Lecture on

Semiconductors / 半導体(Physics of semiconductors)
2021.4.7 Lecture 01
Institute for Solid State Physics, University of Tokyo

Shingo KatsumotoHow the lecture will go on?
 Powerpoint (converted to pdf) file will be uploaded in the corresponding ITC-LMS site,(https://itc-lms.ecc.u-tokyo.ac.jp/lms/course?idnumber=202135603-00290F01)
by the day before the lecture.
 The lecture notes (in Japanese, English) will be uploaded in the site
https://kats.issp.u-tokyo.ac.jp/kats/semicon4/by the end of the lecture week.
 Small amount of problems for your exercise at home will be given in the last of thelecture in every two weeks. Submission deadline of the solutions is two weeks later. I
hope I can collect them through LTC-LMS but if that is difficult I will prepare my own
web script.

 In the very last of the lecture in July, the problems for your report will be given. The
deadline for the submission of the report will be notified then.
 The lecture is recorded on the cloud. I hope I can upload the video for one or two
weeks.
 I hope I can find some ways to get questions from you (via chat, etc.?)2

Lecture Plan
Related site: https://kats.issp.u-tokyo.ac.jp/kats/semicon4/
1) Crystal structure and crystal growth１）結晶構造と結晶成長

2) Energy band, effective mass approximation

２）エネルギーバンド，有効質量モデル

3) Carrier statistics and chemical doping３）半導体キャリア統計とドーピング

4) Optical properties

４）光学的性質

5) Semi-classical treatment of charier transport

５）電気伝導の半古典論

6) Homo/hetero junctions, semiconductor devices６）ホモ・ヘテロ接合，半導体デバイス（光，電子）(optical, electrical)

７）微細構造技術による量子構造（量子井戸，細線，

7) Quantum structures (quantum wells, wires,

ドット）dots) by nanofabrication techniques８）量子輸送の基礎

8) Basics of quantum transport９）電流磁気効果，量子ホール効果

9) Galvanomagnetic effects, Quantum Hall effects１０）スピン物性 (スピントロニクス)

10) Spin-related phenomena (spintronics)

１１）トポロジカル効果

11) Topological effects

3

Characteristics of semiconductors• Not metal• Middle range band gap• Weak divergence of resistivity with lowering
temperature
Structure sensitive (conduction) properties• Drastic changes in electric conduction withultra-small amount of impurities• Changes in electronic and optical properties
with quantum confinement structures likequantum wells, wires, and dots

Yu & Cardona,“Fundamentals of
Semiconductors”

4Chapter 1 Crystal structures and crystal growths

http://www.geologyin.com/2014/11/crystal-structure-and-crystal-system.html

Crystal structurepolysinglecrystals
Uniform solids
amorphous

Crystals: Spatially periodic structures
Unit of spatial repetition Primitive cell: unit of spatial repetition with smallest number ofatoms
Unit cell: unit of spatial repetition taken as for human to findex) GaAs
symmetry of the crystal

face centered cubic (fcc)
Unit cell

Primitive cell

6

Bravais lattices

7

Lattice, reciprocal lattice (1)ex) diamond structure (fcc)

Lattice: spatial repetition of the unit structure.

a0

𝑙𝑖 : integers, 𝒂𝑖 : primitive (translation) vector
𝑹: lattice vector

a3

z

a2
a1

y

Lattice potential 𝑈(𝒓)𝑮: reciprocal lattice vector

a0

a0

x

8

Lattice, reciprocal lattice (2)b2ex) diamond structure (fcc)
reciprocal lattice: bcc
kz

b3

ky
b1

primitive reciprocal vectors

kxFirst Brillouin
zone
G/2

G

Plane that cuts G at G/2 vertically→ unit cell in the reciprocal lattice:

Brillouin zone

Points withhighsymmetries:
G, X, L, W
9

Inorganic crystals often used as semiconductors: Group IV
Diamond structure (fcc)

diamondsiliconII

III

germanium

IV

V

VI

(a-Sn)SiC
SixGe1-x

10Inorganic crystals often used as semiconductors: Group III-V
Zinc-blende (fcc)

GaAs

II

III

IV

V

VI

(ZB)

GaN
Wurzeit (hexagonal)

(WZ)

CdTe
(ZB)

11

Energy gaps and lattice constants of representative (cubic) semiconductors
Room temperature

Energy gap (eV)Colored stripes:
Lattice matching groups→ Heterojunctions are available
Lines: Mixed crystals
GaN (Wurzeit)
a = 3.19 Å, c = 5.19 Å
3.4 eV (RT)

Lattice constant (Å)

12Various methods for semiconductor crystal growth
Vertical
Bulk growthCzochralski
Bridgman
Floating zone melting

Horizontal Bridgman
Horizontal
Zone melting
Chemical Vapor Deposition(Vapor Phase Epitaxy)

Thermal decomposition
Plasmachlorideorganic
metal

Liquid Phase Epitaxy
Thin film growth

Molecular Beam Epitaxy
Atomic Layer Deposition
Laser ablation“Tree-like” classification is
actually difficult.
13Crystal growth: Czochralski method
Liquid encapsulated
Czochralski (LEC)(GaAs, InP, etc. high vapor
pressure materials)

B2O3 etc.

Czochralski method forsilicon (Wikipedia)

14

Bridgman methods
Bridgman-Stockbarger method

Horizontal Bridgeman (HB) process

From Wikipedia

temperature gradient freezing method
15

Floating zone method

https://www.youtube.com/watch?v=jPijg8NIamo
16

Chemical vapor deposition (CVD), metal-organic CVD (MOCVD)
Thermal decomposition

MOCVD(organometallic vapor phase epitaxy, OMVPE)

Plasma CVD

Organic metal gases
17

Molecular Beam Epitaxy (MBE)
Ultra-high vacuum evaporation

Refractive high energy electron diffraction (RHEED)

heating
Substrate
RHEED

Ga

Mn In

As

shutter
control

HRTEM
intensity (arb.)

Sb

1.5

1

0.5
0

time
2

4

6

8

10

18

Chapter 2 Energy bands, effective mass approximation

Bing Concert hall at Stanford University
https://www.deccaeurope.com/Case-Studies/bing-concert-hall-at-stanford-university-california

Bloch theorem and nearly free electron model
Bloch theorem
Eigenstates in lattice potential:

𝑹: Lattice translation vector
n: band index

One-dimensional system with a weak periodic potential

Perturbation is important from 𝑘 ± 𝑘𝑤
Energy crossing between |𝑘 and |𝑘 − 𝑘𝑤 occurs around 𝑘 = 𝑘𝑤 /2

Hamiltonian around 𝑘 = 𝑘𝑤 /2 in the space formed with |𝑘 and |𝑘 − 𝑘𝑤

20

Nearly free electron model (2)Bloch theorem
Standing wave
Energy gap: ∆𝑘 = 0 → 2𝑉0

Lattice periodic function
Points 𝑘𝑤 /2 and −𝑘𝑤 /2 are equivalent

Shifts from these points can berenormalized into 𝑢𝑛𝑘 (𝑟)→Reduced zone expression
Overall dispersion

Energy gap due to the phases ofstanding waves

21

Nearly free electron model (3) Reduced zone expression

extendedzone

reducedzone

22

Empty lattice approximation
𝐸(𝑘)1st Brillouin zone
The free space has a lattice periodicity.

𝑘
4𝜋
−
𝑎

2𝜋 𝜋
−
−
𝑎
𝑎

1st Brillouinzone
of fcc lattice

0

𝜋
𝑎

2𝜋
𝑎

Empty lattice
expression
for fcc4𝜋𝑎distanc Points
e

num
ber

0

(0,0,0)1

3(1,1,1),…

8

2(2,0,0),…

6

8(2,0,2),…

12

11(3,1,1),…

24
23

Empty lattice approximation and more realistic band structure

Empty

Si pseudo potential calculation24

Tight-binding approximation
Single atom on single unit cell
Single atom Hamiltonian:

𝑇: kinetic energy, 𝑢: atomic potential

𝜙𝑛 : eigenfunctions for 𝑅𝑖 = 0

Lattice periodic function: Bloch form
25

Tight binding approximation (2)Crystal field contribution
Hopping integral

26

Tight binding approximation (3)𝑡𝑛 nearest neighbor only = 𝑡

Cosine band with the width of 4𝑡.

27

Lecture on

Semiconductors / 半導体(Physics of semiconductors)
2021.4.14 Lecture 02
Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of last week
Chapter 1 Crystal structure and crystal growth
Crystal structure

•
•
•
•

Basis, primitive cell, unit cell
Lattice, Bravais lattice
Reciprocal lattice
Brillouin zone

Semiconductor materials

•
•
•
•

Crystal growth• Czochralski• Bridgman• Floating zone

Bulk

Group IV: C, Si, Ge, Sn, SiC, SixGe1-x
Group III-V: GaAs, InP, AlAs, InAs, GaSb, …
III-N: GaN, InN, …
Group II-VI: CdTe, HgTe, …
• MOCVD
Thin film• MBE

Chapter 2 Energy bands, effective mass approximation
Nearly free electron model, empty lattice approximation2

Tight-binding approximation (TBA)
Single atom on single unit cell
Single atom Hamiltonian:

𝑇: kinetic energy, 𝑢: atomic potential

𝜙𝑛 : eigenfunctions for 𝑅𝑖 = 0

Lattice periodic function: Bloch form
3

Tight binding approximation (2)Crystal field contribution
Hopping integral

4Tight binding approximation (3)𝑡𝑛 nearest neighbor only = 𝑡

Cosine band with the width of 4𝑡.

5

Methods for obtaining band structure
Experiments

•
•
•
•
•

Hot electron transport
Optical absorption
Electroreflectance
Cyclotron resonancePhotoemission spectroscopy

Empirical calculation• Pseudo-potential approximation• 𝑘 ∙ 𝑝 perturbation

ab-initio calculation• Local density approximation• Augmented plane wave
• Generalized gradient approximation
6

Angle resolved photoemission spectroscopy (ARPES)

GaAs

T. C. Chang et al.
Phys. Rev. B21,
3513 (1980).

see, e.g. for short review B. Lv, T. Qian, H. Ding, Nature Reviews Physics 1, 609 (2019).

7

Angle resolved photoemission spectroscopy (ARPES) (2)Soft-X ray ARPES of GaAs
Bands below Fermi energy can be detected
Strocov et al., J. Electron Spectroscopy and Related Phenomena 236, 1 (2019).

8

Plane wave expansion
Crystal Schrodinger equation:(1)

Bloch function (omit band index)(2)

Fourier expansion(∵ lattice periodicity)(3)(2), (3) → (1)(4)

Each term in the sum over G iszero in (4)(5)

For (5) to have non-trivial solution
9

Pseudo-potential calculation method→ We need 𝑉𝑮
Pseudo potential method: 1. Only consider valence bands and conduction bands around the
Fermi level. Effect of core electrons in renormalized into
periodic potential.
2. Replace real potential with pseudo potential which gives
similar tailing of wavefunction.
V(r)
Vp(r)
uk(r)

ukp(r)

band structure: almost determined in skirt characteristics
10

Pseudo potential calculation method (2)Replacement withpseudo potential𝑍𝑒
𝑉 𝑟 =−
𝑟
simplest example

Crystal pseudopotential: vectors pointing nuclei in the unit cell

Fourier transform:
𝑁: unit cell number
Ω: unit cell volume: form factor (Fourier transform of 𝑊𝑝 (𝑟)) depends only on potential form
: structure factor depends only on internal structure of unit cell11

Empirical pseudo potential calculation for fcc semiconductorsex) GaAs

Distance from the origin and numberof points in reciprocal lattice
distance

Pointsnumber0(0,0,0)1

3(1,1,1),…

8

2(2,0,0),…

6

8(2,0,2),…

12

11(3,1,1),…

24

Form factors obtained from fitting to optical experiment12Band structure of Si and GaAs
conduction valley energy surface

Si
001

010
100

spin-orbit interaction is not taken into account

GaAs

13

Definition of effective massGroup velocity of wavefunction withenergy eigenvalue 𝐸𝑛 (𝑘)Acceleration
With definition: inverse effective mass tensor:effective mass tensor

14

Energy surface measurement (cyclotron resonance)
Motion of charged particle in magnetic field:
cyclotron motion in the plane perpendicular
to the magnetic field

Cyclotron frequency
Landau quantization

Optical pumping → microwave absorptionGe𝜔𝑐 → cyclotron mass 𝑚c

B𝜃

Energy surface: ellipsoid

Electron-hole distinction ← circular polarization(24 GHz microwave)
Dresselhaus, Kip, Kittel, Phys. Rev. 98, 368 (1955).
15

Cyclotron resonance16

k·p perturbation
Crystal Schrodinger equation:(1)

Bloch function(2)

Equation for lattice periodic
function(3)

Perturbation by 𝒌-dependent term
Good approximation for small k → band edge information(a) In the case of no degeneracy

17k·p approximation (2)(b) In the case of n-fold degeneracy in 𝑢00 (𝒓)orthogonal

Approximate the perturbed wavefunction
Substitute to the equation for u
Taking inner product with |0𝑖(4)For eq.(4) to have non-trivial solution

18

Spin-orbit interaction
Spin-orbit Hamiltonian
spin-operator

From identity

eigenequation19

G-band edges of diamond and zinc-blende semiconductorsEunperturbed equation:

electron

Diamond, zinc-blende: formed from 𝑠𝑝3 orbitalsEckperturbation

heavy hole
Ev1
Ev2

non-zero element

light holespin-split off

𝛼: spin coordinate

Ev3

20

G-band edges of diamond and zinc-blende semiconductors (2)21G-band edges of diamond and zinc-blende semiconductors (3)Eigenvalue equation

EelectronEc

Ignoring the term 𝑃 2 𝑘 2 we finally obtain:kheavy hole
Ev1
Ev2
light holespin-split off

Ev3

22

Band warping, a conventional way to get band parametersEkconstant-energy
surfaceheavy hole
Ev1
Ev2

light hole

Summary of 𝑘 ∙ 𝑝 second order perturbation

23

Table of band parameters

Lundstrom “Fundamentals of Carrier Transport” (Cambridge, 2000).

24Graphene: A two-dimensional material (another example of TBA)
Graphite

Graphenevan der Waals

covalentGraphene lattice/reciprocal lattice structure
Atomic orbitals

Honeycomb latticeGraphene lattice/reciprocal lattice structure
Lattice: unit cell

Reciprocal latticeTight binding model
Sublattice wavefunction
tight-binding
Linear combinationtight-binding
Hamiltonian equation

Eigenvalues:Sublattice transition term

Take the nearest neighbor approximation:

Dirac points in k -space

A Dirac point𝑘𝑥𝑘𝑦Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.4.21 Lecture 03
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of lecture in the last week
Chapter 2 Energy bands, effective mass approximation
Energy band calculation

Nearly free electron approximation
Tight-binding approximation(empirical) Pseudo-potential calculation method
k·p perturbation method

Energy band measurement

Angle-resolved photoemission spectroscopy (ARPES)
Cyclotron resonance

Example of tight-binding approximation: Band structure in graphene2

Contents today
Envelope function (effective mass approximation)
Chapter 3 Carrier statistics and chemical doping

Density of states
Definition and properties of valence band hole states

Carrier distribution in intrinsic semiconductors
Shallow hydrogen-like impurity states
Shallow impurity states in Si
Doping and carrier distribution

3

Envelope function (effective mass approximation)
Inverse effective mass tensor:
Problem: Non-uniform perturbation potential
Schrödinger equation
Expand 𝜁(𝒓) with Blochfunction 𝜓𝑛𝒌 𝒓 = |𝑛, 𝒌𝑛′ , 𝒌′ | →Fourier transform of 𝑈(𝒓)Fourier expansion of4

Envelope function (2)Ω0 : unit cell space,𝑣0 : unit cell volume

Assumption: 𝑈(𝒓) varies little inthe scale of the lattice constant𝑈(𝒓) is weaker than the latticepotential: Elements between
different n are negligible
5Envelope function (3)Assumption:

Effective mass equation

6

Derivation of effective mass equation with Wannier function
Wannier function (WF):
Fourier transform of Bloch function

Expansion by Wannier functions

WF tends to localize around the lattice points.
WF are orthogonal.

Effective mass equation
Effective mass approximation

7

Ekimov et al., Nature 428, 542 (2004).

Diamond
Boron doped diamond

Chapter 3 Carrier statistics and chemical dopingDensity of states
Number of states per energy (per volume)

E

E

E
9

Electrons and holes

Fermi (electron) distribution function

vacuumsingle hole state

𝐸(𝑘)

𝐸(𝑘)
𝑇 → 0 Step function
vacuum
c.b.

c.b.

E
k

k

v.b.

Electric field E

vacuum total current
single empty
state at k invalence bandv.b.

All the electrons in the v.b. move in k-space in this way. So does theempty state.

Equation of motion of the empty state10

Definition and properties of valence band hole states𝐸(𝑘)Definition: single holevalence band electrons with a single empty Bloch state

c.b.

(i)

Becausein the vacuum state.

k

(ii)

Energy measured from the valence top

v.b.

(iii)

𝜀

hole band

(iv)
k
v.b.
11

Carrier distribution in intrinsic semiconductors
Hole distribution function

Numbers of electrons and holes exist between 𝐸 and 𝐸 + 𝑑𝐸

Approximate density of states with those of free particles

12

Carrier distribution in intrinsic semiconductors (2)Maxwellian approximation

𝜌(𝐸): effective density of states
13

Carrier distribution in intrinsic semiconductors (3)Mass-action law
The charge neutrality condition𝑛𝑖 : intrinsic carrier density

General expressions

14

Doping and carrier distribution
Si

Si

Si

Si

Si

Si

III

Sb

Si

Si

Si

Si

II

Si

IV

V

VI

B

Si

Si

donor
emits an electron

acceptoremits a hole

Donor concentration is higher: n-type
Acceptor concentration is higher: p-type
Donors and acceptors compensate each other.
For silicon

Donors: P, As, Sb
Acceptors: B, Al, Ga

15

Impurity levels in Si and GaAs

S. M. Sze, Physics of semiconductor devices

16

Shallow hydrogen-like impurity states
Conduction mass is isotropic and unique.

001

Effective mass equation
010

We can readily use the results of the hydrogen atom with replacing the massand the dielectric constant.

100

GaAs

1s wavefunction𝑓(𝑟)Effective Rydberg constant:

Effective Bohr radius:
17How good is the approximation18

Shallow impurity states in Si
Donor biding energy in Si (meV)
DopantLi

ThermalMeasurement

Optical

32.8

P

As

Sb

Bi

44

55

39

69

45

53.7

43

70.6

For [001] spheroid

Si

Effective mass equation

Variational method
a (10-8 cm)

b (10-8 cm)

E (1s) (meV)

Si

25

14.2

29

Ge

64.5

22.7

9.2

Not sufficient agreements.
Need more accurate calculations.
19

Doping and carrier distribution
Uniform donor concentration 𝑁D
EntropyHelmholtz free energy
Starling approximation
Donor level

For acceptors
note: the formula is symmetric if we introduce captured hole concentration
20

Doping and carrier distribution
𝐸F is given from n or p as

In the case of n-type semiconductor with compensation21

Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.4.28 Lecture 04
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of lecture in the last week
Envelope function (effective mass approximation)
Chapter 3 Carrier statistics and chemical doping

Density of states
Definition and properties of valence band hole states

Carrier distribution in intrinsic semiconductors
Shallow hydrogen-like impurity states
Shallow impurity states in Si
2

Contents today
Doping and carrier distribution
Temperature dependence of carrier concentration

Exciton

Chapter 4 Optical properties (bulk)
Quantization of electromagnetic field
Number state, coherent state
Optical response of two-level system
Optical absorption with inter-band transition
3

Doping and carrier distribution
Uniform donor concentration 𝑁D
EntropyHelmholtz free energy
Starling approximation
Donor level

For acceptors
note: the formula is symmetric if we introduce captured hole concentration
4

Doping and carrier distribution
𝐸F is given from n or p as

In the case of n-type semiconductor with compensation

5

Temperature dependence of carrier concentration(I) Impurity regime I: Temperature is very low.

(II) Impurity regime II:
T is a bit higher.

(III) Exhaustion regime:(IV) Intrinsic regime: direct excitation between
the v.b. and the c.b. is not negligible.
6Degenerate semiconductors
sparse impurities: insulator

dense impurities: metal

Empirical metal-insulator criterion

7

Excitons (Wannier type)Free excitons
hole

Binding energy

electron

Reduced mass
Exciton kinetic energy
E(k)continuous

n =2

Energy for exciton
creation

n =1

Excitonic complexes
Ebx

photon

Eg

k

+
+
+
+
+

+
+
+
+

H2+
HH2+
- - H2- - H2+ : donor + : hole

- : electron

Excitonic ionsExcitonic molecule(bi-exciton, trion)
8

Chapter 4 Optical properties (bulk)

Luminescence from CdTe quantum dots (Sigma-Aldrich)Quantization of electromagnetic field
1-d harmonic oscillator
up/down operators

Eigenenergy
Starting point: Electro-magnetic field is a set of harmonic oscillators (Jeans theorem)

Quantization:
Creation/annihilation
operators
10Quantization of electromagnetic field (2)11Properties of number state, coherent state
Number state
Expectation value ofelectromagnetic field is zero
Quantum fluctuation is nonzero even for |0

Coherent state
Probability of n-photons

Poisson distribution

If we write 𝛼 = 𝛼 𝑒 𝑖𝜙Expectation value: classical electromagnetic field

12

Optical response of two-level system|𝑏
Threefundamental
processes|𝑏(a) absorption

ℏ𝜔0

ℏ𝜔0

ℏ𝜔0
|𝑎

|𝑏

|𝑎
(b) spontaneous emission|𝑎(c) stimulated emission

Hamiltonian with electromagnetic field

Light absorption process
assumption
Perturbation part in
13

Optical response of two-level system (2)Schrödinger equation:
t=2

t = 1.5
t=1
t = 0.5

Ignore 𝒌p

Energy conservation
14

Rabi oscillation
Rotating wave approximation(drop 𝑒 −𝑖𝜔𝑡 )solution
initial condition
Rabi oscillation
Rabi frequency

Oscillator strength and selection rule
Oscillation strength for |0 → |Φex
Application to an electron-hole localized system (with main, angular momentum quantum number)(wavefunction)=(lattice periodic)×(envelope)
Envelope functions varies slowly→ the momentum can be ignored
Selection rule
Oscillation strength(absorption, stimulated emission)
For spontaneous emission(photon number part → 1/2)
16

Light absorption and luminescence in semiconductors
Application to extended states

direct absorption

indirect absorption

photoluminescence17

Optical absorption with transition from valence to conduction
Plane wave vector potential

Poyinting vector
Definition of absorption coefficient 𝛼
W: number of photons
absorbed per unit time
perturbation
Conduction electron |𝑐𝒌 ,valence hole |𝑣𝒌′
Transition probability is
18

Optical absorption with transition from valence to conduction (2)Bloch electrons

Absorption coefficientfor direct absorption
joint density of states
G point
19

Optical absorption with transition from valence to conduction (3)Minimum at 𝒌0

Change of variables

20

Optical absorption with transition from valence to conduction (4)Absorption coefficient

Oscillator strength
Indirect gap
semiconductors

Si

ExampleGaAs
21

Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.5.12 Lecture 05
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of lecture in the last week
Doping and carrier distribution
Temperature dependence of carrier concentration
Exciton

Chapter 4 Optical properties (bulk)
Quantization of electromagnetic field
Number state, coherent state
Optical response of two-level system
Optical absorption with inter-band transition2

Contents today Optical absorption with inter-band transition
 Photon emission from inter-band transition
 Optical absorption with exciton formation
 Photon emission from exciton recombination
 Concept of exciton-polariton

3

Light absorption and luminescence in semiconductors
Application to extended states

direct absorption

indirect absorption

photoluminescence

Photon momentum ℏ𝑘𝑝 can be ignored in most cases.

4

Optical absorption with transition from valence to conduction
Plane wave vector potential

Poyinting vector
Definition of absorption coefficient 𝛼
W: number of photons
absorbed per unit time
perturbation
Conduction electron |𝑐𝒌ۧ,valence hole |𝑣𝒌′ۧ
Transition probability is
5

Optical absorption with transition from valence to conduction (2)Bloch electrons

Absorption coefficientfor direct absorption
joint density of states
G point6

Optical absorption with transition from valence to conduction (3)Minimum at 𝒌0

Change of variables

7

Optical absorption with transition from valence to conduction (4)Absorption coefficient

Oscillator strength
Indirect gap
semiconductors

Si
ExampleGaAs
8

Luminescence by inter-band transition
E

Electron-hole recombination:

Radiative recombination
Non-radiative recombination

Classification of luminescence with excitations• Photoluminescence• Electroluminescence• Thermoluminescence• Cathode luminescence• Sonoluminescence• Triboluminescence• Chemiluminescencek9Pseudo-Fermi level
Planck distribution
Semiconductor under irradiation
Introduction of pseudo-Fermi levels:

Electron distribution function
in conduction band
Electron distribution function
in valence band
optical absorption
spontaneous emission
stimulated emission
balance equation

Einstein relation
10

Relation with phenomenological approach
So far: Optical response of two-level system?

Inter-band absorption
Extended states
Other effects : refractive index

Macroscopic phenomenological approach
Starting point:
Maxwell equation

Non-magnetic dielectric
Wave equation
Effect of polarization
11

Relation with phenomenological approach (2)Linear response approximation
Relative dielectric function 𝜀𝑟Below we consider isotropic crystal: response function tensor → scalar
The effect of polarization isnormalized into the term oftime-derivative→

Polariton equation
Absorption: imaginary part of response function: complex dielectric function, orcomplex refractive index
absorption coefficient

12Phenomenological approach: Lorentz model
Electromagnetic field in Materials:
set harmonic oscillators (𝑚, 𝑒, 𝜉)energy dissipation
oscillator concentration N
eigenfrequency
long term stable
solution

relative dielectric
function
Multimode:
ration of mode j→ 𝑓𝑗

𝑓𝑗 : oscillator strength13

Optical absorption by excitons
Exciton wavefunction(effective mass
approximation)

r: electron-hole relative coordinate
R: center of mass coordinate
exciton local wavefunction

Fourier transform

exciton total wavelength
ground state

excitation

Transition probability

14

Optical absorption by excitons (2)BecauseBecause the sum will be taken over ke
is large only forwhile

is almost constantwhich isFermi’s golden rule:
For 𝜙(0) not to be 0, 𝜙must be an s-state
Imaginary part of thecomplex relative dielectric
function(spin degree of freedom: factor of 2)

15

Optical absorption by excitons (3)Exciton absorption peaks in GaAs.
Fehrenbach et al., J. Luminescence 30, 154 (1985).

in simplest form
identity
comparison gives

Kramers-Kronig relation

Sturge, Phys. Rev. 127, 768 (1962)16

Photoemission by exitons
Photoemission: reversal process
Bound exciton emission peaks in Cu2O

Jang et al., Phys. Rev. B 74, 235204 (2006)

17

Exciton-polariton
Concept of exciton-polariton
Chain of photon-exciton
1 cycle ～ few fs
coherent propagation in solids

transverse wave:

polariton equation

Longitudinal wave:

18

Exciton-polariton (2)Resonance
Dispersion relation

19

Bose-Einstein condensation of exciton-polaritonsJ. Kasprzak et al., Nature 443, 409 (2006).
20

Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.5.19 Lecture 06
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of lecture in the last week Optical absorption with inter-band transition
 Photon emission from inter-band transition

 Optical absorption with exciton formation
 Photon emission from exciton recombination
 Concept of exciton-polariton

2

Contents today
Concept of exciton-polariton (continued)

Chapter 5 Semi-classical treatment of transport
Transport coefficientClassical transport: Boltzmann equation
Currents: particle flows
Drude formula, Diffusion current, Hall effectVarious scatterings
Heat transport, Thermoelectric effect3

Exciton-polariton
Concept of exciton-polariton
Chain of photon-exciton
1 cycle ～ few fs
K
coherent propagation in solids

transverse wave:

K
K

K

K

K

K

polariton equation

Longitudinal wave:
4

Exciton-polariton (2)For transverse wave

Real-imaginary comparison

Resonance
Dispersion relation

5

Bose-Einstein condensation of exciton-polaritonswkJ. Kasprzak et al., Nature 443, 409 (2006).
6

Chapter 5 Semi-classical treatment of transport
Streintz
NernstArrhenius

HieckeKlemencic
Aulinger

Ettingshausen
Boltzmann

Hausmanninger
Ludwig Boltzmann
1844 - 1906
From Wikipedia

Classical, semi-classical transport, transport coefficientelectrons : most electric devices
Transport in condensed matter: Charge, heat, spin carriersions : batteries, sensors

quantum mechanical

Quantum transport

Classical, semi-classical transport
nature in transport

Semi-classical: quantum mechanics affects energy distribution function
Classical semi-classical boundary Fermi degenerate temperaturefor 3-dimensional systems
for 2-dimensional systems

External perturbation → Linear response: Transport coefficient

conductivity tensor

Conductance, Resistanceresistivity tensor
8Classical transport: Boltzmann equation𝒑(𝒓, 𝒑) 6-dimensional phase space
Distribution function 𝑓(𝒓, 𝒑, 𝑡)
No collision:𝒓
Introduction of collision:

𝑑𝒓𝑑𝒑

Boltzmann equation
9

Currents: Particle flows
Boltzmann equation

Relaxation time approximation:
𝒑: Anisotropic distribution = Current

the collision term leads
𝑓0 : isotropic in p-space →to current: time derivative of the distribution, zero for steady states: velocity times spatial gradient in the particle density → Diffusion current: force on the particles times gradient of f in p-space → Drift current

10Drift current by electric field
𝑓(𝑘)
𝑘
𝑘𝑦
−𝑘𝐹

𝑘𝑥
Density of states:
Kinetic energy:
law of equipartition of energy
11Drude formula
For metals (𝑇𝐹 ≫ 300 K)

Paul Drude (1863 – 1906)

Low temperature approximation:

Drude formula for metals

For Maxwell distribution

Drude-like formula12Diffusion current
No external force:
Relaxation time approximation:

Take the x-direction to that of 𝛻𝑓:

Einstein relation:𝑒𝜏
𝜇 = ∗ : mobility𝑚

13

The Hall effect
Galvanomagnetic effect: Force on electrons ← Lorentz force

𝐴𝑡 term creates 𝑗𝑦 hence Ε𝑦 : Hall voltage (electric field)
The Hall coefficient is defined asWith cyclotron frequency
In caseMobility is defined and expressed as

14

Carrier scattering mechanisms
Scattering mechanisms
Defect scattering
Crystaldefects

Impurity

NeutralCarrier-carrier
scattering

Alloy

IonizedPhonon scattering

Intravalley
Acoustic

Deformation
potential

PiezoelectricIntervalley
Optic

Nonpolar

Acoustic

Optic

Polar

The parameter which represents the scattering mechanism= averaged time interval of scattering
Scattering time:15

Matthiessen’s rule and effect of scattering on the electric transport
Matthiessen’s rule (series connection of scattering)

Reduction of
impurity scatteringby modulation
doping structure

Fletcher et al., J. Phys. C 5, 212 (1972)

Walukiewicz et al. Phys. Rev. B 30, 4571 (1984).

16

Heat transport, thermoelectric effectHeat flux density:
Temperature gradient

Carrier thermal conductivity
J: current, V: voltage, T: temperature, Q: heat flow

A

B

Seebeck effect
B

Peltier effect

J

B
B

V

A

J

B

x
Thomson effectJA

Seebeck coefficientJ

Peltier coefficientThomson coefficient
can be obtained from single material
17

The Kelvin relations
Unit charge

𝑇

𝑇𝑚
B

𝑇 + Δ𝑇A

𝑇𝑚
Quasi-static process →Thermodynamics can be applied

B
𝑉𝐴𝐵

First law of thermodynamics

Second law of thermodynamics
Taking ∆𝑇 → 0, these two become
The second equation becomes

The Kelvin relations are obtained as
The absolute Seebeck, Peltier coefficients can be obtained from the relations.

18Seebeck coefficient as material constantMaterial specific (absolute) constant canbe experimentally obtained from

Then for other materials

Thermocouple

A

Δ𝑇

B

19

Boltzmann equation and thermoelectric constants
For the thermoelectric effect, we need to consider (only) 𝛻𝑇
The distribution function in lhs is
replaced with unperturbed one.

With
From the above we can rewrite

Then the Boltzmann equation gives
Substituting the above andinto the current expression

20

Boltzmann equation and thermoelectric constants (2)𝑗𝑥 = 0 means the balancing of the drift current and the diffusion current

Then the Seebeckcoefficient is calculated as

Maxwell approximation
Energy dependence ofrelaxation time

Seebeck measurement provides information on EF and scattering mechanisms

21

Peltier device
Sign of the coefficient changes
with carrier charge

22Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.5.26 Lecture 07
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of last week
Concept of exciton-polariton (continued)

Chapter 5 Semi-classical treatment of transport
Transport coefficientClassical transport: Boltzmann equation
Currents: particle flows

Drude formula, Diffusion current, Hall effectVarious scatterings
Heat transport, Thermoelectric effect2

Boltzmann equation and thermoelectric constants
For the thermoelectric effect, we need to consider (only) 𝛻𝑇
The distribution function in lhs is
replaced with unperturbed one.

With
From the above we can rewrite
Then the Boltzmann equation gives
Substituting the above andinto the current expression

3

Boltzmann equation and thermoelectric constants (2)𝑗𝑥 = 0 means the balancing of the drift current and the diffusion current

Then the Seebeckcoefficient is calculated as

Maxwell approximation
Energy dependence ofrelaxation time

Seebeck measurement provides information on EF and scattering mechanisms

4Peltier device
Sign of the coefficient changes
with carrier charge

5Physics in spatially structured semiconductors
Our apparatus:

 Band structure Effective mass approximation
 Carrier statistics

 Electron-photon couplings
 Thermodynamics

 Semi-classical transport (Boltzmann
equation)

6Chapter 6 Homo and hetero junctions

Bell laboratories ～1984

Bell laboratories 90’s
Lucent Technologies

2002 Shoen scandal
2006 Merging of Lucent and Alcatel

2008 Official announcement on
quitting from physics!
2016 Alcatel-Lucent Bell labs
→ Nokia Bell labs

pn homo junctions
Fabrication of pn junctions:

Epitaxial growth
Counter doping
Masked, focused doping

Diffusion current:
Entropy increase
Balance:
Minimize Free energy

Equilibrium

Drift current:Internal energy decrease Carrier depletion layer (space charge layer) Built-in potential
10pn junction thermodynamics
Consider electronsattached to a vacuum
without work function

Vacuumfor electronse+ e-diffusion+ e- +

𝐹 = 𝑈 − 𝑇𝑆
Voltage (internal energy cost)

-

+
+
+
+
+
+
evoltage (polarization) → energy costdonors+ e-

Minimization of 𝐹 → Built-in (diffusion) voltage 𝑉𝑏𝑖
Diffusion (entropy)

number of sites: N

number of sites: N

particle number: 𝑁1

particle number: 𝑁2

11

Estimation of built-in potential
Notation of carrier concentrationelectrons

holes

n-layerp-layerNumber of cases:
Stirling approximation::Mixing entropy

12Estimation of depletion layer width
E

−𝑤𝑝
𝑁A

𝑤𝑛
𝑁D

x

Charge neutrality:

Electric field:

Built-in potential is13Current-voltage characteristics
Electrons

Equilibrium

Current balance

External voltage V
Forward bias (against 𝑉𝑏𝑖 ) : lowers barrier for diffusion current 𝑛𝑛

Electron, holesummation
14

Injection of minority carriersJ

pn junction circuit
symbolminority carrier current

Jbarrier overflow

Forward bias region: minority carrier injection

Diffusion equation:

0V

0Minority carrier diffusion length:

Fate of injected minority carriers

Radiative recombinationℎ𝜈light emitting
diodeNon-radiative recombinationelectron scattering
phonon15

Solar cells (external injection of minority carriers)Jℎ𝜈

𝑒𝑣𝑛 Δ𝑛𝑝

electron-hole
pair creation

dark

0

illuminated0
V
Minority carriers which diffuse to thejunction region are swept out to theother side.

Voltage for J =0

Voc

Current for V =0

Jscexternal injection

Filling factor (FF) =Gerald Pearson, Daryl Chapin
and Calvin Fuller at Bell labs. 195416

pn junction (bipolar) transistors
Bipolar junction transistor

Field effect transistor
p

n p n

n

Bipolar transistor structures and symbols
collector (C)John Bardeen, William Shockley,
Walter Brattain 1948 Bell Labs.

base (B)

p
n
p

C

JC
n
p
n

B
JB
emitter (E)PNP type

C JC
B
JB

E JE

E JE
NPN type
17Base-Collector, Collector-Emitter characteristics
p

n

E

B

C

n
E
−𝐽C
𝑉BC

p
B

n
C

𝐽𝐶

Collector-Emitter

Base-Collector

𝐽E

n

18

Current amplification: Linearization with quantity selection
𝐽𝐶 = ℎ𝐹𝐸 𝐽𝐵
Emitter-common
current gain

How a bipolar transistor amplifies signal?
JB

A

Diffusion
e+
e+ e+
e+
e+
e+
EmitternVECe- e-e- eee-

e-

e-

e-

Base
p

Collector
n
JC
A
19

Expression of hFE
Sweeping out of minority
carriers at the depletion edge
Diffusion current in thebase: constant: linear in x

Device cross section A
The law of mass action

Recombination current:

20

Example of an amplification circuitDepletion layer width with reverse bias voltage
𝑉𝑏𝑖 + 𝑉

Poisson equation

charge:under conditions
++
p
n
+
+
𝑤𝑛
−𝑤𝑝

potential boundary:

The integration gives

22Effective capacitance and reverse bias voltage

This gives a way for the doping profiling.

−𝑉𝑏𝑖

Varicap diodecircuit example

V

KB505

Frequency modulation
Phase lock loop23

pn junction field effect transistor (JFET)

Circuit symbols
D

D
G

G

S

S
n-channelp-channel

pn junction FETLGp+
wd (y)-NDe
S

Vch(y)

2wt

D

n
p+

G

Vg

y

conductivity

channel widthelectric field

pinch off (internal) voltage:
Only valid for wd < wt/2.

I-V characteristics of JFET
JDS

JDS
linearJDS vs VDS
saturation
VGS =0

VGS

|VGS|

VDS

Example: 2N5457
n-channel
depletion-type26

Application of JFET
Low linearity → linearization with feedback with high gain
High input impedance, low bias current (operation at the reverse bias region): fit the input stage of operational amplifier

Example: OPA827

Inverting amplifier27

Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.6.02 Lecture 08
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of the last week

Chapter 6 Homo and hetero junctions
pn homo junctions
Solar cells
Bipolar transistors

John Bardeen, William Shockley,
Walter Brattain 1948 Bell Labs.

Gerald Pearson, Daryl Chapin
and Calvin Fuller at Bell labs. 19542

Current amplification: Linearization with quantity selection
𝐽𝐶 = ℎ𝐹𝐸 𝐽𝐵
Emitter-common
current gain

How a bipolar transistor amplifies signal?
JB

A

Diffusion
e+
e+ e+
e+
e+
e+
EmitternVECe- e-e- eee-

e-

e-

e-

Base
p

CollectornJCA3

Expression of hFE
Sweeping out of minority
carriers at the depletion edge
Diffusion current in thebase: constant: linear in x

Device cross section A

The law of mass action

Recombination current:

4

Example of an amplification circuitDepletion layer width with reverse bias voltage
𝑉𝑏𝑖 + 𝑉

Poisson equation
charge:under conditions
++
p
n
+
+
𝑤𝑛
−𝑤𝑝

potential boundary:

The integration gives

6Effective capacitance and reverse bias voltage

This gives a way for the doping profiling.
−𝑉𝑏𝑖

Varicap diodecircuit example

V

KB505

Frequency modulation
Phase lock loop7

pn junction field effect transistor (JFET)

Circuit symbols
D

D
G

G

S

S
n-channelp-channel

pn junction FETLGp+
wd (y)-NDe
S

Vch(y)

2wt

D

n
p+

G

Vg

y

conductivity

channel widthelectric field

pinch off (internal) voltage:
Only valid for wd < wt/2.

I-V characteristics of JFET
JDS

JDS
linearJDS vs VDS
saturation
VGS =0

VGS

|VGS|

VDS

Example: 2N5457
n-channel
depletion-type10Application of JFET
Low linearity → linearization with feedback with high gain
High input impedance, low bias current (operation at the reverse bias region): fit the input stage of operational amplifier
Example: OPA827

Inverting amplifier
11

Schottky barrier (metal-semiconductor junction)
metal

semiconductor𝑒𝜙𝑆 𝐸c

𝑒𝜙𝑀

𝐸F

𝐸D

𝐸F

Walter Schottky
1886-1976

Q: space charge

electrostatic
potential
Charge balance:

𝐸v

0

xd

Voltage V --> barrier height e(Vs-V)
x

𝐸c

-Q
𝐸F

𝑤𝑑
𝐸v

barrier overcoming current
No minority carrier injection
12

MES-FET
In most cases, the surface barrier is dominated by the pinning of Fermi
level with the high density surface states.
MES (metal-semiconductor) FET: depletion layer device.
Complementary circuit is difficult.

Data are given in S-parameters.

Saturation effect is a bit weak.

NE76084

13

MOS FETenhancement

depletion

Complementary metal-oxide-semiconductor (CMOS) circuit

PMOS

NMOS

Simplified
CMOS inverter
circuitinversion

Low leakage current
Single gate input both on/off switch
14

FinFET
Improvement of MOS FET𝜅High -𝜅 materials for dielectrics other than SiO2

Low voltage action requirement:
Multi-gate structure to wrap up the conduction channel
FinFET structure

3.9

HfO4Si

11

Si3N4

7

Al2O3

9

ZrO2

25

HfO2

25

Pinch-off the channel with wrapping gate
Less than 1 ps
Less than 1 V
Inversion conductive mode

TSMC 7 nm
transistors

TSMC 30 nm gate (2012)15

Heterojunction and envelope function
f(r): envelope function

Effective mass approximation

This holds for spatially slow perturbation U(r).
Then how about heterointerface?
𝜓 𝐴 (𝒓)𝜓 𝐵 (𝒓)

1. For simplicity we assumez0Then continuity condition at z =0 becomesIn xy-plane, the Bloch theorem tells
For z-freedom, we apply 𝑘 ∙ 𝑝 perturbation.

envelope function for z

16

Heterojunction and envelope function (2)withWhen the band mixing effect isignorable, we can also apply theeffective mass approximation
for the heterojunctions.

Continuity condition:

continuity in derivative

discontinuity
17Band discontinuity
Band discontinuity parameters

Anderson’s rule: affinity from the vacuum level
determines the alignment

R. L. Anderson, IBM J. Res. Dev. 4, 283 (1960).
18

Heterojunction types

Type-I

Type-II

Type-III(Type-II staggered)

19

Chapter 7 Quantum Structure(Quantum wells, wires, dots)

Quantum well (elementary quantum mechanics)
𝑉(𝑥)Outside the well:

𝑉0

States localized inside the well: 𝐸 < 𝑉0
𝑥
−𝐿/2

𝐿/2

Inside the well:

Envelope function
connection21Quantum well

22

Optical absorption of quantum wells

Envelope functions

Lattice periodic functions

Direct transition rate:
Transition energy:
𝐸g
Two dimensional density of states:

hh
lh

23

Optical absorption of quantum wells

24Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.6.09 Lecture 09
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo KatsumotoBand discontinuity
Band discontinuity parameters

Anderson’s rule: affinity from the vacuum level
determines the alignment

R. L. Anderson, IBM J. Res. Dev. 4, 283 (1960).
2

Heterojunction types

Type-I

Type-II

Type-III(Type-II staggered)3

Chapter 7 Quantum Structure(Quantum wells, wires, dots)

Quantum well (elementary quantum mechanics)
𝑉(𝑥)Outside the well:

𝑉0

States localized inside the well: 𝐸 < 𝑉0
𝑥
−𝐿/2

𝐿/2

Inside the well:

Envelope function
connection5Quantum well

6

Optical absorption of quantum wells

Envelope functions

Lattice periodic functions

Direct transition rate:
Transition energy:
𝐸g
Two dimensional density of states:

hh
lh

7

Optical absorption of quantum wells

8

Excitons in quantum wellholeelectron

Schrödinger equation
Variable separation
Radial wavefunction

m: magnetic quantum number

E(k)

e

Power series expansion
The series to be stopped
at a finite length

k
hh
lh

9

Excitons in quantum well

GaAs bulk

3

Absorption coefficient (10 /cm)

20

300 K

hh
lh

n =2
10

n =1
GaAs/AlAs superlattice
8.3 nm/9.3 nm

0
0.6

0.7

l (mm)

0.8

0.9
10

Quantum barrier
𝐴1 (𝑘)1𝐵1 (𝑘)𝑄

𝐴2 (𝑘)2𝑀𝑇

𝐵2 (𝑘)

Simpler way to consider tunneling through energy barriers
Generally𝑣𝑔 𝜓 Transfer matrix: T-matrix
 Scattering matrix: S-matrix

momentum conservation→ relation between wavefunctions

Transfer matrix: 𝑀𝑇

𝑀𝑇 for a barrier width 𝐿 height 𝑉0

Inside the barrier
Boundary condition: value
derivativeThen
is obtained as
11Transfer matrix for rectangular barrierL
𝐴1 (𝑘)𝐵1 (𝑘)𝐴2 (𝑘)
1

𝑉0

2

𝐵2 (𝑘)

𝑀𝑇t, r : complex transmission and reflection coefficients
Transmission coefficient 𝑇 = 𝑡 2 , reflection coefficient 𝑅 = 𝑟 2
Then the transfer matrix is expressed as
12Application of transfer matrix: double barrier transmission
T-matrix for well

𝑀𝑇

𝑀𝑊

𝑀𝑇

Calculation of transmission coefficient13

Double barrier transmission

Resonant transmission condition: zero points of cosine term

14

Transport experiment of double barrier conduction
Measurement scheme

Sample structure

Result at 4.2 K
SourceTransmission coefficient

𝑒𝑉𝑠𝑑

STEM image

Drain

light holeheavy hole

𝐸/𝑉0
Calculated transmission coefficient15

Application of T-matrix (2): Semiconductor superlatticeunit cell

Schrödinger equation
d =W+L

Bloch theorem
Kronig-Penny potential: 𝑉𝐾𝑃 (𝑥)Unit cell transfer matrix
𝑎𝑖
𝑏𝑖

𝑎𝑖+1
𝑏𝑖+1

Eigenvalues 𝑒 ±𝐾𝑑 (𝑀𝑑 : unitary)

Theorem: Tr 𝐴 = σ(eigenvalue)

The relation between k (free electron wavenumber) and K (crystal wavenumber)
16

Semiconductor superlattice

𝛿-function series with the coefficient C.
effect of superlattice potential

Raphael Tsu and Leo Esaki, 1975

:no solution → band gap
Around 𝑘𝑑 = 𝑛𝜋 (𝑛 = 1,2, ⋯ )STEM image of
AlAs (30 nm)/GaAs (30nm)
superlattice17

Modulation doping and 2-dimensional electrons
equation
AlGaAs:Si

AlGaAsDonor potential

GaAs
Electric field of
sheet charge at z’Heterointerface
potential
Poisson-Schrödinger scheme
potential
Schrödinger equation
Boundary condition18

Electron mobility in MODFETMatthiessen’s rule (series connection of scattering)

Reduction of
impurity scatteringby modulation
doping structure

Fletcher et al., J. Phys. C 5, 212 (1972)

Walukiewicz et al. Phys. Rev. B 30, 4571 (1984).

19Formation of quantum wires: split gate
Metal gates
Depletion layer

Two-dimensional
electrons
Two-dimensional electrons are pinched with depletion layers from Schottky gates to a one-dimensional system.
Electric field along z-axis can be approximated as
The bottom part of the confinement potential can be approximated by harmonic potential.

20

Self-assembled nano-wires

G. Zhang et al.
NTT technical
Review

http://iemn.univ-lille1.fr/sites_perso/
vignaud/english/35_nanowires.htm
21Core-shell nanowire transistor

Vds= 1 V

Vds=-2 V

L. Chen et al.,Nano Letters 16, 420(2016).
22

Carbon nanotube

23

Quantum dots: zero-dimensional system
Quantum dots with
nano-fabrication
techniques

split gate
wrap gatewith charge detector

vertical type24

Formation of quantum dots: self assemble
MBE growth modes
Frank-van der Merve

RHEED

Volmer-Weber

AFM

Stranski-Krastanow

STM
25

Formation of quantum dots: Colloidal nano-crystals

26

Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.6.16 Lecture 10
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of last week Band discontinuity at heterojunction
Chapter 7 Quantum Structure (Quantum wells, wires, dots)
 Quantum wells
 Excitons in quantum wells
 Quantum barriers

2Modulation doping and 2-dimensional electrons
Donor potential

AlGaAs:Si

AlGaAsGaAs
Electric field of
sheet charge at z’Heterointerface
potential
Poisson-Schrödinger scheme
potential
Schrödinger equation
Boundary condition3

Electron mobility in MODFETMatthiessen’s rule (series connection of scattering)

Reduction of
impurity scatteringby modulation
doping structure

Fletcher et al., J. Phys. C 5, 212 (1972)

Walukiewicz et al. Phys. Rev. B 30, 4571 (1984).

4Formation of quantum wires: split gate
Metal gates
Depletion layer

Two-dimensional
electrons
Two-dimensional electrons are pinched with depletion layers from Schottky gates to a one-dimensional system.
Electric field along z-axis can be approximated as
The bottom part of the confinement potential can be approximated by harmonic potential.

5Self-assembled nano-wires

G. Zhang et al.
NTT technical
Review

http://iemn.univ-lille1.fr/sites_perso/
vignaud/english/35_nanowires.htm
6Core-shell nanowire transistor

Vds= 1 V

Vds=-2 V

L. Chen et al.,Nano Letters 16, 420(2016).
7

Carbon nanotube

8Quantum dots: zero-dimensional system
Quantum dots with
nano-fabrication
techniques

split gate
wrap gatewith charge detector

vertical type9

Formation of quantum dots: self assemble
MBE growth modes
Frank-van der Merve

RHEED

Volmer-Weber

AFM

Stranski-Krastanow

STM
10

Formation of quantum dots: Colloidal nano-crystals

11Window layer in solar cells
p+GaAsAlGaAsnGaAsQuantum efficiency

Optical devices with minority carrier confinementwith Window
without Window

𝐸𝑔1

Photon energy

𝐸𝑔2

(b)
Panasonic HIT

𝐸𝑔2
𝐸𝑔1

(a)

𝐸𝑔2 > 𝐸𝑔1(c)

12

Light emitting diodes (LEDs)
Emission spectrum
Quantum
efficiency
non-radiative

radiative

Nick Holonyak Jr.

Minority
carrierdiffusion
Carrier recombination in
depletion layer
injection efficiency

Internal quantum efficiency

13

External quantum efficiency
absorption

FresnelOptical lossesqx

Absorption lossqc

Fresnel loss

total reflection

Device example

Optical efficiency: 𝜂opt

Total reflection loss

External quantum efficiency
InGaN-based
YAG
Narukawa et al.,JJAP 41, L1431(2007).

Windisch et al., APL 74, 2256 (1999).

GaN/InGaN/GaN

Textured surface AlGaAs/GaAs
Fujii et al., APL 84, 855 (2004).
𝜂𝑒𝑥𝑞 >30 %14

Double heterojunction (DH) LED
p-(Al, Ga)Nn-(Al, Ga)N

p-GaN

n-GaN

High internal quantum efficiency
n-SiC
substrate

InGaN
--

c.b.
- -

Advantages of DH LED

 Narrow active region → high np product No need for doping in active layer → lessconcentration of non-radiative
recombination center Diffusion of minority carrier to surface,recombination centers is reduced.
Low absorption loss Energy of emitted photons is lower thanthe band gaps of the top and the bottom
layers.

+ +

+

++

v.b.
15

Laser diode
LASER: Light Amplification with Stimulated Emission of Radiation|𝑏

Coherent state: Classical oscillating electromagnetic fieldℏ𝜔0|𝑎(c) stimulated emission

Probability of stimulated emission

Energy absorption of media from the light (coherent state):
This means if a state with 𝑁𝑏 > 𝑁𝑎 is realized, 𝜀 < 0, i.e., the energy is absorbed from the media to light
Increase of amplitude of the coherent state: Bosonic stimulation16

Laser diodeADL-65074TL-1 - Laser
Diode 655nm 7mW

17

Chapter 8
Basics of Quantum Transport

M. A. Topinka et al. Science 2000;289:2323-2326

18

Quantum entanglement 1 2

  A  B

Direct product

A

B

A 1

B 1

1

A 2

B 2

2

    A 1  A 2  B 1  B 2

Maximally entangled state  A 1  B 2
von Neumann entropy (entanglement entropy)

Quantification of Entanglement?

Density matrix 𝜌 = |𝜓 𝜓|
𝑆 = tr 𝜌ln𝜌Boundary between classical and quantum
𝜒1𝜓1
Environment wavefunction: 𝜒should associate with electron paths

𝜓2
𝜒2

Then the interference term is𝜒1 𝜒2 = 1: Full interference𝜒1 𝜒2 = 0: No interference Particle-Environment maximally entangled
Electron transport: Electron – Phonon inelastic scattering
Electron – Electron inelastic scattering
Electron – Localized spin scattering
20

Length limit quantum coherence (Coherence length)
Monochromaticity: Thermal length

Energy width:

Diffusion length:

Phase width:

Thermal diffusion length
Ballistic thermal length(Some) inelastic scattering time: 𝜏inel

Ballistic transport:
Diffusive transport:
21

Conductance quantum
L, R : Particle reservoirs
Thermal equilibrium:
well defined chemical potentials
Instantaneous thermalization:
particles loose quantum coherence

𝐿: wavefunction normalization length

Conductance quantum22Conductance quantum as uncertainty relation

Wave packet:
Fermion statistics: electron charge concentration =Energy width:

Wave packet width in time:

Conductance quantum comes from fermion statistics of electrons

23Quantum point contact (QPC)
gate

𝑊(𝑥)
depletion
layer

2DEG
gate

Transmissible one-dimensional system: Conductance Channel
24Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.6.23 Lecture 11
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of last week Two-dimensional electrons at heterointerface
 Quantum point contacts, quantum wires
 Core-shell nanowires

 Two dimensional systems → quantum dots
 Self assembled quantum dots

 Colloidal quantum dots
 Optical devices with minority carrier confinement
Solar cells, DH LEDs, Laser diodes

2

Chapter 8
Basics of Quantum Transport

M. A. Topinka et al. Science 2000;289:2323-2326

3

Quantum entanglement 1 2

  A  B

Direct product

A

B

A 1

B 1

1

A 2

B 2

2

    A 1  A 2  B 1  B 2

Maximally entangled state  A 1  B 2
von Neumann entropy (entanglement entropy)

Quantification of Entanglement?

Density matrix 𝜌 = |𝜓 𝜓|
𝑆 = tr 𝜌ln𝜌Boundary between classical and quantum
𝜒1𝜓1
Environment wavefunction: 𝜒should associate with electron paths

𝜓2
𝜒2

Then the interference term is𝜒1 𝜒2 = 1: Full interference𝜒1 𝜒2 = 0: No interference Particle-Environment maximally entangled
Electron transport: Electron – Phonon inelastic scattering
Electron – Electron inelastic scattering
Electron – Localized spin scattering
5Length limit quantum coherence (Coherence length)
Monochromaticity: Thermal length

Energy width:

Diffusion length:

Phase width:

Thermal diffusion length
Ballistic thermal length(Some) inelastic scattering time: 𝜏inel

Ballistic transport:
Diffusive transport:
6

Conductance quantum
L, R : Particle reservoirs
Thermal equilibrium:
well defined chemical potentials
Instantaneous thermalization:
particles loose quantum coherence

𝐿: wavefunction normalization length

Conductance quantum7Conductance quantum as uncertainty relation
Space coordinate-wavenumber

Wave packet:

Fermion statistics: electron charge concentration =Energy-time

Energy width:

Conductance quantum
comes from fermion
statistics of electrons

Wave packet width in time:
Fermion anti-bunching effect:

ℎ/𝑒𝑉

e

e

e

e
8

Quantum point contact (QPC)
gate

𝑊(𝑥)
depletion
layer

2DEG
gate

Transmissible one-dimensional system: Conductance Channel
9Scanning tip conductance measurement
scanning tip

gate

gate

Tip image potential scatters electrons→ conductance shifts from quantized value
Scattering amplitude ∝ 𝜓 2
M. A. Topinka et al.,Nature 410, 183 (2001)
10

Shot noise reduction on the conductance plateaus
Flow of single electron: Time domain: d-function approximation

Current fluctuation density for infinitesimal band 𝑑𝑓

Flow of N-electrons

M. Hashisaka, et al., J. Phys.: Conf. Ser. 109, 012013 (2008): Poissonian noise11

Microwave and electron waveguides
Quantum point contact

Microwave waveguide
Quantum point contacts orquantum wires can beviewed as “electron
waveguides.”12

Landauer formula for two-terminal conductance
𝑇=1

reservoir

A

Scattering

𝑇𝑖𝑗 ≤ 1reservoir
SampleV
Transition to otherchannels

Electron spin

Fermion antibunching

Waveguide connection
Rolf Landauer
1927 - 1999

13

Scattering matrix (S-matrix)
𝐴1 (𝑘)
T-matrix

𝐵1 (𝑘)1𝑄

𝐴2 (𝑘)2𝑀𝑇

𝐵2 (𝑘)

Transfer matrix: 𝑀𝑇

𝑎1 (𝑘)
S-matrix𝑏1 (𝑘)1𝑆2

𝑎2 (𝑘)

incoming

𝑏2 (𝑘)

outgoing

Complex probability density flux
14

Series connection of S-matrix

15

S-matrix

A

Multi-channel

Unitarity

Reciprocity(time-reversal symmetry)BOnsager reciprocity
Complex conjugate andScattering solution: Sc 𝑎 → 𝑏

Lars Onsager
1903-1976

Landauer-Büttker formula

sampleMakus Büttiker
1950-2013Landauer-Büttker formula: Application to 4-terminal measurement
𝐽
1

2
𝑉24

3

4

𝐽

Onsager reciprocity

Onsager reciprocity in AB ring
30
Onsager's law for non-local
four-terminal (30 mK)
I = 4-6, V = 7-5
I = 7-5, V = 4-6

25

R ()

20
15
10

4

5

6

0

5

7

-0.04

0.00

0.04

Magnetic Field (T)Magnetoresistance: Universal conductance fluctuation including AB oscillation

S-matrix: Application to Aharonov-Bohm ring

SAB

Sw

St

SAB
St

St
Sw

21

Bunching and anti-bunching of particles
Two-particle wavefunction:(+: boson, -: fermion)

Probability of finding twoparticles at the same position

Boson: bunching, bosonicstimulation → laser oscillation,
Bose-Einstein Condensation

22

Waveguide for exciton-polariton
exciton-polariton

Chain of photon-exciton (photon-dressed exciton)
1 cycle ～ few fs
coherent propagation in solids

photon→ cavity photon

dispersion relation: light effective mass ~ 10−4 𝑚exciton

23

Mach-Zehnder interferometer (voltage-type)
Kinetic phase shift with electric field:
𝛿𝐸: energy shift due to the depletion of quantum welljunction-FET type waveguide2 mmp-electrodep-electrode
AlGaAs1
AlGaAs2GaAspolyimide
polyimide

AlGaAs2
AlGaAs1n-electrodeKatsuyama, Hosomi, Micro. Eng. 63, 23 (2002).

n-electrode

Voltage control of optical output
through interference24Mach-Zehnder interferometer 2 (optical control)
Kinetic phase shift with electric field:
𝛿𝐸: energy shift due to the barrier by optically excited carriers (quasi-Fermi levels)

Sturm et al., Nature Comm. 5,3278 (2014)25

Lecture on

Semiconductors / 半導体(Physics of semiconductors)

2021.6.30 Lecture 12
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of the last lecture

Chapter 8 Basics of Quantum Transport Boundary between classical and quantum (coherence length)
 Conductance quantum
 Quantum point contact

 Landauer formula for two-terminal conductance Scattering matrix (S-matrix)
 Onsager reciprocity
 Landauer-Büttker formula for multi-terminal conductance2

Aharonov-Bohm effect

1
F

1

D

D

F


C
2

2

(a)

(b)

Onsager reciprocity in AB ring
30
Onsager's law for non-local
four-terminal (30 mK)
I = 4-6, V = 7-5
I = 7-5, V = 4-6

25

In the case of two-terminal measurement
R ()

20
15
10

4

5

6

0

5

7

-0.04

0.00

0.04

Magnetic Field (T)Magnetoresistance: Universal conductance fluctuation including AB oscillation

S-matrix: Application to Aharonov-Bohm ring

SAB

Sw

St

SAB
St

St
Sw

8

Disappearance of electrons?

?

Bunching and anti-bunching of particles
Two-particle wavefunction:(+: boson, -: fermion)

Probability of finding twoparticles at the same position

Boson: bunching, bosonicstimulation → laser oscillation,
Bose-Einstein Condensation

Fermion: anti-bunching,conductance quantization, shotnoise reduction

10

Waveguide for exciton-polariton
exciton-polariton

Chain of photon-exciton (photon-dressed exciton)
1 cycle ～ few fs
coherent propagation in solids

photon→ cavity photon

dispersion relation: light effective mass ~ 10−4 𝑚exciton

11

Mach-Zehnder interferometer (voltage-type)
Kinetic phase shift with electric field:
𝛿𝐸: energy shift due to the depletion of quantum welljunction-FET type waveguide2 mmp-electrodep-electrode
AlGaAs1
AlGaAs2GaAspolyimide
polyimide

AlGaAs2
AlGaAs1n-electroden-electrode

Voltage control of optical output
Katsuyama, Hosomi, Micro. Eng. 63, 23 (2002). through interference12Mach-Zehnder interferometer 2 (optical control)
Kinetic phase shift with electric field:
𝛿𝐸: energy shift due to the barrier by optically excited carriers (quasi-Fermi levels)

Sturm et al., Nature Comm. 5,3278 (2014)

13

Exciton-polariton condensation

Deng, Haug, Yamamoto, Rev. Mod. Phys. 82, 1489 (2010).

14

Exciton-polariton condensation2

Byrnes, Kim,
Yamamoto, Nat. Phys.
10, 803 (2014),

15

Single electron effect-e---CMillikan’s oil droplet experiment

Coulomb blockadeRole of power sources
J

C

−𝑒/2𝐶
A

𝑒/2𝐶

V

V

Power sources: Automatically supply energy.
Energy → Enthalpy

𝐻 = 𝑈 − 𝑃𝑉Constant interaction model, capacitor model
mN+1

Constant interaction: U
EFElectron number: N
Interaction energy

EF
mN

Chemical potential

mN-1

Charge relations

−𝑁𝑒
dot

𝑄2

Electrostatic energy

𝑄1
Enthalpy

Chemical potential

C

𝑉𝑑

𝐶g
𝑉g

Coulomb oscillation
𝐻(𝑁, 𝑉g )
Vg

EF

m N+2

EF

m N+1

N =0

mN

𝐽

1

2

𝑉g

m N -1J𝑉g𝑉g

Coulomb diamond
V sd

(d)

𝐽
(a)

(c)

𝑉g
𝑉sd
Vg
(b)

𝑉g

(e)

Coulomb oscillation and diamonds

Quantum confinement

𝐻(𝑁, 𝑉g )Zero-dimensional confinement to a quantum dot gives
shifts in Coulomb peak positions.

𝜖1 , 𝜖2 , ⋯ , 𝜖𝑁 , 𝜖𝑁+1 , ⋯

𝜖𝑁+2
𝜖𝑁+1
𝜖𝑁

Enthalpy shift by quantum
confinement
Chemical potential shift

Shift in gate voltage

∆𝜖𝑁+1

∆𝜖𝑁

𝜖𝑁+1

∆𝜖𝑁

𝜖𝑁
𝑁𝑒

𝑁+1 𝑒
𝐶𝑠
∆𝜖𝑁
𝑒

𝐶g 𝑉g

Quantum confinement effect in a vertical quantum dot

Two-dimensional harmonic potential
Potential shape:
Easy solutions from 1d
harmonic potential
Eigen energies:

𝑛𝑡 + 1 degeneracy
With spin degeneracy: 2, 4, 6, 8, ⋯𝑁 = 2, 6, 12, 20, ⋯, 𝑛 + 1 𝑛 + 2 ,⋯

Quantum dot in magnetic field

Hamiltonian with 𝑩 = (0,0, 𝐵)

Expansion of the kinetic energy term

Definition of cyclotron frequency and composite
harmonic confinement potential frequency
The Hamiltonian is rewritten as
Fock-Darwin state eigen energies

Quantum dot in magnetic field
12

10

2nd

E ( n r, l )/ h w

1st

8

6

4
N=12
2

Lowest Landau
012

w c /w

3

4

5

Fock-Darwin states
12

Level crossing points

22
20

-0.5
10

18

2nd

16

𝑛𝐿 : Landau index= 1, 2, ….

8

6

4
N=12

14
Vg Peak Position ( V)E ( n r, l )/ h w

1st

10

-1

8
6
4
2

-1.5

2

Lowest Landau
0

12

1

2

w c /w

3

4

・: Solutions
5

N=0
0

1

2

Magnetic Field (T)3Lecture on

Semiconductors / 半導体(Physics of semiconductors)
2021.7.7 Lecture 13
10:25 – 11:55

Institute for Solid State Physics, University of Tokyo

Shingo Katsumoto

Review of the lecture in the last week Aharonov-Bohm effect and quantum transport Bunching and anti-bunching of particles (bosons and fermions) Waveguide propagation of exciton-polaritons
 Bose-Einstein condensation of exciton-polaritons Single electron effect in quantum dots

2

Review: Single electron effect in transport through quantum dots
Enthalpy

𝐻(𝑁, 𝑉g )

Vg

EF

m N+2
m N+1
mN
m N -1

Vg

N =0

1

2

m N+2
EF

EF

𝑉g

EF
m N+1

J

mN
m N -1

𝑉g
3

Quantum confinement

𝐻(𝑁, 𝑉g )Zero-dimensional confinement to a quantum dot gives
shifts in Coulomb peak positions.

𝜖1 , 𝜖2 , ⋯ , 𝜖𝑁 , 𝜖𝑁+1 , ⋯

𝜖𝑁+2
𝜖𝑁+1
𝜖𝑁

Enthalpy shift by quantum
confinement
Chemical potential shift

Shift in gate voltage

∆𝜖𝑁+1

∆𝜖𝑁

𝜖𝑁+1

∆𝜖𝑁

𝜖𝑁
𝑁𝑒

𝑁+1 𝑒
𝐶𝑠
∆𝜖𝑁
𝑒

𝐶g 𝑉g

Quantum confinement effect in a vertical quantum dot

Two-dimensional harmonic potential
Potential shape:
Easy solutions from 1d
harmonic potential
Eigen energies:

𝑛𝑡 + 1 degeneracy
With spin degeneracy: 2, 4, 6, 8, ⋯𝑁 = 2, 6, 12, 20, ⋯, 𝑛 + 1 𝑛 + 2 ,⋯

Quantum dot in magnetic field

Hamiltonian with 𝑩 = (0,0, 𝐵)

Expansion of the kinetic energy term

Definition of cyclotron frequency and composite
harmonic confinement potential frequency
The Hamiltonian is rewritten as
Fock-Darwin state eigen energies
Degree of degeneracy at B = 0

Quantum dot in magnetic field
12

10

2nd

E ( n r, l )/ h w

1st

8

6

4

2

Lowest Landau
012

w c /w

3

4

5

Fock-Darwin states
12

Level crossing points

22
20

-0.5
10

18

2nd

16

𝑛𝐿 : Landau index= 1, 2, ….
=𝑛𝑟 + ( 𝑙 + 𝑙)/2

8

6

4
N=12

14
Vg Peak Position ( V)E ( n r, l )/ h w

1st

10

-1

8
6
4
2

-1.5

2

Lowest Landau
0

12

1

2

w c /w

3

4

・: Solutions
5

N=0
0

1

2

Magnetic Field (T)3Review: the Hall effect
Edwin HallV

1885-1938

Longitudinal
voltage+++++𝑩++++++++++++++++
𝑞𝒗 × 𝑩+++++

Current

𝑞
----𝑦

V

𝒗

𝑞𝑬H
- ---- ---- - ---------

Transvers voltage
Hall voltage𝑥(longitudinal voltage)(longitudinal 𝑅𝑥𝑦 =𝑅𝑥𝑥 =(current) resistance)(Hall voltage)(current)(Hall resistance)

Integer Quantum Hall Effect
1

0.3
n=1

B
Vy
Jx

0.2

Vx

0.6

2DEG 5 0mK
n=2

0.4

Rxx (h/e2 )RHall (h/e2 )

0.8

0.1
3

5

0.2

4

6
0

2

4

6
B (T)

8

0
10

Birthday of quantum Hall effectKlaus vonKlitzing

IQHE and Landau quantization

From Wikipedia

Two dimensional electrons under magnetic field
B

Hendrik Lorentz
1853 - 1928

Lorentz force (magnetic field only)
Cyclotron motion

E

This can be viewed as a motion in harmonic potential.

With electric field
R: Moves vertically to E with constant velocity E/B
Quantum mechanical Hamiltonian (no external electric field)

Landau quantization (two-dimensional)
Commutation relation
Magnetic length
Lev Landau
1908 - 1968

Space coordinate operator
Guiding center operator

down/up operator
Remember:
1-d harmonic oscillator

down/up operators

Landau quantization: Landau gauge
Diagonalize X : Landau gauge 𝑨 = (0, 𝐵𝑥)
Schrödinger equation

Plane wave solution along y

Harmonic oscillator solution
x-direction Gaussian centery-direction group velocity = 0

Landau quantization: forms of wavefunctions

Diagonalize X

Diagonalize 𝑋 2 + 𝑌 2← Symmetric gaugeShubnikov-de Haas oscillation
Number of states in 𝑆 = 𝑊𝑥 × 𝑊𝑦
“Distance” of k-values in y-direction: 2𝜋/𝑊𝑦: Filling factor (number of Landau levels filled with electrons)
E
5 4

3

2

1

𝐸𝐹 0
𝑛=0
𝐸 = ℏ𝜔𝑐 𝑛 +

5 4

3

𝜈=2

B

1
2
𝜈=1

(peak position)

SdH oscillation (example)(peak number)Localization/delocalization of wavefunctions
D (E)
山
Peak

delocalized
非局在Localized
局在谷
ValleyEhwc

𝜉(𝐸)−1 = 𝛼 𝐸 ∝ 𝐸 − 𝐸𝑁 𝑠
Numerical simulation s = 2 for N = 0
Aoki & Ando, PRL 54, 831 (1985).

Edge mode explanation of IQHE𝑚𝜔𝑐(𝑥 − 𝑋)2
2

U(x)

V(x)

In an edge mode, the group velocity appears because
the energy levels varies with x.
E 2x
E 1x

Current brought by a Landau edge mode

E 0x
- W/2

W/2
y

試料部分

Sample(X, Y)

x(X, Y)B

One dimensional system:
Landauer formula is applicable

x

E
mB

mA

Chiral edge mode: No backscattering!

Explanation from topological aspect
Bloch electrons under magnetic field: tight binding modelTranslational operator:
Hamiltonian:→ simultaneous diagonalization → Bloch states

does not have translational symmetry
Magnetic translation operator
Symmetric gauge

Magnetic Brillouin zone
HoweverMagnetic unit cell: unit vectors 𝒂, 𝒃 → magnetic unit vectors (𝑞𝒂, 𝒃)
Lattice vector : 𝑅 ′ = 𝑛 𝑞𝒂 + 𝑚𝒃

𝑇𝐵𝑹′ : elements commute

𝜓: simultaneously diagonalizes ℋ and 𝑇𝐵𝑹′

Magnetic Brillouin zone:

Magnetic Bloch function:

Magnetic Bloch function

Remember k∙p approximation

Schrodinger-like equation for 𝑢𝑛𝒌 (𝒓)k- dependent HamiltonianKubo formula for 𝜎𝑥𝑦
Electric field along y-axis: E

Unperturbed state

Ryogo Kubo
1920 - 1995

Magnetic Bloch function (II)
Velocity operator:

Normalization:𝑗 = 𝑥. 𝑦Kubo conductivity calculated with magnetic Bloch functions

Vector field:

Berry connection

Berry curvatureTKNN Formula
Existence of zero or anomaly
Magnetic Brillouin zone
On the boundary 𝜕𝐻

2p / a
HII

HI

ky

0

H

: Chern number (integer)
Topological invariantkx

2p / qa

Thouless-Kohmoto-Nightingale-den Nijs (TKNN)
FormulaLaughlin’s discussion
R. Laughlin, Phys. Rev. B 23, 5632 (1981).

B

Φ
Robert LaughlinyxLandau gauge
Magnetic flux Φ : X shift

Chern number =1

Summary of “topological aspect”(a) In 2D system under magnetic field: magnetic Bloch functions,magnetic Brillouin zone(b) Kubo formula for Hall conductivity: matrix elements of velocity
operator(c) From (a) and (b) Hall conductivity is obtained as the integration of
Berry curvature over magnetic Brillouin zone(d) TKNN formula: Chern number (topological invariant) times
quantum conductance(e) Chern number is integer (due to single-valuedness of atomic part)and non-zero in quantum Hall system (Laughlin’s discussion)
31

Bulk-Edge correspondence

Hasan & Kane, Rev. Mod. Phys. 82, 3045 (2010).

Transition between bands with different Chern number only can attained
through energy gap collapse.

Fractional quantum Hall effect
Laughlin stateLecture on
2021.7.14 Lecture 14

Semiconductors / 半導体

10:25 – 11:55(Physics of semiconductors)

Institute for Solid State Physics, University of Tokyo

Shingo KatsumotoWhat we have seen
Semiconductor basics








Band structure
Effective mass approximation
Carrier statistics
Electron-photon couplings
Thermodynamics
Semi-classical transport (Boltzmann equation)

Spatial modulation basics








Modulation doping: pn-junctions
Schottky junctions, MOS junctions
Hetero-junctions
Quantum confinement
Quantum wells, wires and dots
Minority carrier confinement

Quantum physics insemiconductors

 Fermion transport: Landauer (-Büttiker) formalism
 T-matrix, S-matrix
 Boson transport, Bose-Einstein condensation
 Quantum dots: Single electron effect, quantum confinement
 Quantum Hall: Edge mode, topological number2

Part of topics
Laser diode

Charge (kinetic) freedom
e+
e+

e+

Quantum confinement

e- ee- e-

e+

e-

e+

e-

𝐽𝐸

𝐽𝐶
𝑉BC

Semiclassical transport
1
n=1

B
Vy
Jx

Vx
2DEG 50mK

n=2

0.4

Rxx (h/e2)

RHall (h/e2)

0.8
0.6

Quantum wells, wires, dots

0.3

0.2

0.1
3

5

0.2

4

6
0

2

4

B (T)

6

8

Quantum Hall and topology
in solid state physics

100

Si technology: FinFET
Quantum dot: single electron, quantum confinement

Chapter 10a Spintronics I

Two current model
Spin injection

Spin degree of freedom: A new paradigm
Charge (kinetic) freedom
e+

e+
e+

e-

ee-

e-

e+
e-

e+

Spin degree of freedom

e-

𝐽𝐸

𝐽𝐶

Giant magnetoresistancespin valve
Spin injection

𝑉BC

Spin-manipulation ofquantum information

1

0.6

B
Vy
Jx

Vx

2DEG50mK
n=2

0.4
0.2
6
0

5 4
2

Rxx(h/e2)

RHal (h/e2)

0.8

0.3
n=1

0.2

0.1

3

4

B(T)

6

8

100

Topological insulators

Nobel laureates

2000

1956

2000 (Chemistry)

Charge (kinetic) freedom
Spin degree of freedom
1973

1985

2016

2010

2007
1998

2014

6

The two current model

Nevill Mott

Divide a current to the one with ↑ spin and the one with ↓ spin.

1905-1996

Drude:
Condition: spin diffusion length 𝜆𝑠 ≫ 𝑙 mean free path (or other lengths)
Spin polarized current:

drift

Einstein relation for metals:

𝜖𝑠 : local Fermi energy, 𝛿𝜖𝑠 : Shift from thermal equilibrium

Spin-dependent chemical potential

diffusion

Spin current
Remember Boltzmann equation
Because spin carriers are dipoles it is difficult to apply forces (needs magnetic field gradient)→Diffusion current only
Spin current (simplest) definition
Angular momentum conservation
With spin relaxation
cf. Charge conservation
Steady statespin diffusion equation
spin diffusion lengthSpin injection

𝑗𝑐
𝜇↑
NM

FM

𝜇↓
𝜇

𝐸

𝜌↑ (𝐸) 𝜌↓ (𝐸)

𝜇0

FM

𝜇
𝐸

𝜌↑ (𝐸) 𝜌↓ (𝐸)

𝜇↑

NM

𝜇0
𝐸

𝜌↑ (𝐸) 𝜌↓ (𝐸)𝐸𝜇↓𝜌↑ (𝐸) 𝜌↓ (𝐸)M = F, NSpin injection and detection
𝑗𝑐
𝜇↑
𝜇↓
FM1

NM
𝜇↑

FM2
𝜇0F2

𝜇0F1
𝜇0N
𝜇↓

Jedema et al. Nature 410, 345 (2001).

Spin precession
Zeeman Hamiltonian

From Heisenberg equation:
Solutionz
Larmor frequency

𝜔0yx

Spin precession experiment



Mg O
FM

FM

NM

DV (mV)

V

jc
NM





SC

-
Gate

-
-

-

-

-



H (Oe)

H









Chapter 10b Spintronics II
Spin-orbit interaction
Spin Hall effectTopological insulator (quantum spin Hall effect)

Spin-orbit interaction (in electron motion)
Pauli approximation of Dirac equation:: Spin-orbit interaction

III

V
BIA: Bulk inversion asymmetry
SIA: Structure inversion asymmetrySIA-SOI Rashba-type SOIBIA SOI(Actually through the valence band)

𝐸±

𝑚∗ 𝛼
− 2
ℏ

𝑚∗ 𝛼
ℏ2
𝑘

SOI and SdH oscillation

6
V g= - 1.0V

T =0.4K

rxx (arb.)

- 0.7
4

- 0.3
0
0.3

2

0.5
1.5
0
1

B (T)

2

Nitta et al., Phys. Rev. Lett. 78, 1335 (1997).

Spin Hall effect𝑘𝑦Effective magnetic field
𝑘𝑦
spin
𝑘𝑥

𝑘𝑥

effective fieldkHow we understand the quantum Hall effect?
Edge mode transport

Topological discussion

Landau quantization

Magnetic Bloch function
Magnetic Brillouin zone

Edge mode transport

Kubo formula

Landauer formula

TKNN formula
Topological invariant
Hall conductance quantization

Bulk-edge correspondence18

Spin Hall effect in an insulator
Remember k∙p approximation

Consider the case these are not zero. Then the discussion is in parallel with the TKNN formula.

Anomalous velocity and quantum spin Hall effect
Wave packet:

Bloch wave expansion

Anomalous velocity
TKNN

Spin-subband
Chern number

Spin Chern number

Topological insulator: helical edge state
𝐸
𝐸𝐹

Chargeconservation:𝑘yOrdinary insulator
Extra spin flow at the edge

0

Helical edge mode:
Topological insulator
Edge mode number = Chern number

Topologically insulating quantum well
König et al., Science 318, 766 (2007).

7.3nm
(a)

(b)

(c)

Summary
Charge (kinetic) freedom
e+
e+

e+

e-

ee-

e+
e-

e+

Spin degree of freedom
Giant magnetoresistancespin valve
Spin injection

ee-

𝐽𝐸

𝐽𝐶

𝑉BC

Spin-manipulation ofquantum information
1

0.6

B
Vy
Jx

Vx

2DEG50mK
n=2

0.4
0.2
6
0

5 4
2

Rxx(h/e2)

RHal (h/e2)

0.8

0.3
n=1
0.2

0.1

3

4

B(T)

6

8

100

Topological insulators