor concentration is ND . We
make the vacuum levels in the both sides fit to each other and extrapolate the bulk band structures to the interface toobtain the band alignment shown in Fig. 6.9(a). Here the Fermi level in the semiconductor places higher than that in themetal causing flow of carriers from the semiconductor to the metal. The carrier flow generates charge accumulation at
the interface creating an electric field perpendicular to the junction plane. The metallic side is also charged up but it has
much higher charge concentration, which screens the electric field within the screening length less than a lattice constantmaking the band bending negligible in this side. Let the accumulated charge in the metal side per unit area −Q, in thesemiconductor side (x > 0, interface at x = 0), the electric field at x is (eND x − Q)/ϵϵ0 and the potential difference
between 0 and xd is∫ xd
ϕ(xd ) =0

1(eND x − Q)/ϵϵ0 dx =
ϵϵ0
E8-1

(

)
eND 2
x − Qxd .
2 d(6.30)

ef S
ef M

ED

Ec
E F e( f - f )
M

EF

S

Ec

Ec
ef surf

EF

EF
ED

Ev

ED

wd
(a)

(b)

(c)

Fig. 6.9 (a) Virtual band alignment, in which a metal and a semiconductor are connected as the vacuum levels forthem agree. (b) Band bending effect to make EF constant throughout the junction is superposed to the alignment in(a). The situation corresponds to an ideal interface without surface states at the semiconductor side. (c) Illustrationof Fermi level pinning by surface states. The surface potential ϕsurf is determined by the position of the dominantsurface states from the band edge Ec . This usually has nothing to do with the difference between the work function.

Let the space charge (depletion) layer width be wd . The condition that electric field outside the depletion layer should be
zero, gives wd = Q/eND . On the other hand, the condition eϕ(wd ) = ϕM − ϕS also gives Q as√√√
2ϵϵ0 (ϕM − ϕS )2ϵϵ0 Vs
Q = 2ϵϵ0 ND e(ϕM − ϕS ), ∴ wd =
≡
.
eND
eND

(6.31)

Here we write eVs ≡ ϕM − ϕS . Now we can illustrate the band structure for electrons (holes for p-type) around themetal-semiconductor interface as in Fig. 6.9(b), showing a potential barrier, which is called Schottky barrier.
An external voltage V is mostly bared in the semiconductor side, and the height of the barrier changes to e(Vs − V )while the height from the metal side remains as eVs . To be more accurate, we need to consider the kinetic energy
distribution in the semiconductor and count the number of electrons which go over the barrier. But here for simplicity
we assume the kinetic energy of electrons in the semiconductor is a constant. Then the equation for thermal electron
emission from metallic surface can be applied to obtain
)
(
)]
(
)[
(
)
]
[
(
−eVs
−eVs
eV
e(V − Vs )
2
2
− exp
= eAT exp
exp
−1 .
J = AT exp
kB T
kB T
kB T
kB T

(6.32)

Here A is the Richardson coefficient. The first term is current from the semiconductor side, the second is that fromthe metal side. The current-voltage characteristics is similar to that of a pn-junction with the Schttkey barrier heightcorresponding to the built-in potential.
In the above the surface of semiconductor is too much idealized for it to have no surface states. However in real
metal-semiconductor junctions, current-voltage characteristics are similar to eq.(6.32). One big difference is in eq.(6.32),
the barrier height should change with changing the metal species but in reality, the barrier height is almost constant for
semiconductor species and independent of metals. This is due to the surface states on the semiconductors. The surface
states have narrow energy widths, very high density of states pinning the Fermi level to the center of them. Hence theband bending exists even before the connection to metals and the alignment is accomplished between the metal EF andthe surface states. This is called pinning of Fermi level by the surface states.
Once the Fermi level is pinned by the surface states, the band bending is determined by semiconductor species. Hencewhen n-type Schottky barrier can be formed for a semiconductor for example, p-type is not available for the same
semiconductor. The other way around. Actually, for GaAs, p-type Schottky barrier is not available while for InP, n-type
Schottky barrier is difficult. This makes it difficult to obtain complementary devices which utilize Schottky barriers.
In the case of metal-oxide-semiconductor (MOS) devices, an inversion layer formed by e.g., pushing down a band of
a p-type semiconductor and turning it to an n-type channel, can be used for complementary device. This is, however,impossible for Schottky devices.
E8-2

6.3.4 MES-FET
Among III-V semiconductors, GaAs is frequently used for electric devices as well as for optical devices. But it isdifficult to form good quality oxide layers on the surfaces, hence no MOS type device for GaAs is available. Instead,
MEtal-Semiconductor FET (MES-FET) structure has been frequently adopted. GaAs has light electron mass, high mobilities. And the effective capacitance of Schottky diode can be small. Hence GaAs MESFETs are often used forhigh-frequency application.
As shown in the left figure, the structure of MES-FET issourcesimple. The conduction channel thickness is controlled with

gate

drain

the reverse bias voltage (gate voltage) through that of depletion layer. The device action, characteristics are similar to
those for JFET. Schottky junctions have larger leak current

depletion layer

in gate characteristics, only single carrier type is available

conduction channel

and complementary circuits cannot be composed with them.
These properties are great obstacles for large scale integration.
MES-FETs are still widely used as high frequency devices for e.g., microwave.

6.3.5 MOS structure
As named, a thin oxide film for insulation is inserted between a metal and a semiconductor in a Metal-OxideSemiconductor (MOS) structure. Needless to say, most frequently used Si has SiO2 as the oxide layer, which is very
stable and has good insulation characteristics. An SiO2 film can be easily formed with thermal oxidation onto a Si. Bothp-type and n-type channels can be controlled and Complementary MOS (CMOS) circuits are easily realized. Also withlow gate leakage current, high on-conductance, off-resistance, the power consumption in logic circuits jumped downwith the CMOS circuits hence increased degree of integration. Now CMOS is doubtlessly the king of semiconductor
circuits. A few decades ago high speed logic circuits were mainly composed with Emitter Coulpled Logic (ECL) of BJT
but the requirement of large scale integration and the increase of cut-off frequency in CMOS circuit have made drasticchange and now, even so called supercomputers are using CMOS circuit in CPU.
MOSFET structure also resembles to JFET and the essential difference to MESFET is the existence of thin oxide
layer between the semiconductor and the gate metal. In a depletion type device, the conduction channel is pinched bydepletion layer while in a enhancement type device, the band is pushed down with gate electric field to form conduction
channel. An oxide layer bears much higher voltage than a Schottky barrier, hence with a strong bending, e.g., formationof an n-type two-dimensional conduction channel below a p-type semiconductor surface (inversion layer).
Si-MOS structures are now used not only in integrated circuits but also for power devices. Recently however, SiC iscollecting wider interest for power devices because of the lower ON-resistances. And for high-frequency power devices,
Metal electrode
Oxide film

VG

Metal electrode

Oxide film

VD

Gate

+
n source

Conduction channelp-Si

Fig. 6.10 Schematic view of a MOSFET device. In
fabrication holes are opened on thermally oxidized films
with lithography. The dopants are diffused through theholes. The structure like this often appears due to theprocess.

+
N drain

Oxide

E8-3

the weight of developement is shifting to GaN-related materials.

6.3.6 FinFET
In the beginning of 21st century, in semiconductor integrated circuits, fierce competition for improving thedegree of integration continued, centered on CMOS.
Compared to other logic circuit schemes, CMOS isoverwhelmingly advantageous in terms of power consumption. Even for such CMOS scheme, what limits
the degree of integration is in-chip heating due to power
consumption. In order to solve this, device driving at low voltage has come to be required. A way is to replace SiO2
with some other insulating thin film that has higher dielectric constant. With this, ON/OFF action of the channels would
be available. The letter κ is often used as the symbol of the dielectric constant and such dielectrics are called “high-κ”
or simply “high-k” materials. From various restrictions, now hafnium silicate, hafnium oxide, zirconia are used for suchhigh-κ materials.
Furthermore, the FinFET, in which the channel shape is changed from the planer type to the fish-fin shaped, has been
now widely used. As illustrated in the figure, in a FinFET, a thin channel is covered with the gate and the depletion layer/
the inversion layer grow over the channel from both sides of the “fin”, resulting in faster switching rate (less than 1 ps)
and higher ON-conductance than those of planer structure. Also the device density can be higher. Now they are the main
structure for the logic LSIs.

6.4 Heterojunction
The two materials on the sides of a junction have similar properties, lattice structures, etc. to each other in semiconductor heterojunction in comparison with Schottky junctions or with MOS structures. As a result, in semiconductor
heterojunctions, sharp changes in the effective potential can be realized, the quantum coherence of electrons is kept over
the junctions. Therefore they can be used for the building block of the devices which utilize quantum effects such as
electron tunneling. And with heterojunctions, one even can create new periodic structures in solids and modify the band
structure. This is called band engineering.
Because this lecture is for the physics in semiconductors, we begin with how to treat such hetero-interface physically.

6.4.1 Effective mass approximation at hetero-interfaces
As in the textbook[1] written by myself or that by Bastard[2], let us consider hte application of effective mass approximation for a hetero-interface under simplest case.
6.4.1.1 Hetero-interface
We consider the situation in which semiconductors A and B (A：z < 0，B：z > 0) are connected at z = 0 (xy-plane).
In each region, the Bloch theorem is appled to writeψ (A) (r) =

∑

(A)

fl

(A)

(r)ulk (r),

ψ (B) (r) =

l

∑
l

E8-4

(B)

fl

(B)

(r)ulk (r),(6.33)

(A,B)

where l is the band index, ulk

are functions with the lattice periodicity. For simplicity, the lattice periodic part of the

Bloch functions and the band dispersions are the same other than the positions of band bottoms and tops.
(A)

(B)

ulk (r) = ulk (r),

(A)

∂ϵl

(B)

/∂k = ∂ϵl /∂k.

With this simplification, the continuity condition of wavefunction at z = 0 gives
(A)

fl

(B)

(rxy , 0) = fl(rxy , 0),

where rxy is a vector in the xy-plane. For the freedom of rxy , the Bloch theorem tells
(A,B)

fl

1
(A,B)
(z),
= √ exp(ikxy · x)χlS√
where 1/ S is the partial normalization factor of plane wave in xy-plane, χl (z) is the envelopefunction along z-direction.
For the freedom along z-direction, we consier the k · p perturbation. That is, first we obtain the lattice periodic function
and the discrete levels for k = 0 and the wavefunctions for k ̸= 0 are obtained by the hybridization of these wavefunctions
caused by the perturbation Hamiltonian, which is proportional to k · p. We write down the equation for χ = {χj } as
D

(0)

(
)
∂
z, −iℏ
χ = ϵχ,∂zwhere the N × N matrix of operators D (0) is
]
(
) [
2
ℏ2 kxy∂ℏkxyℏ2 ∂ 2iℏ∂(0)
Dlm z,= ϵl (z) +δlm +−· ⟨l|pxy |m⟩ −⟨l|pz |m⟩∂z
2m0
2m0 ∂z 2
m0
m0
∂z
with

(A)

ϵl (z) = ϵl(B)

(6.34)(6.35)(z ≥ 0).

(6.36)

z < 0 (z ∈ A)z ≥ 0 (z ∈ B).

(6.37)(z < 0), ϵl

Here we write |um0 ⟩ as |m⟩, etc.
Emphasizing “band-discontinuity potential,” we write{0
Vl (z) ≡ (B)(A)ϵl − ϵlThen we reach the simultaneous equation of {χl } as *1
N
∑
m=1

{[

2
ℏ2 kxyℏ2 ∂ 2(A)
ϵm0 + Vm (z) +−2m0
2m0 ∂z 2

]

}
iℏ
ℏkxy
∂
δlm −
+
⟨l|p̂z |m⟩· ⟨l|p̂xy |m⟩ χm = ϵχl .
m0
∂z
m0

(6.38)Let us consider the continuity condition of the envelope function χl of band l. Because we have assumed that ul iscommon for A and B, χl should be continuous at the interface. On the other hand, the integration of (6.38) over theinterface and the continuity of χl leads to the condition
A (A) χ(A) (z0 = 0) = A (B) χ(B) (0),where
Alm = −

[
]
ℏ2
∂
2i
δlm
+ ⟨l|pz |m⟩ .
2m0
∂z
ℏ

(6.39)(6.40)

It is now clear that the band-hybridizing term ⟨l|pz |m⟩ from the k·p perturbation breaks the simple continuity of derivativeof the envelope function.

*1 If we go up to the second order in k, we have many other terms, which makes the equation very complicated. We thus have omitted them.

E8-5

6.4.1.2 Joint of envelope function
Next we do not equate u nor the band dispertion (effective mass) but only the single band is considered. The effective
mass equation is a second-order differential equation, and the general boundary connection conditions are as follows.
() (χ(A) (0)t= 11t21∇A χ(A) (0)

t12
t22

)(

)
χ(B) (0),∇B χ(B) (0)(6.41)

where, taking a as the common lattice constant,∇A,B =m0 ∂.
mA,B a∂z(6.42)

TBA = {tij } is called interface matrix.
The particle current density along z is determined by the envelope function as
[
]
ℏ
∂χ ∂χ
∗
χ
(z)
−
χ(z)
.
j(z) =
2im∗∂z∂z(6.43)

From the particle-number conservation, j(z) in A and B regions should be the same. The condition is equivalent to
det TBA = 1.

(6.44)

Because this condition is fulfilled when TBA is the unit matrix I, the simplest envelope function approximation is to
put TBA = I. In this case, the envelope function can be treated just the same as the real wavefunction. In the case of
GaAs-(Al,Ga)As interface, the interface matrix obtained for a one-dimensional tight-binding model indicates the envelope
function approximation works well.
In such a case, we can consider the step function potential at the boundary with the height of band discontinuity
which is determined by the combination of the materials. And the envelope function can be viewed as ordinary quantum
wavefunction. On the above basis, we now can use methods to design quantum systems such as one-dimensional potential
by thin film growth technique.

(a)

(b)
Fig. 6.11 (a) Diagram displaying symbols for the band alignment parameters at a junction of crystals A and B. (b)Anderson model, in which the relative positions of bands are determined by the affinities from the vacuum level.

E8-6

6.4.2 Anderson’s model
Figure 6.11 illustrates a long-used Anderson model *2 for heterojunctions[3]．In the model, as shown in Fig. 6.11(a),the bands in the bulk continue to the interface. The effect of charge transfer is taken into account as built-in potential justlike the treatment of pn homo-junctions.
An important point in this model is the relative band position at the heterojunctions. In the Anderson model, as shownin Fig. 6.11(b), this is determined from the quantity called “affinity”, which is the lowering in the energy of electrons
with condensation into the crystal state. Then in this model, the connection of the bands is determined by the species of
the crystals In the figure, the affinities of A and B are χA and χB respectively.
The model, in itself, has many problems, many of which are on the “affinity.” Can the affinities be well-defined? Can we
calculate them? Are they measurable? We do not have time to go into the problems and furthermore, the experiments haveshown that such simple modeling does not work at the level of device designing, in which we need detailed informationof band-discontinuity.
We will have a brief look at the junction types and summarize theoretical approachs to the band-discontinuities inAppendix 6C.

6.4.3 Classification of heterojunctions
Semiconductor heterojunctions are classified phenomenologically by the alignment of bands at the interface. Figure 6.12 shows three types of band alignment. (a) is most frequently found and called type-I. On the larger gap side,
the conduction bottom is higher and the valence top is lower. In type-II, as shown in (b), the conduction bottom and thevalence top shift to the same direction when an electron passes the interface. There is a common energy gap region for
A and B in the case of Fig. 6.12 (b). When the missalignment is larger and the energy gap at the interface is closed as in(c), in Japan they call the alignment type-III and in other countries staggered type=II. For example, in ref. [4], the authors
call (c) as type-II.

DE c
E gA

DE c

E gA

E gB

E gA

E gB
DE v
A
(a)

DE v

B

A

B

(b)

DE c

DE v

E gB

A

B

(c)

Fig. 6.12 (a) Type-I: To the wider gap semiconductors, the conduction bottom goes up, the valence top goes down.
(b) Type-II: The conduction bottom and the valence top moves to the same direction when going through the junction.
Also called broken-up or misaligned．(c) Type-III: (Special classification in Japan) Same as type-II but there is nooverlapping in the energy gap. Instead there is an overlapp between the valence band on one side and the conduction
band on the other side. Also called staggered.

*2 This “Anderson” is a different person from novel laureate Philip W. Anderson in Bell lab. P. W. Anderson is famous for his “Anderson model”

of impurities and R. L. Anderson was in IBM Watson. Bit confusing.

E8-7

6.5 Formation of heterojunctions
We have already had a look on the epitaxial growth technique. Here I just mention about the lattice matching andenergy variation.

6.5.1 Epitaxial growth
Most popular method to form heterojunctions of semiconductors is epitaxial growth already presented in the lecture by
Prof. Akiyama. Epitaxial growth methods can be classified into liquid-phase epitaxy, vapour-phase epitaxy, and vacuum
deposition. In liquid-phase epitaxy, precipitation onto crystal substrates from melts of ingredients is used. The growths
occur in states close to equilibrium and high quality crystals can be obtained while it is hard to obtain sharp interfaces.
When one needs sharp interfaces and precise control of layer thicknesses, usually the latter two methods of epitaxy are
adopted.
An important point in the formation of heterojunction is the lattice matching in lattice constants and crystal systems.
In Fig. 6.13, we plot representative compound semiconductors and elemental semiconductors on the plane of lattice
constant and energy gap. Most of the plotted semiconductors have a common crystal system, FCC bravais lattice. Verticalgray bands indicate possible groups of lattice matched heterostructure growth though these combinations are not always
available in practical growths. Besides these semiconductors, heterojunctions of GaN family are important for industrial
demands. They usually have Wurtzite structure (hexagonal close-packed, HCP) and need high temperature treatments,
the heterostructures thus are mostly composed within nitride families.
Even with considerable lattice mismatch, a misfit-dislocation free growth to a certain film thickness is possible. An
estimation of the thickness given as a balance point of the strain energy concentrated on dislocations and that within wholegrown film, is called Matthews’ critical thickness[5]. Because actual crystal growths are carried out under some nonequilibrium condition, the total free energy not necessarily takes the minimum, the process is generally non-adiabatic.
Hence the Matthews’ thickness is just a rough estimation. In many cases we need to keep substrate temperatures high

4

ZnS
ZnSe

3

Energy gap (eV)

AlP
2 GaP
Si

Zn 0 . 5 Mn 0. 5 Se
ZnTe Cd 0 . 5 Mn 0 . 5 Te

CdS
AlAs

CdSe
InP

GaAs

1

AlSb

CdTe

GaSb
Ge

InSb

InAs
0

HgS
5.4

5.6

5.8

HgSe

HgTe

6.0
6.2
6.4
Lattice constant (Å)

E8-8

6.6

Fig. 6.13 Plots of the lattice constants
and the energy gaps of II-VI, III-V compound semiconductors and IV elemental
semiconductors. The lines connecting
the points indicate possible mixed crystals. Vertical gray bands indicate possiblegroups of lattice matched heterostructure
growth.

Fig. 6.14 Conceptual illustration of van der
Waals heterostructure, which is produced bystacking various two-dimensional materials.

enough during growths and the difference in coefficients of thermal expansion in the two materials sometimes causes
dislocations or strains. Many points should be taken into account in actual growths[6].

6.5.2 van der Waals heterostructure
Recently van der Waals heterostructure, which is formed in completely different way, is collecting attentions[7]. Thatis a mechanical stacking of two-dimensional materials like graphene as shown in Fig. 6.14 (graphene will be introduced
later as a two-dimensional electron system without heterointerface). Sometimes epitaxial growth like CVD is adopted but
in many cases mechanical stacking of exfoliated two dimensional materials creates high-quality heterostructure, which
implies possible completely new formation method of heterostructure.

E8-9Chapter 7 Quantum structures (quantum wells, wires, dots)

So far, we have looked at low-dimensional systems such as graphene, which is a two-dimensional substance. These
were, so to speak, natural low-dimensional systems. By using heterojunction, metal joining, and microfabrication technology, human hands have come to enter the design of material systems to a certain extent. In this chapter, we look at
three typical quantum structures, two-dimensional, one-dimensional, and zero-dimensional systems.

7.1 Quantum well
A region with lower potential sandwiched with two heterojunctions to higher potential materials is quantum well.
The readers should be familiar with it since introduction of elementary quantum mechanics. In other words, however,the semiconductor heterojunction technology has made the quantum well as a real substance from just an exercise forstudents.

7.1.1 Discrete quantum levels in a quantum well
Let the well width be L, the barrier height V0 . In x ≤ −L/2, L/2 ≤ x (outside the well) Schrödinger equation is
[
]
ℏ2 d2
+
V
−
0 ψ = Eψ.
2mdx2
Let us put κ ≡

(7.1)

√
2m|E − V0 |/ℏ and let C1,2 , D1,2 be constants specific to the regions, the solution outside the well can{be written as
ψ(x) =

C1 exp(iκx) + C2 exp(−iκx)

E ≥ V0 ,D1 exp(κx) + D2 exp(−κx)

E < V0 .

(7.2)

In the case of E < V0 , the wavefunction should be localized around the well and zero for x → ±∞, then
L/2 < x で D1+ = 0,x < −L/2 で D2− = 0.

Superscript ± distinguish the regions positive/negative of x. Inside the well, letting C1 , C2 be constants, we write thewavefunction with plane waves as√ψ = C1 exp(ikx) + C2 exp(−ikx),

k≡

2mE
,
ℏ

(7.3)

where for simplicity, we assume the effective mass m is common for inside and outside the well. The boundary condition
at x = ±L/2 where the potential is discontinuous is now applied. Continuity and differentiability at the potential
boundary x = 0 require{
Continuity{
Differentiability

C1 exp(ikL/2) + C2 exp(−ikL/2) = D2+ exp(−κL/2),
C1 exp(−ikL/2) + C2 exp(ikL/2) = D1− exp(−κL/2),

ikC1 exp(ikL/2) − ikC2 exp(−ikL/2) = −κD2+ exp(−κL/2),
ikC1 exp(−ikL/2) − ikC2 exp(ikL/2) = κD1− exp(−κL/2),

E8-10

y
n=4

V0

3p

E3

y 3 (x)E2y 2 (x)E1y 1 (x)y = kL n=3
2p
n=2
p
n=1
0

0

k0

k

(a)

-L/2 0

L/2

x

(b)

Fig. 7.1 (a) A plot for graphical solutions of k which satisfy eq.(7.4). The crossing points of the functions−2 arctan(k/κ) + nπ and kL give the solutions of (7.4). (b) Bound eigenstates for n = 1, 2, 3 under the condition l = 8. The baselines for the wavefunctions are the eigenenergies E1,2,3 measured with V0 (for l = 8 there are
only three bound state solutions, which is different from the situation in the left figure).

respectively. Erasing the constants the following condition is obtained.
(

)2
(
)
κ − ikkexp(2ikL) == exp −4i arctan,
κ + ikκk
2mV0∴ kL = −2 arctan √ 2
+ nπ,
κ20 ≡
, n = 1, 2, · · · .
2ℏ2
κ0 − k

(7.4)

Let us take kL as a positive value without loosing generality because the solutions contain −k equivalently, and werestrict the value of arctan(x) between 0 and π/2. As shown in Fig. 7.1(a), the crossing points of the curves and the line,√−2 arctan(k/ κ20 − k 2 ) + nπ and kL give the values of k, which satisfy (7.4). As easily guessed from the analogy withthe case of infinite barriers, even numbers of n correspond to odd parity wavefunctions, while odd numbers correspondto even parities.
In Fig. 7.1(b), we show the form of wavefunctions for the bound states in the case of l = 8.

7.1.2 Optical absorption in quantum wells
We would like to have a short look at optical absorption in quantum wells. As usual we take z-axis vertical to the wellplane. We write the envelope functions for electrons and holes as ϕe (z) and ϕh (z) respectively and then approximate thetotal wavefunction asψe (r) = ϕe (z) exp(ikxy · rxy )uc (r),}ψh (r) = ϕh (z) exp(ikxy · rxy )uv (r).

(7.5)

uc , uv are lattice periodic parts of the Bloch eigenfunction with k = 0. Direct type inter-band optical absorption
probabilities are proportional to⟨uc (r)|∇|uv (r)⟩∫ ∞−∞dzϕe (z)∗ ϕh (z).

(7.6)

In the case of infinite height barriers, the envelope functions are written as sin(nπz/L), cos(lπz/L) (n = 2, 4, · · · ,l = 1, 3, · · · ) and the latter integration over z in (7.6) is finite only between electron envelope function and hole envelope

E8-11

Absorption coefficient

Absorption coefficienthhlh

E 1 (hh, lh)

E 2 (hh) E 2 (lh)E 3 (hh)

1

i=1
hh
lh

i=2
i=3
hh lh

0.5

GaAs/AlAs
MQW
L =7.6nm
01.6

Photon energy

(a)

lh

hh

(b)

1.8

2

2.2

Photon energy (eV)Fig. 7.2 (a) Illustration of theoretically proposed optical absorption spectrum, in which both the coupling densityof states and the exciton density of states in the quantum well are taken into account. The approximation that thetransition exists only between electrons and holes with the same quantum index. In the valence band of fcc semiconductors we have heavy and ligh holes and transitions with the two bands are considered in the figure. (b) Opticalabsorption spectrum of a AlAs/GaAs multiple (40 layers) quantum well with width 7.6 nm. The finite barrier heightcauses transitions between the levels with different quantum indices, which appear in exciton peaks.

function with the same quantum index (n or l in this case). For finite heights, this orthogonality breaks leaving parity
selection rule but still elements between different quantum indices are small and we only consider the transition between
the states with the same index. The energy associated with the transition is
E = Eg + ∆En(eh) +(eh)where ∆En

ℏ2 2
k ,
2µ xy

(7.7)

is the sum of the energies for electron and hole in n-th energy levels, 1/µ = 1/m∗e + 1/m∗h is the reduced

mass. The last term for two-dimensional kinetic energy indicates that there should be continuous absorption spectrum(eh)above ∆En

corresponding to the two-dimensional density of states.

From E = (ℏ2 /2m∗ )k 2 and n = πk 2 /(2π)2 = (E/4π)(2m∗ /ℏ2 ), the two-dimensional density of states can be
written as

m∗
dn
=
H(E)
dE
2πℏ2

(H(x) : Heaviside function).

(7.8)

This is constant for energy and with (7.7), we expect a staircase like optical absorption spectrum.
Formaion of excitons appears in optical absorption as peaks at energies lower than the fundamental absorption edge.
Such peaks for excitons in quantum wells are illustrated in Fig. 7.2(a). Only the ground states (n = 0) of the excitons
are considered. And coupling density of states between electrons and holes with different subband quantum indicesis ignored assuming that the barrier is high enough. Figure 7.2(b) shows an experimental result on an AlGaAs/GaAs
multiple quantum well with width 7.6 nm. The lineshape of the absorption spectrum can be understood as an overlap ofstaircase-like shape reflecting the two-dimensional density of states (7.8) and absorption by excitons indicated as hh or lh.
Because the barrier height is finite in the experiment, peaks due to the transition between states with different quantum
indices are also observed. The effect of low-dimensionality is observable in increases of binding energy of excitons,
which results in wider separation of exciton peaks from absorption edges and the peaks persist up to higher temperatures.
Now we can see that the optical absorption spectra can provide experimental determination of band-discontinuities∆Ec , ∆Ev . In the combination of GaAs-Alx Ga1−x As, reseachers could not separate lh and hh peaks in very earlyexperiments presumably due to low quality of samples. The result once led them to a wrong conclusion of ∆Ec : ∆Ev =85 : 15 because ∆Ev should be too small to accomodate the lh level. After the revised experiments, it was establishedE8-12

that ∆Ec : ∆Ev = 57 : 43 is a good empirical law.

Appendix 6C: Trials for simple theory to find band discontinuities
The capacity of computers has increased dramatically, and first-principles calculations that require a lot of computerresources as LAPW, can now be performed relatively easily. Even though, the researchers are still trying to construct atheory to obtain band discontinuity from a small number of experimental parameters using simple physical principles.
I will introduce such researches so far, but it has been found that many cannot withstand the subsequent criticisms,
experiments, and first-principles calculations.

6C.1 Common anion ruleThis “common anion rule” is considered for compound semiconductors[8] which have finite inonicity. The claim is asfollows. Because the valence band is mostly composed of p-orbitals of anion, ∆Ev ≈ 0 for the semiconductors with acommon anion. This is a surprizingly rough theory. The prediction is far from experiments and from other models.

6C.2 Pseudo-potential theory
The quantitiy “affinity” can be formally calculated from the first principles. In the era of Anderson’s research, ∆Ec are
determined from the experiments, and the affinity is obtained from the fitting. Here we see a theory, which is constructedby Frensley and Krömer[9] aiming at finding band discontinuity from bulk parameters.
The calculation goes as follows. First with the self-consistent pseudo-potential method, the relative positions of bulk
bands are calculated in the electrostatic potential inside the crystal[10]. Next from the electronegativity of consisting
atoms and from the band structure, the electrostatic potential at the interface is calculated and the relative positions ofbands are obtained. They claimed the agreement with exeriments[9].

6C.3 LCAO theory
W. A. Harrison applied his linear combination of atomic orbitals (LCAO) theory to the heterostructure in ref.[11, 12].
In Harrison’s theory, LCAO forms the bands. Most of bands in semiconductors can be expressed with the combinationof the single s-orbital and the three p-orbitals. The valence band top is composed of p-orbitals and expressed as
ϵcp + ϵap
Ev =
−
2

[(
]1/2
)
ϵcp − ϵap 2
2
+ Vxx
,
2

(6C.1)

where ϵc,ap are the energies of p-orbitals of cation and anion respectively on their own sites, Vxx is the matrix element
between neighboring p-orbitals. According to the theory[11], Vxx is approximated as
Vxx = 2.16ℏ2 /md2 ,(6C.2)

where m is the electron mass and d is the bond length. The number 2.16 is obtained by fitting the results for Si and Ge tothose of other band calculation[13].
In this method, the valence band discontinuities for many semiconductor can be easily calculated and often used for theestimation. It is said to give fairly good agreements with experiments in many cases though with many exceptions e.g.,in the case of GaAs-AlAs the Harrison theory gives ∆Ev =0.04 eV for about 0.5 eV in the experiments[14]. Actually themain difference in ref. (6C.1) from the common anion rule is just ϵcp and still the approximation is rough.

E8-13

6C.4 Interface dipole theory
This theory had been pur forward by Tersoff and Harrison[15, 16, 17]. First Tersoff criticized Harrison’s LCAO theory
that the theory is not reallistic in that the charge transfer between the semiconductors is ignored and no dipole exists at
the interface[15]. Then they collaborated in constructing LCAO theories with electric dipoles[17].
In the Tersoff’s original idea, on the surface of a semiconductor (an insulator), an energy level of “charge neutrality”
can be defined in the band gap. The charge neutrality level is given at the point where the contributions of the valenceorbitals and those of the conduction orbitals are balanced. In a metal-semiconductor Schottky junction, this charge
neutrality level should be matched to the Fermi surface in the metal*3 . The charge transfer through the interface is over a
very short distance, only single lattice constant. The transfer length scale is very different from the space-charge createdso as to match EF with the bulk value. When two semiconductors are jointed, there is no charge transfer for matchedcharge neutrality levels. Otherwise the transfer occurs to match the charge neutrality levels. Hence, if we can calculate
the position of charge neutrality level, the band offset can be derived from that.

(a)

(b)
Tab. 6C.1 (a) The in-gap state EB obtained by equating the contributions from the both bands to (6C.3) and thepositions of EF measured in the Schottky diodes with Au and Al as the electrodes.

For the calculation of the charge neutrality level (or, metal-induced gap states, MIGS), the contributions from thevalence and the conduction to the real-space averaged Green function∫
G(R, E) =d3 r∑ ψ ∗ (r)ψnk (r + R)nknk

E − Enk

=

∑
nk

eik·R
E − Enk(6C.3)

are equalized to give MIGS EB . EB obtained with this method and the positions of EF obtained in Schottky diodes
with Au and Al electrodes[?] are listed in Tab. 6C.1(a). Already in this table, the agreement is not very good. And after
the publication, there occured many criticisms including the experiments. In conclusion, the theory is convenient in thediscussion of chemical trend but it is hard to say that it can be used for device design.

6C.5 Example of first principles calculation
Wei and Zunger proceeded with the so-called first-principles calculation of the interface, and the low accuracy of thecommon anion law and even the simple LCAO theory is due to the roughness of the bulk band calculation rather thanb
isand the surface contribution ∆EVBM
to thethe effect of the interface dipole[19]. That is, the bulk contribution ∆EVBM

energy difference ∆EVBM at the valence band maximum (VBM) are in the relationb
is∆EVBM = ∆EVBM+ ∆EVBM.

(6C.4)*3 In the most of real Schottky junctions, there are defect levels with very high densities and the Fermi levels are pinned there. At the heterointerface

with small defect densities, the situation is different.

E8-14

is
b
According to their claim, ∆EVBM
is small and the problem in LCAO theory rather lies in the estimation of ∆EVBM.

That is in Harrison theory, only s and p orbitals are considered but particularly the contribution from the d orbitals of
cation is comparatively large and the most of disagreement with experiments can be explained with this (calculation was
done by all-electron generalized linear augmented plane wave method[20]).

Tab. 6C.2 ∆Ev for semiconductors with Te and As as anion. The results of simple LCAO, experiments, andall-electron first principles calculation.

The calculated results are summarized in Tab. 6C.2. Now the LAPW method can be rather easily utilized in the formof convenient packages like HiLAPW or VASP though still consumes large calculation resources and the jobs are heavy).
The method is only for periodic systems and in the case of heterointerface, the unit cell is taken large along verticaldirection to the interfaceso as to contain two intefaces and the periodic boundary condition is applied. This is, in a sense,
calculation of a superlattice band structure and can be used to check the staircase approximation of the heterointerface.

Appendix 6D: Recombination current and ideality factor
In the discussion of current-voltage characteristics of pn junctions in the text, we only considered the diffusion current.
In reallistic pn-junctions, various other factors contribut the current. Here we have a brief look at the current caused bycarrier recombination in the depletion layer at the junction interfaces.
First we consider direct gap semiconductors, in which the interband recombination rate is much higher than those inindirect ones. Let the interband recombination rate be Re , this should be proportional to the carrier concentrations n andp. Thus Re is proportional to the product pn. Let Rrc be the coefficient, then
Re = Rrc pn.

(6D.1)

Re equals to the thermal activation rate Gth of the electron-hole pair in the dark and in equilibrium. Then the law of massaction gives
Rrc =

Gth
Gth
= 2 .
pn
ni

(6D.2)

When there is optical activation or minority carrier injection by the external current, the activation rate and the recombination rate are not balanced and the difference is the net recombination rate U . In n-type semiconductors, the variationin the hole concentration is the main factor. If we write pn = p0 + ∆p, nn ≈ ND , thenU = Re − Gth = Rrc (pn − n2i ) ≈ Rrc ∆pND ≡

∆p
,
τp

(6D.3)

where we define the minority carrier lifetime as
τp =

1
.
Rrc ND(6D.4)

Similarly the electron lifetime in p-type semiconductors are written as
τn =

1
.
Rrc NA

E8-15(6D.5)

In contrast, in the indirect gap semiconductors like Si or Ge, the carrier recombination is via the localized traps. In thatcase, the net recombinatin rate is, according to so called Shockley-Read-Hall statistics[21] written asU=σn σp vth Nt (pn − n2i )[][],Et − Ei
Ei − Et
σn n + ni exp+ σp p + ni exp
kB T
kB T

(6D.6)

where Nt is the trap density, σn , σp are the capture cross-sections for electrons and holes respectively, Et is the trap level,
Ei is the Femi level of the intrinsic semiconductors. And vthe is the thermal velocity of the minority carrier
√
3kB T
vth =
.
m∗

(6D.7)

In eq.(6D.6), U takes the maximum at Et ≈ Ei . Though actually Et distribute over the band gap, the trap levels close to
Ei contribute lartgely to U *4 , then as a coarse approximation, we consider only single species of traps and put Et = Ei
thenσn σp vth Nt (pn − n2i ).
σn (n + ni ) + σp (p + ni )U=

(6D.8)

Just like in the case of interband transition, we write U as ∆p/τp or ∆n/τn , giving
1,
σp vth Nt

τp =

τn =

1
.
σn vth Nt(6D.9)

Now we use quasi-Fermi levels introduced in eq.(6.4) and from eq.(3.13) the np product is written as
np = n2i expµe − µh.
kB T(6D.10)Substitutin the above into (6D.6), we obtain[]eV
σn σp vth Nt n2i exp
−1
k T
[
]
].
[B
U=
Et − Ei
Ei − Et
σn n + ni exp+ σp p + ni exp
kB T
kB T
Then again we put Et = Ei , and for further simplicity, we assume σn = σp = σ to obtain
]
[
]
[
eV
eV
σvth Nt n2i exp
σvth Nt n2i exp
−1
−1
kB T
kB T
].
U=
= [
Ei − µhµe − Ein + p + 2ni
ni exp
+ exp
+2
kB T
kB T

(6D.11)(6D.12)Further, when µe , muh are position dependent, U takes the maximum in the case Ei places in the middle between µe andµh . Then (6D.12) reduces to[]eVσvth Nt n2i exp
−1
1
eV
kB T
[
] ≈ σvth Nt ni exp
U≈
eV
2
2kB T
2ni exp+22kB T

eV > kB T.

(6D.13)

Because the above is the maximum the estimation should be a bit large but the current density due to the recombination∫ wd

can be written as
jrc =

0

qU dx ≈

qwd ni
eV
exp
.
2τ
2kB T

(6D.14)In eq. (6D.14) in comparison with eq. (6.11), the voltage term in the exponential has an extra factor 1/2. To put it
plainly, this is because the energy exchange when recombination occurs in the trap is half that of the case where the*4 This means that the lifetime of minority carriers is determined by a deep level, especially in indirect semiconductors. In “solar grade” Si, in

conparison with those for LSI (impurity −10 ∼ 10−11 ), the purity can be a bit lower while the deep level concentrations should be reduced.

E8-16

current is generated by overcoming the bandgap (6.11). In this way, different processes in which current flows generally
have different voltage coefficients. Then in experiments, the forward current is written as
JF ∝ expeVηkB T(6D.15)and the factor η (ideality factor) is fit to the experiment. When η is close to 1, the diffusion current is dominant and thejunction is close to the ideal case. When it is close to 2, the recombination current inside the depletion layer is dominant.
In the laboratories, η sometimes goes over 2 and still takes higher values. In the case of pn-junctions, the interface
comes to the middle of depletion layer and there is some interdiffusion of dopants, thus the factor 2 is frequently obtained
while it is usually close to 1 in the case of Schottky junctions.

References[1] 勝本信吾「半導体量子輸送物性」(培風館，2014).
[2] G. Bastard, “Wave Mechanics Applied to Semiconductor Heterostructures” (Editions de Physique, France, 1990).
[3] R. L. Anderson, IBM J. Res. Dev. 4, 283 (1960); Solid-State Electronics 5, 341 (1962).
[4] C. Liu et al., Phys. Rev. Lett. 100, 236601 (2008).
[5] J.E. Matthews, A.E. Blakeslee, J. Crystal Growth 27, 118 (1974).
[6] H. C. Casey, Jr., M. B. Panish, “Heterostructure Lasers” Part B (Academic Press, 1978).
[7] A. K. Geim and I. V. Grigorieva, Nature 499, 419 (2013).
[8] A. G. Milnes and D. L. Feucht, “Heterojunctions and Metal Semiconductor Junctions” (Academic Press, 2012).
[9] W. R. Frensley and H. Kroemer, Phys. Rev. B 16, 2642 (1977).
[10] M. L. Cohen and J. R. Chelikowsky, “Electronic Structure and Optical Properties of Semiconductors” (Springer,1989).
[11] W. A. Harrison, J. Vac. Sci. Tech. 14, 1016 (1977).
[12] W. A. Harrison et al., Phys. Rev. B 18, 4402 (1978).
[13] D. J. Chadi and M. L. Cohen, phys. stat. solidi (b) 68, 405 (1975).
[14] 竹田 美和, 応用物理 67, 1077 (1998).
[15] J. Tersoff, Phys. Rev. B 30, 4874 (1984).
[16] J. Tersoff, Phys. Rev. Lett. 56, 2755 (1986).
[17] W. A. Harrison and J. Tersoff, J. Vac. Sci. Tech. B 4, 1068 (1986).
[18] S. M. Sze and K. K. Ng, “Physics of Semiconductor Devices” 3rd ed. (Wiley, 2008).
[19] S.-H. Wei and A. Zunger, Phys. Rev. Lett. 59, 144 (1987).
[20] S.-H. Wei, H. Krakauer, and M. Weinert, Phys. Rev. B 32, 7792 (1985).
[21] W. Schockley and W. T. Read, Phys. Rev. 87, 835 (1952); R. N. Hall, Phys. Rev. 87, 387 (1952).

E8-17

Physics of Semiconductors (9)Lecture note on
9th June (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

We have seen the effect of confinement with heterojunctions. When the barrier width is finite, the transport across thebarrier with quantum tunneling. These can be viewed as elementary quantum mechanics though it is an important stepthat such phenomena can be observed in real systems in the very beginning of semiconductor quantum physics. Actually
the heterojunction technique leads to the prosperity of the field and many novel devices have been created.

7.1.3 Excitons in two-dimensional systems
In the previous section, in the absorption spectrum of a quantum well, we observed peak structures around the absorption edges. They are from the excitons explained in Sec. 3.3.2 thought the lowering of the spatial dimension results insome quantitative differences from the excitons in the bulk. We would like to have a brief look at the excitons in both two
and three dimensions.
Let us treat it as a problem of a hydrogen atom then we treat Schrödinger equation with a Coulomb-type central force
potential Vc (r),

(
)
ℏ2 2
− ∗ ∇ + Vc (r) ψ(r) = Eψ(r),
2m

(7.9)

in lower dimensions. Here m∗ is the electron-hole reduced mass. And we need to change the potential form as
Vc2d (r) = −e2,
4πϵϵ0 |r|Vc1d (r) = −e2,
4πϵϵ0 (|z| + 0.3r0 )(7.10)

particularly for one-dimensional (along z-axis) systems. This is because simple transformation of eq.(7.9) into onedimension causes anomalous behavior including divergence of binding energy. The potential form in eq.(7.10) is given
as an empirical formula which well fits to a practical numerical calculation on confinement into a finite width quantum
wire (a cylinder with radius r0 ). Below, we rapidly see the solutions, which are nothing but hydrogen atom solutions.
Under variable separation hypothesis, the solutions for eq.(7.9) can be written in the forms
ψ 3d = ρl e−ρ/2 R(ρ)Yl,m (θ, φ), ψ 2d = ρ|m| e−ρ/2 R(ρ)eimφ ,ψ 1d = R(ζ).

(7.11)

ρ and ζ are dimensionless variables, which correspond to radial variable and z variable respectively. The definitions are√−8m∗ E
ρ = αr, ζ = α(|z| + 0.3r0 ), α =.
(7.12)ℏ
R(ρ), R(ζ) are the solutions of the following equations.
 ( 2
)
∂
∂


+ q R(ρ) = 0 :3-, 2-dimensional,
ρ 2 + (p + 1 − ρ)


∂ρ
∂ρ
√
)
( 2

∂
λ
e2
m∗
∂


+
+
R(ζ)
=
0,
λ
≡
−
:
1-dimensional,

2
∂ζ
∂ζ
ζ
4πϵ0 ℏ
2E
where p, q are{p=

2l + 12|m|(3-dimensional),(2-dimensional)

{
q=

λ−l−1
(3-dimensional),
λ − |m| − 1/2 (2-dimensional)

where l is angular momentum quantum number and m is magnetic quantum number.
E9-1(7.13)(7.14)

Fig. 7.3 Exciton absorption peaks appeared in the absorption spectrum of a GaAs(8.3 nm)/AlAs(9.3 nm) superlattice
at room temperatre (red line). The coupling between thequantum well in the superlattice is weak. n in the figure is thesubband index and different from the one of excitonic states
index. The peak at the ground subband (n = 1) shows the
splitting into lh and hh[1]. The n = 2 peak is considered asfrom hh reduced mass.

For three and two dimensional systems, R(ρ) in eq.(7.13) is expanded as follows.
R(ρ) =∑β ν ρν ,βν+1 = βννν−q.
(ν + 1)(ν + p + 1)

(7.15)For this R(ρ) to be finite, this expansion should stop at a finite number, which condition requires νmax = q. The main
quantum number q then is defined as follows.
n ≡ λ = νmax + l + 1 (3-dimensional),n≡λ−1= νmax + |m| (2-dimensional).
2(7.16)The exciton energy levels for three- and two-dimensional systems can be expressed as follows.
3d
Ebn
=−

E0
n2

n = 1, 2, · · · ,(7.17)

2d
Ebn
=−

E0
(n + 1/2)2n = 0, 1, · · · .

(7.18)

Here the energy unit E0 is
E0 =e2
4πϵϵ0 ℏ2
∗
,
a
=
,
0
8πϵϵ0 a∗0m∗ e2

(7.19)

where a∗0 is the effective Bohr radius. From eq.(7.16), we see that n = 0 is available for two-dimensional systems andthe ground bound state energy is −4E0 . This means the binding energy is four times larger than that in three-dimensional
systems where the ground state energy is −E0 . In the process of an exciton formation, spatial confinement increases the
kinetic energy due to the uncertainty in momentum. In three-dimensional systems, the enhancement occurs for all three
dimensions while in two dimensional systems, the confinement along the direction perpendicular to the plane has alreadybeen included into the shift of band edge and the binding energy is measured from the edge. Hence it is qualitatively
easily understood that the exciton binding energy becomes larger with lowering the system dimension.
Generally radial wavefunction is expressed with Laguerre bi-polynomial and exponential functions. In three dimen3d
2d
sional systems, 1s wavefunction is written as ψ1s∝ exp(−r/a∗0 ). Similarly let ψ1s∝ exp(−r/a∗2d0 ), (7.13) へ

l = m = 0 and substitution into Schrödinger equation gives a∗2d
= a∗0 /2. The spatial size of excitons in two-dimensional0systems is half of that in three-dimensional systems in accordance with increment in the binding energy.
E9-2

In Fig. 7.3, we show the absorption coefficient of a GaAs(8.3 nm)/AlAs(9.3 nm) superlattice (red line) and taht of ahigh purity bulk GaAs (blue line). In the bulk line, a shoulder structure at the absorption edge is observed. On the other
hand, the exciton absorptions take clear peak structures. Furthermore, the peak at the absorption edge of 1st subband(n = 1) shows a clear splitting due to two effective masses lh and hh in the valence band, which result in the two reduced
masses. The exciton peak at the second subband edge is also clearly observed and considered as from hh reduced mass.
These observations are possible by the above enhancement in the binding energy due to the confinement.

7.2 Quantum barrier“Upside down” of a quantum well potential gives a quantum barrier potential. In the quantum well problem, the focus
was on the bound states inside the well while in quantum barriers we see characteristic tunneling phenomena in theupside-down states of resonant scattering.

7.2.1 Transfer matrix
Let us consider a region Q in a one-dimensional space and as shown in Fig. 7.5(a), and incoming wavefunction A(k)with wavenumber k from the left hand side (LHS), outgoing wavefunction A2 (k) to the right hand side (RHS), andB2 (k), B1 (k) for the other way around. Here we take the momentum k to be common for the momentum conservation.
The suffices 1 and 2 indicates the boundaries 1 and 2.
Let us take for an example that a rectangular barrier with width L, and height V0 . We define κ ≡−κxwavefunction inside the barrier be Vi (κ) + Wi (κ). V , W correspond to e

,e

κx

√

2mV0 /ℏ. Let therespectively and from the Schrödinger

equation, ∂Vi /∂x = −κVi , ∂Wi /∂x = κWi . The suffix i indicates positions in real space, just as above, putting 1 and 2
to the left and the right edges of the barrier andV2 = V1 e−κL ,W2 = W1 eκL .

Now the boundary condition can be written as ∂A1,2 /∂x = ikA1,2 , ∂B1,2 /∂x = −ikB1,2 , hence,
A1 + B 1 = V 1 + W 1 ,ik(A1 − B1 ) = κ(−V1 − W1 ),A2 + B2 = e−κL V1 + eκL W1 ,ik(A2 − B2 ) = κ(−e−κLV1 + e(7.20)
κL

W1 ).

(7.21)For short expression, k, κ for A ∼ V are not shown.
First we erase V1 , W1 , then (A2 , B2 ) and be expressed with (A1 , B1 ). Because of the linearity, the solution can be(

written in a matrix form as

A2
B2

1

)

(
=

m11
m21

m12
m22

)(

A1
B1

)

(
≡ MT

)
A1
.
B1

2

A 1( k )A 2( k )Q

B 1( k )B 2( k )Fig. 7.4 Scheme of T-matrix

E9-3(7.22)

Then matrix {mij } is obtained as[]
k 2 − κ2
m11 = cosh(κL) + isinh(κL) ,


2kκ

k 2 + κ2m=−isinh(κL),
12


2kκ


 m = m∗ ,m22 = m∗11 .
21
12(7.23)

Specific form of MT surely depends on shape of potential though the relation between input and output can always be
written in the matrix form as in (7.22) guaranteed by the linearity of Schrödinger equation. A matrix like MT is called
transfer matrix (T-matrix).
In Eq.(7.23), MT has the symmetry of m21 = m∗12 , m22 = m∗11 , which comes from the time-reversal symmetry andthe even symmetry in the potential shape.
Let B2 = 0, and the ratio of transmission wave A2 and reflection wave B1 to the incident wave A1 can be given from
(7.22), (7.23) ast≡|m11 |2 − |m12 |2
1
2ikκ
A2
=
= ∗ = 2
,
A1
m∗11
m11
(k − κ2 ) sinh(κL) + 2ikκ cosh(κL)
B1
m21(k 2 + κ2 ) sinh(κL)
r≡
=−
= 2
.
A1
m22
(k − κ2 ) sinh(κL) − 2ikκ cosh(κL)(7.24)(7.25)

t, r are called imaginary transmission coefficient and imaginary reflection coefficient respectively. They are related
to the transmission and reflection coefficients asTransmission: T = |t|2 ,Reflection: R = |r|2 ,|t|2 + |r|2 = 1,

(7.26)and the T-matrix MT can be expressed with them as
(
MT =

1/t∗
−r/t

)
−r∗ /t∗.
1/t(7.27)

7.2.2 Transmission through double-barrier structure
Let us consider the transmission through the double barrier potential illustrated in Fig. 7.5. Quantum well and quantum
barrier are upside down to each other and the double barrier may have the position in between them. Let the boundaries
be 1∼4 as in the figure and the wavefunctions also as A1−4 and B1−4 . For the left barrier the setup is the same as thatin the previous section and (7.23) is applcable. Next in the well part between the barriers, a particle gains a kinetic phasefactor exp(ikW ) during the traverse. Hence as T-matrix for this part we can adopt()
exp(ikW )
0
MW =
.
0
exp(−ikW )(7.28)The right barrier is just the same as the left. The expression of T-matrix does not depend on local coordinates and MT
can be used as it it.

L

L

W

A1

A2

A3

A4

B2

B3

B4

V0
B1
1

2

3

Fig. 7.5 Schematic illustration of double barrier
potential.

4
E9-4

Then the total T-matrix MDW of the double barrier structure is, as obvious from the definition, obtained as the productof all T-matrices as
(
MDW =

m11
m21

m12
m22

) ( ikW
e
0

)(

0

e−ikW

m11
m21

m12
m22

)

(
≡

T11
T21

)
T12
.
T22

(7.29)The transmission coefficient is, from (7.29),
T11 = m211 exp(ikW ) + |m12 |2 exp(−ikW ) (∵ m12 = m∗21 ).
The interference effect due to the double barrier structure appears in the second term. Let the argument of m11 be φ, andwriting m11 = |m11 | exp(iφ) we get∗
T11 T11= ((|m11 |2 e2iφ eikW + |m12 |2 e−ikW )(|m11 |2 e−2iφ e−ikW + |m12 |2 eikW )= (|m211 − |m12 |2 )2 + 2|m11 |2 |m12 |2 (1 + cos(2(φ + kW )))= 1 + 4|m11 |2 |m12 |2 cos2 (φ + kW ).
The the transmission coefficient is obtained as
T =

1
1
=
.
|T11 |2
1 + 4|m11 |2 |m12 |2 cos2 (φ + kW )(7.30)

The final form of transmission coefficient is then in combination obtained with (7.23).
Figure 7.6(a) shows thus calculated transmission coefficient T for various barrier widths L as a function of energy
of incoming wave. The relation between the barrier width and the well width is fixed as W = 2L. Here L and E are√transfomed into dimensionless parameters l ≡ ( 2mV0 /ℏ)L and E は ϵ ≡ E/V0 respectively. The points where thetransmission coefficient hits 1 are due to resonant scattering and the condition is written as()
1
φ + kW = n −π (n = 1, 2, · · · ),2

(7.31)

from (7.30), where φ is witten from (7.23) as[ 2]
k − κ2
φ = arctantanh(κL) ,2kκ

(7.32)

1

2.5

2

1.5

0.8

n=1

l=1

n=2

0.6

l

Transmission coefficient T

where we restrict the region to −π/2 < φ < π/2. With this, n should take a natural number.

0.4

T
0.2

(a)

0

0.2

0.4

0.6

e= E / V 0

0.8

1

(b)

e

Fig. 7.6 (a) Transmission coefficient T calculated on (7.30) as a function of the energy of incoming wave for various
barrier widths. Well width - barrier width relation is fixed to W = 2L. (b) The same results are plotted in a grayscale as a function of the incoming energy and the barrier width. White broken lines indicate the resonance condition(7.31), (7.32).

E9-5

In Fig. 7.6(b), the same data are plotted in a gray scale versus a plane of energy and barrier width. White broken lines
indicate the resonant scattering condition in the above equation. With increasing l, the peaks become sharper, which
tendency is due to the elongation of time for staying inside the well, that makes the life width determined from theuncertainty relation smaller. If we take the limit L → ∞ keeping W finite, the system becomes a quantum well with afinite barrier height and the resonant scattering condition approaches to that for bound eigenstates.

7.2.3 Transport of double barrier diodeDouble barrier diode is a device, which realized the double barrier structure with hetero-inferfaces. Here we introduce
an experiment on such a device with GaAs-AlAs hetero-interfaces, p-type doped electrodes. Hence the device works
as a double barrier for holes. The band discontinuity is ∆Ev =0.47 eV. There are two species of holes at the top ofvalence band in GaAs with effective masses 0.51m0 and 0.082m0 , which are called “heavy” and “light” holes (hh andlh) respectively. We ignore the mass difference in AlAs for simplicity (actually the difference is not small but does notaffect the result significantly). The potential prepared has, as shown in the upper panel of Fig. 7.7(a), widths of 5nm bothfor the barriers and the well. The barriers and the well parts do not have any doping. Figure 7.7(a) shows a photographof the sample cross section taken by a scanning transmission electron microscope, STEM.
The transmission coefficient T thus calculated with the above parameters and the structure shown in Fig. 7.7 is displayed as a function of energy in Fig. 7.8. Because the effective masses of holes are comparatively heavy and the barrierheight is high, the transmission peaks are very sharp. We thus can see the behavior of tail only in the semi-log plot. We
see below the barrier threshold, 5 heavy hole resonance peaks and 2 light hole peaks. Figure 7.7(a) shows the positionsof resonance levels in the well numerically calculated from eq.(7.31).
In order to see the behavior of tunneling, usually source-drain voltage Vsd is applied as illustrated in Fig. 7.7(b).
Inside the source and the drain, highly concentrated holes screen the electric field and the applied voltage should mainly
consumed across the double barrier regions. In actual situation, however, the contact resistances also cause significant
voltage drops.

0.55eV
0.46

0.32
0.30

L2
H4

0.17
0.083
0.077
0.019
GaAs

H3
L1
H2
H1
GaAs

5nm
(a)

Souce

AlAs

AlAs

H5

e
Drain

eV sd

(b)

Fig. 7.7 (a) Upper panel: Potential diagram of the double barrier
diode prepared for the present experiment. The energy base is taken
to the top of the valence band and the energy of holes is positive in thisplot. H1 − H5, L1, L2 are the positions of resonant levels for heavy
holes and light holes respectively. Lower panel: Scanning transmission electron micrograph of the sample. Darker regions are AlAs. (b)Schematic potential diagram for a source-drain biased double barrier
diode.

E9-6

100
Light hole

Transmission coefficient

10-5
10-10
10-15
10-20

Fig. 7.8 Energy dependence of transmission coefficient for the
double barrier structure with parameters given in the text andwith eq.(7.30). The peaks hit 1 actually but too narrow to besampled.

Heavy hole
10-25

Energy (eV)

0

0.2

0.4

We ignore the distortion of the originally rectangular-shaped potential due to the applied electric field. Then, as in theillustration, the energy of an injected hole is in accordance with the resonant level when the applied voltage reaches twiceof it. The transmission coefficient takes a peak at that time, that is the amount of holes passing through the barriers withina unit time, namely the current should take a peak (see Appendix E for more realistic current lineshape).
A measured current-voltage curve in a double barrier diode (the one in Fig. 7.7) is shown in Fig. 7.9(a). Several current
peaks appear versus the voltage. To clarify the peak positions the absolute value of voltage-derivative the current with a

0.2

0

0.5

1

1.5

2

V sd (V)C -d I /d V (I sd ( m A)

50

C Ó P Ê)

0.4

H5

25
H2,
H1 L1 H3

L2

C Ó P Ê)

10-2

-0.1

1

L2

H5

0.1

0

H4

10-3

H4
0

H1

d V /d I (G l  M [  Ê(eV)constant bias C is plotted in a semi-log scale in Fig. 7.9(b) *1 .

H2, L1 H3
2

0

1

2

V sd (V)V sd (V)(a)(b)
Fig. 7.9 (a) Current-voltage characteristics of the double barrier diode introduced in Fig. 7.7. Resonant levels corresponding to the peaks are indicated by the arrows. The inset indicates peak positions of energy levels on the voltage
axis. (b) Emphasis is on the peak positions with differentiating the current with the voltage and the absolute value
being plotted in semi-log scale. The inset is enlargement around the origin.

*1 This transformation is just for the clarity in sight.

E9-7

7.2.4 Superlattice
The next step, in the course of quantum mechanics, we have double quantum well, which is very important as a qubit.
We skip it, to my regret, for the shortage of time. I would like to remind you we have lectures on “nano-quantum
information” in the applied physics department (but in Japanese). Here I would like to give a short introduction ofheterojunction superlattice, which was proposed by Leo Esaki and Raphael Tsu and has provided rich physics. The
basic idea of heterojunction superlattice is realization of Kronig-Penny type potential, illustrated in Fig. 7.10. This, ina sense, recovers spatial translational symmetry of the lattice lost by the introduction of the interface but in a different
manner.
Let us express a Kronig-Penny type potential as VKP (x) and write down the Schrödinger equation as[]ℏ2 d 2−+V(x)
ψ(x) = Eψ(x),VKP (x) = VKP (x + d).
KP
2mdx2

(7.33)

According to Bloch theorem, we write the eigenstate wavefunction as a product of a plane wave and a lattice periodic
function with d = L + W as the lattice constant.
ψK (x) = uK (x)eiKx ,uK (x + d) = uK (x),

K≡

πs
.
Nd

(7.34)

s takes an integer from −N + 1 to N − 1. The transfer matrix Md corresponding to the unit cell of the system is
( ikW
)(
) (
)
e
0
m11 m12m11 eikW
m12 eikW
Md (k) ==.
(7.35)
m21 m220e−ikW
m21 e−ikW m22 e−ikW
As before, we write the input/output in the left hand side of i-th cell as (ai , bi ), then from (7.34),
(
)
( )
( )
ai+1
a
a
= Md i = eiKd i
bi+1
bi
bi

(7.36)

should hold, that is, this is a problem of engenvalue eiKd of matrix Md . From the unitarity of Md , or from “reversed”
equation of (7.36), the two eigenvalues e±iKd are obtained. We re-use {mij } in (7.23) to get to the equations
eiKd + e−iKd = 2 cos Kd = TrMd = 2Re(e−ikW m∗11 ),cos [K(L + W )] = cosh(κL) cos(kW ) −k 2 − κ2sinh(κL) sin(kW ).
2kκ

(7.37)(7.38)

By use of φ in (7.32), expressioncos(Kd) = |m11 | cos(kW + φ) =1
cos(kW + φ)|t|

(7.39)

is available.
Transforming the Kronig-Penny potential to a series of δ-function potentials can be attained with taking limits L → 0,
W → d, V0 → ∞(V0 L = C(constant) to obtain the conditioncos(Kd) = cos(kd) +mCsin(kd).
ℏ2 k

d

W

L

V0
Fig. 7.10 One dimensional rectangular potential (Kronig-Penny type potential)

E9-8(7.40)

mC
sin( kd )h 2kcos( kd )+50cos( Kd )

0

5

kd

10

Fig. 7.11 RHS of (7.40) as a function of kd.
Here mdC/ℏ2 is taken to be 13. The gray belts
indicate “allowed bands”.

15

Figure 7.11 shows the RHS as a function of kd. The solution K for (7.40) exists for the RHS to be in [−1, +1] corresponding to the gray bands namely the energy bands.
Let us simplify the energy dispersion relation of a single band as
E(K) =

Enw(1 − cos Kd).
2

(7.41)The group velocity and the effective mass arevg (K) =Enw d
sin Kd,2ℏ

m∗ (K) =ℏ2
Enw d2

sec Kd.

(7.42)The equation of motion of an electron in a periodic potential under a uniform electric field Em is written as
m∗

dv
dK
=ℏ
= F = eEm .
dt
dt(7.43)We see an effective mass in a periodic potential can be negative.
An acceleration according to (7.43) results in K = eEm t/ℏ. Now we put a wave packet with zero-velociy at the origin
x = 0, and observe the time evolution. From (7.42),
(
)
eEm d
Enw d
sin
t ,
vg (t) =2ℏℏ

x(t) =

)]
[
(
Enw
eEm d
t .
1 − cos
2eEmℏ(7.44)The result indicates that in spite of the constant acceleration, the wave packet oscillates in space. The phenomenon iscalled Bloch oscillation, an observation of which in an actual lattice is almost impossible due to various scattering. In a
superlattice, however, the super-period devides the large original band into “mini-bands” and the acceleration to the topof a mini-band before scattering. The Block oscillation was thus observed in superlattices in optical measurements.

7.3 Modulation doping and two-dimensional electrons
The most popuar artificial structure made with heterojunctions is the two-dimensional electrons with modulation doped
heterojunctions (two-dimensional electron gas, 2DEG). As is illustrated in Fig. 7.12, in a single heterojunction, doping isgiven just in the wider band region. Now let us see what happens here for n-type doping.
Let us take the z-axis vertical to the surface and the hetero-interface plane as in the figure. In a “rigid band” model, theconduction band discontinuity ∆Ec emerges and the carriers re-distribute. Let us take the plain case of the combinationof Alx Ga1−x As and GaAs. Then we can adopt the approximation that the envelope function in the effective mass
approximation as the electron wavefunction itself, and electron-electron interaction can be treated within the HartreeE9-9

Al x Ga 1- x As
-Q +
+Q
+
+
Si +
+
++

N dep

A

GaAs
B
z( z )++ + ++

DE c

EF

Ez

z

Fig. 7.12 Schematic cross sectional view of two-dimensional electrons at a modulation doped Alx Ga1−x As/GaAs heterointerface.

approximation*2 . Then, the Poission-Schrödinger equation including the electrostatic potential formed by ionized donor,
the band discontinuity and the 2DEG itself should be solved self-consistently for obtaining quantilzed energy levels andwavefunction (envelope function) along the direction perpendicular to the 2DEG plane.
z-axis is taken to be perpendicular to the heterointerface plane. As in Fig. 7.12, the surface Schottky barrier creates asurface depletion layer. Let the charge at the surface be −Q and the electric field from the charge should be compensatedwith that from charges at ionized donors (in the figure Si) +Q in the amount and screened from inside. Let us write thenumber of all the residual ionized donors per unit area (integrated along z-axis) as Nddep . The electrostatic potential fromthe charges is, far inside the lattice from the doping region, VD (z) = (4πe2 /ϵϵ0 )Ndep z. Between the doped region andthe hetero-interface, a non-doped region called “spacer ” is often places. The spacer spatially separates the 2DEG and
the ionized impurities, decreases scattering probabilities of two-dimensional electrons, resulting in very high mobility ofelectrons. A too thick spacer, however, lifts up the band depletes the well and throws out the 2DEG.
Let us adopt a variable separation type expression for 2DEG wavefuntion, Ψ(r) = ψ(x, y)ζ(z). ζ(z) is the envelope
function along z-axis. The areal concentration n2d is the function of discretized energy level Ez , which is in other words
the kinetic energy along z-axis for ζ(z). The areal charge density at position z ′ is then −en2d |ζ(z ′ )|2 , the sheet charge of
which creates the electric field −(4πe2 /ϵϵ0 )n2d |ζ(z ′ )|2 |z −z ′ | as calculated from the Gauss theorem. In the Hartree-only
mean field approximation, the potential should include these terms. The potential created by the 2DEG itself is∫ ∞4πe2
V2d (z) = −n2d (Ez )|ζ(z ′ )|2 |z − z ′ |dz ′ .
ϵϵ0−ξHere the integral cut-off ξ should be taken longer enough than the penetration depth of ζ(z) in to AlGaAs barrier. We
write a step potential with discontinuity ∆Ec just at the interface as Vh (z). Now the total potential can be written as
[
]
∫ ∞
4πe2
V (z) = Vh (z) +Ndep z − n2d (Ez )|z − z ′ ||ζ(z ′ )|2 dz ′ .
(7.45)
ϵϵ0−ξ
Schrödinger equation for ζ(z)

[
−

]
ℏ2
∂2
+ V (z) ζ(z) = Ez ζ(z)2m∗ (z) ∂z 2

(7.46)

should be solved self-consistently to obtain (consistent) ζ(z). The effective masses m∗ are different in the two species ofsemiconductors and the boundary condition should beζ(0)(A) = ζ(0)(B) ,1 dζ (B)
1 dζ (A)
= ∗
.
∗
mA dz
mB dz

(7.47)In the Poisson-Schrödinger procedure, one should solve the equations from (7.45) to (7.47) consistently. The above only
treats the Hartree term. In general, the Fock term, or the correlation effect is also important in mean field theory. However,
it is known that the correlation effect does not affect ζ(z) or Ez so much and here we ignore it for simplicity.
*2 Even within the mean field theory, the interaction term contains the Fock term (exchange), but the contribution was calculated to be small.

E9-10

It is comparatively easy to solve Poisson-Schrödinger equation numerically for a simple band with small spin-orbit
interaction, like the conduction band in GaAs. For more complicated cases, e.g., multiple valleys, strong spin-orbit
interaction, etc., the scale of numerical calculation increases. If one needs to expand the calculation to other quantities
with obtained ζ(z) for such a case, approximate formulas with simple mathematical forms are convenient. For example,
in Fang-Howard approximation, the formula√ζ(z) =()
bzb2
z exp −
2
2

(7.48)

is used as the trial function with b as a parameter for variational calculation. The result of the variational calculation is
given as48πme2b =ϵϵ0 ℏ2(3)
11
n2d + Nd .
32

(7.49)In this approximation, penetration of wavefunction into the barrier (spacer) is ignored. Another approximation form
which takes such penetration into account is given in, e.g. ref.[3].

7.4 Fabrication of quantum wires
Nowadays we have so many methods to fabricate quantum wires and reviewing in this narrow space is impossible.
Here we have a short look at a few examples of them.

7.4.1 Split gates, other physical approach
The split gate method starts from a 2DEG. Metallic films on the surface form Schottky barriers and deplete the electrons
underneath them. Then we can build potentials with various shapes through those of the metals.
In the split gate method, enlargement of depleted regions with reverse (negative) bias voltage V as in Fig. 7.13(a) isoften used. Let us consider a simple model illustrated in Fig. 7.13(b), where two half-infinite metals are placed with adistance w. The line density of charge σ, created by applied gate voltage is assumed to be uniform. The electric field
formed by these charges has the z-component Ez (x, d) as,
)
(
)]
[
(
−σ
x + w/2
x − w/2Ez (x, d) =− arctan.
π + arctan
2πϵϵ0
d
d

(7.50)(7.50) depends on d, but as a coarse approximation, we ignore the dependence within the depth η of the two-dimensional
electron gas (2DEG) potential and the potential modulation due to the split gate is summarized as Vsg (x) = eηEz (x, η).

(a)

(b)

Fig. 7.13 (a) Schematic illustration of micro-fabrication by split-gate method. (b) Electrostatic potential formed by a split gatecalculated on a simple model in the text.

E9-11

3mm

Fig. 7.14 (a) Scanning electron micrograh of InAs nanowires along[111] grown by vapor-liquid-solid
phase method with metal particles
as catalyst on an InAs(111)B substrate. (b) GaN-InGaN core-shell
type nanowires.

500nm

(a)

(b)

An example of Vsg (x) is shown in the lower panel in Fig. 7.13(b). As indicated by the broken line, a parabollic potential
well approximates Vsg (x) around the bottom *3 .
The gate electrodes in Fig. 7.13(b) hence give confinement along x in addition to z-direction heterinterface confinement
potential. The kinetic energies for these two directions are quantized and quantum wires are realized. Density of states in
one-dimensional systems has a divergence at the bottom of band, which corresponds to the descrete energy levels in thepotential in Fig. 7.13(b).
Another way to form quantum wires with physical methods is “cutting” of 2DEG into thin and long shapes with wet(i.e. in some solvent) or dry (i.e. in some plasmas) etching method. There are also several ways such as regrowthof heterostructures onto a cleaved edge of another heterostructure to form T-shaped thin-line potential, ion-inplantation
inactivation, etc.

7.4.2 Self-assembled nanowires
After putting some “seeds” onto semiconductor substrates by electron beam irradiation, etc., crystal growth onto it
causes nanowire growth at the seeds under some conditions. In such a growth, with changing “flying” materials heterostructures or doping can be installed in the wire. Figure 7.14(a) shows InAs nanowire along [111] grown on an
InAs(111)B substrate with Au nanoparticles as the catalyst by vapor-liquid-solid phase method. Fiugre 7.14(b) shows
GaN-InGaN core-shell type nanowires.
Well-known carbon nanotubes, which are rolled up graphenes with nanometer-size diameters, may not be classified
into one-dimensional systems, are also a kind of “self-assembled systems”.

7.5 Fabrication of quantum dots
Fabrication methods of quantum dots are also classified into physical methods, self-assemble methods, and their combinations. I would like to introduce just a part of them.

7.5.1 Physical method
Quantum dots (QDs) are expected to have a very wide range of optical applications, and quantum dot lasers are alreadyon the market. Here, however, we restrict ourselves to the QD for the study of transport. To measure the electric
conduction, we need to touch electrodes to the dots. As the electrode material we consider a normal metal with an
ordinary Fermi surface. And as the “connection”, we consider tunneling junctions, through which electron can transmit*3 Anyway a rounded bottom of symmetric potential can often be approximated by a parabolla because the leading term in the power expansion isusually of the second order.

E9-12with quantum tunneling. At least a single electorode is required. And to compose structure for measurement of transport
between two particle reservors just like quantum wires, FET, two electrodes, hence two tunneing junctions are required.
As shown in Fig. 7.15, these electrodes are called just like FET, source and drain.
In quantum dots, the density of states is like a series of δ-functions. The electric conduction is determined by thetunneling probability of junctions and the relative positions of the δ-function like density of states and the Fermi levelsin the drain and the source. If we place another metallic electrode close to the dot without tunneling probability, theelectrochemical potential of the dot can be controlled with the electric field from the electrode. The electrode is called
gate. Figure 7.15(a) shows the total schematic of the quantum dot for conduction measurement.
The split-gate method introduced for the quantum wire can also be applied to form quantum dots. Figure 7.15(b)illustrates a possible pattern of Schottky electrodes. For the tunnel junctions, quantum point contacts near the pinch-off
condition are used. For the gates, also Schottky electrodes are used in the reverse bias condition. As can be imaginedfrom the figure, because the reverse bias voltage enlarges the depletion layer to squeeze the dot region, the size of thequantum confinement potential is diminished that widens the level intervals other than the single electron effect, which
will be discussed later. Hence the gate is sometimes called “plunger” gate. In this kind of configuration, the source, dot,drain line up side by side along the two-dimensional electrons, hence called “lateral” quantum dots. When the numberof electrons in a quantum dot is reduced by the gate voltage, the dot size also becomes smaller, it is spatially separated
from the source and drain, the tunnel probability becomes smaller, and conduction becomes unmeasurable. This once
considered as a difficult problem but has been overcome by the remote charge detection. The structure in Fig. 7.15(c)is made from the double barrier structure The structure shown in Fig. 7.15(c) is made by cutting out a double barrier
structure into a cylinder shape. Then Schottky gate electrodes are deposited on the side of the cylinder. This is called a“vertical” quantum dot. The tunnel couplings are determined by the double barrier structure, not affected by the electron
number. The property makes the structure apropriate for the experiments for small number of electrons. There is aproblem in connection with external quantum circuits, which requires various devicing.

7.5.2 Self assembling method
The epitaxial growth has various “mode” in the growth process. The layer-by-layer growth mode is called Frank-van
der Merve (FvdM) mode (Fig. 7.16(a)). When the interface energy accumulation between the thin film and the substrateis large due to the combination of materials, the deposited material is repelled from the substrate in the beginning of thegrowth and a three dimensional growth begins. As a such growth mode, the Volmer-Weber (VM) mode is illustrated
in Fig. 7.16(b). In Fig. 7.16(c) we show the Stranski-Krastanow (SK) mode, in which the growth is two-dimensionalat the very beginning but changes into three-dimensional due to the lattice distortion inside the film. With such threedotsourcedot

drainsourcedrain
dottunnelingbarriergate(a)tunnelingbarrier(b)gatesource

drain

gate

(c)

Fig. 7.15 (a) Schematic of quantum dot structures for transport measurement. Two electrodes for examining conduction, a source, and a drain are connected via a tunnel barrier across the quantum dot, and a gate electrode that
controls the potential of the dot is arranged at a distance. (b) Illustration of a “lateral” quantum dot. Nano-fabricatedmetallic gates on two-dimensional electrons are used. (c) Illustration of a “vertical” quantum dot. The dot layer isbetween two barrier layers and the doped upper and lower layers are the source and drain. The layers are cut to a piller
and the metallic gate is deposited surrounding it (in the figure a part of the gate is drawn).

E9-13dimensional growth, the structures with dimensions less than two can be obtained.
(a)

(b)

(c)

Fig. 7.16 Various epitaxial growth modes. Blue-gray circles represent atoms in the substrate, while open circles do the atoms in
the fim. (a) Frank-van der Merve. (b) Volmer-Weber. (c) Stranski-Krastanow.

A representative example of self-assembling of low-dimensional systems is the growth of quantum dots with SK mode
growth. Such examples are shown in Fig. 7.17. When some amount of a semiconductor InAs, which has the lattice
constant 7% larger than that of GaAs, is deposited on a GaAs substrate and kept at some high substrate temperature forsome time, the indium atoms on the substrate migrate and accumulate to form quantum dot structures.
In the case of InAs, dots are self-assembled because when a relatively small number of In atoms are present on thesubstrate, it is more energetically stable to perform three-dimensional growth to escape from the interface, which getsstrong lattice distortion from the substrate. The indium atoms first form a two-dimensional wetting layer with thicknessof a few lattice constant then dots are randomly formed in the shape that depends on the crystal direction of the surface.
The quantum dots produced by the SK mode growth have random sizes and positions. On the other hand, the SK dots
are with high crystal qualities and with high densities, and thus widely used for optical devices like quantum dot lasers.
In addition, since InAs has low junction resistances with metals, conduction measurement is also performed by attaching
metal electrodes. A method often adopted as a combination of self-assembling and physical methods is to attach gatesand barrier electrodes to self-forming nanowires by lithography to make dots. In particular, the self-assembling of InSb
or InAs, to which heterojunction technique is difficult (though not impossible) to apply, is used to form quantum dots andother structures with many gate electrodes. There is also a method to form quantum dots with implementing the barrier
layers into nanowire during the growths.

5nm

3mm

100nm
(a)

(b)

(с)

Fig. 7.17 (a) InAs quantum dot growth on GaAs (001) substrate. Upper panel: RHEED pattern of two-dimensional
growth at the beginning of InAs growth. Lower penel: RHEED pattern of three-dimensional growth of quantum dot
structure. (b) Atomic force micrograph of self-assembled quantum dots. (c) Scanning tunneling micrograph of aquantum dot. The lattice image can be seen, manifesting that the whole dot is a single crystal.

7.5.3 Colloidal quantum dots
In recent years, the colloidal manufacturing method has come to be widely used to form optical quantum dots for opticaluse. As shown in Fig. 7.18, this is a method of obtaining quantum dots by injecting a dot material called a ”precursor”
into a solvent, dissolving it, making it supersaturated by a temperature change, and precipitating a part of it. From therelationship between surface area and volume, when the degree of supersaturation falls below a certain value, dots that
continue to grow and dots that redissolve are separated, so dots with relatively uniform sizes can be obtained. This isE9-14

called Ostwald reipening. After the growth reached saturation, surface covering of the grown dots with another material
is possible by adding a different precursor into the solvent. Such covered quantum dots obtained in this way are called
core-shell type quantum dots. The luminescence wavelength of the quantum dots can be tuned by their size and hence it
is possible to form high efficiency luminous materials. They are already applied to, e.g. quantum dot displays.

Fig. 7.18 Left (A): Illustration of quantum dot formation with precipitation from supersaturated solvent. The timedependence of the density of precursor is plotted in the graph. Left (B): Illustration of experimental setup of precipitation method. Right upper: Schematics of core-shell type quantum dots. [4] Right lower: Illumination from CdS-based
core-shell quantum dots and TEM images. (Ocean Nanotech. web site. https://www.oceannanotech.com/)

References[1] T. Ishibashi，S. Tarucha and H. Okamoto, Proc. GaAs and Related Compounds, p. 587 (Oiso, 1981).
[2] L. Esaki and R. Tsu, IBM J. Res. Dev. 14, 61 (1970).
[3] T. Ando, J. Phys. Soc. Jpn. 51, 3893 (1982).
[4] C. B. Murray, C. R. Kagan, and M. G. Bawendi, Annu. Rev. Mater. Sci. 30, 545 (2000).

E9-15

Lecture note on

Physics of Semiconductors (10)16th June (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

I would like to introcude some of optical devices with hetrojunctions though the confinement scales are not in quantum
regime.

7.6 Confinement of injected minority carriers and optical devices
The minority carriers injected with pn junctions or with optical excitations, are transported by diffusion currents ordrift currents in solids. Spatial geometries, injection currents etc. are used for the control of diffusion currents. A
typical example is the bipolar transistor. Drift currents can be controlled through internal potentials introduced by hetero,
Schottky, MOS junctions, and through bias voltages, and gate voltages. A simple example is the window layers of solar
cells. As illustrated in Fig. 7.19(a), a window layer is placed on the top layer of a pn-junction solar cell. It should have alarger band gap than that of the material for the pn-junction.
One of the factors of lowering the conversion effciency of solar cells, is the non-radiative recombination of injectedminority carriers via the highly dense surface states, which also cause the pinning of the Fermi level in Schottky junctions.
The current through the device is driven by minority carriers swept out by the built-in potential of the pn-junction.
Minority carriers created inside the semiconductor have a random initial momentum and diffuse also to the surface. Manyof them are lost at the surface with non-radiative recombination and thier energies either as heat. When the surface hassome decolation to prevent reflection, the increase of the surface area reults in the enhancement of surface recombinatn
rate.

Fig. 7.19 (a) Upper: Illustration of a solar cell with a window layer. An example of AlGaAs/GaAs pn junction.
Lower: Schematic band-diagram of the solar cell shown in the upper panel. (b) The schematic diagrams of quantum
collection efficiencies of cells with (red line) and without (blue line) the window layer. (c) Illustrations from brochureof HIT solar cells, which demonstrates the enhancement in the conversion efficiency with the heterojunctions.

E10-1

If the cell has the layer with a larger bandgap Eg2 on the top as illustrated in Fig. 7.19(a), the diffusion of minority
carriers to the surface is blocked with the heterojunction barrier. The materials are chosen so as to have a good connection
at the junction and not to have in-gap recombination centers as AlGaAs-GaAs in the figure. Then the minority carriers
reflected at the junction diffuse back to the pn junction and contribute to the photocurrent. Figure 7.19(b) illustrates thequantum efficiency spectra η(hν) of pn junctions (energy gap Eg1 ) with and without the window layer with the energy gap
Eg2 . hν is the energy of photons. ν is the photon frequency throuout this section. The quantum efficiency is defined as
the ratio of the number of electrons in the photocurrent to that of the photons in incoming flux. Without the window layer(blue line), η(hν) decreases with increasing the photon energy due to the increase of minority carrier creation close to
the surface and hence the increase of surface recombination. With the window layer (red line), the surface recombinationis reduced and the value of η is kept close to 1 up to around hν ∼ Eg2 . Above Eg2 , due to the absorption in the window
layer, the surface recombination increases and the efficiency decreases. If we can choose Eg2 around the energy above
which the mode density of sun light is small, we can expect large enhancement in the conversion efficiency. The window
layer can be viewed as an example of reducing the diffusion current with a kind of drift current caused by barriers atheterojunctions.
An ingenious example of the application of above technique to market-selling devices is the solar cells named HIT(heterojunction with intrinsic thin-layer), which were developed in SANYO and now are produced and sold in Panasonicbland. The base is a crystal Si solar cell but they utilized the fact that amorphous Si has a larger effective band gap.
In the structure a Si active layer is sandwitched by clad amorphus layers, which cause confinement of minority carriers
inside the crystal Si. HIT still has a top-class conversion effciency but unfortunately, it has been announced that it will be
discontinued due to various reasons.

7.6.1 Light emitting diodes
To take an important example of confinement of minority
carriers, we consider luminescent devices with pn-junctions
as the injectors of minority carriers. Such electroluminescentdevices are called light emitting diode (LED).
There are various processes of photon emission by the recombination of injected minority carriers, but here we restrict
ourselves to the direct recombination of electrons in the conduction band and holes in the valence band. The luminescentintensity I(ν) is written as]−(hν − Eg )
.
exp
kB T
[

I(ν) ∝ ν (hν − Eg )2

1/2

(7.51)

Figure 7.20 shows an example of luminescent spectra from a
GaAs pn-junction. With decreasing the temperature, the band
gap Eg widens mainly due to the variation of lattice constant.
As a result, the luminescent peak narrows and shifts to highenergy (blue-shift). As for the second peak in the spectrum at
77 K, the authors of Ref. [1] commented only the existence,but it looks like the luminescence from the impurities.
Fig. 7.20 Electroluminescence spectra of a GaAs
pn-junction. From [1].

Important parameters of LED characteristics are the wavelength and the efficiency. In the case of homo pn-junction

luminous layer, the wavelength is almost determined by the band gap as in Eq. 7.51. The quantum efficieency ηq is
E10-2

defined as the ratio of the radiative recombination processes Rr to the total recombination processes R of the injected
carriers as
ηq ≡

Rr
τnr
τtot
=
=
,
R
τnr + τr
τr

1
1
1
≡
+ ,
τtot
τnr
τr

(7.52)

where τnr , τr are the lifetimes limited by non-radiative recombination and radiative recombination respectively. τtot is
the total lifetime of minority carriers obtained from Matthiessen’s rule. The interband radiative recombination probability
is proportional to the electron-hole density product, that is
Rr ∝ np.

(7.53)

Under the injection of minority carriers, the law of mass action naturally does not hold: np ̸= n2i .
The current density of minority carriers is, as seen in Eq. (6.11), gvien by the sum of the electron flux density, the hole
flux density,[

De np0Dh pn0je + jh = e
+
Le
Lh

][

(
exp

eV
kB T

)

]
−1 ,

(7.54)and the recombination rate inside the depletion layer (width wd ) expressed in the form of current as[()]
eni wd
eV
jR =
exp
−1 .
2τ0
2kB T

(7.55)The recombination in the depletion layer mainly occurs at mid-gap deep levels resulting in the factor 1/2 in the term eVjust as in Eq. (6D.13). We take the case of an n+ p junction, in which the n-side is heavily doped and the luminescence ismainly by electron-hole recombination in the p-layer. Then the injection efficiency of the junction isγ=je.
je + jh + jR

(7.56)The internal quantum efficiencyis thus defined from Eq. (7.52) and Eq. (7.56) as
ηiq = γηq .

(7.57)

There are various limiting factors of the internal quantum efficiency, some of which are related to the crystalinity as deeplevels. The device structures also affect the efficiency through the surface recombination, etc.
The external quantum efficiencyhas the ultimate importance for the LEDs. As we can see from Eq. (7.51),
the energy of photons emitted by direct interband transition has a peak slightly above the energy gap.

Then

the reabsorption of light by the crystal itself occurs and the absorbed photons result in the absorption loss.
As shown in the left figure, we take x as the distancefrom the surface to the emission point, θ as the angleof the ray from vertical to the surface. Let α be theabsorption coefficient and we have the absorption loss
ζabs = 1 − exp(−αx/ cos θ).

(7.58)

When a light pass through the interface between the materials with refractive indices n̄1 and n̄2 , we have thereflectance(Γ=n̄2 − n̄1n̄1 + n̄2

)2.

(7.59)The loss at the surface by the reflection is called Fresnel loss. Because the reflactive index inside semiconductors is
generally larger than that in the vacuum or in the air, when θ exceeds the critical angle θc , the surface causes totalreflection, which reults in the optical loss. The ratio of the number of photons finally emitted from the surface nf tothat of photons once produced inside the crystal is called optical efficiency. The ratio of nf to the number of injected
E10-3

Fig. 7.21 Conceptual diagram of double-heterojunction LED.
InGaN is taken as the material for the active layer.

carriers is called external quantum efficiency. Let ηopp , ηexq be the optical efficiency and the external quantum efficiency
respectively, then from the definitions
ηexq = ηopp ηiq .

(7.60)

Generally, simple pn-junctions have very low external quantum efficiency less than few %.
As in the solar cells, surface textures to cause multiple reflection are effective to reduce the Fresnel loss and the totalreflection loss. And also like the case for solar cells, double-heterojunction(DH) is frequently used to enhance theinternal quantum efficiency and to reduce the absorption loss. The concept is shown in Fig. 7.21. The radiative layer
is inserted between the cladding layers of materials with larger band gap than the active material. In the figure, thechemical dopings are just done in the cladding layers. The elimination of chemical doping in the active layer reduces therecombination in the depletion layer. Injected minority carriers are confined into the thin active layer, resulting in highnp product and in high internal quantum efficiency. Furthermore, the energy of emitted photon is less than the band gapof cladding layers and the absorption does not occur there. Mirror-like layers are often placed at the back planes to reflectforward the photons emitted backward. In Ref. [2], the authors reported the external quantum efficiency of 77% in YAG
active type LED with InGaN-LED activation. ηexq ∼ 30 % were the highest then, and the case of YAG is extraordinary.
But now the technology is widely used for LED illumination.

7.6.2 Laser diodeThe light emission in LEDs is by spontaneous emission drawn in Fig. 4.1(b). Now we consider the stimulated emission
drawin in (c). |A0 |2 in the transition probability is proportional to the energy density of electromagnetic field. We write
the energy density as nλ ℏωλ /V , where V is the system volume, nλ is the number of photons in mode λ. Such a coherent
electromagnetic field excites electric dipole moment µ in the material, creating the transition element between |a⟩ and |b⟩(see Fig. 4.1(c) for the two-level system). If we write r = r0 cos(ω0 t), then p = mωλ r. We rewrite ⃗e · p as (ωλ m/e)⃗e · µE10-4FeedbackMedia withinverted distribution
Feedback

AlGaAsGaAs

AlGaAs

p+

p

n+

E
-

-

-

-

-

E g1

(a)

E g2
+

(b)

+

++

(c)

eV

+

x

A

A
B

B

cleavedsurfaceE g1

cleavedsurfaceA

Fabry-Perot typeA(d) Distributed Feedback typeFig. 7.22 (a) Confinement of photons with parallel mirrors. Also can be viewed as the feedback by mirrors. (b)Illustration of double heterojunction laser diode structure. AlGaAs and GaAs are taken as representative materials.
With forward bias, the distribution inversion is realized. (c) Fabry-Pérot type lase structure, which is constituted ofparallel mirrors of cleaved surfaces. Material B is sandwitched by material A with larger band gap. (d) Distributed
feedback (DFB) type laser structure, in which corrugation type grating is built in at the A-B interface.

and put ω = ω0 = ωλ . Then the probability of stimulated emission with the transition |b⟩ → |a⟩ isPba (t) =ωλt2|⟨a|⃗e · µ|b⟩|2 nλ ,ϵϵ0 ℏV
2(7.61)

which is proportional to nλ . The symmetry of Eq. (7.61) tells that the probability of light absorption with transition|a⟩ → |b⟩ is Pab = Pba . Equation (7.61) tells that the more photons in a mode the higher the probability of stimulated
emission to this mode. The phenomenon can be interpreted as a Bosonic stimulation, which is the origin of BoseEinstein condensation. And the photonic state is described as a coherent state[3]. The coherence can be understood in aclassical picture that µ is excited coherently by the electromagnetic field.
As a model of the medium of photon propagation, we consider a set of such two-level systems. Let Na , Nb be theconcentrations of the two-level systems at the state |a⟩, |b⟩ respectively. When the light of ωλ propagates the media, theenery absorbed by the media in unit volume is written as
E = (Na − Nb )Pba (τ )ℏωλ ,(7.62)

where τ is the averaged interaction time of light with each two-level system. If the state Nb > Na is realized E < 0,namely the light absorbs energy from the media and the light is amplified. The light is in coherent state with a common
phase of photon. Such amplification of photons (increment in the photon number in the same mode) and the device(apparatus) to realize the phenomenon is called light amplification by stimulated emission of radiation, LASER.
E10-5

A laser diode (LD) is a light emitting element that uses a pn junction like an LED, but uses a double heterojunction (ora stronger confinement structure) to create an inversion distribution and to cause laser action. In order to strongly amplify
light, it is necessary to advance the light in the population inversion medium, but the light is also strongly amplified byconfining it in the resonator using a mirror surface and reciprocating in the same medium(Fig. 7.22(a)). Figure 7.22(b)shows an LD structure in the beginning of the research. An example of the simplest Fabry-Pérot type cavity of the laser
oscillation is illustrated in Fig. 7.22(c). In Fig. 7.22(d), the structure called distributed feedback (DFB) type laser diodeis illustrated. A corrugation is introduced at the hetero-interface as a grating and to make the structure a cavity.

E10-6Chapter 8 Basics of quantum transport
Let us go into quantum transport, which is one the major subjecs in semiconductor physics, because one-dimensional
systems are the best for the construction of theoretical models.

8.1 Classical transport and quantum transport
We treated electrons as particles in the secton of “classical transport” besides counting the number of cases while in thesections of heterojunction and quantum confinement, they were treated as quantum mechanical waves. The difference inthe treatment of electrons in the same material comes from the scales of energy and spatial range. Until now we havetreated pn-junctions classically and double barrier diodes quantum mechanically, but these two are actually marginalcases. In a pn junction, the deletion layer becomes thinner with increasing the doping concentrations. When both the player and the n-layer are highly doped, the depletion layer is very thin and the Fermi level penetrates into the conduction
band in n-layer and the valence band in p-layer. Now in the both layers the density of states exist around the Fermi level
and they are very close with a thin separation. Then a quantum tunneling, which is nothing but a quantum phenomenon
occurs through the depletion layer. The structure is called “Esaki diode”, a representative quantum device. On the other
hand in some double barrier diodes, depending on the materials and the structures, no resonance peak can be observed
e.g. at room temperatures. Let us briefly discuss here in what case we need to treat a phenomenon quantum mechanically.
We already had a very short discussion in the beginning section of classical transport. Let us go a bit deeper here.
Now in what case does quantm coherence appear in transport? The “length” which expresses the criteria is quantum
cohrence length *1 . In very short, when an electron travels in a solid, the averaged length over which the electron
propagates with quantum coherence, is called “quantum coherence length” and often written as lϕ .
Whether we can observe quantum coherence in experiments or not depends not only on the essential quantum coherenceof each particle but also on the coherence between the particles i.e., statistical fluctuation of the interference. The former
is, in very short, due to quantum entanglement with a large number of surrounding freedoms (environment).
Let us consider a double slit experiment shown in the left. The interfer-

scattering
centers

ence pattern on the screen isc1|ψ|2 = |ψ1 + ψ2 |2 = |ψ1 |2 + |ψ2 |2 + 2|ψ1 ||ψ2 | cos θ,y1y2

where the third term in the RHS is the quantum interference. Consider thec2

situation that from the starting point to the screen there exist some scatterings, at which the electron has interactions with a quantum mechanicalstate χ (quantum mechanical freedom other than the electron). The interaction should be different for the two paths 1 and 2, then
ψ1 → ψ1 ⊗ χ1 ,

ψ 2 → ψ2 ⊗ χ2 .

As a result, the interference term changes into
2|ψ1 ||ψ2 | cos θ⟨χ1 |χ2 ⟩.
*1 The word “coherence length” is used in various ways in different meanings. In condensed matter physics, for example, it appears in treatingsuperconductivity for multiple meanings.

E10-7

Hence if χ1 ⊥ χ2 the inner product is zero and the interference term vanishes. In such a state, two freedoms ψ and χ
are in maximally entangled state (Appendix F). In other words, the quantum coherence length in this case is the length
over whch the electron (freedom) makes up a maximally entangled state with another degree of freedom (envirnment).
A little question here is that χ1 and χ2 may be orthogonal at one-moment but the time evolution after that may restore
the interference killing the orthogonality. There are, of course, many such setups *2 . Here we adopt that in not-speciallydesigned quantum systems, with time evolves the entanglement spreads over many other freedoms and disentanglement
never occurs.
There is another kind of “dephasing” in experiment. Even if each particle is able to interfere with itself, when the
wavelengths of particles are widely distributed, in other words monochromaticity is not high enough, the interference
patterns are also distributed and averated out. Let us estimate the characteristic length, over which the difference in thepatterns is small enough for them to survive after averaging. Electrons being fermion, the energy of movable electron
at absolute zero is EF , i.e., they are completely monochromatic. The energy width appears at a finite temperature T as
∆E = kB T . The difference in the electron phase accumulated during time τ is 2π∆f τ = 2π∆Eτ /h = 2πkB T τ /h. A
criterion in time can be the time for the phase difference becomes 2π, that is
h
.
kB T
√
In diffusive transport, the diffusion length is written as l = Dτ , and this determines a kind of coherence length lth as
√
hD
,
(8.1)
lth =
kB T
τc =

which is called thermal diffusion coherence length. In ballistic transport, the electrons get few scatterings during thetraversal through the sample. They get through the sample with the velocity vF and
lth =

hvF
.
kB T

(8.2)

After traversal over the above themal length, the coherence is lost from the result of averaging even though intrinsic
quantum coherence survives. Attention should be paid for lth particularly in experiments.
After knowing lϕ , what are the conditions for the quantum mechanics to appear in transport? Firstly we should list up
the case that the sample size is shorter than lϕ . For even shorter sample size, shorter than the representative de Broglie
length (i.e. the Fermi wavelength), as we already saw, quantum confinement effect (descretized energy levels) emerges,
into which we do not go into here. Secondly, we often have some characteristic lengths in transport besides the sample
size. A representative is the magnetic length, which appears when an external magnetic field is applied. The magnetic√
length, or minimum cycrotron radius is written as lB = h/eB for magnetic flux density B. When lB ≤ lϕ , there
appear various quantum effects in magneto-transport.
We finish up this very short section for quantum coherence and decoherence here. Below we go into quantum coherent
transport.

8.2 Landauer formula
In this lecture, I would like to introduce one view point, from which we view the transport in quantum systems as the
conduction in “quantum circuits”*3 . In this section we see the most basic part of it.
*2 This can be experimentally verified. With this fact, some people claim that the theory of decoherence based on environmental freedom is wrong.

But this is, of course, misunderstanding. In the theory of decoherence from the environmental freedom, “intrinsic decoherence” does not exist
besides the thermodynamic limit. In real systems no thermodynamic limit has been achieved though the results of statistical mechanics apply. In
the same way, with progress of entanglement with many degree of freedoms, the disentanglement becomes more difficult and impossible at last.
*3 Here “quantum circuit” is different from what we use in schematization of quantum information manipulation.

E10-8

The Kubo formula is an ultimate form of linear response, which was studied from the beginning to middle of 20th
century in Bell labs and other places. It is now an indispensable tool for theoretical studies in condensed matter physics.
On the other hand in practical analyses of experiments, the Landauer formula, which can be derived as a one expressionof the Kubo formula[4], is often used. I hope students refer to other lectures e.g., statistical physics, for the introductionof the Kubo formula and here we go into quantum transport with the simplest introduction of the Landauer formula.

8.2.1 Conductance quantization
The lowest dimension in which “transport” exists is one. We thus
LR

first consider the conductance of a one-dimensional fermion system.

S

Here we adopt an ingenious modeling by Rolf Ladauer, illustratedmL

in the left figure. A one dimensional conductor without scattering iseVconnected to two particle reservoirs, in which the chemical poten-mLtials are well defined as they have huge (infinit) number of particles
and are in thermal equilibrium. Let the chemical potentials of left
and right reserviors as µL , µR respectively. The current brought bya state with wavenumber k can be written as
j(k) =ee dE(k)
vg =
,
L
ℏL dk

(8.3)

where L is the length for normalization, thus e/L is the charge density. The total current J then is
∫ kR
J=

j(k)
kL

e
L
dk =
2π
h

∫ µL
dE =
µR

e
e2
(µL − µR ) = V.
h
h

(8.4)

The conductance is finally obtained as
G=

J
e2
=
≡ Gq ≡ Rq−1 .
G
h

(8.5)

This is the conductance for one-dimensional conductor without scattering and called conductance quantum. If weconsider the spin degree of freedom, and when the spin can be treated as just a double degeneracy of quantum states, we
simply multipy Gq by two and may call 2e2 /h a conductance quantum. Rq is called quantized resistance.
The above discussion is, in a sense, a paraphrase of te uncertainly principle. Let us see that in a more transparent
form. The problem is equivalent to that we pack wavepackets with a width ∆k in k-space into a one-dimensional fermion
system. The highest charge density in the system is e/∆x for wavepackets with a width ∆x in the real space. The velocityof the packet is ∆E/ℏ∆k, giving the current as
J=

e2
e ∆E
= V,
∆x ℏ∆kh(8.6)

which is the same result as before. Here we write ∆x∆k = 2π, ∆E = eV .

8.2.2 Quantum point contact and concept of conductance channel
One dimensional fermion system discussed above is in other expression quantum wire (QW) or quantum point
contact (QPC). A way to realize them in semiconductor structures is to confine a two-dimensional electron gas (2DEG)
into a narrow region.
In the case of QPC, “a narrow region” means, as in Fig. 8.1(a), a narrow short region gradually squeezed from a wide
2DEG. As is easily imagined, such a structure can be realized with the split gate technique introduced in Sec.??. Thiscan be modelized as in Fig. 8.1(b). x-axis is taken to longitudinal to the QPC “waveguide”. Here we assume adiabatic
E10-9

y
E1 E2 E3

gate

2DEG

depletionlayer

V eff ( x )
E3

gate

E2
E1
(a)

V(y)

x

(b)
Fig. 8.1 (a) Schematic of quantum point contact. (b) Simplified model of a QPC. Upper panel: Electrons are
excluded from hatched regions and confined in white region, one dimensionally at the narrow gap. The right figure
shows the confinement potential along the cross section at the broken line. Descrete eigen energies E1,2,3 correspondto the three effective potential drawn in the lower panel. Lower panel: Illustrates effective potentials Veﬀ (x) ineq.(8.8).

propagation of electrons through a QPC, that is, the total energy of an electron E = Ekx + Eky does not change during
the traversal though Ekx , Eky transform each other.
Though harmonic potentail like in Fig. ??(b) is generally a good approximation for such kind of confinement, here
we take, for simplicity, the hard-wall approximation illustrated in Fig. 8.1(b). With W being the width of confinement,
the wavefunction in y-direction is written as φn (y) = cos(nπy/2W ) (n: an odd number ), sin(nπy/2W ) (n: aneven number). We assume that change of W for x is slow enough so that we can separate x and y dependencies asψ(x, y) = φn (y)ϕ(x). Then, the equation is)∂2∂2+ 2 φn (y)ϕ(x)∂x2∂y(( nπ )2 )ℏ2∂2= φn (y)+ϕ(x) = Eφn (y)ϕ(x).
2m ∂x22Wℏ2Hψ(x, y) =
2m

(

(8.7)

(8.7) depends on x, and the assumption of adiabaticity requires it holds for each x-position. This can thus be viewed as apotential problem with effective potential along x-direction
Veﬀ (n, x) =

ℏ2
2m

(

nπ
2W (x)

)2
.

(8.8)

The situation is illustrated in the lower panel of Fig. 8.1(b). The effective potential Veﬀ (n, x) has index n, which is forthe descrete quantized energy along y. The partitioning of total energy is, then,Etot = Ekx (n, x) + Veﬀ (n, x),(8.9)

and we can treat a propagating state as a one-dimensional one indexed with n. Such a one-dimensional state is called√
conductance channel, the density of states to which is proportional to 1/ E − Ec . When the system is in equilibrium,
EF is, of course, constant over the system though the effecive potential Ekx (n, x) is channel dependent and thus the
Fermi wavenumber kxF , the density of states should be determined for each channnel.

E10-10Conductance ( e 2 / h )

14
12
10

8
6
4
2
0
-1.0

-0.8

-0.6

Gate voltage (V)(a)(b)
Fig. 8.2 (a) Atomic force microscope of a QPC gate structure. White raised regions are the gate electrodes placed onan AlGaAs/GaAs two-dimensional electrons. (b) Conductance of a QPC at 30 mK as a function of the gate voltage.

8.2.3 Transport experiments in QPCs
Let us see some experimental results on transport through real QPCs. With increasing the negative voltage to the gateelectrodes, the effective potential in (8.8) becomes higher due to the narrowing of W (x) and the number of conduction
channels whch can go over the potential top decreases.
Figure 8.2(a) shows an AFM image of a split gate structure fabricated with nano-fabrication techniques. In Fig. 8.2(b),we plot the electric conductance G of the QPC as a function of the gate voltage Vg . G shows staircase-like variation
versus Vg with a constant height of stairs about 2e2 /h. Namely G is quantized to an integer times 2e2 /h. The systemholds the time-inversion symmetry and the spin degeneracy. Hence the experiment confirms the result of Eq. (8.5).
In the experiment shown in Fig. 8.3, the conductance of a QPC is adjusted on the plateau of n × 2e2 /h (n: integer)
and the tip of an atomic force microscope (AFM) is placed just on the surface close to the QPC. Then the image potentialof the tip in the two-dimensional electrons causes weak scattering of the electron wave resulting in a small shift ofthe conductance from the quantized value. With scanning the tip the shift is plotted versus the tip position, then as inFig. 8.3(b), on the plateau of n = 2, we observe a wave with two anti-node is flowing out from the QPC. The numberof anti-node is three for n = 3 and one for n = 1. The above results shows the number of anti-node of standing wave
along y, that is the number of channels transmitting through a quantum wire is equal to the quantization number n of the
conductance *4 .

8.2.4 Conduction channel and transmission probability
In the above we have introduced the concept of conduction channel referring to the QPC experiments. The shortness
of QPC is to escape from the scattering and longer structure is available if the mean free path exceeds the size. Actually,in longer quantum wires made of high-mobility two-dimensional electron systems, the quantization of conductance has
been observed. Next we consider the widening of the quantum wire. With the increase of width, the level spacing ofquantization along the width narrows, the number of states below the Fermi level EF increases if the position of EF from*4 Some of you may think that if the number of channels is, say 3, then the waves with antinodes 1, 2, and 3 should be overlapped. The argument√
is correct. However, the density of states of one-dimensinonal systems is expressed as 1/ ϵ − ϵ0 with ϵ0 as the band edge. Then in actual
measurement, the amplitude of wavefunction with highest channel is detected. Also, the electrons traversing on the highest channel have thelowest the kinetic energy along x and easily scattered by the probe potential, detected in the experiment.

E10-11

tip
gate
gate

x
y

A

V

(a)

(b)
Fig. 8.3 (a) Illustration of experimental setup to measure the wavefunction amplitude with a scanning probe microscope (SPM). With measuring the conductance of the QPC, the tip is scanned over the conduction channel. (b) The
image of the shift in the conductance from the quantized value measured with this setup. The center part is drawn
from the topography obtained by AFM measurement. The upper is measured on the nch = 2 conductance step. The
lower is for nch = 3. (The data are taken from Topinka et al., Science 289, 2323 (2000))

the bottom of the band is fixed. We take the limit of infinity in width and the system is now a two-dimensional. We write√
the electron density as n2D then we find the number of channels per unit length is n2D .
So far we have considered systems without scattering. What we are treating here is coherent quantum transport andrandom inelastic scatterings by phonons etc. are out of scope. However, the potential scatterings by impurities or latticeimperfections do not break quantum coherence and they can be taken into account. The scatterings are transitions between
the propagating states, or from the view point introduced here, transitions between the conduction channels. Hence we
express the scattering centers with points as in Fig. 8.4(b) and through the points electrons enter different channels.
Note that they are quantum mechanical scattering and the electrons do not completely “change” their tracing lines atthe scatters. Instead the electron waves are devided at the scattering points and continue propagation. The conduction
channels play the role of waveguide for microwaves. With such waveguides and joints in various shapes, we can separate
or join microwaves. At such joints there also should exist reflection which reverse the direction of propagation. The sameshould happen at the scattering centers. When the number of scattering center increases and the system can be viewed asa “disordered metal,” the system in the channel expression is like a cobwebs as illustrated in Fig. 8.4(c). At first sight,
it looks difficult to treat. But instead of treating inside, we just pay our attentions to the channels at the inlet and at theoutlet. We write the transmission probability of electron propagation from i-th channel at the inlet to j-th channel as Tij .
From the fact that the single channel without scattering has the conductance of e2 /h with the transmission probability
T = 1, the conductance G of a conductor that has the matrix of transmission probability {Tij } is (with consideration of
the spin degree of freedom 2)
G=2

e2 ∑
Tij .
h i,j

Equation (8.10) is called Landauer formula for 2-terminal conductance.

E10-12(8.10)

outlet

inlet
(a)

i

(b)

j

(c)

Fig. 8.4 (a) A two-dimensional conductor is expressed as a set of one-dimensional channels. Two-terminal configuration. (b) Introduction of a scattering center, which causes transitions between the conduction channels. (c)Disordered conductor with multiple scattering centers.

8.3 S-matrix
We have introduced scattering centers (joints) through the comparison of conduction channels with waveguides for
microwave. Actually researchers often call conduction channels electron waveguides. Also, we often use interference
circuits, in which quantum wires are joined/devided at some points just like joints of waveguides. For the treatment ofsuch joints, scattering matrix, S-matrixis often used as is in the case of microwave circuits. As in Fig. 8.5(b), we write
the wavefunctions coming into a scatterer from left and right as a1 (k), a2 (k) respectively, and the same for outgoing ones
as b1 (k), b2 (k). The S-matrix representing the scatterer is defined as
(
)
(
) (
)(
)
b1 (k)
a1 (k)rL tR
a1 (k)=S=,b2 (k)a2 (k)tL rR
a2 (k)(8.11)

where tL,R , rL,R are complex transmission and reflection ratios from left and right respectively. They bare phase shifts
occurring at the scattering in their complex phases. Here we adopt the lower case expression for the “wavefunction flows”
in order to distinguish them from Ai (k) etc. so far used because the directions of the flows are different by definition.
There are the relations to transmission and reflection probabilities TL,R , RL,R as
TL,R = |tL,r |2 = 1 − RL,R = 1 − |rL,R |2 .

(8.12)

Unlike T-matrices, S-matrices cannot have the output as the next input because the channels are mixed in the operand
vectors. On the other hand, as seen in Eq. (8.11), each element has clear physical meaning and the parameters of thescatteing can be readily extracted.
In the above, in a sense, we have considered a connection of two channels with the same wavenumber. If we consider
the extension to more general cases, we need to take care that each channel has different wavenumbers, dispersions. Evenin the simplest case of a QPC, when it is on the plateau of G = n × 2e2 /h, it has n conduction channels and the Fermi

A1

A2

a1

MT
B1
(a)

a2
S

B2

b1

b2

(b)

Fig. 8.5 (a) Conceptual diagram of T-matrix MT . (b) Conceptual diagram of S-matrix S.

E10-13

wavelengths are different for different channels. In such a case, we cannot simply use wavefunctions for a1 (k). Instead,we writeai (k) =√vFi ψai (kF ),(8.13)

where ψai (kF ) is the wavefunction (the same for bi ). Under this definition, the norms of input/output vectors represent thestrengths of “probability density fluxes.” We call t as a comlex transmission probability and |t|2 = T is (real) transmission
probability. Then in this way, we can call ai (k) etc. in (8.13) as complex probability flow.

8.3.1 Connection (joint) of S-matrices
For the series connection of T-matrices, as we did in the double barriers, we can simply take the product of them, which
procedure simplifies the calculation and saves the trouble. On the other and for the series connection of S-matrices, as in
the figure shown below, the eight lines for input/ouput should be in cross connection and the results should be expressedin terms of a new S-matrix. For the calculation we first write( )( ) ( (A) (A) ) ( ) ( )( ) ( (B)
b1
a1
rL
tR
a1
b3
a3
rL
= SA
= (A)
,
= SB
= (B)
(A)
b2
a2
a
b
a
2
4
4
tL
rR
tL

(B)

tR
(B)
rR

)( )
a3
.
a4

(8.14)

By using the boundary conditions

a1

a2

a3

a4

SA
b1

b2 = a3 , a2 = b3 ,

SB
b2

b3

b4

(8.15)

we drop these variables from the final simultaneous
equations, to get the single S-matrix. The result is thefollowing S-matrix SAB .



)−1
(A)
(A) (B)tL
I − rR rL
(
)−1
(A)
(B)
(A) (B)tLtL
I − rR rL((A) (B)(A)
rL + tR rL

SAB = 


)−1
(
(B)
(B) (A)tR
I − rL rR

(
)−1
.
(B)
(B)
(A) (B)(A) (B)rR + tLI − rR rL
rR tR
(A)
tR

(8.16)At the first sight, it looks just complicated and you may wonder why we need to take such a way for calculation.
However, the expression shows the behavior of wave propagating over two series scatterers. To see that we take out (1,1)
element of Eq. (8.16) and expand the second term as((A) (B)I − rR rL)−1(A) (B)(A) (B)(A) (B)= I + rR rL + (rR rL )2 + (rR rL )3 + · · · .

(8.17)

This clearly shows that the second term is the summation of the processes,

b1 a1each of which is a reflection including multiple reflections between thetwo scatterer A and B. Elements of S-matrices have clear meanings as in
Eq. (8.11) and are easy to be interpreted. And because the inputs and theoutputs are separated, we can easily apply them for multiple channels ora2b2electrodes.
In the above, we did not consider the evolution of phase when the wave

S

propagetes between the scatterers. This can be taken into account by inserting T- or S-matrix to express the phase rotation. With this we can treat
the case that the transmission line works as a resonator.

a3

b3

aj

bj

We have redefine the input/output as in Eq/ (8.13), and expansion tomulti-channel systems can be done with increasing the dimension of input/output vectors. When we write down (8.17), the care is taken for theE10-14

order of product, the denomination is expressed as the multiplication of the inverse, and 1 is expressed as I. These arefor the expansion to multi-channel systems with converting a1 etc. to vectors, r (A)r etc. to matrices.
We do not consider any interaction (scattering) between the channels on the transmision line. Therefore, in the case ofmultiple channel, assignment of the channels to the transmission lines is not a crutial problem in the S-matrix treatment.
Hence, as illustrated in the figure, the wire connection can be done regardless of the lines. In that sense, S-matrices are“nodes” of the lines. T-matrices shceme is not so easy to be applied to channel multiplication. We utilize both methods
to treat the electron waveguide circuits.

8.4 Onsager reciprocity
An important property of S-matrices is the unitarity. From the definition of complex probability flow in (8.13) andthe conservation of probability requires |a|2 = |Sa|2 . Then it is almost trivial that S-matrices should be unitary. From
the unitarity, a very important property in the symmetry called Onsager reciprocity is derived. The Onsager reciprocity,which holds generally in the transport phenomena, is expressed in the form of S-matrix as
S(B) = tS(−B) (Smn (B) = Snm (−B)),(8.18)

where B is the external magnetic field.
The derivation is as follows. The problem here is essentially the potential scattering described by the Schrödinger
equation[](iℏ∇ + eA)2
+ V ψ = Eψ.
2m(8.19)

We take the complex conjugate of (8.19) and revert the direction of the magnetic field with A → −A to get][(iℏ∇ + eA)2
+ V ψ ∗ = Eψ ∗∴ {ψ ∗ (−B)} = {ψ(B)}.
2m(8.20)

This means ψ(B) and ψ ∗ (−B) forms the same set of solutions (here {· · · } means the set of · · · ). Remember that ψ(B)is a scattering solution of Schrödinger equation (8.19). Let us express a scattering state as Sc{a → b}(a is the incoming
wave to the S-matrix, b is the scattered wave).
Sc{a(B) → b(B)} ∈ {ψ(B)},i.e.,(8.21)

b(B) = S(B)a(B).

(8.22)

We take the complex conjugate of (8.22) asb∗ (B) = S ∗ (B)a∗ (B).

(8.23)

Now to take the complex conjugate of a propagating wave exp(±ikr) corresponds to the inversion of direction of propagation*5 . That is, by taking the complex conjugate, the incoming wave and the scattered wave are exchanged.
Sc(b∗ (B) → a∗ (B)) ∈ {ψ ∗ (B)}∴B → −B results ini.e.

From (8.26)∗∗(8.24)∗

Sc{b (−B) → a (−B)} ∈ {ψ (−B)} = {ψ(B)}∗∗a (−B) = S(B)b (−B).
b∗ (B) = S −1 (−B)a∗ (B),

(8.25)(8.26)(8.27)*5 Schrödinger equation (8.19) does not depend on time, and then taking the complex conjugate of means the sign reversal of ikr keeping the signof iωt.

E10-15

and the comparison with (8.23) givesS ∗ (B) = S −1 (−B) = S † (−B) (∵ unitarity SS † = S † S = I)
∴ S(B) = t S(−B).

(8.28)Q. E. D.
The following simple symmetric property is derived for the case of so far discussed two-terminal transport, in which
the system only has single inlet and single outlet, and the resistance (ρxx ) is defined as the ration of the voltage drop
between the electrodes to the current.
ρxx (B) = ρxx (−B).

(8.29)In the above proof, the linearity of the transport coefficients is assumed. Hence in non-linear devices, the reciprocity
is broken under finite bias. Even for the non-linear devices, if the I-V characteristics is symmetric to the origin, thereciprocity recovers with including reversing the bias.

8.5 Landauer-Büttiker formula
So far we have treated coherent transport in two-terminal conductors. As in the S-matric scheme, experiments ofcoherent transport can be seen as a kind of scattering experiments. The terminals correspond to the detectors catching
the scattered wave, and the number of terminals can be larger than two in general transport measurements. The scattering
theory, which treats many terminals with equal footings, is the Landauer-Büttiker formalism.
Let us index the terminals with p, q (Fig. 8.6). Terminal p is connected to the particle reservor which has the chemical
potential µp = −eVp . The net current Jp which flows from terminal p into the sample is obtained as follows. We
consider the sum of the electron fluxes times −e flowing into p from other terminals to p. And we subtract the sum fromthe electron flux times −e flowing from p to the sample.
Jp = −2e ∑[Tq←p µp − Tp←q µq ].
h q(8.30)

With expressing Tp←q etc. in the form of matrix T as1
m1 = -eV1
J1

Tpq ≡ Tp←q (p ̸= q),∑Tq←p ,q̸=p2
J 2 m2 = -eV2

mq = -eVqq

Tpp ≡ −

and with writing J =t (J1 , J2 , · · · ), µ =t (µ1 , µ2 , · · · )(column vectors), we can express

Jq

J=

Sample

2e
T µ.
h

Also
2e2
µq
,
Gpq ≡
Tp←q とおくと
−e
h
∑
Jp =
[Gqp Vp − Gpq Vq ].
(8.31)

Vq =

Jp

p

m p = -eVpq

The above is the essence of Landauer-Büttiker formal-

Fig. 8.6 Model to derive LB formalism.

ism but it still needs some constraints as below.
First, the current conservation tells that∑

Jq = 0.

q

E10-16(8.32)

Next, when all the terminals are at the same potential, the currents should be zero, i.e.
∑
[Gqp − Gpq ] = 0.

(8.33)q

Further, for the external magnetic field B, the Onsager reciprocity
Gqp (B) = Gpq (−B)(8.34)

holds. This can be proven from the Onsager reciprocity of S-matrix. The above is Landauer-Büttiker formalism (LB
formalism) of electron transport.
Let us apply the LB formalism to a sample with four terminals. We take the origin of energy so as for the fourthchemical potential to be zero, i.e. µ4 = −eV4 = 0. Then we can write down
  
 
J1
G12 + G13 + G14
−G12
−G13
V1
J2  =  V2  .
−G21
G21 + G23 + G24
−G23
J3
−G31
−G32
G31 + G32 + G34
V3(8.35)

Now we consider the boundary condition
J1 = −J3 ,J2 = −J4 ,(8.36)

which is called Casimir problem. The problem is reduced to ordinary situation of four probe measurement with J2 = 0 inthat the current is flowing through 1 and 3 while the voltage between 2 and 4 is measured without current. With writing
Vij ≡ Vi − Vj , the solution of this problem is given as
( ) (
J1
α11
=
J2
−α21

−α12
α22

)(

)
V13
,
V24

(8.37)

where
α11 = 2Gq [−T11 − S −1 (T14 + T12 )(T41 + T21 )],−1(8.38a)(T12 T34 − T14 T32 ),(8.38b)α21 = 2Gq S −1 (T21 T43 − T23 T41 ),(8.38c)α12 = 2Gq S

α22 = 2Gq [−T22 − S−1(T21 − T23 )(T32 + T12 )],S = T12 + T14 + T32 + T34 = T21 + T41 + T23 + T43 .

(8.38d)(8.39)In Eq. (8.37), the current is expressed with the voltages, but in real experiments often the current is given by the external
circuit and the voltages (chemical potentials) V1 ∼ V3 are rearranged to fulfill the condition (8.36).
The reciprocity (8.34) gives the constraint
α11 (B) = α11 (−B),

α22 (B) = α22 (−B),

α12 (B) = α21 (−B)(8.40)

to the solution (8.37). We apply this to ordinary four-termial problem and assign 1 and 3 to the current probes, 2 and 4 tothe voltage probes and write the resistance obtained from LB formalism as R13,24 . Then we see
R13,24 =V2 − V4α21=,J1α11 α22 − α12 α21(8.41)

which does not show the symmetry to the magnetic field inversion like (8.29) though each matric element fulfills theOnsager reciprocity. On the other hand, the resistance measured with current-voltage exchanged terminals is
R24,13 =α12,α11 α22 − α12 α21(8.42)

which is, from Eq. (8.40) symmetric to the reversal of magnetic field.
Generally from
Rmn,kl = Rq

Tkm Tln − Tkn Tlm,
D
E10-17

D ≡ Rq2 (α11 α22 − α12 α21 )S,(8.43)

the reciprocity
Rmn,kl (B) = −Rkl,mn (−B)(8.44)

holds. The minus sign is just due to the order of terminals.
The above results bring intresting information in measuring magnetoresistances of four terminal samples in quantum
coherence. That is, generally in four terminal measurement, the magnetoresistance is not symmetric to B = 0 (ρ4t (B) ̸=
ρ4t (−B). However, if we exchange the set of voltage probes and that of current probes and reverse the field direction
B → −B, then the resistance is unchanged.

References[1] W. N. Can, IEEE Trans. Electron Dev., ED-12, 531 (1965).
[2] Y. Narukawa et al., Jpn. J. Appl. Phys. 46, L963 (2007).
[3] 本格的に学ぶには例えば，R. Loudon, “The Quanutm Theory of Light” (3rd ed., Oxford, 2000); P. Meystre and M.
Sargent III, “Elements of Quantum Optics” (Springer, 1990); 松岡正浩 「量子光学」 (裳華房, 2000) など．
[4] 早川尚男 「非平衡統計力学」 (サイエンス社，2007).
[5] S. Datta, “ElectronTransport in Mesoscopic Systems” (Cambridge Univ. Press, 1995).
[6] 勝本信吾 「メゾスコピック系」 (朝倉書店，2002)

E10-18

Lecture note on

Physics of Semiconductors (11)23rd June (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

8.5.4 Aharonov-Bohm ring
As an application of S-matrix, we consider the transmission coefficient of an Aharonov-Bohm (AB) ring. The channel
configuration is shown in Fig. 8.19(a). We write the S-matrix for the two junctoins with three channels as *1
√
√ 

0√
−1/ 2 −1/ 2
St = −1/√2
1/2−1/2  .
−1/ 2 −1/2
1/2
The AB phase is taken into account by inserting
(
)
0
eiθAB
SAB = −iθAB,e0θ ≡ 2πϕe= ϕ (ϕ is the flux piercing the ring)ϕ0ℏ(8.51)(8.52)

into one of the parallel paths. We insert the S-matrix
(
Sw =

0

eiθ0

eiθ0
0

)
(8.53)

into the counter arm to express the phase difference between the two paths. The phase shift θ0 from the path difference
does not depend on the direction of the propagation while the sign of θAB is inverted with inversion of propagation. The
Onsager reciprocity (8.18) is kept with these mathematical settings.
From the total S-matrix, the complex transmission coefficient of the ring is obtained as[3]t=

4 sin θ0
.
iθ
iθ
AB
1+e
(e AB + eiθ0 − 3e−iθ0 )(8.54)The transmission coefficient T = |t|2 shows AB oscillation of the period ϕ0 in ϕ (the magnetic flux piercing the ring). T
also oscillates versus θ0 with the period of 2π. |t|2 is symmatric for ϕ = 0, which is due to the reciprocity induced on(8.54) by the Onsager reciprocity introduced into S-matrix (8.52).
The phase of the oscillation with period ϕ0 varies on θ0 as a rectangular wave with amplitude π. The amplitudeof the oscillation disappears around the phase jumps, which does not mean the disappearance of the magnetoresistanceoscillation and the ϕ0 /2 period and higher frequency components increase in the amplitudes. As above, the ϕ0 -oscillation
only takes the phase offset of 0 or π, which property is called “phase rigidity[4].” The phase rigidity means that we cannot
detect the phase shift over the quantum dot inserted into one of the arms of an AB ring*2 ．

8.6 Qauntum transport and particle statistics
As a transport phenomenon in a semiconductor, the transport of electrons (charge and spin) is often considered, butsome quasiparticles baring transport behave differently from electrons. We will have a look how we apply the quantum
transport theory (or not apply).
*1 This form is frequently adopted though it is completely symmetric and a bit special in that sense.
*2 If we consider multiple conductance channels and also restrict the region of magnetic field, the phase looks smoothly changes with flux[5]though this does not mean the breaking of the Onsager reciprocity.

E11-1

a4

b2
S AB

a 1 =1

St

b1

a2
b3

b4
a5

2p
b6

St

Sw

a 6 =0

b5

a3

0

(a)

(c)

-2p
-1

0
f/f0

1

2.8
2.2

0
1

|t|2

1.6

q0

1.0

f/f0
(b)

2p 0

q 0=0.4
(d)
-1

0
f/f0

1

Fig. 8.19 (a) S-matrix modeling of an AB ring. (b) The transmission coefficient of the AB ring |t|2 is plotted (surfaceplot) as a function of the phase shift from the path difference (θ0 ) and the magnetic flux piercing the ring ϕ/ϕ0 . (c)Color plot of the same calculation over a bit wider region. (d) The same transmission coefficient as a function of
ϕ/ϕ0 with θ0 as a parameter. The AB oscillation of period ϕ0 once disappears around θ0 = 1.6 and then the reverted
oscillation, that is, with π-shift in the phase appears.

8.6.1 Bunching, anti-bunching
In the previous section, we have introduced the Landauer formula to treat the electric conduction in semiconductor
quantum structures. In the discussion, we assumed the transport of electrons and used the unit charge and the Fermi
distribution in the derivation. And the conductance quantization is derived from the anti-bunching of fermions on quantum
wires. On the other hand, in order to calculate the transmission coefficient Tij , we have introduced T-matrices and Smatrices scheme. These are only to calculate the transmission and reflection of waves without the relation to the particle
statistics. Hence the method should be applicable regardless of particle statistics.
Let us have a look on bunching and anti-bunching properties. The wavefunction for identical two particls is in real
coordinate representation as1
ψ(r1 , r2 ) = √ [ϕ1 (r1 )ϕ2 (r2 ) ± ϕ1 (r2 )ϕ2 (r1 )].
2
In the double sign, + corresponds to bosons, and − to fermions.

E11-2(8.55)

Fig. 8.20 (a) Schematic probability densities in coordinate representation |ϕ1 (r)|2 , |ϕ2 (r)|2 of one-particle wavefunctions. The two wavefunctions have a partial overlap. (b) In the case of (a), the spatial probability density ofparticle 2 is plotted as a function of coordinate r2 with taking the position of the particle 1 r1 as the origin in the
two-particle wavefunction ϕ in (8.55).

As shown in Fig. 8.20(a), we consider gaussian shaped wavefunctions ϕ1,2 (r) with partial overlap. Figure 8.20(b)shows the probability density of the two-particle wavefunction versus the relative position. As we can easily see by
putting r2 = r1 , the probability densities are{
2|ϕ1 (r1 )|2 |ϕ2 (r1 )|2|ψ(r1 , r1 )| =02

(boson),(fermion).

(8.56)

That is, in the case of bosons, the probability density is twice the case of single particle while in the case of fermions thedensity is zero. This means that the bunching occurs for bosons while the anti-bunching occurs for fermions.
The discussion of (8.3)∼(8.5), which leads to the Landauer formula can be understood in this context of anti-bunching.
That is, the difference in the chemical potentials of two electron reservoirs, eV is the energy window to be used to form
the wave packets of elctrons i.e., ∆E ∼ eV . The time for such a wave packet to go through a point in the real spaceis, from the uncertainty relation, ∆t = h/∆E = h/eV . From the anti-bunching property or Fermi statistics, a single
wave packet can accomodate a single elecrtron (if the spin degree of freedom is taken into account, two electrons). Thenthe current flowing though the one-dimensional system is J = e/∆t = (e2 /h)V (with spin freedom, (2e2 /h)V ), which
results in the same conclustion in the previous section.
The above discussion is the same calculation of that done in the k-space though it gives an important insight of theflow of electrons on quantum wires. When the conductance is quantized in such a quantum wire, the electrons flow withthe time interval of h/eV . As is describe in the shot noise section of Appendix 8A, the current flow is approximatedas a periodic series of delta-functions and the shot noise disappears. In the Landauer’s discussion, the conductancequantization is the consequence of the fermion’s anti-bunching property and the above consideration means that can beexperimentally confirmed through shot noise measurement.
We expand the above to quantum wires with transmission coefficients T less than 1 to get G = T Gq . In this case,there appear free spaces between the wave packet due to the electron reflection and the packing of electrons becomes
stocastic to some degree. This states can be viewed as follows. A perfectly ordered series of wave packets are occupied
by electrons with probability T , by holes with probability 1 − T . Identical electrons (also holes) cannot be distinguishedand the number of cases for vacancies, i.e. the degree of randomness is proportional to T (1 − T ). In the limit T → 0,
this goes to Eq. (8A.3) and with using the relation J = 2T Gq V for voltage V , the noise power spectrum is given as[?]S≡⟨(δJ)2 ⟩2e2= 2e
V T (1 − T ).
∆fh(8.57)

The above S is suppressed from SPoisson in Eq. (8A.3) by factor 1 − T . Generally, we refer to Fano factor as the ratioof variance to the average. In the present case that corresponds to the ratio of the shot noise to the Poisson noise and is
1−T.
E11-3

Fig. 8.21 Shot noise measured on a quantized conductance plateau of a QPC, and at around transition regions. (a)QPC conductance as a function of the gate voltage. Positions for noise measurements are indicated by color-framed
open symbols. (b) Shoit noise measured at the three points indicated in (a) as a function of the source-drain voltage.
(c) Fano factor (red circles) as a function of the conductance of the QPC. The blue line is obtained from a simple
model, in which Fano factor should be 1 − T . From [6].

Figure 8.21 shows an example of shot noise measurement of a QPC. In panel (a), the conductance is shown as afunction of the gate voltage and the current noise (spectral density of the square of current fluctuation) were measured at
the three points indicated by color-framed symbols (on a conductance plateau and at neighboring transition regions). In
panel (b), the current noise is plotted versus the source-drain voltage. In the transient regions, the noise increases withthe voltage indicating the appearance of shot noise while around the center of plateau, the increase of noise is very small,indicating the reduction of noise. In panel (c), the noise data are converted to Fano factor and shown as a function of the
conductance. The blue line shows the consequence of a simple model, in which the Fano factor should be 1 − T . The
data distribute a bit above the model line confirming the noise reduction with the conductance quantization.

8.6.2 Transport of exciton-polaritons
As a bosonic quasiparticle, we consider exciton-polariton (E-P), which is introduced in Sec. 4.4.2. An E-P is a composite of photons and excitons created as a result of strong coupling of light and matter. Being pairs of fermions, excitons
obey Bose statistics but the effective mass is the sum of those for electrons and holes as me + mh . On the other hand,as can be seen from the dispersion relation in Fig. 4.7, an E-P has a very small effective mass around k ≈ 0. This makes
the control of phase of E-Ps easier and the reseachers are trying to apply E-Ps for optical integrated circuits. The lighteffective mass makes the critical temperature of BEC (8B.10) very high. Actually, the observations of BEC have beenreported.
8.6.2.1 Cavity exciton-polariton
In the section of laser diode, we have seen a structure of two-dimensional cavity. In Fig. 8.22(a) we show a transmission
line made by cutting the two-dimensional cavity into a thin mesa structure. Here, the structure is such that GaAs is used asa quantum well, which is sandwiched between GaAs / AlGaAs superlattices (SL), and a GaAs clad layer is placed on theoutside. The effective refractive index of the SL part is lower than that of GaAs, and photons are confined in this region.
On the other hand, excitons are confined in the central GaAs quantum well because the SL regions work as barriers dueto the band discontinuity. The excitons in this case are confined in the two-dimensional plane of the cavity and as we saw
in Sec. 7.1.3 the binding energy of the excitons becomes larger and they are stabilized. With the above devicing, E-Ps canpropagate the waveguide but for stable propagation, we need to prepare some low temperature environments. The limit oftemperature is extimated from the gap between the upper and the lower branches in the dispersion relation in Fig. 8.22(c)and that can be go over the liquid nitrogen temperature.

E11-4

reflactive index

10 m mphotonElectrode

GaAs/AlGaAs SL
GaAs quantum well
GaAs/AlGaAs SL
GaAs clad

potential

GaAs clad

exciton(b)(a)

Fig. 8.22 Example of transmission line for excitonpolariton. (a) Cross sectional view of the transmission line, in which a GaAs quantum well is inserted
into AlGaAs/GaAs superlattices. (b) Upper: Confinement of photons with distribution of reflactive index. Lower: The cladding superlattice parts form
the confinement potential. (c) Dispersion of cavity
exciton-polarioton in a different cavity from (a) and(b). From [10].

(c)

The dispersion relation in Fig. 4.7 is made up from photons in crystals and excitons. In the present E-P waveguide,
the photons are strongly confined into the micro-cavity and the photon dispersion relation changed from that in bulk.
Figure 8.22(c) shows the avoided crossing between the cavity photon and the exciton caused by the strong coupling oflight and matter. The structure of cavity here for the calculation of the dispersion is shown in Fig. 8.24.
In the left panel of Fig. 8.23, a conceptual figure of a Mach-Zehnder (MZ)-type interference device composed of thecavity transmission line shown in Fig. 8.22(a). As mentioned in the section of exciton, because the response to the electricfield is opposite for electrons and holes, if is is a comletely single-body composite particle, the effect of the electric field
can be hardly observied. However, in the structure of Fig. 8.22(a), the binding energy of excitons can be varied by theelectric field and with that, the wavenumber varies as][√√
2m(Ek − δE)
2mEk.
∆φ = L−ℏℏ(8.58)

δE represents this variation in the kinetic energy and L is the length of the gate region. This gives modulation in the
output of the two-path circuit shown in the left panel of Fig. 8.23. Finally the output is transposed into light at the edgeof the transmission line and the output can be detected as the light strength. The transmission circuit in Fig. 8.23 is calledin the paper[7], as an MZ interferometer though, because it has a single output line, some reflection exists at the joint, it
should be called as a two-path AB-type (in the present case, the AB phase does not exist and “ring-type” may be a better
expression). As shown in Fig. 8.23(a), (b), with the voltage the light ouput power can be controlled by over 10 dB andvoltage-light switching function is realized.

E11-5

Fig. 8.23 Left panel: Concept of Mach-Zehnder (MZ)-type interference device. (a) Variation in the ouput of MZ
interference device versus gate voltage (4.5 K). (b) The same for 77 K.

Fig. 8.24 Left panel: Schematic view of the cavity transmission line for the observation of exciton-polariton condensation. Upper figure shows the refractive index. Right panel: (a) Calculated reflection coefficient of hte left cavity. (b)Structure of reflactive index in the cavity. The red line is the distribution of electric field along z-axis for the localized
mode showing the sharp resonant dip around the center of the spectrum shown in (a). From [10].

8.6.2.2 Condensation of exciton-polariton
While electrons are the representative of fermions flowing through quantum circuits, as we have seen above, E-Ps
in microcavities is a system, with which we can explore the boson flow through quantum circuits experimentally. The
consequence of Fermi statistics on fermion flow in quantum circuits is the conductance quantization and the reduction ofshot noise. On the other hand, bunching of the identical particles is the characteristics of the Bose statistics, as we haveseen in Sec. 8.6.1. As a result, Bose-Einstein condensation (BEC) or similar phenomenon with condensation occurs.
The stimulated emission is also a phenomenon similar to the boson bunching, and photons in a cavity of a laser can beviewed as a kind of condensation though the lasing occurs in non-equilibrium open systems while BEC is a phenomenonin equilibrium. There is thus a clear difference[8, 9]．
Figure 8.24 shows the diagram of refractive index in the cavity prepared for the observation of BEC. This figure also

E11-6

(a)

(b)
Fig. 8.25 (a) Conceptial diagram illustrating the “cooling” process for micro-cavity E-P to cause BEC. (b) Ditribution of E-Ps in the space of wavenumber and the energy measured from the optical emission. Pth is the criticalexcitation power to create E-Ps with the critical particle density for BEC. from [9].

shows the energy density of electric field along z-axis in the mode localized at the center of cavity, which is calculatedwith T-matrix method. T-matrice and S-matrices are thus used to calculate various quantities related to wave propagations.
A short summary on BEC of three dimensional ideal boson gas is given in Appendix 8B. As the expression of thecritical temperature Tc in Eq. (8B.10), the lighter the effective mass is, and also the higher the particle density is, thehigher Tc becomes. Conversely, when the mass and the temperature are given, the critical particle density for BEC toappear is defined.
Figure 8.25(a) illustrates the process of creation of a BEC with laser light irradiation on the cavity system. In the
beginning of the process, many E-Ps with high energies are excited with the lase pulse. They emit energies as phonons tothe crystal and are cooled down. If the laser power is higher than the critical value Pth , as cooling, a BEC is created anda macroscopic number of E-Ps fall into the lowest energy state. In Fig. 8.25(b), the wavenumber-energy distributions of
E-Ps measured from light leakage, are given around the BEC critical power.
Here we need to be careful about the meaning of “BEC.” The present E-P system is composed of modes confined to a
2-dimensional plane and 3-dimensional BEC in App. 8B cannot be directly applied. In the space with dimension lower
than or equal to 2, no infinitely long range order does not exist as mentioned in Mermin-Wagner theorem[11]. Instead,
Berezinskii-Kosterlitz-Thouless (BKT) transition occurs and the order decays with some power of the distance[12]. Actually, the existence of BKT transition was evidenced in detailed analysis of experiments. And the observation of vortexpair is announced. There are so many reports on the BEC of E-P systems and the research is active both in theory andexperiment.

E11-7

Appendix 7A: Laser diode and waveguide
Here a short supplement on the structure of waveguide for laser diode (LD) is given. Let us consider the Fabry-Pérot
type LD with waveguide (cavity) length L. Let mj , n̄ and λ be an integer, the reflactive index, and the wavelength in thevacuum respectively, then the condition of resonance is
mj

λ
= 2L.
n̄

(7A.1)

Therefore the interval in the resonant wavelengths and that in the resonant frequencies are
∆λ =

λ2
,
2Ln̄

∆ν =

c
,
2Ln̄

(7A.2)

respectively. In usual systems, λ ≪ L. When the amount of carrier injection is large and the luminescence is broad inwavelength, precise determination of l is not requied monstly and multi-mode oscillation around a center wavelenth isobserved.
In the above, we write the light intensity simply as I0 exp(−α′ z). Then α′ can be expanded as I(z) = I0 exp((g−α)z),
where g is the optical gain, α is the material specific absorption coefficient. Let us write the reflection ratio of the twomirrors as R1 and R2 respectivly, then the condition for the amplification to occur isR1 R2 exp[(g − α)2L] > 1.
Thus the threshold optical gain gth for the total amplification is()
11
gth = α + ln
.
L
R1 R2

(7A.3)The reflactive index n̄1 is common for both sides of the homo pn-junction, while the reflactive index in the active layern̄2 is larger than n̄1 . z-axis is taken as in the figure and we consider the electromagnetic wave propagate along the z-axis.
The propagation mode is transverse electric (TE), i.e., the electric field along z axis is absent (Ez = 0). Also the mode isassumed to be uniform in y direction. Thus we only need to consider the elecric field in y direction, which is determined
[

from

]
∂2
∂2
∂2
+ 2 − µ0 ϵ0 ϵ 2 Ey = 0.
∂x2
∂z
∂t

(7A.4)

The change of magnetic permeability in semiconductors from the vacuum is little then we use µ0 here.
p-AlGaAs

GaAs

L

n-AlGaAs

p-AlGaAs

R1

GaAs
R2

n-AlGaAs

Fig. 7A.1 Left panel: Schematic band diagram of a pn-junction for LD. The active layer is non-doped GaAs and thedoping layers with larger band gap than that of the active layer, are AlGaAs. Right panel: Substrate edges are formedby cleaving and work as half mirrors, which make the active GaAs layer into a cavity.

E11-8Normalized electromagnetic
energy density

1.0

Al 0.10.2
dFig. 7A.2 Distribution of normalized electromagnetic energy density of fundamental mode on x (origin at the centerof active layer) in a cavity with an active (i-)layer of GaAs,
The parameter is the content of Al.

0.4

0
-1.0

-0.5

0
x ( m m)0.5

1.0

We treat the system as a waveguide and look for a solution of standing wave on x and propagating wave on z. Then wefind a solution inside the active layer (|x| ≤ d/2) as
Ey (x, z, t) = A cos(κx) exp[i(ωt − βz)],whereκ2 = µ0 ϵϵ0 ω 2 − β 2 = n̄22 k02 − β 2 ,

k0 =

ω
.
c∗ n̄2(7A.5)(7A.6)

And outside the active layer (|x| > d/2), the solution should decay with |x| → ∞. The condition of connection at
x = ±d/2 での Ey restricts the solution to(
Ey (x, z, t) = A cos

κd
2

)

[
(
)]
d
exp −γ |x| −exp[i(ωt − βz)].
2(7A.7)

Equation (7A.4) requires
γ 2 = β 2 − n̄21 k02 .

(7A.8)

From Maxwell equation, the continuity in z-component of magnetic field at x = ±d/2 requires√( )β 2 − n̄2 k 2κdγtan= = √ 2 2 1 0.
2κn̄2 k0 − β 2(7A.9)The values of κ, γ and β are determined from the above equations. Because tangent is a π-periodic function, there are
multiple solutions, each of which forms a descrete mode.

Appendix 8A: Shot noise
We express information as a time-varying physical quantity and transmit it using various transport phenomena. Every
physical quantity has fluctuations and among them time varing ones are called noise *3 . The noise can be classified into
external noise and intrinsic noise. While the former comes from “outside” of the system, the latter is included in thephysics of the quantity itself. Particularly in the case of electric current by electron flow, the representative intrinsic
noises are thermal noise (Johnson-Nyquist noise) that caused by random thermal motion of electrons and shot noise thatoriginates from the particle nature of electrons and the randomness in the flow.
Let us consider first the current by a single electron observed at time tp , expressed as Jp (t) = eδ(t − tp ). From theFourier expansion∫ ∞Jp (t) = e−∞

exp[2πif (t − tp )]df = 2e

∫ ∞cos[2πf (t − tp )]df,(8A.1)

0*3 Fluctuation means the distribution of observed values in multiple identical measurements and the parameter of sampling is not restricted to time.

An example of non-time dependent fluctuation is aperiodic conductance oscillations in disordered mesoscopic conductors(P. A. Lee and A. D.
Stone, Phys. Rev. Lett. 55, 1622 (1985)).

E11-9

we see that the current itself has an amplitude of 2e independent of frequency. In the infinitesimal frequency width df
at frequency f , we take the average ⟨· · · ⟩ over one period. Let us write the integrand in (8A.1) as jp and the current√√
flusctuation is δJp = ⟨jp2 ⟩df = 2edf .
Next we consider an electric current by two electrons observed at tp and tq , Jpq = e[δ(t − tp ) + δ(t − tq )]. In the
Fourier transform of Jpq , there is a phase difference ϕ = f (tq − tp ) between the two Fourier components from the twodelta-functions. The phase difference appears in the square of Fourier transformed function as an interference term:
2
jpq= jp2 + jq2 + 2jp jq cos ϕ.

(8A.2)The interference terms, however, cancel out when we add up many such two-electron currents and take the average√
2 = 2( 2e)2 . A current by many electrons randomized(represented as · · · due to the randomness in tq − tp , i.e. jpq
on time is equivalent to this many sampling. Hence, let N be the time averaged number of flowing electrons then the
averaged current is J = eN and the current fluctuation over the bandwidth ∆f is⟨(δJ)2 ⟩/∆f (≡ SPoisson ) = N × 2e2 = 2eJ.

(8A.3)The square of current fluctuation is proportional to the average of current corresponds to the fact the variance of Poisson
distribution is the average (the number of electrons per unit time N ). This case of complete randomness is called Poissonnoise.
On the other hand, when the electrons flow with a constant interval, there is no fluctuation(timing of sampling would
result in shifts of e in counted charge, but this is not a random variation). This can be understood from Fourier analysisof the current. Let us write the regular series of delta function with interval as τ as δτ (t). Because δτ (t) is a τ -periodic
function, the Fourier series expansion on the region [−π/τ, π/τ ] is possible as follows.
(
)
∞
1 ∑
2π
δτ (t) =exp −in t .
τ n=−∞
τ(8A.4)

Then the Fourier transform is written as
]
[ (
) ]
∫ ∞[ ∑
∞
∞ ∫ ∞
1
1 ∑
2π
−in(2π/τ )tiωt
F {δτ (t)} =ee dt =
exp i ω − nt dt
τ n=−∞ −∞τ−∞ τ −∞
(
)
∞
2π ∑
2π
2π
=
δ ω−n
=
δ2π/τ (ω), (8A.5)
τ n=−∞
ττthat is, it is also a regular series in ω space and there is no continnum spectrum, which is the sign of random variation.
This means the disappearance of shot noise.

Appendix 8B: Bose-Einstein condensation
The Bose-Einstein Condensation (BEC) *4 is called a phase transitoin that is not due to the interaction between freedoms(quantum statistical phase transition). Though phase transitions caused by interaction beween some freedoms can be
intuitively understood, there are different types of phase transitions, in which the transitions are caused as the results ofcompetition between various factors. A representative is BEC.
In the case of bosonic systems, in spite of the absence of “force” betwen the particles, there exists the tendency forthem to occupy the same quantum state originating from their statistical property. Let us see that for the case of two*4 The acronym of BEC is applied to both Bose-Einstein Condensation and Bose-Einstein Condensate. In actual use, the confution is not serious.

E11-10particles. We write a solution of the wave equation for two particles as ψ(x1 , x2 ). For the composition of wavefunctionsof the system Ψ(x1 , x2 ) that reflects the statistical property of bosons, the symmetrization of ψ results in1
Ψ(x1 , x2 ) = √ [ψ(x1 , x2 ) + ψ(x2 , x1 )] .
2

(8B.1)

Hence the probability of finding the system at (x1 , x2 ) is|Ψ(x1 , x2 )|2 =]1[|ψ(x1 , x2 )|2 + |ψ(x2 , x1 )|2 + ψ(x1 , x2 )∗ ψ(x2 , x1 ) + ψ(x1 , x2 )ψ(x2 , x1 )∗ .
2

(8B.2)

This reveals that the last two interference terms intensify the probability of finding the system under the condition of
x1 = x2 . Let us write the de Broglie wavelength as λ and the averaged distance between the particles as l. Then at low
temperatures λ ∼ l, this tendency of bosons makes many of them to occupy the state of k = 0, which behavior leads to
BEC. The above discusstion is expressed as
p2= kB T,
2M

Ek =

∆p ∼

√

∴λ=

M kB Thh∼√.
∆p
M kB T(8B.3)√
λ elongates as 1/ T with lowering the temperature. And with growing of the overlapp between the single particle
wavefunctions makes them undistinguishable and the symmetrization of the wavefunction cause the condensation to theground state in the phase space (r, p). The phase transition to the condensate at a certain temperature is BEC.

8B.1 Bose-Einstein condensation of ideal gas
Let us consider spin 0 ideal Bose gas. For the Bose distributionf (ϵ) =

1(β ≡ (kB T )−1 )e(ϵ−µ)β − 1(8B.4)

we define the point of µ = 0 as follows. At T = 0, from (8B.4) all the particles fall into the ground state, there we defineµ(T = 0) = 0.

(8B.5)

At finite temperatures, let N be the number of particles in the system:N=∑f (ϵi ).

i∫

In the usual case we can writeN→f (ϵ)D(ϵ)dϵ.

(?)

Here the number of particle at the ground state N0 should be
N0 =

1
1
kB T
kB T
∼
=−
→µ∼−
.
e−µβ − 1−µβµN0(8B.6)

If we calculate the particle distribution on this line, for three dimensional ideal gas

Therefore

V m3/2N=√
2π 2 ℏ3

ϵ(k) =ℏ2 k 2
m3/2 V √ϵ.
then D(ϵ) = √2m
2π 2 ℏ3

∫ ∞

√

0

ϵ

e(ϵ−µ)β − 1dϵ =(mkB T )3/2√
V
2π 2 ℏ3

E11-11∫ ∞0(8B.7)√xdx,
ex−α − 1(8B.8)

where x ≡ ϵβ and α ≡ µβ. We write the definite integral term as I(α), then I is
√ ( )
3
x
π
dx =
ζ
∼ 2.6,
x
e −1
2
2

∫ ∞ √
I(0) =
0(8B.9)

which decreases with increasing of the absolute value of α < 0. Then, in this logic, with T → 0 the maximum numberof N determined from (8B.8) goes to zeo. It is apparent that we have dropped something from the counting. That is, ofcourse, the macroscopic number of particles fall into the ground state.
From Eq. (8B.8),√

I(α) =

2π 2 ℏ3 N.
(mkB T )3/2 V

When this excessds (8B.9) at low temperatures the anomaly (increase in the particle number at the ground state.) occurs.
This critical temperature Tc is

[
]2/3
2πℏ2
N
T < Tc ≡.
mkB ζ(3/2)V(8B.10)Here l ≡ (V /N )1/3 is the average distance between the particles and Eq. (8B.10) is interpreted asl=h√∼ λ(T = Tc ).
ζ(3/2) 2πmkB Tc(8B.11)This confirms the statement that the BEC takes place when the average de Broglie wavelength is comparable with theaverage particel distance.
Below Tc , we add the number of ground state particles N0 to Eq. (8B.8):
V m3/2N=√
2π 2 ℏ3∫ ∞√0e(ϵ−µ)β − 1ϵdϵ + N0 .

From Eq. (8B.6), N0 becomes a macroscopic number fro T < Tc , then µ = 0. Therefore[[]( )3/2 ]∫ ∞ √TV m3/2ϵV (mkB T )3/2√
N0 = N − √dϵ = N 1 −I(0) = N 1 −.
N
Tc
2π 2 ℏ3 0 eϵβ − 1
2π 2 ℏ3(8B.12)(8B.13)This is just like a spontaneous magnetization rapidly grows to finite values below the critical temperature in the ferromagnetic transition.
The total energy of the system for T < Tc is calculated as
V m3/2E=√
2π 2 ℏ3

∫ ∞
0

ϵ3/2
dϵ
eβϵ − 1
∫ ∞ここで T < Tc では(8B.14)
√ ( )
3 π
5
x3/2
dx
=
ζ
より
ex − 1
4
2

0
( )(
)
3/2
3
5
m
V (kB T )5/2 .
E= ζ
2
2
2πℏ2

Then the heat capacity at constant volume is calculated as
( )(
15
5
m )3/2
5/2
Cv =ζV kB T 3/2 .
4
2
2πℏ2

(8B.15)(8B.16)

Cv shows a cusp at Tc indicating that this is the phase transition.

8B.2 Bosonic stimulation
Here we have a look at bosonic stimulation for N particles, which is, though, essentially the same as what has been
mentioned on the case of two particles in Sed. 8.6.1. As we have seen, the bosonic stimulation works as if it is a driving
E11-12

C v / Nk B

2

1

0

1
T/Tc

Fig. 8B.1 Specific heat at constant volume of three dimensional ideal Bose gas as a function of temperature. Tc
is the critical temperature of the BEC.

2

force in BEC or laser oscillation. Let us consider a identical boson system the case a particle in state φini gets perturbationand transitions to other single particel state φfin . Now the problem is the difference in the transition probabilities to thestate occupied with N particles and to the empty state. We write the initial state as(i)

ψ+ (r1 , · · · , rN +1 ) = √N∏
1
R̂m,N +1 det(+) {φi (rj )}φini (rN +1 ).
∏(N + 1)N ! l nl ! m=1(8B.17)The symbol det(+) represents permanent, which is obtained by making the signs of all the terms into +. The final(f )state ψ+ is obtaned by exchanging φini with φfin . Let the matrix elements of perturbation Hamiltonian be a, i.e.
⟨φfin |Ĥ1 |φini ⟩ = a.
(f )

(i)

Assuming that φi (i ≤ N ) is orthogonal to φfin , among ⟨ψ+ |Ĥ1 |ψ+ ⟩, number of terms that give non-zero a is
∏
(i)
(f )
(N +1)N ! l nl !. This is equal to the sqare of the denominator in normalization constant. Then finally ⟨ψ+ |Ĥ1 |ψ+ ⟩ =a.
On the other hand, assuming all of φi (i ≤ N ) are φfin , we can write(i)ψ+ = √N∏

1(N + 1) m=1

R̂m,N +1 φfin (r1 ) · · · φfin (rN )φini (rN +1 ).

(8B.18)All of the N ! terms in det(+) are φfin (r1 ) · · · φfin (rN ) and devided by N ! in the denominator of normalization constantto 1. However the final state is(f )ψ+ = φfin (r1 ) · · · φfin (rN )φfin (rN +1 ).

(8B.19)√Then we get ⟨φfin |Ĥ1 |φini ⟩ = a N + 1, and from the Fermi’s golden rule, the transition probability should be N + 1
times larger.

References[1] S. Datta, “ElectronTransport in Mesoscopic Systems” (Cambridge Univ. Press, 1995).
[2] 勝本信吾 「メゾスコピック系」 (朝倉書店，2002)[3] Y. Gefen, Y. Imry, and M. Ya. Azbel, Phys. Rev. Lett. 52, 129 (1984).
E11-13

[4] A. Yacoby, R. Schuster, and M. Heiblum, Phys. Rev. B 53, 9583 (1996).
[5] A. Aharony, O. Entin-Wohlman, T. Otsuka, H. Aikawa, S. Katsumoto and K. Kobayashi, Phys. Rev. B 73, 195329(2006).
[6] M. Hashisaka, Y. Yamauchi, S. Nakamura, S. Kasai, K. Kobayashi, and T. Ono, J. Phys.: Conf. Ser. 109, 012013(2008).
[7] K. Ogawa, T. Katsuyama and H. Nakamura, Phys. Rev. Lett. 64, 796 (1990).
[8] 山本喜久，宇都宮聖子，日本物理学会誌 67, 96 (2012).
[9] T. Byrnes, N.-Y. Kim, and Y. Yamamoto, Nat. Phys. 10, 803 (2014).
[10] H. Deng, H. Haug, and Y. Yamamoto, Rev. Mod. Phys. 82, 1489 (2010).
[11] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966).
[12] P. Minnhagen, Rev. Mos. Phys. 59, 1001 (1987).

E11-14

Lecture note on

Physics of Semiconductors (12)30th June (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

8.7 Single electron effect and quantum confinement
In the end of the chapter, we will have a look on single electron effect and quantum confinement to zero-dimensional
system in quantum dots.

8.7.1 Single electron effect
In the transport through quantum dots the first importantce is on the single electron effect. The single electron effectis in very short, the the electrostatic energy of an electrostatically isolated system changes with adding(extracting) anelectron, and when this increase is larger than the thermal fluctuation, the tunneling of the electron is prohibited. Thiseffect is called Coulomb blockade. The electrostatic energy of a quantum dot is described with a capacitance C of the
dot and the electrostatic energy of charging by a single electron is Ec = e2 /2C, which is finite and even can be large forsmall C. As a first approximation we separates the electronic states into two: states inside the dot and those outside thedot. Hence the number of electrons in the dot takes an integer (descrete value). There are two possible simplest transport
processes of single electrons from a source to a drain: the dot catches an electron from the source then releases one to thedrain, and conversely the dot releases one to the drain first, then catches one from the source.
Let us take the simplest constant interaction model, in which any pair of electrons in the dot has the same (constant)
Coulomb interaction energy U . Then the total Coulomb energy in the dot with N electrons is
EcN =NC2 U =N (N − 1)U
U (N − 1/2)2
U
=
− .
2
2
8

(8.59)The variation in the Coulomb energy with the transition N → N + 1 is
∆E+ (N ) = (N − 1)U.

Vg
m N+1
EF

U

(8.60)

G

m N+2
EF

EF

EF
m N+1

mN
mN
m N-1
(a)

Vg

(b)

m N-1

(c)

Vg

Fig. 8.26 (a) Schematic diagram of chemical potentials in the constant interaction model, in which the chemical
potentials are descrete with the same distance U (the amplitude of two-electron interaction). At zero-bias, when noneof the descrete chemical potentials meets the Fermi level EF in the source and drain, a finite energy is required for anelectron to tunnel, which prohibits the tunneling (Coulomb blockade). (b) When the origin of the descrete chemical
potentials is shifted by the gate voltage Vg and one of them hits EF , the tunneling thus the electric conduction becomes
possible. (c) When Vg is swept, a repetition of processes (a) and (b) results in the series of sharp peaks with a regularinterval in the quantum dot conductance G (Coulomb oscillation).

E12-1

If we ignore other kinds of energy, ∆E+ (N ) should be the electrochemical potential of N -th electron and from Eq./ (8.60)
we see that the electrochemical potentials are ordered with the same distance being proportional to N .

8.7.2 Coulomb oscillation, Coulomb diamond
Let us write the electrochemical potential of the dot with N -electrons µN , and let N0 be the electron number when
the dot is in equilibrium with the electrodes, then µN0 < EF < µN0 +1 . When µN0 is equal to EF (Fermi energy in theelectrodes), electrons can go into or out from the dot with tunneling from the electrodes, then at zero source-drain voltage(Vsd = 0) the electric conductance (G(0)) takes a finite value. When µN0 does not hit EF , the tunnleing of an electron
between the electrodes and the dot requires a finite energy and is prohibited (Coulomb blockade). As in Fig. 8.26(a),(b), that condition of finite G(0) appears with a constant interval. Hence G(0) forms regular peaks for a sweep of Vg as
shown in Fig. 8.26(c), which is called Coulomb oscillation.
The constant interaction model can also be described as a simple circuit model illustrated in Fig. 8.27(a). Here thecharge of an electron is −e.
Q1 + Q2 = −eN,

Q2 = Cg (Vd − Vg ),Q1 = CVd ,(8.61)and the charging energy isE=

1
1
CVd2 + Cg (Vd − Vg )2 ,2
2(8.62)

in which the second term is the integral of the work done by the power source connected to the gate electrode fromvoltage 0 to Vg . When we thermodynamically treat the problem whether the process proceeds or not under the condition
that some system outside automatically provides energy, we need to consider enthalpy, which in the case of the pressureof atomosphere, written as H = U − P V . Here P V , the product of pressure and volume, corresponds to the automaticenergy supply corresponding to the second term in Eq. (8.62). Then from (8.61) and (8.62),
H(N, Vg ) =(N e − Cg Vg )2.
2(C + Cg )(8.63)

If we plot this as a function of Vg , as shown in Fig. 8.27(b), parabollas are lined up corresponding to N and the Coulomb
peaks appear at the crossing points of the prabollas.
Next we consider the case that the gate voltage is fixed, the drain is grounded, and the source voltage is swept. The
simplest model for such situation is shown in Fig. 8.28. At the positions of Coulomb peaks, the topmost chemical
potential of the dot hits the Fermi level in the source and drain electrodes. The number of electrons differs by 1 at (a)and at (c). When Vg is at (b), the current is blocked at zero bias. Then if we increase the source voltage (decrease EF in
the source), and the chemical potential position used for the conduction at (a) goes in between the Fermi levels of source
H(N,Vg)

-Ne
Q2
dot
Q1
C

Vd C
g

N=0

1

2

Vg

Vg
J

(a)

(b)

E12-2

Fig. 8.27 (a) Simple model of the single electron charging in a quantum dot
described with a self capacitance C anda gate capacitance Cg . (b) Enthalpy
H(N, Vg] ) calculated in the model as afunction of Vg .

and drain, a finite conductance appears. Then a conduction appears outside the yellow parallelogram, inside which theconduction is Coulomb blocked. The parallelogram is called Coulomb diamond *1 .
The color plot in Fig. 8.29 shows an example of measured Coulomb diamonds. The sizes of the diamonds are not thesame mostly because of the quantum confinement effect explained in the next section. There also should be the variationin the dot size by Vg and the variation in the effective capacitance. Various other effects should affect the sizes of the
diamonds and conversely from the size we can know various physical propaties of the dot[2].
We see some line structures outside the diamonds, which come from the quantum confinement and the level descreteness (next section). There are also vague tile-like structures outside the diamond, which is called Coulomb staircase anddue to the increase in the number of possible chemical potential levels. In the experiment, we also see that the verticalboundaries have a bit slanted. This is due to the capacitance between the source electrode and the dot. The capacitancemediates some of the electric force lines from the source to the dot.
V sd

(d)

(a)

(c)

Vg
(b)

2

Conductance (e /h)(e)Fig. 8.28 Simple description of Coulomb diamonds. In
the present case the bias voltage is applied to the source
electrode (the drain is grounded). Yellow colored region isa Coulomb diamond (Coulomb blocked region). (a), (c) At
zero-bias condiction, electric conduction occurs with tuning the dot chemical potential through the gate voltage. In(b), the system is out of the above resonance condition andthe conduction is Coulomb blocked. At finite bias voltages
on the source, the conduction appears, in (d) with the use of
the chemical potential position that used in (a), vice versain (e). Finally, the conduction is prohibited in the yellow
colored region.

0.010

Vsd (meV)

0.000
1.0

2

0.03

0.5
0.02

0.0
0.01

-0.5
-1.0
-0.225

Conductance (e /h)

0.005

0

-0.220-0.215-0.210Vg (V)
Fig. 8.29 Coulomb diamond structure appeared in the transport through a quantum dot made from 2DEG at a heterointerface. The abscissa is the gate voltage Vg . The upper panel shows the zero-bias conductance, which shows
Coulomb oscillation. In the lower, the conductance is color plotted on the plane of Vg -Vsd . Clear diamond structures
are observed. The parallel lines outside the diamonds are from the conduction through excited states in the dot.

*1 The parallelogram becomes a diamond for symmetric configuration of voltage sources.

E12-3

I

Source ( n -GaAs)(b)

V

Dot (InGaAs)

(a)
(c)
Gate

N=2
6

Drain ( n -GaAs)
10

2

N=0
0

4

6

-1.5

9

12

-1.0

16

-0.5

DE ( N ) (meV)

Current (pA)

20

4
12

9

2

0

D=0.5 m m

6

4

0Gate Voltage (V)

5

16

10

15

20

20

Electron Number ( N )Fig. 8.30 Coulomg oscillation in a vertical type quantum dot. (a) Dot current as a function of the gate voltage. Thereis no Coulomb peak for further negative Vg than the region indicated as N = 0. The inset illustrates the sample
structure. (b) Distances between the Coulomb peaks as a function of the electron number. The data are from [5].

8.7.3 Quantum confinement
Next we consider the case we cannot ignore the descreteness of the orbital energy due to quantum confinement. We
number the orbital energy levels from the ground state (different numbers are assigned to all the degenerate states). Let
the energy of i-th state be ϵi . We also ignore the terms that do not have relation with N . Then enthalpy H is
H(N ) =(N e − Cg Vg )2+ ϵN .
2Cs(8.64)

The crossing points are obtained as
)
{(
}
e
1
e − Cg Vg + ∆ϵN∆H(N, N + 1) = H(N + 1) − H(N ) =
N+
Cs
2
{(
)
}
1
1
Cs
VgX (N, N + 1) =
N+
e+
∆ϵN ,
Cg
2
e

∆ϵN ≡ ϵN +1 − ϵN
(8.65)

which has a shift from the Coulomb peak position ∆ϵN as in Eq. (??). From the shift we can get the energy spectrum inthe quantum dots. This method is called addition energy spectroscopy. Because in the case of degeneracy, ∆ϵN = 0 andfrom the position of Kramers degeneracy, we can perform quantitative spectroscopy with this as a standard.
Let us have a look on a famous example, in which the researchers realized a two-dimensional harmonic potential. In
this experiment, a two-dimensional quantum well was inserted into barrier layers and metallic doped “electrode” layers(source and drain) were placed at the top and the bottom of a cylindrical specimen (vertical type quantum dot). The
confinement along vertical direction is strong and we only consider the ground state for this direction. Figure 8.30(a)shows the Coulomb oscillation and there is no peak in the left side (negative Vg ) of a small peak at about −1.6 V, which
fact indicates that the dot is empty in this region. Then we can assign N = 0 to this region and then we can also assign
the other number of electrons to blockade regions.
We review two-dimensional harmonic oscillator shortly. The two dimensional coorinate is take to xy. The in planeconfinement potential V (x, y) and the descrete eigenenergies can be written with the parameter ω0 representing thestrength of the potential as
V (x, y) =mω02 2(x + y 2 ),2Enh = ℏω(nh + 1) (nh = 0, 1, 2, · · · ),E12-4

(8.66)

which have an equidistance. Bound eigenstates of an isotropic potential can be indexed by the quantum number of angularmomentum l and the radial quantum number nr . In the above case nh = 2nr + |l|. The number of possible combinations(nr , l) is nh + 1, then with spin degeneracy, Enh has 2(nh + 1) fold degeneracy.
As a simplest analysis of the data in Fig. 8.30(a), the peak intervals are plotted versus the number of electrons N in
Fig. 8.30(b). Clear peaks are observed at N = 2, 6, 12. This reflects the fact that the bound states in two dimensional∑nhharmonic potential take shell structures at j=02(nh + 1) = (nh + 1)(nh + 2). In the simplest model, the shift of
Coulomb peaks should correspond to ℏω0 , hence the peak height in Fig. 8.30(a) should be common. In the experimental
data, however, there is a strong tendency that the peak interval decrease with the number of electrons. The tendency is
considered to be mainly due to the increase in the effective capacitances. We also see small peak structures at the middleof the clear peaks N = (nh + 2)2 (4,9). This comes from Hund’s rule, which tells that the states should be occupied
by electrons as to maximize the total spin due to the exchange energy. With quantum dots we can perform experiments
knowing the number of electrons and informaiton on potential, quantum dots are sometimes called as “artificial atoms.”
Let us see the effect of magnetic field vertical to the 2d-plane. Here we ignore the Zeeman effect. The effect ofmagnetic field on the orbitals appears in two terms in Hamiltonian. First is the inner product of angular momengum l
and the field flux density vector. Second is the confinement into two-dimensional harmonic potential due to the cyclotron
motion. The second effect modifies the effective confinement potential as√mΩ2 2
Veﬀ (x, y) =(x + y 2 ),Ω ≡ ω02 + (ωc /2)2 ,2(8.67)

where ωc = eB/m is the cyclotron frequency for magnetic flux density B. Then the energy corresponds to (nr , l) is
E(nr , l) = ℏΩ(2nr + |l| + 1) + ℏωc l/2.

(8.68)For general finite fields the orbital degeneracy is lifted by the angular momentum. The eigenstates with energies in (8.68)
are called Fock-Darwin states. The energies in Eq. (8.68) vary with magnetic field as plotted in Fig. 8.31(a). We write
12

10

22
20

-0.5

2nd

18
16

8

14
V g Peak Position (V)E ( n r, l )/ h w

1st

6

4
N=12

12
10
-1
8
6
4
2

2
-1.5

Lowest Landau
N=0

0

1

2

3

4

5

0

w c /w

1

2

3

Magnetic Field (T)

(a)

(b)

Fig. 8.31 (a) Energy levels of Fock-Darwin states (from the ground state to 10th excited state at zero field) as a function of magnetic field. They converge into Landau levels at high fields. Line colors are assigned from corresponding
Landau levels. Thick red line is the trace of ground state for electron number N = 12. (b) Coulomb peak positions of
the quantum dot in Fig. 8.30 versus vertial magnetic field. Black dots are calculated from Eq. (8.69), which represents
the position of last crossing point for fixed N . The potential parameter ω is determined from the peaks N = 3 ∼ 6.

E12-5

nL ≡ nr + (|l| + l)/2 and take the limit B → ∞, to obtain E(nr , l) → ℏωc (nL + 1/2). That is, they converge into
Landau quantized levels.
As shown in Fig. 8.31(a), the levels depend on magnetic field with many crossings. The ground state of electrons with afixed number is given by packing electrons from lower levels. The line of topmost level accommodating electrons shouldhave kinks at such crossings. In Fig. 8.31(a), one of such a line is indicated as a thick red line for N = 12. The seriesof kinks ends up at the field where all the electrons are accomodated into the states corresponding to the lowest Landaulevel. The last crossing is between the line going to the lowest Landau level for N and the line for (nr , l) = (0, 1).
Because of the spin degeneracy (ignoring Zeeman splitting), the former is given as (nr , l) = (0, −int(N/2)) (int(x) isthe largest integer smaller than or equal to x). This condition is given as2ℏΩ + ℏωc /2 = ℏΩ(int(N/2) + 1) − ℏωc int(N/2)/2( ω )21c∴= int(N/2) − 2 +.
ωint(N/2)(8.69)In the first approximation the Coulomb peak distance is constant and ignoring the last term for large N , the last crossing
points depends on magnetic field prabolically. Actually such behavior is observed in Fig. 8.31(b). If we determine ω tofit Eq. (8.69) to kinks of N = 3 ∼ 6, and put dots to the predicted end points of kinks, they agree nicely with the data up
to N = 14.

8.8 Quantum dots and quantum circuits
Quantum dots (QDs) can be connected with quantum wires to form quantum circuits. A QD affects the circuit conductance though the transmission probability and the phase shift as characteristics of resonant scattering. Here the effectof single electron charging is only on the positions of chemical potentials for resonant scatterings. Hence in the simplest
approximation, QDs are treated simple resonators and we do not consider the single electron effect explicitly.

8.8.1 Quantum dot and scattering experimentTransport in mesoscopic systems can be viewed as scattering experiments in solids. We can see that clearly in quantum
circuits

QDs.

with

For example, in the scattering of electrons with an atom, the Fano5resonance occurs as the interference between the incident wave with
q=2

continuous energies and the wave scattered by discrete atomic lev-

4

T (ε)

els. And the same effect is observe in circuits with QDs. In the Fano
effect, the scattered wave gets rapid phase shift by π over the reso-

3

nance position and the interference results in distorted lineshape in
2

1

0
-10

1

the resonance. There the energy dependence of transmission coeffi-

0.5

cient is given as

0

-5

0

ε =(E−E0)/Γ

T (ϵ̃) ∝510(ϵ̃ + q)2
E − E0, ϵ̃ ≡.
2ϵ̃ + 1Γ(8.70)

Here q is called Fano parameter, which determines the lineshape asin the left figure. The larger absolute value in q results in the larger
asymmetry and q = 0 gives a symmetric dip (anti-resonance).

Here we do not go into the derivation of (8.70)[2]. Rather we modelize the circuit with S-matrices and obtain thelineshape numerically. One-dimensional band of quantum wires are assigned to “continuum states” and quantum confined
E12-6

d

f/f 0

ikd

0
e
ikd
e
0

1

0.5
p/4

Phase shift

Transmission coefficient

p/2

1

0.5

0

-p/4

0

2

3

k

-p/2

4

(a)

0
(b)

k (corresponds to gate voltage)

5

k

1

(c)

Fig. 8.32 (a) Upper: Double barrier model. Lower: Transmission coefficient (red) and phase shift (blue) of themodel in the upper panel. k corresponds to gate voltage. The reflaction coefficient of a barrier is 0.7. (b) Colorplot of conductance of the system shown in the right upper inset (AB ring+QD) versus plane of k (gate voltage) andmagnetic flux ϕ piercing the ring. The conductance is higher for black → red → yellow. In the AB ring model in
Fig.8.19(a), Sw is replaced with the S-matrix obtained in (a), and a finite transmission coefficient is introduced into
SAB . Reflection at the dot barrier is 0.7 and that in the reference arm is 0.82. (c) Transmission of (b) is plotted versus
k for ϕ/ϕ0 =0, 0.01, 0.19, 0.29, 0.38, 0.48 from down to up.

discrete states in a QD are assigned to “discrete states.” To have interference between incident wave and scattered wave,
the incident wave is devided into two, one of which goes through a QD and the ohter directly goes to the outlet. A QD is
formed as a one-dimensional double barrier structure. The model is described by S-matrices as in Fig. 8.32(a). That is,the barrier and the dot S-matrices are
(
Sb =

cos θ
i sin θ)i sin θ
,
cos θ

(
Sd =

0
eikd

)
eikd
.
0

(8.71)

Here k is the wavenumber representing kinetic energy, which corresponds to a gate voltage. In this model, the transmission coefficient and the phase shift are calculated from the composite S-matrix as shown in Fig. 8.32(a), where π phaseshift at the resonance peak is clearly observed. This is common for resonance. Resonance is a response of system, inwhich a pole exists in the region Re(z) < 0 on the complex z-plane. The angle from the pole to a point on the real axischanges from −π to 0 with the movement of the point from −∞ to +∞.
As in Sec. 8.5.4, an S-matrix for two junctions with three channels is written as
√
√ 

0√
−1/ 2 −1/ 2
St = −1/√2
1/2−1/2  .
1/2−1/ 2 −1/2
Also for the AB phase, we insert an S-matrix
(
)
0
eiθAB
SAB = −iθAB,e0θ ≡ 2πϕe= ϕ (ϕ is flux through the ring.)
ϕ0ℏ(8.72)(8.73)

to one of the parallel paths. And the QD represented in (8.71) is inserted into the other path. Though thus obtained analytical form of transmission coefficient is complicated, numerical calculations shows clear Fano effect as in Fig. 8.32(b),(c). The direction of the lineshape distortion (parameter q in (8.70)) changes with the period ϕ0 as shown in (c). Thisis natural consequence of the interference and evidences that the distortion comes from the rapid π change in the phaseshift appeared in Fig. 8.32(a).
E12-7

0.925

(b)
0.920
B (T)

0.45

0.40

φ=0

2

G (e /h)

0.40

0.915

0.35

0.35

φ0/2

0.30

φ0/4

0.30

0.910

2

-0.08

-0.06
Vg (V)-0.04(a)

G (e /h)

-0.08

-0.06
Vg (V)

-0.04

-0.02

(c)

Fig. 8.33 (a) Color plot of conductance of an AB ring with a QD on one of the paths versus gate voltage and magnetic
field. (b) Scanning electron micrograph of the sample. The gates with × mark are not used. (c) Coulomb oscillations
at three representative magnetic fields[1]．

Figure 8.33 shows results of an experiment. The Fano lineshape and Fano parameter vary aginst the flux piercing theAB ring as expected.

8.8.2 Qauntum dot and the Kondo effect
The Kondo effect in QDs originates from the indirectinteraction between electrons in electrodes via localized states.
We do not go into the details due to time limitation. In Appendix 8C, very short summary is given. At low temperatures
most of the freedoms die out and conductors are like empty cavities (electron “waveguides” are also cavities just likemicrowave wavefuides). An exceptional case comes from the existence of energy-degenerated freedoms just at the
Fermi level. Fermi spheres themselves are such exceptions but if there exists another degenerated freedom exists and thefreedom has quantum entanglement with electrons at the Fermi level, the Kondo effect appears. The Kramers degeneracy
due to time-reversal symmetry, i.e. spin degeneracy is an easiest example of such degenerated freedom. Hence QDs with
odd number of electrons are convenient for the experiments because the topmost level is occupied by a single electron
and has spin 1/2.
The Kondo effect first appeared as increase of resistance in diluted magnetic alloys with decreasing temperature. Jun
Kondo gave theoretical solution to this problem and simultaneously found the divergence in the second order perturbation.
This Kondo problem became a big problem of physics beyond the frame of solid state physics. For the problem,
Anderson impurity model was proposed, renormalization group theory was developed. The renormalization grouptheory was applied to quark confinement problem in particle physics and led to the concept of asymptotic freedom andthe establishment of quantum chromodynamics.
As shown in App. 8C, in very short, the Kondo effect in QDs is anomalous enhancement of tunneling probability
from Hamiltonian HT by many body resonance between degenerate freedom and the Fermi surface. In the case of
QDs, the process expressed by HT is the transmission of electrons through the QDs. That is, enhancement of HTmeans enhancement of conductance. If we consider the anaoly with double barrier resoance, the Kondo many bodyresonance anomalously enhances the conductance and even when the original conductance is very small due to the

E12-8

2.0

2 e 2/ hConductance ( e 2 / h )1.5

Conductance

lnT

1.0

0.5

0.0

TK

-450

T

-425

-400

-375

Gate voltage (mV)

(a)

(b)
Fig. 8.34 (a) Illustration of electric conductance through a quantm dot as a function of temperature with the emergence of the Kondo effect. Solid blue line is for odd number of electrons in the QD. In this case the Kondo effectemerges and at high temperatures the conductance is enhanced in proportion to ln T as the temperature decreases,
whereas below TK the enhancement is saturated to reach the unitary limit 2e2 /h. Green broken line is for even number of electrons and the conductance goes to zero with the Coulomb blockade. (b) The Kondo effect appeared in the
conduction experiment of a QD. The parameter is temperature and the arrows indicate the direction of lowering thetemperature, with which the conductance increases in the valleys with odd nubmer of electrons and decreases in thosewith even number of electrons[6].

Coulomb blockade, the final transmission probability should be 1 (unitary). In this case, the conductance is, from theLandauer formula, the universal value 2e2 /h.
A characteristic feature of the Kondo effect is that it is always in resonance with the Fermi surface. Therefore roughly
speaking in the Coulomb valleys with odd number of electrons the conductance is 2e2 /h and in those with even number ofelectrons the conductance is zero. In Fig. 8.34, we show conceptual behavior and actual observation of QD conductancearound the temperature characteristic of the Kondo effect (Kondo temperature, TK ).

Appendix 8C: The Kondo effectWe consider a QD with the impurity Anderson model, in which only one electrons exists in the dot (n = 1) as theground state. The dot has spin 1/2, can be viewed as a kind of magnetic impurity. We write the energy required to add anelectron as ∆E + , that required to extract one as ∆E − . Then
∆E − = µ − µ1 = µ − ϵ0 .

∆E + = µ2 − µ = ϵ0 + U − µ,(8C.1)

These energies give the state-allowance-times h/∆E ± from the uncertainty relation for the excited states. There shouldbe, then, second order tunneling processes with HT by utilizing these excited states as the intermediate states. The
probabilities of such processes are−γL∗ γR,∆E −γL∗ γR.
∆E +(8C.2)Such tunnel processes of higher order is called co-tunneling. The Kondo effect can also be regarded as a phenomenon inwhich the tunnel probability amplitude due to co-tunneling increases anomalously.
First, the Hamiltonian
H = Hleads + Hdot + HT

E12-9

(8C.3)

is unitary-transformed as{∗
ckσ = (γL∗ cL,kσ + γR
cR,kσ )/γc̄kσ = (−γR cL,kσ + γL cR,kσ )/γ2, γ 2 ≡ γL2 + γR

(8C.4)

Then the tunnel Hamiltonian is transformed as∑
HT =[(γL c†L,kσ + γR c†R,kσ )dσ + h.c.]
k,σ

=

∑

[γc†kσ dσ + h.c.],(8C.5)k,σin which we can ignore c̄kσ because it has nothing to do with the coupling to the dot. This transformation renormalizes
the two electrodes model of QD into “a QD and a system with a Fermi surface” model. It formally equalizes a QD fortransport experiment with electrodes to a magnetic impurity in a metal *2 ．
The transformed Anderson impurity model Hamiltonian is written asH=∑

ϵk c†kσ ckσ +∑ϵd d†σ dσ +

σ

kσ

∑

(γc†kσ dσ + h.c.) + U d†↑ d↑ d†↓ d↓ .

(8C.6)kσ

The condition for having single electron in the ground state of the dot is
ϵd < EF < ϵd + U.

(8C.7)

Under the condition, we regard the interaction term (the third term with Vkd ) of the conduction electron (s-electron) in
the electrode and the dot electron (d-electron) as a perturbation. The first order of perturbation does not exist because itchanges the number of d-electron, and the leading order is second. This means we need to consider co-tunneling process
as of the leading order.
There are following four perturbation processes on the state in which the d-electron has up-spin ↑. The contraint is that
only ↓-electron is allowed to enter the dot by Pauli principle. We write the unperturbed state as ψ↑ .
1) k ↓→ d ↓→ k ′ ↓2) k ↓→ d ↓，d ↑→ k ′ ↑ (down-spin electron goes into the dot then up-spin electron goes out)

k

1)

d
k’

k

2)

d
k’
k

3)

d

d

k’

k’

4)

d

k
d
k’

*2

k’

Fig. 8C.1 2nd order possible tunneling processes
for the initial states of an up-spin electron inside thedot These 1)∼4) correspond to 1)∼4) in Eq. (8C.8),(8C.9) respectively.

There is a difference in the physical meaning. In the case of magnetic impurities, c†k′ σ ckσ means impurity scattering. In the case of QDs,
on the other hand, it represents transmission and reflection via co-tunneling. Another difference is that in the case of impurities, k should be
three dimensional vectors. This can be, however, transformed to one-dimensional problem with partial wave expansion of scattered wave, andmathematical equivalence is kept. This method is used to find the exact solution based on the Bethe ansatz.

E12-103) d ↑→ k ′ ↑，k ↑→ d ↑ (up-spin electron goes out then up-spin electron goes into the dot)4) d ↑→ k ′ ↑, k ↓→ d ↓ (up-spin electron goes out then down-spin electron goes into the dot)
Effective Hamiltonians for these processes are
γ2 †
γ2 †
†
c
2)
→
−
c ′ d↑ d†↓ ck↓ ,′ ↓ d↓ d↓ ck↓ ,k∆E +∆E + k ↑
γ2 †
γ2 †
†
†
3) →
d
c
c
d ck↓ c†k′ ↑ d↑ .
d
,
4)
→
′
k↑
k ↑ ↑
∆E − ↑∆E − ↓

1) → −(8C.8)(8C.9)

Just same as above, four perturbation processes exist for the case that ψ↓ is the initial state. The effective Hamiltonians
for these processes are obtained by the replacement ↑↓→↓↑. These terms are summed up to be∑ γ2∑ γ2 †d†σ dσ +c ′ ckσ−∆E∆E + k σ
kσ
kk′ σ
)
∑ ( 1
1
2
(c†k′ ↑ ck↑ d†↑ d↑ + c†k′ ↓ ck↓ d†↓ d↓ + c†k′ ↑ ck↓ d†↓ d↑ + c†k′ ↓ ck↑ d†↑ d↓ ).
+
γ
+
∆E +
∆E −
′

(8C.10)kk

The first term represents process 3) for the case of k = k ′ . Because k is outside the Fermi surface, at low temperature
under Fermi degeneracy condition, we assume ck c†k = 1, c†k ck = 0. The second term is for process 1). To obtain thisterm we use the fact that from d↓ ψ↑ = 0, we can write d↓ d†↓ = 1, d†↓ d↓ = 0. The residual part of process 3) and those of2) and 4) are expressed in the third term.
Here we transform the above toc†k′ ↑ ck↑ d†↑ d↑ + c†k′ ↓ ck↓ d†↓ d↓ =1 †1(c ′ ck↑ − c†k′ ↓ ck↓ )(d†↑ d↑ − d†↓ d↓ ) + (c†k′ ↑ ck↑ + c†k′ ↓ ck↓ )(d†↑ d↑ + d†↓ d↓ ).
2 k↑2Because the spin operator of the dot Ŝ is expressed as
Ŝz =

1 †
(d d↑ − d†↓ d↓ ), Ŝ+ = d†↑ d↓ , Ŝ− = d†↓ d↑ ,2 ↑the summation of the second and the third term in (8C.10) is rewritten to the summation of the following two Hamiltonians(Hd ,Hsd ):
Hd =

Hsd =

∑

[
γ2

kk′

∑
kk′ σ

[
γ

2

1
1
−
+
∆E
2

(

1
1
+
+
∆E
∆E −

)]

c†k′ σ ckσ ,

][
]
1
1
†
†
†
†
+
Ŝ
c
c
+
Ŝ
c
c
+
Ŝ
(c
c
−
c
c
)
.
′
′
′
′
+
k↑
−
k↓
z
k↑
k↓
k ↓
k ↑
k ↑
k ↓
∆E +
∆E −
(

Let us define J as

(8C.12))J = γ2

1
1
+
∆E +
∆E −

Hd =

∑( J) †−
ck′ σ ckσ
2
′

then

(8C.11),(8C.13)(8C.14)kk

is ordinary potential scattering, which does not depend on spin. On the other hand,]∑[
Hsd = JŜ+ c†k′ ↓ ck↑ + Ŝ− c†k′ ↑ ck↓ +Ŝz (c†k′ ↑ ck↑ − c†k′ ↓ ck↓ )
kk′

]
∑[
=J
(Ŝx + iŜy )(ŝxj − iŝyj ) + (Ŝx − iŜy )(ŝxj + iŝyj ) + 2ŝzj Ŝz(8C.15)

j

= 2J

∑

ŝj · Ŝjis expressing the exchange interaction between spin of conduction electrons sj and spin on the dot. This is often called
sd-Hamiltonian, which originally expresses interaction between electron spin (s) and localized spin (in many cases, inE12-11

k’

k

d

d

Fig. 8C.2 Diagram representing electron scattering k → k′ by a dot
in the first order of HT . The time flows from the left to the right. d
represents the dot and the up/down arrows indicate spins.

HTd-orbital, then d-spin) in diluted magnetic impurity system. Now with unitary transformation in (8C.4), we can also apply
the sd-Hamiltonian to QD-electrode systems.
Then as transmission Hamiltonian HT we ignore potential scattering Hd and take only Hsd . Then with adding theelectorons in electrodes, the effective Hamiltonian
]
∑ †
∑[
Heﬀ =
ϵk ckσ ckσ + J
Ŝ+ c†k′ ↓ ck↑ + Ŝ− c†k′ ↑ ck↓ +Ŝz (c†k′ ↑ ck↑ − c†k′ ↓ ck↓ ) .

(8C.16)kk′kσ

is obtaiend (Schrieffer-Wolff transformation).
Kondo calculated the scattering amplitude by the effective Hamiltonian (8C.16) to the second order in Born approximation. That is, he treated J in (8C.16) as the parameter of perturbation and calculated up to the quadratic term of J (theforth term of γ). The operator of transition between the left and right electrodes is given asT̂ = HT + HT

1
HT + · · · .
ϵ − H0 + iδ

(8C.17)

The tunnel probability of L → R is formally written as
ΓL→R = 2

∑ 2π
k,k′

ℏ

2

⟨Rk ′ |T̂ |Lk⟩ δ(ϵRk′ − ϵLk )f (ϵLk − µL )[1 − f (ϵRk′ − µR )].

(8C.18)Let us treat the scattering |k ↑⟩ → |k ′ ↑⟩. Perturbation to the first order of J is expressed in the diagram shown in
Fig. 8C.2 and calculated as⟨d ↑; k ′ ↑ |T̂ (1) |d ↑; k ↑⟩ = J/2.

(8C.19)The conduction process HT requires two consecutive tunnelings and this corresponds to the second order of γ (J is thus
proportional to γ 2 ), and co-tunneling process in (8C.2).
There are three types of processes in the second order of J ⟨d ↑; k ′ ↑ |T̂ (2) |d ↑; k ↑⟩ as shown in Fig. 8C.3 andFig. 8C.4. The first and second processes are not associated with spin flip and they are distinguished as electron process(Fig. 8C.3(a)) and electron-hole pair process (Fig. 8C.3(b)) for the intermediate propagation process *3 ．The contributionof these two terms is calculated as∑ ( J )2
∑ ( J )2
1
−1
[1 − f (ϵq )] +f (ϵq )
2
ϵ
−
ϵ
+
iδ
2
ϵ
−
(2ϵ
− ϵq ) + iδqqq
∑ ( J )2
1
=
2
ϵ
−
ϵ
q + iδ
q( )2 ∫ D
J
1
=
dϵ′ ν
ν : Density of states′ + iδ2ϵ−ϵ−D]( )2 [
D+ϵJ
ν ln− iπ .
=
2D−ϵ(8C.20)*3 Here “hole” state refers to Fermi liquid lacking single electron. This is largely different from the “hole” state defined as the state created byextracting an electron from valence band (Sec. 3.1.2).

E12-12

q

k

k’

k

k’

d
d

d

d

HT

HT

HT

HT

(a)

q

(b)

q

k

k’

k

k’

d
d

(a)

d

d

HT

??

HT
q

(b)

Fig. 8C.3 Non spin-flip
processes for 2nd orderof HT . (a) Process withelectron excitation as theintermediate state. (b) In
the intermedate state of thisprocess, an electron-hole
pair propagates. The hole isannihilated by recombination
with an electron in theelectrode.
Fig. 8C.4 Processes withspin flipped intermediatestats in the second orderof HT . (a) Process withelectron excitation as theintermediate state.
Thisprocess is absent due tothe angular momentum
conservation.
(b) In theintermedate state of thisprocess, an electron-hole pair
propagates.

The result does not show any anomaly as a consequence of electron-hole symmetry. Here as for the electronic states inmetal, we have adopted a rough (or abstract) approximation that a band spread over [−D, D] on energy with a uniform
density of states ν. Such a “toy” model is often good to see the essence of phenomenon.
In the processes shown in Fig. 8C.4, spin flips occur in the intermediate states. However in (a), the dot spin should be
3/2 for the conservation of angular momentum, and this process is prohibited. In (b), an electron-hole pair propagates in
the intermediate state and the contribution is calculated as
∑
q

J2

1
f (ϵq ) = J 2 νϵ − ϵq + iδ{−J 2 ν ln |ϵ|/D≈−J 2 ν ln kB T /D∫ D

1f (ϵ′ )dϵ′
−D ϵ − ϵ + iδ

|ϵ| ≫ kB T,|ϵ| ≪ kB T.

(8C.21)This term diverges logarithmically with temperature lowering or smaller ϵ. This is the anomalous term found by Kondo.
And various phenomena originate from this anomaly are called the Kondo effect.
Let us consider the origin of this term. In the case of non-spin-flip processes, the anomalous terms cancel each otherdue to the electron-hole symmetry. That is, if we look electrons or holes separately the anomaly exists regardless of
spin-flip and the origin is the existence of Fermi surface, which represents huge asymmetry. At absolute zero, states
inside a Fermi sphere are fully occupied while those above the Fermi surface are completely empty with almost infinite
degeneracy. In the processes without spin flip, the electron-hole symmetry perfectly cancels this huge asymmetry. On theother hand in the processes with spin flip, conservation of spin angular momentum prohibits electron propagation process
in the intermediate state *4 , and the asymmetry at the Fermi surface appears as the anomaly.
Because the perturbation leads to the divergence, the perturbative treatment itself is in failure for the condiitons close*4 Mathematically this comes from non-commutativity of Ŝ+ and Ŝ− (the commutation relation yields Ŝz ), hence one can say that this is due to aquantum mechanical effect.

E12-13

to the divergence. Treatment of this problem thus requires various methods other than simple perturbation. In order forhandling this Kondo problem, a number of methods including renormalization group, have been developed. Here we do
not go into the detail of such methods. Instead we have a look on the results at lowest temperatures obtained from yearsof research.
In sd Hamiltonian (8C.15), there is antiferromagnetic interaction between the dot spin and the conduction electron
spin. That is, the dot spin attracts electrons with anti-parallel spins and repels those with parallel spins through exchangeinteraction arises from co-tunneling. As a result, seen from a distance, a cloud of spin polarization of conduction
electrons appears to cling to the dot spin. The Kondo problem indicates the anomalous enhancement of the above effect.
The cloud like state of spin polarization created as above is called Kondo cloud. On the other hand, the spin polarization
surrounding the dot spin reaches complete screening of the dot spin, further polarization stops. This is called unitary
limit.
This phenomenon can be viewed as many-body resoance. A representative of single-body resonance (resonace thatcomes from potential) is the resonant tunneling through double barrier structures. In the double barrier structure, nomatter how high the barrier height is and how small the tunnel probability is, where the energy of the incident wave isin resonance, the reflected and transmitted waves are infinite sum up of coherent reflection and transmission by the twobarriers. And finally the reflections cancel each other out, the total transmittance is 1. This resonance energies are close
to the bound state energies of an imaginary quantum well composed by making the barrier thicknesses infinite. When thebarrier thickneeses are finite, resonance makes average staying time anomalously long and the modes are called quasibound states. Even if an electron enters the quasi-bound state, it eventually leaks to the outside, so it is in a resonancestate with the free electrons in the electrodes. When the Fermi level hits resonance, the transmission probability reaches
a peak value. An example is shown in Fig. 8C.5.
The Kondo effect has many common features with double barrier phenomenon. While single-body resonance is basedon infinite number of reflections, the Kondo resonance occurs as a result of infinite degeneracy at Fermi surface. In
potential resonance, orbital effect results in non-uniform probability distribution. In the case of the Kondo effect, theforce works among spins and no charge inhomogeneity appears. Instead, localized spin polarization occurs as Kondo
cloud. The biggest difference is that the Kondo cloud is always in resonance with Fermi surface.
Electrons stay in quasi-bound states for finite times. Let τa be the average staying time and the resonance has lifetime
broeadning h/τa (= ℏΓ, Γ is tunneling frequency). Does the Kondo cloud have width? If it does, how large is that?
When the thermal broadening of Fermi surface is larger than the resonance width, the temperature dependence of thecontribution of resonance to conduction should be weak. On the contrary, when the thermal width is narrower than
the resonance width, the temperature dependence also disappears. The energy scale representing resonance width is, in
temperature unit, called Kondo temperature, for which symbol TK is often used.

1

a

b

0.8
0.6

T

E

0.4

V0
0.2

0

x

0.2

0.4

E/V0

E12-14

0.6

0.8

1

Fig. 8C.5 Left: Schematics of double barrier potential. Vertical axis isenergy, horizontal axis is space coordinate. Broken lines indicate thepositions of quasi-bound state energies, i.e. positions of resonant tunneling. Real wavelengths of incident
waves are much longer than the illustration. Right: Example of transmission probability under condition√
2mV0 /ℏ and k0 b=5.0.
of k0 ≡
Solid line is for k0 a=0.5 and broken
line is for k0 a=2.0.

For correct estimation of Kondo temperature, perturbation in (8C.21) is not enough and the effect of higher order terms
should be taken into account in some way. Though, roughly speaking, TK can be estimated as the temperature-dependent
term in (8C.21) is comparable with J at high temperature side. That is, from−J 2 ν ln kB TK /D ∼ J,
TK is given as

kB TK ∼ De−Jν .

In the above rough estimation, with increasing the anti-ferromagnetic coupling strength J, TK dcreases exponentially.
Larger J corresponds to higher barriers in double barrier, narrower lifetime width with decreasing tunneling probability.
The same for the density of states ν. On the other hand, widening of band D makes resonance wider, TK higher. Actuallyin the present simple model, J and D are not independent. But with ignoring that, widening of the resonance by increasing
D is interpreted as the increase of contribution from deeper inside the Fermi sphere.

References[1] K. Kobayashi et al., Phys. Rev. Lett. 88, 256806 (2002); Phys. Rev. B 68, 235304 (2003).
[2] 勝本信吾，「半導体量子輸送物性」 (培風館，2014)．
[3] 近藤 淳，「金属電子論」(裳華房，1983)，芳田圭，「磁性」(岩波書店，1991)[4] Yu. V. Nazarov and Y. M. Blanter, “Quantum Transport” (Cambridge, 2009).
[5] S. Tarucha et al., Phys. Rev. Lett. 77, 3613 (1996).
[6] W. G. van der Wiel, S. De Franceschi, T. Fujisawa, J. M. Elzerman, S. Tarucha, L. P. Kouwenhoven, Science 289,
2105 (2000).

E12-15

Lecture note on

Physics of Semiconductors (13)7th July (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Chapter 7 The Quantum Hall effect
In many cases, transport in higer dimensions can be understood as that in networks of one-dimensional quantum wires.
On the other hand in the case of Landau quantization, mixing of two-dimensional freedoms is important and it is easierto treat the system as continuous two-dimensional space (see Sec.[?] for discrete treatment)..

10.1 Two-dimensional electrons under magnetic fieldLet us write the Hamiltonian asH =πx2 + πy2
(pc + eA)2
π2
m 2
v =
≡
=
.
2
2m
2m
2m
π ≡ pc + eA,(10.1)(10.2)

where π is dynamical momentum, corresponding to real space velocity as v = π/m∗ . π has commutation relations
among themselves and with space coordinates asℏ2[πα , β] = −iℏδαβ , (α, β = x, y), [πx , πy ] = −i 2 .
l(10.3)We see that x and y components of the momentum do not commute. The fact corresponds to the classical circulating
orbits, which mix up the x and y coordinates, in other words they are no longer independent. l is called magnetic length√

defined as
l≡

ℏ
=
eB

√ √
1 ϕ0
,
2 πB

(10.4)√
which is 1/ 2 times the radius of circle for single flux quantum (ϕ0 ≡ h/e). l is also called minimum Landau radius.
√
The factor 1/ 2 corresponds to the zero-point energy term ℏωc /2 in Eq. (10.9), which we will see later.
We define the operator R̂ of guiding center coordinate (X, Y ) asr̂ = R̂ +l2(πy , −πx ),ℏ(10.5)

where r̂ is the real space operator of electrons. The second term in the right hand side is from the classical solution (notin this note). From the commutation relation between πx and πy , we get[X, Y ] = il2 .

(10.6)The Hamiltonian does not depend on (X, Y ), thus (X, Y ) is a constant of motion while from the commutation relationin (10.6), there is an uncertainty between X and Y . Now we see that as a set of canonically conjugate variables of thesystem we can take R, π other than (r, pc ).

E13-1

10.1.1 Landau quantization
As in (10.1), the Hamiltonian is quadratic for π and in the form of harmonic oscillator *1 , by introducing down/up
operators aslla = √ (πx − iπy ), a† = √ (πx + iπy ),2ℏ
2ℏ(

it can be written as†

H = ℏωc[a, a ] = 1,

1
a a+
2
†

(10.7)).

This is in the harmonic form and the engenenergies are given as)(1(n = 0, 1, 2, · · · ).
En = ℏωc n +2(10.8)(10.9)

This is interpreted as the discretization of (angular) momentum with quantum confinement by magnetic field. Suchquantization of orbitals by magnetic field is called Landau quantization.

10.1.2 Guiding center
Because R commutes with Hamiltonian (10.1), the eigenenergies in Eq. (10.9) do not depend on R, thus they are degenerate as the degree of freedom in R. Two dimensional systems under perpendicular uniform magnetic field still keeps
spatial translational symmetry. In the set of eigenfunctions which have the guiding center as an index, the translational
symmetry is kept through the freedom in R. The Landau levels have large degeneracy and the basis can be taken invarious form. The uncommutability between the components of R brings large variaty in the outlooks of the basis.
Let us find the basis that diagonalizes X. For that Landau gauge A = (0, Bx, 0) is convenient. Form Eq. (10.1),
Schrödinger equation is given by[)2 ](−1 ℏ2 ∂ 2ℏ∂
(pc + eA)2
ψ=
+ eBx
ψ(r)
− −i
Hψ=
2m
2m ∂x2∂y[]1∂=−ℏ2 ∇2 − 2iℏeBx+ e2 B 2 x2 ψ(r) = Eψ(r).
2m
∂y
y 4
2

n=2

N =0
m j =0(10.10)

1.0
x

-4

-2

2

0.8

4

-2
n=1

0.6

-4
y 4

N =0
m j =2

0.4

y
n=0
x
-1
(a)

0

x
-4

0.2

4
0

1
(b)

-4

Fig. 10.1 (a) Gray scale plots of probability
densities |ψnk (r)|2 in eigenstates (10.12) withthree values of n, which diagonalize X. The
unit of length is lB , the width along x is about√
2n + 1lB . (b) The same for the basis, which diagonalizes X 2 + Y 2 (not mentioned in the text).
In the case of N = 0, the distribution is around the√circle with the radius | 2|mj |lB at the origin.

*1 It is written as a sum of π 2 and π 2 . πx and πy are canonically conjugate operators.
x
y

E13-2

This Hamiltonian does not contain operator y and y-dependent part of the wavefunction should be a plane wave. Thus
we substitute variable separable form ψ(r) = u(x) exp(iky) into the above equation to obtain
[
(
)2 ]
[
]
ℏ2 d2
(eB)2
ℏ
ℏ2 d2
mωc2
2 2
−
+
x+
k
u(x) = −+(x + l k) u(x) = Eu(x).
2m dx22meB
2m dx22(10.11)This is an equation of a one-dimensional harmonic oscillator that has the center at x = −l2 k. The eigenvalues are given
in Eq. (10.9), and the eigenfunctions are written as()()
x − xk(x − xk )2
ψnk (r) ∝ Hn
exp −exp(iky) (xk ≡ −l2 k),l
2l2(10.12)where Hn is n-th order Hermite polynomial. In each of these states, X is fixed as X = xk = −l2 k = −l2 py /ℏ while
it is a plane wave on y spreading over whole space, namely Y is fully uncertain. In the states given by Eq. (10.12), theenergy does not depend on k. Though the states are extended along y, hence the group velocity is zero (∂E/∂k = 0).
On the other hand ∂X/∂k is not zero, then if some x-dependent potential is added to the system, the states gain a finitegroup velocity and motions along y.
Figure 10.1(a) shows gray scale plots of probability density in Eq. (10.12). We see they are uniform along y while
one-dimensional harmonic oscillators along x. We are not showing the functional form here but the eigenstates can bechosen so as to diagonalize X 2 + Y 2 . In this case, as shown in Fig. 10.1(b), the probability densities are localized bothfor x and y. The reason why their outlooks are so different in spite of the fact that they have the same eigenenergy, is ofcourse it has large degeneracy and also the degeneracy comes from the freedom in R, which is the freedom in the real
space.

10.2 Integer quantum Hall effect10.2.1 Shubnikov de Haas (SdH) oscillation
Let us consider the process of increasing magnetic field applied perpendicular to a two-dimensional electron system.
With Landau quantization (10.9), the energy levels are as in Fig. 10.2, spead radially from the origin (Landau fan, fandiagram). How the electrons occupy those Landau levels when the system is connected to particle reservoirs as intransport experiments? The external particle reserviors make the Fermi level EF constant but if we impose this condition,
the origin in Fig. 10.2 should shift with magnetic field. The origin in Fig. 10.2 is defined as the zero-point of kinetic
energy in xy-plane, namely energy levels quantized along z-axis. In the simple approximation in Sec. 7.3, the positionof EF is determined to screen the electrostatic potential formed by ionized potential with arial density Ndep . Thenwith variation in the density of states for kinetic freedom in xy plane, the distribution of occupied states also varies tocompensate the potential from the impurities. This leads to the shifts in self-consistent potential and the position of thelowest level (origin). If we look the Landau fan from the coordinate in which the origin is fixed, EF varies with magnetic
field. Below, we adopt this coordinate (constant 2DEG arial density ns ).
Let us find the arial density of states per single Ladau level nL . For that we count the number of possible wavefunctions
in Eq. (10.12) in the area of Wx × Wy in xy-plane. The function in Eq. (10.12) is a plane wave along y and the “distance”
2of the states in k-space in 2π/Wy . On the other hand, the section 0 ≤ X ≤ Wx corresponds to −Wx /lB≤ k ≤ 0 in

k-space for the wavefunctions. Hence the number of of states in the area S = Wx Wy is
2
S
Wx /lB
=
2
2π/Wy
2πlB

∴ nL =

1eB
B
2 = h = ϕ ,2πlB
0(10.13)that is the number of quantum flux in the flux density. The number of Landau levels occupied by electrons isν=

ϕ0 n s,B

E13-3(10.14)

5 4

3

2
1

E

E F (0)n =01ö
æE = hw c ç n + ÷
2ø
è

5 4

3

n=2
B

n=1

Fig. 10.2 Landau levels in (10.9) as a function of magnetic field. The broken line indicates the potision of EF in(0)
this frame under the condition of constant ns . EF is forzero magnetic field.

which is called filling factor
At absolute zero, electrons occupy Landau levels from the lowest one and EF is locked to the highest occupied Landaulevel. With increasing magnetic field, the changes of “topmost occupied level” take place at the points ν hit integers,
where EF shifts from E = ℏωc (ν + 1/2) to ℏωc (ν − 1/2). To summarize, in Fig. 10.2, EF oscillates as indicated bybroken line. This oscillation and the resultant oscillation in the electric resistance is called Shubnikov-de Haas (SdH)
oscillation.

10.2.2 Localization of wavefuntion
I believe there is no rigorous proof but it is widely believed that in two-dimensional systems with some potential disorder, time-reversal symmetry, no spin-orbit interaction, all the particle states (wavefunctions) localize spatially (Anderson
localization). Magnetic field breaks the time reversal smmetry and the Anderson localization is simultaneously broken.
However, with further increase in magnetic field, the cyclotron radius becomes shorter than the characteristic length ofthe potential disorder, localization appears due to a bit different mechanism.

delicalized

Peaklocalized

BottomE(a)(b)

Fig. 10.3 (a) Schematic diagram showing how the Landau level wavefunction is localized by the impurity potential
in a strong magnetic field. Wavefunctions in the form of Fig. 10.1(a) are bound on equipotential lines of disorderedpotential. The lines that depict drifting while rotating are classical orbits. (b) Landau level energies are broadenedby disordered potential and localized as shadowed. Delocalized states exist around the centers of Landau “bands,”
corresponding to the concave-convex transition equipotential lines.

E13-4

0.3

0.8
y
T=50mK

0.6

0.4

ν=2

ρxy

0.2
ρxx (h/e2)ρxy (h/e2)

x

3
4

0.2

6

5

0.1
ρxx

0
0

2

B (T)

4

6

0

Fig. 10.4 Example of integer quantum Hall effect. 2DEG at an AlGaAs/GaAs interface is fabricated into the shape of Hall bar shown in the inset. A current is applied to the long thick line (xdirection), ρxx is obtained from the voltage between two probes placed along x while ρxy is obtained from the probes placed face to face alongy.

Such behavior is illustrated in Fig. 10.3(a). Electric field applied to electons in cyclotron motion causes movement
perpenducular to the field. In nonuniform potential as shown in the figure, such movement of electrons results in arounding motion bound on an equipotential line. Then the state as a whole is spatially localized. Such spatial confinement
leads to broadening of Landau levels as we have seen in Fock-Darwin state. Then the delta-function density of statesof original Ladau level gets broadening as illustrated in Fig. 10.3(b). On the other hand, there are a small number of
equipotential lines that do not make a closed loop between potential peaks and dips as in Fig. 10.3(a). Such a state on
non-closed equipotential line should be extended and it is known that each broadened Landau “band” has a single suchextended state at the center. This is also illustrated in Fig. 10.3(b).

10.2.3 Characteristics of integer quantum Hall effect
In Fig. 10.4 we show an example of measured integer quantum Hall effect (IQHE). Increasing magnetic field perpendicular to two-dimensional plane, the Hall resistance ρxy deviates from classical linear dependence on magnetic flux
density B (Eq. (5.15)) and a clear staircase structure emerges. In the IQHE, the heights of the plateaus are exactly
ρxy =1
2.5812 × 104
h 1
=
(R
)
≈
(Ω),
K
e2 n
n
n

(n = 1, 2, . . . ).

(10.15)

As can be guessed in Fig. 10.4, in the plateau regions simultaneously ρxx = 0, that is, finite current flows without
longitudinal voltage. The current here is, like superconductivity, a kind of supercurrent without energy dissipation.

10.2.4 Explanation based on edge mode transportComparing the experiment shown in Fig. 10.4 and the localization/delocalization diagram in Fig. 10.3(b), we see thatthe supercurrent which causes ρxx = 0 flows and ρxy is quantized when EF does not exist in the regions of delocalizedstates. To put this the other way around, ρxy is in a transient region between quantized plateaus when EF exists in
the retions of delocalization. Namely the quantization and supercurrent take place when two-dimensional electrons are
insulating in the bulk.

E13-5

( m * w c /2)( x - X ) 2

U(x)

V(x)
E 2x
E 1x
Fig. 10.5 Two-dimensional electrons understrong magnetic field are confined by “gutterlike” potential V (x). The effective potential including the effect of magnetic field is expressed
as U (x) (sum of V (x) and magnetic confinement
potential). Formation of edge states is indicatedby broken lines. The lower panel illustrates classical skipping orbits.

E 0x
-W/2

x

W/2
y

(X,Y)

(X,Y)
Sample bulk B
x

The “edge state model” explains the phenomenon based on edge mode transport. A sample with a finite width as
illustrated in Fig. 10.4 inevitably has edge states*2 . To model that, we consider a two-dimensional electron gas confinedin x-direction by a well-like potential V (x) with width W , spreading over y. In this model the current is applied iny-direction (for convenience the coordinate is rotated by π/2).
V (x) is added to Eq. (10.10) for the wave equation. Figure 10.5 illustrates the situation, in which the gutter-like
potential and the harmonic potential by magnetic field co-exist. V (x) = 0 deep inside the bulk and ordinary Landau
quantization takes place while in the vicinity of edges, V (x) makes the effective harmonic potential narrower, i.e. effective ωc larger, hence Landau levels go up with approaching the edges. The increase of n-th Landau level begins where√
X-coordinate of guiding center is in the width of wavefunction 2n + 1lB /2 to the edge. In the region of such level
increase,2⟨vy ⟩ = dE/ℏdk = −(lB/ℏ)dE/dX(10.16)

becomes finite, giving spatial motion to Landau quantized electrons. Such mobile states correspond to classical skipping
orbits, which consist of cyclotron motions and collisions to an edge as illustrated in the lower panel of Fig. 10.5. They
are called edge states. In the edge states the direction of electron motion is determined by the sign of magnetic field.
Normalizing the edge mode wavefunction in the length Ly along y, the current brought by the mode is j = (e/Ly )⟨vy ⟩.
A single mode at one-side edge is occupied up to the electrochemical potential µ. We take a base energy E0 lower than µand higher than the bulk Landau level with the same Landau index as the edge mode. The current brought by the electrons
occupying the states from E0 to µ in this edge state is obtained from (10.13) and (10.16) as
∫ Xµ
J=
X0

Ly dX ee
2 L ⟨vy ⟩ = h
2πlB
y

∫
dX

A

B

-W/2

m
A
m
B
W/2

XB

X

XA

dE
e
= (µ − E0 ).
dXhFig. 10.6 Schematic drawing of a Landau level with edge modes. Finite net
current is flowing perpendicular to thefigure (y-direction) and consequently finite gradient in x-direction is given (exaggerated).

*2 There is no edge state along the current if the two edges are connected. Such a structure in a plane is called “Corbino disk.”

E13-6(10.17)

When EF is in regions of localization, in equilibrium the chemical potential is uniform over the sample and the edges
opposite to each other have counter-flowing currents with the same amount, the bulk states are localized and the net
current is zero (circular equilibrium current is flowing at the edge). Now we apply the boundary condition that the net
current Jy flows along y. As in Fig. 10.6, Jy is the difference between currents JA and JB at edges A and B respectively.
Hence from Eq. (10.17), there should be a difference between µA and µB , which leads to the Hall voltage. Thenσxy =e(JA − JB )
e2
Jy
=
= .
Vx
µA − µBh(10.18)This is the conductivity for single Landau level, and for ν levels σxy is ν times of this value, thus the IQHE is explained.
The above derivation is the same as that of the Landauer formula other than crossing of x and y. The quantization is
not so pricise for QPC conductance while surprizingly high precision is achieved for IQHD because of the chirality andthe geometrical effect in the edge modes. In the case of QPC, conductance channels with opposite direction are spatiallyoverlapped and backscattering of electrons can easily occurs. On the other hand in the case of IQHE, there is a macroscopic spatial distance between counter-flowing edge states and the probability of backscattering is astronomically low,
and the transmission coefficient is exactly one. Therefore, the quantization of IQHE should be inaccurate if the sample
width is narrowed and scattering between the edge states is likely to occur, which has been confirmed by experiments.
In the above simple model, we ignore the Hall electric field inside the sample. dE/dX caused by the Hall electric fieldleads to finite bulk current though they cancel each other, does not contribute to Jy and the above discussion still holds.

10.3 Explanation based on topological invariant
We continue theoretica explanation for IQHE. In this section, we need to introduce several new concepts. Below wecontinue along Ref. [4].

10.3.1 Bloch electrons in magnetic field
We expand the concept of Bloch electron to two-dimensional electrons in magnetic field. This way of treatment is close
to tight-binding model while that in Sec. 10.1 is based on two-dimensional free electron. In a two-dimensional squarelattice, we write the translational operator by lattice vector R as TR .
TR f (r) = f (r + R).
By expanding f (r) with plane wave eikr , from TR eikr = eik(r+R) = eikR eikr , TR is written as()
i
TR = exp
R·p .
ℏ(10.19)TR commutes with H0 (lattice Hamiltonian for zero magnetic field) and the Bloch states are defined as the eigenstatesthat diagonalize the two operators simultaneously.
We then proceed to treat a system under a uniform magnetic field.
H =1(p + eA)2 + V (r).
2m

The lattice potential V (r) is invariant for the operation of TR though the vector potential A is not. Generally
A(r) = A(r + R) + ∇g(r).

E13-7(10.20)

The loss of translational symmetry due to the cyclotron motion, which does not conserve momentum. Now we consider
modification of the translation operator. We define magnetic translation operator by replacing p with p + eA in
Eq. (10.19). Under symmetric gage A = B × r/2, the magnetic translation operator TBR is given by
{
[
]
[
]}
e
r
i
ie
TBR ≡ exp
R · p + (r × B) = TR exp(B × R) ·
.
ℏ
2
ℏ
2

(10.21)

TBR commutes with H , and there exists a basis which diagonalizes the two operators simultaneously. Care should
be taken that the magnetic translational operators do not commute each other generally just like the guiding center
coordinates of Landau levels do not. The commutation relation can be represented as a phase factor in
TBRa TBRb = exp(2πiϕ)TBRb TBRa ,

ϕ=

eB
ab,
h

(10.22)where a and b are the lengths of unit vectors. Hence ϕ is the magnetic flux piercing a unit cell in the unit of flux quantum
h/e. When ϕ is a rational number p/q, commutable set of magnetic translational opertors can be prepared as a latticelimits translational vectors into discrete lattice vectors. To have simpler view, we consider a magnetic unit cell, which
is defined from magnetic unit vectors qa, b corresponding to original unit vectors a, b. A magnetic lattice vector R′ isexpressed as

R′ = n(qa) + mb.

(10.23)

Then the flux piercing the magnetic unit cell is p (integer) times a flux quantum and the magnetic translational operators
TBR′ commute each other.
Now we take ψ as a common eigenstate of H and TBR′ . Let Tqa and Tb (we do not write BR′ for simplicity) beelements of the set of TBR′ , then the eigenvalues are written as
Tqa ψ = eik1 qa ψ,(10.24a)Tb ψ = eik2 b ψ,

(10.24b)where k1 , k2 are generalized crystal momenta. In reduced zone representation, k1 , k2 can be limited in the first magnetic
Brillouin zone 0 ≤ k1 < 2π/qa, 0 ≤ k2 < 2π/b. The magnetic eigenstates is written in the Bloch formψnk (r) = eikr unk (r),where n is a band index, k is a generalized momentum. The conditions for unk (r) are as follows.
( πpy )unk (x + qa, y) = exp i
unk (x, y),
( b
)
πpx
unk (x, y + b) = exp −i
unk (x, y).
qaThen if we write unk (r) in the amplitude-phase factor form as unk (r) = |unk(r) | exp[iθk (r)],I1∂θk (r)p (integer) = −
dl ·
,
2π
∂l

(10.25)(10.26a)(10.26b)(10.27)

where the integral route is taken counter clock direction along the edge of magnetic unit cell.

10.3.2 Hall conductivity from linear response theory
In the k·p perturbation for the band calculation, by renormalizing the plane wave part of wavefunction into the Hamiltonian we obtain the equation for the lattice periodic part unk (r). We can go the same way for the tight-binding modelin strong magnetic field. Operation of the Hamiltonian in (10.20) on the magnetic Bloch function in (10.25) can becalculated from peikr = eikr (ℏk + p) as(p + eA)2 eikr unk (r) = eikr (ℏk + p + eA)2 unk (r).
E13-8

We can rewrite the Schrödinger equation as
Hk unk (r) = Enk unk (r),Hk =1(−iℏ∇ + ℏk + eA)2 + V (r).
2m

(10.28)Now we utilize Kubo formula for Hall conductivity in (9B.3). We take the basis as magnetic Bloch funtions and state
indices are taken as (n, k). Velocity operator v can be written as v = (−iℏ∇ + eA)/m, and for the integration in the
numerator we write the matrix element of the operator by using braket representation unk (r) → |n, k⟩ as⟨n, k|v|m, k′ ⟩ = δkk′

∫ qa

∫ b
dx

0

dyu∗nk vumk′ ≡ δkk′ ⟨n|m⟩.

(10.29)0

From the periodicity in k space, the integration just on the magnetic unit cell is enough. The normalization should be
∫ qa

∫ b
dy|unk (r)|2 = 1.

dx
0

0

By using k-dependent Hamiltonian in (10.28), we can write down the matrix elements as
⟨
⟩
1
∂Hk
⟨n|vx |m⟩ =
n
m ,
ℏ
∂kx
⟨
⟩
1
∂Hk
⟨n|vy |m⟩ =
n
m ,
ℏ
∂ky
where k = (kx , ky ). These are further calculated as
⟩
⟨
⟩
⟨
⟩
⟨
∂um
∂Hk
∂un
m = (Em − En ) nm ,n= −(Em − En )
∂kj
∂kj
∂kj

(10.30a)(10.30b)j = x, y.

Substtuting the above to the Kubo formula (9B.3) to obtain]∑ [ ⟨nk|∂Hk /∂kx |mk⟩⟨mk|∂Hk /∂ky |nk⟩
e2 ∑ ∑
σxy = −i−c.c.
f (Enk )ℏ(Enk − Emk )2nkm(̸=n)⟩⟨⟩⟨⟩ ⟨⟩]∑ [⟨ ∂un
∂un
∂un
∂un
e2 ∑ ∑f (Enk )
m
m
m
m
= −i
−
ℏ
∂kx
∂ky
∂ky
∂kx
n
k
m(̸=n)[⟨⟩ ⟨⟩]∂un ∂une2 2π ∑ ∑∂un ∂unf (Enk )=−.
h i∂kx ∂ky∂ky ∂kxn(10.31)(10.32)k

Now we define a vector field Ank with
Ank =

∫

d2 ru∗nk ∇k unk = ⟨unk |∇k |unk ⟩.

(10.33)

We assume T = 0 and that EF is in the localized region. Writing the summation on k as the form of integration, σxy is
given byσxy =∫∫e2 1 ∑
e2 1 ∑
d2 k[∇k × Ank ]kz =d2 k[rotk Ank ]kz .
h 2πi
h 2πi
MBZ
MBZ
En <EF(10.34)

En <EFThe integration is over the magnetic Brillouin zone.
Because at the edges of a magnetic Brillouin zone, kx = 0 and kx = 2π/qa, ky = 0 and ky = 2π/b are the same
points, topologically the zone is two-dimensional torus T 2 = S 1 × S 1 . When Ank is single-valued on this torus, σxy
calculated from (10.34) is zero as known from the Stokes theorem. That is, for σxy ̸= 0, Ank should have non-trivial
topology. Here it is important that the magnetic Brillouin zone is a torus, which cannot be squeezed continuously tosingle point. If such squeezing is possible, Ank defined on the manifold cannot have non-trivial topology.
To see the topology of Ank , we consider local gauge transformation. A solution of Schrödinger equation (10.28)
uk (r) can be transformed with an arbitrary continuous function f (k) to another solution
u′k (r) = exp[if (k)]uk (r).
E13-9(10.35)

2p/a
H II

HI

ky

0

¶H

kx

2 p / qa

Fig. 10.7 Illustration of phase of wavefunction when it has zero in
the magnetic Brillouin zone.

u and u′ are physically the same. From the definition in Eq. (10.33), this transformation corresponds toA′nk = Ank + i∇k f (k).

(10.36)

To eliminate the above uncertainty originated from the gauge undertainty, we assume fixing the phase of unk (r) at one
point. With this, though, we cannot fix the entire phase over the whole magnetic Brillouin zone. We assume unk (r) is
zero at a point k0 . As shown in Fig. 10.7, the magnetic Brillouin zone is devided into retion HI that contains k0 and
residual region HII . If HI contains a zero, the phase must “rotate” around the zero as in the figure. On the other hand
HII should be connected at the edges as a torus and the wavefunction should have different structure in phase. Hence weneed to take different gauges in the two regions.
For simplicity we consider the contribution of band n only and n can be omitted. The integrals in (10.34) are, byapplying Stokes’ theorem to the two regions, given by
[∫
] I
∫
dk
1
d2 k[rotA]kz +d2 k[rotA]kz =(AII − AI ) ·
.
I=
2πi I
2πi
II
∂H

(10.37)The integral over circumference of region II cancels out due to the torus boundary condition (equivalent to “back andforth” integration over a single line). On the boundary ∂H, with gauge transformation the relation of the wavefunction isexpressed asiθ(k)uIk = uII.
ke

From the definition (10.33), the integral should be
I
[ II
] dk
∆∂H θ
II II
II
II
=
.
I=
⟨uk |∇k |uIIk ⟩ + (i∇k θ)⟨uk |uk ⟩ − ⟨uk |∇k |uk ⟩ ·
2πi
2π
∂H

(10.38)(10.39)The phase evolution over single circulation on the boundary ∆∂H θ should be an integer times 2π and I is limited to an
integer. Let νC be that integer. And let nB be the number of bands lower or at the same level as EF , we finde2,
(10.40)hwhich tells that the Hall conductance should be an integer times e2 /h. Equation (10.40) is called Thouless-Kohmotoσxy = nB νCNightingale-Nijs (TKNN) formula[5]. νC is called Chern number and known to be 1 for the Landau bands. The above
gives the same result as Eq. (10.18).
Chern number is the number of anomalies in the phase of wavefunction, equivalently the number of zeros. It is a kindof topological invariant. The origin of Chern number is in the topological property of energy bands. In order to turn atorus into a sphere, we should once tear up the surface around the hole then sew the surfaces together and finally erase
the hole. Similarly to change the band structure into the one with different topology (Chern number), we need to crush
the band gap once. For this reason, the Hall conductance found in TKNN formula is stable and precise regardless of thevariety of sample properties.
Here Ak is a Berry connection, rotAk is a Berry curvature in Appendix 9A. We will revisit them in the section of
topological insulator.
E13-1010.3.3 Laughlin’s gedankenexperiment
Robert Laughlin considered a sample in which a 2DEG is rolled into a cylinder with a radius of R and a circularelectrode is attached to the end of the cylinder (Fig. 10.8)[6]. The magnetic field is emitted outward from the core of thecylinder and is applied perpendicularly to the 2DEG. x and y axes are taken as in the figure. There is no edge because
the current is applied along x and the sample is closed along y. Further, a thin, long solenoid is placed at the core and an
applied current creates a magnetic flux Φ through it. The flux does not touch the 2DEG directly but gives an AB phaseon orbits going around the cylinder. The vector potential for the perpendicular magnetic field and that for the field by thesolenoid are in Ladau gauge(
A = (0, Bx),

AΦ =

Φ
0, −
2πR

)
.

(10.41)

We write down the wavefunction in the form of Eq. (10.12). Because the system is circular in y-directoin, the wavefunction should go around the circle.
The current in y-direction is
jy =

1 ∂Et
.
Lx ∂Φ

(10.42)

Et is the total energy on the cylinder per the normalization length Lx *3 . The vector potential in Landau gauge is
a = (0, Bx − Φ/Ly , 0). We take the unperturbed Hamiltonian H0 as the one of 2DEG under magnetic field. Then theeffect of solenoid flux is taken into account by the transformation
(
)
2π Φ
h
ky → ky −
,
ϕ0 ≡
Ly ϕ0
e

(10.43)in the Hamiltonian. This transformation corresponds to the variation in X-coordinate of the guiding center as
( )
Φ Lx
X→X+
.
ϕ 0 Nϕ(10.44)

Then the variation in penetration magnetic flux Φ → Φ + ∆Φ appears as that in X ∆X = (Lx /Nϕ )∆Φ/ϕ0 . If ∆Φ is2/Ly , which is q times the distance in x between the eigenstates.
an integer (q) times ϕ0 , then ∆X = qLx /Nϕ = 2qπlB

Namely the states shift to q-th next eigenstates and the variation in Φ is absorbed into the phase of wavefunction. When

R
F

F+DF

y
x

B
(a)

V

(b)

V

Fig. 10.8 (a) Rolled up two-dimensional system used in Laughlin’s gedankenexperiment. The magnetic field B isemitted outward from the core of the cylinder and is applied perpendicularly to the 2DEG. A thin, long solenoid is
running at the core of the cylinder giving an AB phase. The electric field is applied in x-direction. (b) Schematic
drawing of the variation in wavefunction when the flux by the solenoid is increased from Φ to ∆Φ.

Let L be the inductance of the cylinder. A state with current J has the magnetic energy
EH = L J 2 /2 = Φ2 /2L , which means ∂EH /∂Φ = Φ/L = J. Let Lx be the normalization length. From J = Lx jy we reach theequation.

*3 Here we prove the equation simply as follows.

E13-11

N Landau levels are completely occupied, that is ν = N , and there is an electric field of E in x, the variation of ∆Φ = ϕ0
causes a variation in the energy of ∆E = −eE∆XNe (Ne = νNϕ = N Nϕ ).
Here we assume as follows. In the quantum Hall state, the current does not depend on the boundary condition in
y-direction. In other words the current does not depend on the absolute value of Φ. Then we replace the derivative in(10.42) with the finite differece to find
1 ∂Et
1 ∆Et
1
jy =
=
=
Lx ∂Φ
Lx ∆Φ
Lx

(

Lx
−eE
Nϕ

)

Ne
e2
= N E.
ϕ0h(10.45)That is the Hall conductance σxy = jy /Ex is quantized as an integer times e2 /h. Because e2 /h is the conductance forsingle band, this is an indirect proof that the Chern number of single Landau level is one.

Appendix 9A: The Berry phase
A common classic example of the Berry phase is the parallel displacement of a vector on a sphere as shown on the left. When the vector is translated in three-dimensional
space on an appropriate trajectory and return to the original point, the vector does notchange. However if we apply a constraint that the vector should be in-plane during the“parallel displacement” (or the vector should be in the tangent plane of the sphere),then as in the left figure, the direction of the vector generally changes when the vector
origin comes back to the starting point. The angle between the starting vector and thereturning vector corresponds to the Berry phase.
Let α be the angle of variation in the direction of the vector, C be the tranjectory,
then α can be expressed as a line integral on C of a vector A. This A is called Berryconnection (Berry connection depends on the constraints on the vector movement). From Stokes’ theorem α can also be
written as the integration over an area S rimmed by C asI∫α=
A · ds =rotA · dσ,
CS

where rotA is called Berry curvature.
Let us go to quantum mechanics. We consider a time-dependent Hamiltonian H(t) and write the eigenvalue equationas
H(t)|n(t)⟩ = En (t)|n(t)⟩.

(9A.1)Taking time-derivatibe and operating the eigenfunction ⟨k| from left we get⟩⟨1∂Hn(t) .
k(t)En (t) − Ek (t)∂t∴ ⟨ṅ|n⟩ + ⟨n|ṅ⟩ = 0 ∴ Re(⟨n|ṅ⟩) = 0.

⟨k(t)|∂|n(t)⟩/∂t ≡ ⟨k(t)|ṅ(t)⟩ =(9A.2)(9A.3)

Let ψ(t) be a solution of the Schrödinger equation composed of H(t). ψ(t) is expanded by |n(t)⟩ as|ψ(t)⟩ =)(∫i t ′ ′ ′En (t )dt , (En′ (t) ≡ En (t) − ℏηn (t),cn (t)|n(t)⟩ exp −
ℏ 0
n

∑

ηn (t) = i⟨n|ṅ⟩).

(9A.4)

Substituting this into the Schrödinger equation we find
] ∑
[
]
)
[
∫
∫
i ′
i t ′ ′ ′i t ′ ′ ′iℏ ċn |n⟩ + cn |ṅ⟩ − En cn |n⟩ exp −E (t )dt =cn H|n⟩ exp −E (t )dt .
ℏ
ℏ 0 n
ℏ 0 n
n
n

∑

(

E13-12(9A.5)

Operating ⟨k| from the left, from Eq. (9A.2) we obtain[ ∫ t]∑ ⟨k|∂H/∂t|n⟩
dck
i
′ ′
′ ′
′
=
exp
(Ek (t ) − En (t ))dt cn .
dt
Ek − Enℏ 0(9A.6)

n̸=k

We consider variation of H(t) slow enough for the variation of the wavefunction to be adiabatic. We take the starting
point of the wavefunction ψ(0) = |m(0)⟩ and the adiabatic change means |ψ(t)⟩ = |m(t)⟩ with no mixing of othereigenstates. Let us express the time evolution of H as that in a set of parameters {Ri (t)}, which can be written in thevector form R(t). We consider a loop trajectory in R-space starting R(0) at t = 0 and coming back to R(0) at time T .
[
]
[
]
∫
∫
i t ′ ′ ′i t|ψ(t) = |m(R(t))⟩ exp −Em (t )dt = |m(R(t))⟩ exp −Em (t′ )dt′ eiγm (t) ,
ℏ 0
ℏ 0
∫ t
∫ t
where γm (t) =ηm (t′ )dt′ = i⟨m(R(t′ ))|ṁ(R(t′ ))⟩dt′ .
0(9A.7)(9A.8)

0

As known from (9A.3), γm is a real number. For a loop trajectory, with variable transformation t → R,
∫ T

I
⟨m(R(t))|∇R m(R(t))⟩ · Ṙ(t)dt = i

γm (T ) = i0⟨m(R(t))|∇R m(R(t))⟩ · dR(t))⟩dR = γm (C).

(9A.9)C∇R is the gradient operator in R-space. Below we omit the subscript R. The above equation means with a loop variationof Hamiltonian associated with adiabatic transition of the state, Berry phase γm (C) is added to the wavefunction. Further
by using Stokes’ theorem,Iγm (C) = −Im∫⟨m(R)|∇m(R)⟩ · dR = −Imc[∇ × ⟨m(R)|∇m(R)⟩] · dS

(9A.10)S

is obtained.

Appendix 9B: Kubo formula for Hall conductivity
The bf Kubo formula is the ultimate form of linear response theory developed from the first half to the middle of the20th century. There are various mathematically equivalent expressions in the Kubo formula, but here we introduce what
is called Nakano-Kubo formula. We consider a two-dimensonal electrons under perturbation eEy of electric field E in
y-direction. First order perturbed states |α′ ⟩ are written by unperturbed eigenstates |α⟩ as|α′ ⟩ = |α⟩ +∑ ⟨β|eEy|α⟩Eα − Eβ

β̸=α

|β⟩.

(9B.1)

To consider the Hall conductance we need to sum up the contributions from each |α′ ⟩ to the current along x-direction.
Then the current density in x-direction to the first order of perturbation is written asjx =∑ ⟨α|(−evx )|β⟩⟨β|eEy|α⟩
1 ∑1 ∑f (Eα′ )⟨α′ |ĵx |α′ ⟩ = 2
f (Eα )
+ c.c.,
2L α
L α
Eα − Eβ(9B.2)

β̸=αwhere f (E) is the Fermi distribution function, L2 is the area of normalization. Because the perturbation term is odd
function, there is no first order energy correcltion, and Eα′ = Eα . Fromii⟨β|vy |α⟩ = ⟨β|ẏ|α⟩ = − ⟨β|[y, H ]|α⟩ = − (Eα − Eβ )⟨β|y|α⟩,ℏℏthis ⟨β|y|α⟩ is substituted into Eq. 9B.2 to obtain
σxy =∑ ⟨α|vx |β⟩⟨β|vy |α⟩jx
e2 ℏ ∑+ c.c..
= 2
f (Eα )EiL α(Eα − Eβ )2βE13-13(9B.3)

Appendix 9C: Fractional quantum Hall effects
In the quantum Hall effect, various novel phenomena and ideas have been found. Among them we have a very shortlook at the fractional quantum Hall effect.

9C.1 Experiment on fractional quantum Hall effects
Fractional Quantum Hall Effect (FQHE) was found in transport experiment in a high-mobility 2DEG. In IQHE, theHall conductance plateaus appear at σxy = nGq (n is an integer) while in FQHE the conductance plateaus appear at
σxy = f Gq ,f=m(n : odd integer, m : integer).
n(9C.1)

Figure 9C.1 shows a representative mesurement of FQHE. The result contains IQHE though the widths of the plateaus
are not prominent and rather the behavior is on the classical line. And at the positions in (9C.1), narrow plateaus areobserved. On the other hand, the behavior of ρxx against the magnetic field is dramatic. Even for narrow plateaus at
positions (9C.1), ρxx goes to zero or becomes very small. Hence fine and steep oscillation is observed. Even in the highmagnetic field region where no IQHE is observed (filling factor ν < 1), fine oscillation is observed. In particular an
oscillation symmetric to ν = 1/2 is observed.
FQHE is very sensitive to the electron mobility, cannot be observed in low mobility samples. In comparison with IQHE,
FQHE is observed at lower temperatures with activation energy of a few K. Generally FQHE is easier to be observed at
higher magnetic field.
Before going into the physics, we have a short look at the mutual electron interaction and the localization. As wesaw in Sec. 10.2.2, when a 2DEG is under a strong magnetic field, the electrons at the Fermi level are in the edge modeat the equipotential lines of impurity potential. The localized state are the states going around the closed equipotential
lines. The electron-electron interaction gives some fluctuation to the impurity potential and there is a possibility to lift
the localization.

Fig. 9C.1 Representative example of FQHE measurement.

E13-14

9C.2 Laughlin state
It has been clarified by long-term researches that the electronic states causing FQHE is a kind of electron liquid, inwhich the electron mutual interaction is dominating the many-body state. The Laughlin state, in spite of its simpleness,has been proven to be a good approximation to such electron liquids. This is a big event in many-body physics since the
BCS theory.
We again consider a two-dimensional electron system on xy plane in the magnetic field of flux density B. Here forconvenience we take the symmetric gauge A = (−By/2, Bx/2). xy-plane can be expressed as a complex plane. The
spatial length is measured by the magnetic length. That is, a point on the 2DEG plane can be represented as a complex
number z = (x − iy)/l. The Hamiltonian with the electron-electron interaction is written as
H =

∑[ 1
j

2m

] ∑
(−iℏ∇ + eA)2 + V (z) +
j<k

e2
.
|zj − zk |(9C.2)First we make a many-body wavefunction from single-body wavefunctions at the lowest Landau level without potential
and the Coulomb interaction. Then the detail of the many-body wavefunction is determined to minimize the electroninteraction energy. The wavefunction which diagonalizes X 2 + Y 2 , thus the angualr momentum is written as()|z|2
ϕ(z) = p(z) exp − 2 ,l(9C.3)

where p(z) is a polynomial of z. Let Ne be the number of electrons and the many-body wavefunction can be written as()∑ |zi |2
ψ(z1 , · · · , zNe ) = f (z1 , · · · , zNe ) exp −,(9C.4)
4i
where a polynomial f should be anti-symmetric for the exchange in (1, · · · , Ne ) due to the Pauli principle.
∏
The general form of the terms in f is (coefficient)× i zimi . This mathematical form indicates that in the state thisterm represents, the i-th electron is occupying the state with angular momentum mi ℏ. Hence the total angular momentum∑M̂ in this term is i mi ℏ, and M̂ commutes with H . Because M̂ represents a conserved quantity, it is desirable to takeψ as to diagonalize H and M̂ simultaneously. For that f should be a homogeneous polynomial.
Further, to make the interaction energy smaller, we consider two-body correlation. The distance between two electrons
i and j is |zi − zj |. Then we try a functional form that f is given by a product of functions g that only depend on zi − zj ,that isf (zi , · · · , zNe ) =∏g(zi − zj ).

(9C.5)

i>j

From the anti-symmetric property of f , g(z) = z q and q should be an odd number. The above consideration is summarized intoψq (z1 , · · · , zNe ) =∏((zi − zj ) exp −qi>j∑ |zi |2i

4),

(9C.6)

which is called Laughlin state.
It has been clarified that various ground states exist in a two-dimensional electron system under a strong magnetic field
due to strong electron-electron correlation. The Laughlin state is proposed to explain FQHE. As we can guess from thefunctional form, it is composed to electron interaction energy. It is known that it is close to the exact solution in the finite
system obtained by using the exact diagonalization.

E13-15

9C.3 Filling factor of Laughlin states
In the Laughlin state (9C.6), let us consider the polynomial in front of the exponential. The electron coordinate zi has
a maximum power of M = (Ne − 1). This term of maximum power represents the state, in which the electron indexed√
i has the maximum angular momentum M ℏ. The orbit of this state spreads by l on the circle with a radius 2M l. The
area corresponding to Ne Landau levels is 2πl2 Ne and the filling factor of the state represented by the term isν=

2πl2 Ne
Ne
Ne
1
=
=
≈ .
π × 2M l2
M(Ne − 1)qq

(9C.7)

Among many terms in the polynomial, the ones with largest orbital radius are that gives the largest angular momentum
to single electron. Hence the filling factor of this term becomes the filling factor of ψq itself. In other words, the filling
factor determines q of the corresponding Laughlin state.

9C.4 Excited states
Next we consider the excitation from Laughlin state (9C.6). For that we write the state with increased angular momen∏
tum by one for each electron as i zi ψq .


2
∏
∏ ∑
∑
|z
|
m
j 
zi ψq =zi
Am1 ,m2 ,··· z1m1 z2m2 · · · zNeNe exp −(9C.8)
4
i
i
j


2
∑
∑
|z
|
m +1
j 
=
Am1 ,m2 ,··· z1m1 +1 z2m2 +1 · · · zNeNe exp −(9C.9).
4j∏i zi increases the angular momentum of each electron and at the same time4
introduces a zero at the origin * . Around the zero, the amplitude of the wavefunction is small with the scale of l and the

The operation of taking the product withnegative charge density decreases, which can be viewed as a positive charge around the zero. This can be treated as aquasiparticle.
We first take the product with∏qk (zk − z0 ) , which introduces q quasiparticles at a point z0 . Now we put an electronwith spatial size of l at z0 . Then the wavefunction is)(∏∏∑ |zl |2|z0 |2qq−.
(zk − z0 )(zi − zj ) exp −
4
4
i<j
k

(9C.10)lThis is nothing but a uniform Laughlin state with the electron number increased by one. The above operation means qquasiparticles with a positive charge and an electron with the charge −e are canceled out. This indicates we can cosider
that the charge of a quasiparticle is e/q.

9C.5 Composite fermion picture
In a Laughlin state (ν = 1/q), the electrons avoid each other and if we keep our eyes on a single electron, it looks as
if a single electron is in a uniform magnetic field. In the ν = 1 Landau level, single a quantum flux Φ0 is going throughthe area of a single electron. In the case of Laughlin state the number of quantum magnetic flux per an electron is q. Let
us consider such an electron as a “particle” with an even number (2k) of quantum flux. Such a particle obeys, if one goesback to Laughlin wavefunction, the Fermi statistices, hence they are called composite fermion (CF)[8]. The magnetic
field such CFs feels is that of q − 2k times quantum flux.
*4 With∏i (zi − z0 ) zero can be introduced any point z0 .

E13-16

That is, the field of q − 2k = 1 can be seen as IQHE state of n = 1 for CFs, where n is the filling factor of CFs.
Similarly in the case of 1/(q − 2k) = n > 1, IQHE of CFs appears for integer n. Because they express extended state ofCFs, the electron wavefunction is also extended. There, the filling factor ν of the electrons is
ν=

1
1
n
=
=
.
q
2k + 1/n
2kn + 1

(9C.11)For k = 1, this gives an FQHE series of 2/5, 3/7, 4/9, · · · , which is comparatively easy to be observed. Taking thesestates as the starting states, we can explain the next generation of FQH states. The above indicates that the FQHE ofelectron can be interpreted as IQHE of CFs. ρxx looks symmetric to ν = 1/2 and the oscillation can be interpreted as
SdH oscillation of CFs.

References[1] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980).
[2] Z. F. Ezawa, “Quantum Hall Effects: Recent Theoretical and Experimental Developements”, (World Scientifric,2013).
[3] M. Büttiker, Phys. Rev. B 38, 9375 (1988).
[4] M. Kohmoto, Ann. Phys. 160, 343 (1985).
[5] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Phys. Rev. Lett. 49, 405 (1982).
[6] R. B. Laughlin, Phys. Rev. B 23, 5632 (1981).
[7] D. J. Thouless, “Topological Quantum Number in Nonrelativistic Physics” (World Scientific, 1998).
[8] J. Jain, “Composite Fermions” (Cambridge, 2007).

E13-17

Lecture note on

Physics of Semiconductors (14)14th July (2021)

Shingo Katsumoto, Institute for Solid State Physics, University of Tokyo

Chapter 10 Spintronics
This is the final lecture of “Semiconductors.” I would like to close the lecture with a talk on semiconductor spintronics.
Though the lecture time is limited, beginning with semi-classical spintronics, spin-orbit stpintronics then finally we would
like to consider the spin Hall effect and the topological insulators.

10.1 Classical treatment of spin transport
10.1.1 What is spintronics?
So far we have seen tranport phenomena in semiconductors from classical to quantum mechanical. An electron has
a charge and simultaneously a spin, associated magnetic moment and angular momentum. Hence motion of an electron
is associated not only with the charge but with the spin angular momentum. However, due to Kramers degeneracy, spin
angular momentums cancel out each other and the microscopic transport of spin angular momentum does not appear in
the net current. Since electrons are charge monopoles, electronics have made great progress by controlling their flow
and accumulation, and semiconductors, which make this possible, have played a central role. However, now that the
down-sizing, speeding up, and lowering of energy consumption are reaching their limits, spintronics is to use spin, whichis the internal degree of freedom of electrons, for information storage and manipulation[1]. For a long time, magnetic
disks and tapes, which are examples of spins frozen by the many-body effect, have been used for information storage.
Giant magnetoresistance (GMR) devices, which utilize magnetic multilayers or spin-valve structure have been used forthe readout of such information from 1990’s. Since then, the word “spintronics” has been gradually used.
The reason why semiconductors were the center of electronics is, paradoxically, because they are insulators. It is mostimportant that there are no electrons in an undisturbed crystalline state, and it is several times more difficult to cut off
current in a metal with good control than to introduce conduction in an insulator, and in metal electronics. The role ofmetals in electronics is limited to wiring *1 ．On the other hand in spintronics, there is no net spin in normal metals in
equilibrium. They can be considered as “vacuum” for spins. In a sense, normal metals are semiconductors in spintronics,
substances to be treated in semiconductor physics.
The above is by no means a play of words. In electronics, there is almost no electric field, that is, a slope of chemical
potential in metals due to its short screening lengths. On the other hand, when an electric current is passed through aninterface between a normal metal and a ferromagnet, the chemical potential is separated by spin. This is similar to thesituation that in semiconductors under minority carrier injection in that different quasi Fermi levels are associated withelectrons and holes. Also, even inside metals, the spin current causes gradients in chemical potentials that can be detectedby external circuits. Therefore various physics we have seen in semiconductors may be observed as spintronics in normal
metals in modified forms.
*1 However, the establishment of the concept of electronic circuits itself depends on the existence of a substance called metal. For the detail, see

the lecture note of the present lecturer on electronics (in Japanese).

E14-1

10.1.2 Two-current model
In the two-current model proposed by Nevil Mott, the net electric current by electrons is devided into the portions
by up-spin (↑) and down-spin (↓) electrons. The model holds when the scattering time of spin flip is sufficiently longer
than those by various mechanisms (Sec. 5.1.5) dominating conduction. The difference in the resisitivities for ↑ and ↓ isdue to the difference in parameters like density of states, kF , etc. The total resistivity ρ is expressed by those for spin
subband channels ρ↑ , ρ↓ , as 1/ρ = 1/ρ↑ + 1/ρ↓ . In diffusive conductors, regardless of magnetic or non-magnetic, thespin diffusion length is generally longer than the mean free path (λF ≫ lF ), hence the two current model is considered
to work well. On the other hand, in ballistic transport, particularly in the presence of strong spin-orbit interaction, thetwo-current model meets difficulty. In this section we treat classical transport with no or weak spin-orbit interaction.
Hence the discussion is on the two-current model.
Assuming metallic conductors and the two-current model, we apply the Drude conductance to each spin-subband,
which is given by σs = e2 ns τs /m∗s (s =↑, ↓). The net electric current density jc is given by j↑ + j↓ while the spinpolarized portion is given by the difference jp↑ = j↑ −j↓ . The polarization of spin-polarized current density is defincedasPc =jp↑(↓)|j↑ − j↓ |=.
|j↑ + j↓ |jc

(10.1)

Each component of the current is further devided into the drift term and the diffusion term as
jps = σs E − eDs (−∇δns ).

(10.2)

10.1.3 Spin-dependent electrochemical potential
Even when there is nonequilibrium between spin subbands by spin injection(or emission), providing that intra-subband
scattering is sufficiently frequent and local equilibrium inside each spin band is kept, we can define local Fermi energy ϵs
and shift from the equilibrium δϵs for each spin-subband. For simplicity, a scaler σs is assumed for conductance tensor
σs . We write the electrostatic potential as ϕ (E = −∇ϕ). The Einstein relation σs = e2 Ns (EF )Ds (this experssion is
low-temperature, metallic version of Eq. (5.13))，δns = Ns (EF )δϵs gives the following.
[]σsDs ∇δnsσs
jps = −e∇ϕ −=[−e∇ϕ + ∇δϵs ].
e
σse(10.3)In the two-current model, the local electrochemical potential for each spin can be defined asµs = −eϕ + ϵs ,(10.4)

which leads to the expression of current density in each spin-subband
jps = −

σs
∇µs .
−e

(10.5)

Below we write electrochemical potential simply as “chemical potential.”

10.1.4 Spin current
There are several ways to defince spin current, namely flow of spin angular mometum. A representative one isj s (r, t) =ℏ(j↑ − j↓ ).
2(−e)

E14-2

(10.6)

A spin current generally is a tensor formed by local spin density vector and flow vector. In the above for simplicity, we
consider a spin current as a flow of z(direction of magnetization)-component of spin angular momentum. Also moregenerally, there is a type of spin current in which a flow of spin angular momentum is mediated by exchange interaction(e.g. by spin-wave).
With writing local spin angular momentum density as s(r, t), and the z component as sz , spin angular momentum
conservation law is written by∂sz+ div j s = 0.
∂t(10.7)In the presence of spin relaxation, we need to consider the relaxation term in the right hand side of Eq. (10.7). Within therelaxation time approximation, we can write
∂sz
ℏ
ℏ
∂sz
+ div j s =+∇ · (j↑ − j↓ ) =
∂t
∂t
2(−e)
2

(

δn↑
δn↓
−
τ↑
τ↓

)
.

(10.8)On the other hand, the charge (ρ) conservation law is given by∂ρ∂ρ+ divj =+ ∇ · (j↑ + j↓ ) = 0.
∂t∂t(10.9)In a steady state, ∂ρ/∂t = ∂sz /∂t = 0. From the constraint that there should be no total spin flip in the whole system
the relaxation times τ↑ and τ↓ should fulfill the detailed balance condition:N↑ τ↓ = N↓ τ↑ ,(10.10)where N↑,↓ are the spin-dependent density of states at the Fermi level. The above and Eqs. (10.8), (10.9) lead to∇2 (σ↑ µ↑ + σ↓ µ↓ ) = 0,1∇2 (µ↑ − µ↓ ) = F 2 (µ↑ − µ↓ ).
(λsf )(10.11a)(10.11b)

F −2
−2
−2
Averaged spin diffusion length λF
= (λF
+ (λF
, where λF
sf is defined from the Matthiessen’s law as (λsf )
↑)
↓)
↑,

λF
↓ are spin diffusion lengths for up and down spins respectively. Equation (10.11b) takes the form of diffusion equation,
thus called spin diffusion equation.

10.2 Spin injection and relaxation
There are various methods of injecting spins into a paramagnetic material, similar to injecting minority carriers into asemiconductor by irradiating light or applying a forward bias to a pn junction. Here, in particular, spin injection from aferromagnet to a paramagnetic material is described. As with minority carriers, spin injection occurs at the interface andspreads into the bulk and disappears (relaxes).
Let us consider an interface between a ferromagnet (FM) and a normal metal (NM) with a current jc perpendicular to
the interface. We write down the spin dependent local chemical potentials (Eq. (10.5)) for FM and NM regions in thefollowing form:
cM
M
M
µM
s = a + b x ± M exp
σs

(

x
λM
sf

)

(
)
dM
x
± M exp − M ,σsλsf(10.12)where M is F(ferromagnet) or N(normal metal). x-axis is taken to be perpendicular to the interface in the direction to thenormal metal, and the origin is taken at the interface. In the double sign ±, + corresponds to ↑, − does ↓. The sum of thefirst two terms in the r.h.s. is written as µ0 , which does not depend on spin. The function form of the third and the fourthterms comes from the fact that the spin-dependent parts should obey the diffusion equation (10.11). It is straightforward
to confirm that the expression in Eq. (10.12) satisfies Eq. (10.11).

E14-3

m
m0

m

m
m0

FM

NM
l sf Fl sf

NFig. 10.1 Sketch of spatial variation of spin-dependentchemical potential at an FM-NM interface, through whichelectrons flow from the FM side.

m

Coefficient a ∼ d is determined as follows. Under the assumption of two-current model, the local chemical potentialNµs for each spin subband must be continuous at the interface, i.e. µF
s (−0) = µs (+0). µ0 can be discontinuous at theinterface for non-equilibrium states while in |x| → ∞ both in FM and NM the difference between µ↑ and µ↓ approaches
zero. This means dF = 0 and cN = 0. Also the sum of current densities in spin-subband should be the total current
density jc .
From the above, the density of states and the spin polarization in F PF , µM
s is given by( )2 F(−e)jc PF λNxsf (1 − PF )σ
]
[
exp
,
F N
λF
σ λsf
sf
2
F
N
2σs σ 1 + (1 − PF ) N F
σ λsf
)]
[
(
N
(−e)jc(−e)jc PF λsfxN[] 1 ∓ exp − N
.
µs =
x+
σN
λsf
σ F λNσ N 1 + (1 − PF2 ) N sf
σ λF
sf
µF
s =

(−e)jcx∓
σF(10.13a)(10.13b)In the comlex symbol +, − correspond to ↑, ↓ respectively. The origin of energy is taken to the chemical potential inequilibrium (jc = 0). A schematic diagram is given in Fig. 10.1(a).

10.2.1 Spin injection and detection
Semiconductors are appropriate for the control of electric conduction because they are insulators in their intrisic states.
Similarly, non-magnetic materials are appropriate for the control and operation of spin current. For that, however, spin
injection into non-magnetic materials just like minority carrier injection in pn junctions, etc. In the configuration in

jc

Cu

V

31

30

Py2

-600 -400 -200

(a)

4.2K

32

Py1

R (m W )

jc

(b)

0

200

400

600

H (Oe)

Fig. 10.2 (a) Configuration of probes and circuit for non-local detection of spin injection. The electric current jc
goes through the center of the cross made of Cu to the left terminal while the spin current injected from Py1 reaches
Py2, causes separation of µ↑ and µ↓ hence a step in µ0 at the N-F (Cu-Py2) interface, which is detected as a voltage
V . (b) Thus measured non-local resistance. The spin-valve like magnetoresistance comes from the difference in thecoercive force between Py1 and Py2 due to the difference in the shape. The data are from [2].

E14-4

Fig. 10.1, spin current and electric current are overlapped and the electric separation of the two effects is difficult. Therefore in many of experiments on spin injection, non-local configuration of electrodes is adopted.
An example is shown in Fig. 10.2. Current jc through permalloy (alloy of Fe-Ni, Py)1 and the Cu sample causes
separation of µ↑ and µ↓ , which means a spin accumulation at the interfaces. Though no electric current flows between
Py2 and Cu, the spin diffusion occurs independently. The spin current thus also flows to Py2 and there causes differencein chemical potential, which is detected as a voltage. From Eq. (10.13) and the spin diffusion equation (10.11b), thedetected voltage is given by)(ρPy ρCu
1
L
2.
V = ± ejc PPy
exp−2
ρPy + ρCu
λCu
sf

(10.14)

Figure Fig. 10.2(b) shows the result of non-local measurement. Due to the difference in widths of Py1 and Py2,
the coercive forces for the magnetic field along the strips are different, which results in the spin-valve like non-local
magnetoresistance. The analysis of experimental results with various parameters by Eq. (10.14), material parameters likePF and λsf can be obtained.
Figure Fig. 10.3(a) shows a schematic view of non-local four-terminal probe configuration, which is often adopted forthe detection of spin injection into semiconductors. An electric current is applied between the left two electrodes including
a ferromagnet, and the difference in the electrochemical potentials of the two electrodes also including a ferromagnet in
the right is measured as a voltage. In such needle-shaped thin film electrodes, due to magnetic anisotropy from shape, themagnetic field is applied along the needles. On the other hand, a magnetic field perpendicular to the injected spins causes
precession of spin magnetic moment (Appendix 10A). If the electron spins rotates in perfect coherence and the starting
angle is also synchronized, the detected voltage oscillates reflecting the precession. In diffusion process, the distance of
migration largely distributes and in real samples the region of injection has a finite width and the oscillation decays withthe progress of precession. This is called Hanle effect.
If the problem is restricted to one dimensional spin diffusion along x-axis in a normal metal, spin dependence in σs ,Ds and τs can be dropped in Sec. 10.1. In Eq. (10.2), since only the diffusion term is effective for non-local effect, thedrift term is dropped. Applying the relaxation time approximation (10.8), we get the spin diffusion equation:∂sz∂ 2 szsz=D 2 −.
∂t∂xτsf(10.15)

2

P

V

MgO
NM

FM

FM

SC

NM

25K

1

D V ( m V)

jc

0

-1

AP

Gate
-2
-80

(a)

(b)

-60

-40

-20

0

20

40

60

80

H (Oe)

Fig. 10.3 (a) Four terminal probe configuration for detection of spin injection into semiconductors (SCs) with non-local resistance.
Current jc between left pair of ferromagnet (FM) and normal metal (NM) causes spin current to the right, which is detected as thechemical potential difference (voltage). MgO for the potential barrier is used for spin-injection with high efficiency. (b) Hanle signal
measured in a similar structure as in (a). The SC here is Si and the spin rotation is caused by the magnetic field perpendicular to thecurrent plane. From Ref. [3].

E14-5This leads to the expression of Hanle signal:∫ ∞jc Pj2∆V = ± 2
dtφ(t) cos ωt,
e NSC 0
(
)
(
)
d2
t
1
φ(t) = √
exp −
exp −
,
4Dt
τsf
4πDt

(10.16)

where d is the distance between the injection and detection electrodes, Pj is the spin polarization just below the injection
electrode, ω = gµB B/ℏ is the Larmor frequency (Appendix 10A).
Figure Fig. 10.3(b) shows a Hanle signal measured in an experiment, in which spins were injected from an Fe electrodeinto Si. The signal can be fitted by (10.16). The fitting provides the parameters λsf , etc.

10.3 Spin-orbit interaction
Henceforth we go into semiconductor spintronics. Here “Semiconductor” refers to materials defined by the charge
degrees of freedom that we have seen in the previous chapters. The spin-orbit interaction (SOI) has been already introduced when we had a look on k·p perturbattion perticularly for fcc-type semiconductors. In spintronics, the SOI isvery important since it connects spin and orbital degree of freedoms. Below we see an example that the SOI becomes
important even in the conduction bands through the mixing with the valence bands, in which the SOI is strong in intrinsic
bulk states. More specifically, we consider two type of spin-orbit interactions in two-dimensional electron systems.

10.3.1 Spin-orbit splitting due to bulk and structural inversion assymetries
In crystals when the lattice has the spatial inversion symmetry, the states of k and −k are degenerate. For example
in the case of ↑-spin states, E(k, ↑) = E(−k, ↑). The time-inversion operation causes −k → k and simultaneously
inversion in spin. When the crystals have time-inversion symmetry not having magnetism and the spatial inversion
symmetry, E(k, ↑) = E(k, ↓). In other words, in order for a system to have finite spin-splitting at finite k, the systemshould have some inversion asymmetry.
In the primitive cell of zinc-blende type crystals, it is apparent that there is spatial inversion asymmetry along [111],
which gives spin-splittings in energy dispersion due to the SOI. Such inversion asymmetries arising from crystal structures
is called bulk inversion asymmetry (BIA). SOIs caused by BIA is called Dresselhaus spin-orbit interaction. The
Dresselhaus SOI is obtained from k · p perturbation that takes BIA into account[4, 5]. BIA is some variance for k → −k,and then the interaction term should be odd order in k. In the caes of Dresselhaus interaction in three-dimensional systems
is in 3rd order. The form of the Hamiltonian is
3d
HDSO= γℏ2 [kx (ky2 − kz2 )σx + ky (kz2 − kx2 )σy + kz (kx2 − ky2 )σz ],(10.17)

where xyz is [100], [010], and [001]. When a two dimensional electron system (2DES) is formed on (001) surface, due
to the averaging of kinetic freedom in z-direction ([001]), a k-linear term appears.
2d
HDSO= γℏ2 [kx (ky2 − ⟨kz2 ⟩)σx + ky (⟨kz2 ⟩ − kx2 )σy ]= β(ky σy − kx σx ) + γℏ2 (kx ky2 σx − ky kx2 σy ).

(10.18)

While the BIA is from the crystal structure, the inversion symmetry is broken by introducing some structures that break
the periodicity like a hetro-interface. This is called structure inversion asymmetry (SIA). The SOI introcuded from the
SIA at an interface is called Rashba spin-orbit interaction [6, 7]. The Rashba-type interaction Hamiltonican is written
in the form:
HRSO = ασ · (k × ez ) = α(ky σx − kx σy ).
E14-6(10.19)

This comes from the expression Eq. (2.56). Here ∇V (in (2.56)) is an electric-field like term perpendicular to the 2DES
plane (z-direction) *2 . We need to be careful for “electric-field like term ∇V .” Let us assume that this ∇V /e is reallyan electric field for a while. Here, ∇V is the total (including the one from the band discontinuity) force, which confines
conduction electrons into the two-dimensional plane. The fact that the electrons are confined into the two-dimensional
plane means that the averaged expectation value of ∇V is zero due to the Ehrenfest theorem on the motion in z-direction,i.e. ⟨∇V ⟩ = 0[8]. Therefore, the Rashba interaction cannot be introduced from a real electric field ∇V /e.
As ssen in the k·p approximation (Sec. 2.2.6-7), the SOI comes from the mixing of valence-band wavefunction into
the conduction-band wavefunction at finite k. There, we introduced a potential V (Eq. (2.56)) common for conduction
and valence bands. If there is a difference in their band discontinuities, we need to consider different V ’s for them, andif ⟨(∇V )z ⟩ = 0 in the conduction band, (∇V )z ⟩ ̸= 0 in the valence band, then the Rashba interaction survives withthat. Along the above line, we write the potentials in conduction and valence bands as Vc and Vv respectively and derive
the expression for the SOI. Because ⟨(∇Vc )z ⟩ = 0, ∇V term is replaced by ∇Vv . In experiments, many phenomenacharacteristic to the Rashba SOI have been found, e.g. in a 2DES of a narrow gap semiconductor InGaAs. They are
considered to arise from the valence band.
We consider a 2DES which only has k-linear terms in the Rashba SOI (10.19) and the Dresselhaus SOI (10.18). We
take a plane wave with wavenumber k = (k cos φ, k sin φ) as the orbital part to write the SOI Hamiltonian as follows.
()(
0−ik̂x + k̂y0
HSO = α+βik̂x + k̂y0−k̂x + ik̂y
(
)
(
)
−iφ
0
ie
0
eiφ
= αk
−
βk
.
−ieiφ
0
e−iφ 0

E

E-

ky

E+

)

ky

m *a
h2

m *a
h2

k

kx

E

(b)
EF

kx

kx

(c)

k-

ky

(a)

−k̂x − ik̂y
0

ky

k+

k-

k+

4m*a
h2

kx

(d)

ky

kx

(e)

Fig. 10.4 (a) Upper: Energy dispersion relation (10.20) in the presence of the Rashba SOI. The spin at +π/2 istaken as “up.” Lower: Three dimensional wireframe expression of the energy dispersion on kx − ky plane. (b)∼(e)are cross sections at E = EF . (b) Two Fermi circles and the direction of effective field (spins) in the case of β = 0(Rashba model). (c) The same as (b) in the case of α = 0 (Dresselhaus model). (d) α and β are finite, but α ̸= β. (e)The case of α = β.
*2 From this expression, one often falls into the following incorrect explanation. – The existence of electric field means creation of charges both

upper and lower sides of 2DES. From an electron running in 2DES those charges cause a loop current enclosing the 2DES. The loop current
results in a magnetic field, which is nothing but the Rashba effective field. – For the reason why this is wrong, see the text.

E14-7

√
In the case of β = 0 (Rashba model), writing the spin part as t (1, eiϕ )/ 2, we get ϕ = φ ± π/2 from the conditionof eigenfunction. Namely the wavevector and the spin are orthogonal. The eigenenergy E± corresponding to ±π/2 are
obtained in the effective mass approximation as
E± =ℏ2ℏ2 k 2∓ αk =
∗
2m
2m∗

(
k∓

m∗ α
ℏ2

)2
−

m∗ 2
α .
2ℏ2

(10.20)Equation (10.20) indicates that the dispersion shifts in k-space depending on the direction of spin. The dispersion is
described as two spin-dependent parabolas as drawn in Fig. 10.4(a), and in Fig. 10.4(b) in a three-dimensional view. If
we cut the dispersion at E = EF , two cocentric Fermi circles appear. The direction of spin on the Fermi circles indicatedin Fig. 10.4(b) rotates inversely to each other. Similarly for α = 0, ϕ = −φ, −φ + π and the energy dispertion is in thesame form as Eq. (10.20) but with replacing α with β. However, a rotation of k on the Fermi circles causes an inverse
rotation of spin as sketched in Fig. 10.4(c).
Under the coexistence of α and β, generally the spin and the dispersion show complicated forms as in Fig. 10.4(d). In
the special case of α = β, the plane wave and the spin are separated as HSO = α(k̂x + k̂y )(σx − σy ). With rotating the√k ±kwave-vector part and the spin part as k± = y√2 x , χ± = ±t (1/ 2, (i − 1)/2), the Hamiltonian is expressed as
H =

ℏ2
2
(k̂ 2 + k̂−) − 2αk̂+ σz′ .
2m∗ +(10.21)σz′ is a Pauli matrix on the basis of χ± . Since the wave-vectors and the spins are separated, once the eigenfunction of thespin part χ± is determined, the dispersion is ordinary parabola whoes center shifts by ±2m∗ α/ℏ2 depending on χ+ , χ−
Therefore, as in Fig. 10.4(e), the centers of the spin-dependent two parabolas shift to each other. The two center-shifted
Fermi circles have a partial overlap.

10.3.2 Spin-orbit interaction and SdH oscillation
In a 2DES where the Rashba interaction is strong and β can be ignored, two Fermi cirlcles with different kF exist as in
2/(4π). The SdH oscillation in such a system
Fig. 10.4(b). This leads to spin-dependent 2DES sheet density nsσ = kFσshould have two different periods in 1/B plot as in Eq. (9.14) resulting in the beating among the two frequencies. The
difference in the size of the Fermi circles is proportional to α, thus to ⟨∇Vv z ⟩. Therefore the frequency of beatng shouldchange with applying the external electric field to cause the change in ⟨Vv ⟩.

6
V g = - 1.0V

T =0.4K

r xx (arb.)

-0.7
4

-0.3
0
0.3

2
0.5

Fig. 10.5 SdH oscillation observed in a 2DES at aquantum well of In0.53 Ga0.47 As. The parameter is thegate voltage Vg applied onto the surface of 2DES. The
arrows indicate the position of nodes in the beats. From
[9].

1.5
0

1

B (T)

2

E14-8

Figure 10.5(a) shows SdH oscillations measured in 2DES at a quantum well of (In,Ga)As grown on an InP substrate.
Although there is a large lattice mismatch between InAs (narrow gap) and GaAs (relatively large gap), lattice matchingis performed on the InP substrate by mixing crystals and setting the In composition to 0.53. In addition, heterojunctions
can be formed by adding Al to the mixed crystal. A clear beat appears in the SdH oscillation, and the position of the node
indicated by the arrow shifts due to the gate voltage, which is expected for the Rashba SOI.

10.4 Spin Hall effect and topological insulator
10.4.1 Spin Hall effect
When an electric field is applied to an electron system with SOI, a spin current is drived in the direction perpendicular
to the field. This phenomenon is called spin Hall effect. Let Jij be the spin current tensor with spin coordinate index i,flow coordinate index j. For the external electric field E, Jij is written as∑
Jij = σsϵijk Ek ,(10.22)kwhere ϵijk is the completely antisymmatric tensor indicating the mutual orthogonality of spin, flow vector of spin current
and electric field. σs is called spin Hall conductance. The spin Hall effect arises from impurity scattering, or SOI from
special orbital motion due to the band structure. The former is called extrinsic spin Hall effect while the latter is called
intrinsic spin Hall effect.
Due to space limitation here I just introduce an example of experiment. Figure 10.6 shows an experiment, in which aspin accumulation at edges of an n-type GaAs sample is detected by the difference in chemical potentials of ferromagnetic
electrodes. A clear spin signal which reverts with current reversal is detected (no signal for anti-parallel magnetization
configuration). The signal is due to the spin Hall effect. From the temperature dependence, it is concluded that the origin
is the extrinsic effect.

10.4.2 Anomalous velocity and spin Hall effectLet us consider the motion of a wavepacket in a crystal. The wavepacket is expanded by Bloch functions and theeffect of external force F = −eE from electric field E on each Bloch component is examined. We introduce “Bloch

15

jc
a

b

5

c
V

−e

∆ V ( µ V)

H

jc+

−500

500Oe

−15

AP

0
jc−

n-GaAs(a)−5(b) −1000−5000
H (Oe)

500

1000

Fig. 10.6 (a) Probe configuration of an n-type GaAs sample. Spin accumulation at the sample edges is detectedby the Fe electrode pairs placed perpendicular to the current. (b) Spin Hall signal from electrodes at 2 µm from thesample edges. The current density is 5.7 ×103 A/cm2 . The solid and open circles are the results for reversed currentsin parallel magnetization. The solid line is for anti-parallel magnetization. The inset shows the Hanle effect between
the electrodes. The temperature is 30 K.

E14-9

Hamiltonian” HB (k) = e−kr H0 eikr , where H0 is the crystal Hamiltonian, and the wave vector k is treated as aparameter. And the eigenfunction is a lattice periodic function unk (r).
From the space of unk (r), a variation in k can be viewed as that in the Hamiltonian. We have considered suchsituation in the introduction of Berry phase, in that the crystal wavenumber k is taken as the set of parameters R for
adiabatic transition. Then the Berry connection and the Berry curvature in this case are
⟨
⟩
⟨
⟩
∂unk
∂unk
∂unk
An = i unk, Bn (k) = i
×
,
∂k
∂k
∂k

(10.23)

respectively.
Let |nk⟩ be a Bloch function, and we drop the band index n limiting the band to n (single band). We assume thequantities in Eq. (10.23) are not zero, k-representation of the coordinate operator r̂ is⟨k|r̂|k′ ⟩ = (i∇k + A) δ(k − k′ ),where ∇k = ∂/∂k. This corresponds to that the dynamic momentum in a magnetic field is written as −iℏ∇ + eA in thecoordinate representation. Then we obtain⟨k|[x̂, ŷ]|k′ ⟩ = (i∇k × A)z δ(k − k′ ) = iBz δ(k − k′ ).
From the Heisenberg equation dq̂/dt = [q̂, H0 − F · r̂]/iℏ, we write the time evolution of operators x̂, k̂x by F as
⟨
⟩
⟩ [
⟨
]
∂E
dk̂x
dx̂ ′
δ(k − k′ )δ(k − k′ )k =, kk = Fx.
k− (F × B)x
dt
∂kx
ℏ
dt
ℏ
Under the above conditions, a wavepacket f is expanded by Bloch functions as f =∑k ak |k⟩ (ak = ⟨k|f ⟩).

time evolution of the averaged values of f in real and wavenumber spaces r0 , k0 are
⟨
⟩ ∑
dr̂
⟨f |k⟩
1
dr0
=v= f
f =
(∇k E − F × B) ⟨k|f ⟩ ≈ (∇k E − F × B)|k=k0 ,
dt
dt
ℏ
ℏ

The

(10.24a)k

dk0 /dt = F /ℏ,(10.24b)respectively. In (10.24a), the average over the wavepacket is replaced with the expectiation value on k. The second termin (10.24a) is the difference from the effective mass approximation due to the Berry curvature. This is called anomalous
velocity.
When the Fermi level is within a band gap (i.e. the system is a band insulator) and the anomalous velocity exists,
the Hall conductance is quantized as σxy = νe2 /h from the TKNN formula. However in ordinary situation with timereversal symmetry, the Berry curvature B is zero, no anomalous velocity exists, the Hall conductance disappears. As in↑↓
the two current model the system is devided into ↑, ↓ and we consider σxy
in each system. From the definition (10.6),↑↓
j s = (ℏ/(−2e))(σxy− σxy)E. Therefore
s
σxy
=

ℏ
−e ↑
−e
↓
(σ ↑ − σxy)=(ν − ν ↓ ) =νs .
−2e xy4π
4π(10.25)Here ν ↑,↓ is the Chern number for each spin subband and the difference between the two νs is called spin Chern number.
For the spin Chern number to be finite, there should be an effect that gives the same action as the magnetic field and theaction should be reversed with spin reversal because in (10.25) the difference between the spin-subband is taken. Systems
with a k-linear SOI just like Rashba model apparently fulfill the condition, which results in the appearance of the spin
Hall effect.

E14-1010.4.3 Quantum spin Hall effect
The above discussion is for an insulator and we need to be careful about “electric curren of each spin” [10]. Even
when the total net current is zero, current in each spin subband may be finite and in such a case, the Hall conductanceis quantized by e2 /h in each subband. Let us consider the case the region y < 0 in xy-plane is occupied with such
χa two-dimensional insulator. With Heaviside function Θ(x), the current components are jxχ = Θ(y)σxy
Ey , jyχ =χ−Θ(y)σxy
Ex , where χ is ↑ or ↓. Then the charge conservation is written as

dρχ
dρχ
e2
dρχ
+ ∇ · jχ =− δ(y)σxy χEx =− δ(y)ν χ Ex = 0.
dt
dt
dt
h
Taking the difference between the two equations for χ =↑, ↓, and integrating on the entire space of the system, we get−edSz= L νs Ex ,dt
2π
where Sz is the z-component of total spin of the system, L is the length at the edge y = 0. The result tells that we needsome anomaly at the edge to conserve Sz .
This leads to the idea of the edge states as in the quantum Hall effect, but in the present case there should be no net
electric current. Then we consider two edge states (helical edge states) with opposite charge velocity and spin at thebounday. We write their dispersions as Ek↑↓ = ±v(δkx − eEx t) (↑: +, ↓: −，δkx = kx − kF ). Because the variation in
the number of particles in the unit time is δN↑↓ = ±eEx L/2π
dSz
1e
= (δN↑ − δN↓ ) = L Ex .
dt
2
2π
With comparison of the two equations, from the condition dSz /dt = 0 in total, the number of helical edge states shouldbe the spin Chern number.
Such an insulator is called quantum spin Hall insulator or topological insulator. Since spin Chern number is an integer,from Eq. (10.25), the spin Hall conductance of a topological insulator is quantized by e/4π.

10.4.4 Quantum well of a topological insulator
We will conclude this lecture by introducing an experiment that verified a topological insulator with quantum spin Hall
effect for the first time. After this experiment, many topological insulators were discovered in a dozen years, and not
only topological insulators but also Dilac semimetals and Weyl semimetals were found, and a wide range of topologicalmaterials such as magnetic ones were found. The research is now widely going on. In addition, the reason why we namedE

conduction band

ordinary insulatorn s =0(a)(b)
EF

topological insulatorn s =1valence bandE14-11

Fig. 10.7 Concept of the topological insulator. (a) The region y < 0 is a 2dimensional topological insulator and therest is a vacuum, which is an ordinary insulator. A pair of helical edge states exists
at the boundary. The spin Chern number inthe topological insulator is 1 corresponding
to the nubmer of edge states. (b) The energy dispersion diagram. The helical edge
states have a linear dispersion relation[10].

Fig. 10.8 (a) Band structure around Γ-point of HgTe and CdTe. Γ6 , Γ7 , Γ8 are Koster symbols for space groupelements, which indicate the symmetries of the bands. (b) Schematic view of the positions of E1 and H1 subbands.
(c) Longitudinal resistances of CdTe/HgTe/CdTe quantum wells. The gate voltage is measured from the position atwhich the Fermi level comes to the center of the gap. Well widths, I: 5.5 nm; II, III, IV: 7.3 nm. The distance between
the electrodes are 13 µm for I and II, 1 µm for III and IV.

the ”quantum spin Hall effect” is that quantum hole insulators, which have a long history, are also considered to be a typeof topological insulator, so the first topological insulator discovered by humans is called the quantum hall insulator. The
reason why I’ve added “quantum spin Hall effect” is that quantum Hall insulators, which have a long history, are also
considered to be a type of topological insulator, so the first topological insulator discovered by humans should be thequantum Hall insulator and I think we need to mention that clearly.
Figure 10.8 shows the setup of the experiment and the results. HgTe is used as the topological insulator. A thin film of
HgTe is inserted between two CdTe films, which work as barrier layers and form a quantum well. Figure 10.8(a) shows
the band diagrams of HgTe and CdTe zinc blende crystals calculated by 8-band k·p model with SOI. In CdTe, just like
GaAs, the Γ6 (J = ±1/2) conduction band mainly comes from s-orbital while the Γ8 (J = ±1/2, ±3/2) valence bandplus the Γ7 spin-split-off band come from p-orbitals. On the other hand in HgTe, the strong SOI causes a band inversion,that is the Γ8 band floats above the Γ6 band. In a HgTe quantum well, the quantum confinement modifies the band
structure. We name quantum-confined level from the electron-like dispersion band Γ6 as E1, and that from the hole-like
dispersion band Γ8 as H1.
A theoretical model named Bernevig-Hughes-Zhang(BHZ) model was proposed for the HgTe quantum well[11]. According to the model, as long as the order of E1 and H1 bands on the energy axis keeps the inversion (EH1 > EE1 ) the
Chern number (Z2 topological number) is 1 namely the HgTe quantum well is a two-dimensional topological insulator.
The quantum well potential is symmetric for the center of well and there is no SIA hence no Rashba SOI. Hence the wellstructure does not give important change in the SOI. On the other hand, the quantum confinement enhances E1 level andlowers H1 level. With decreasing the well width, then, the order in the level is transformed into the ordinary one at thecritical width and the system goes into an ordinary insulator.
In Fig. 10.8(c) the four-terminal resistance of quantum wells of CdTe/HgTe/CdTe, in which the E1-H1 crossing critical
width is dc =6.3 nm. The well widths is 5.5 nm (less than dc ) for I and 7.3 nm for others. When the well width is lessE14-12

than dc , the resistance is very high around Vg = 0 indicating that the system is an ordinary insulator. On the other handwhen the width is wider than dc and H1 level places above E1, a topological insulator is realized and a helical edge state
appears at the sample edge. The electric conductance through the helical edge state should be 2e2 /h from the Landauer
formula (??). Actually in the samples III (sample width 1 µm), IV (width 0.5 µm), the conductances are 2e2 /h aroundthe gate voltage for the insulating phase, which fact indicates the realization of the topological insulating (quantum spinHall) phase.

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766 (2007).
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Appendix 10A: Motion of small magnetic moment
10A.1 Electron spin in a magnetic fieldLet us consider a single electron spin s in static magnetic field B0 along z-direction. We consider only the Zeeman
energy:H = (eℏ/2m0 )gB0 ŝz = gµB B0 ŝz ,where µB is the Bohr magneton. From the commutatio relatoin of spin operators [ŝj , ŝk ] = iŝl /2 ((j, k, l) are cyclic
replacement of (x, y, z)),[H , ŝx ] = igµB B0 ŝy , [H , ŝy ] = −igµB B0 ŝx , [H , ŝz ] = 0.
Hence the Heisenberg equation of motion tells.
∂⟨sx ⟩gµB=−
B0 ⟨sy ⟩,∂tℏ∂⟨sy ⟩gµB=
B0 ⟨sx ⟩,∂tℏ∂⟨sz ⟩= 0.
∂teg∴ ⟨sx ⟩ = A cos ω0 t, ⟨sy ⟩ = A sin ω0 t, ⟨sz ⟩ = C, ω0 =
B0 ,
2m0

(10A.1)(10A.2)where A2 + C 2 = s2 . Equation (10A.2) is representing precession around z-axis with the Larmor frequency ω0 .
Next we add a rotating magnetic field B1 (ex cos ωt + ey sin ωt) in xy-plane. The time-dependent Hamiltonian withthat is written as
H (t) = gµB (B1 cos ωtŝx + B1 sin ωtŝy + B0 ŝz ).
E14-13

The time evolution of spin-wavefunction χ(t) = u(t)| ↑⟩ + d(t)| ↓⟩ is represented as
( )
(
)( )
∂ u
B0
B1 e−iωtuiℏ= −gµB.
B1 eiωt−B0d∂t d
The solutions of the above simultaneous differential equations are expressed asu(t) = C(Ω ∓ ω0 ± ω)ei(±Ω−ω)t/2 ,(10A.3a)v(t) = ±Cωc e(10A.3b)i(±Ω+ω)t/2

where C is an integration constant, ωc = eB1 /m0 , and Ω =√,(ω − ω0 )2 + ωc2 . Taking the initial condiction as u(t) = 1,d(t) = 0, the solutions are√u(t) =

ω22 − c2 sin
Ω

(

)
ωc
Ωt
Ωt
+ α e−iωt/2 , v(t) =sin eiωt/2 ,2Ω2

where α = arctan(Ω/(ω − ω0 )). Then we obtain|d(t)|2 =

ωc2
Ωt
,
sin2
(ω − ω0 )2 + ωc2
2(10A.4)

which indicates a Lorentz type resonance and the osicllation with the frequency Ω(= ωc ) at ω = ω0 .

10A.2 LLG equation

B

Appling the equation of motion (10A.1) to a general macroscopic magnetic

damping

moment M , we obtain the Landau-Lifshitz equation-M ´ BgµB∂M=−
M × B.
∂tℏ(10A.5)The addition of the relaxation R of M gives

q

gµB
∂M
=−
M × B + R.
∂tℏ

M

(10A.6)

What should be the mathematical form of R? Since M has the lowest energyjat the direction of B, the relaxation should be a force to this direction as shownin the left. The force is perpendicular to −M × B and M . It is natural to infer
that
RLL = −λM× (M × B),|M |(10A.7)

where λ is a constant. This is called the Landau-Lifshitz damping term.
Another idea is that the relaxation rate should be proportional to time-variation of ∂M /∂t, which has the same direction
as −M × B. In this idea the damping term can be written with a constant α as
RG = α

M
∂M
×
,
|M |
∂t

(10A.8)which is called Gilbert damping term. If we substitute the equation of motion (10A.5) into this ∂M /∂t, the term isthe same as the Landau-Lifshitz term. Adopting RG for the damping, we reach the Landau-Lifshitz-Gilbert (LLG)
equation:
∂M
gµB
M
∂M
=−
M ×B+α
×
,
∂t
ℏ
|M |
∂t
which is often used to describe motions of magnetization phenomenologically.
E14-14(10A.9)