Electron spins in nonmagnetic
semiconductors
Yuichiro K. Kato
Institute of Engineering Innovation, The University of Tokyo

Physics of non-interacting spins
Optical spin injection and detection
Spin manipulation in nonmagnetic semiconductors

Physics of non-interacting spins
In non-magnetic semiconductors such as GaAs and Si, spin interactions
are weak; To first order approximation, they behave as non-interacting,independent spins.

• Zeeman Hamiltonian• Bloch sphere• Larmor precession•,, and∗• Bloch equation

2

3The Zeeman Hamiltonian
Hamiltonian for an electron spin in a magnetic fieldmagnetic moment of
an electron spin

: magnetic moment: magnetic field: Landé g-factor(=2 for free electrons): Bohr magneton,)

58 eV/T: electronic charge: Planck constant: free electron mass
: spin operator: Pauli operator

4The Zeeman Hamiltonian
Hamiltonian for an electron spin in a magnetic fieldmagnetic moment of
an electron spin

: magnetic moment: magnetic field: Landé g-factor(=2 for free electrons): Bohr magneton

2

58 eV/T: electronic charge: Planck constant: free electron mass

2: spin operator: Pauli operator

5The energy eigenstates
0,0,

Without loss of generality, we can set the -axis to be the direction of , i.e.,

Spin “up”

1
|↑2

|↑

1
2

Spin “down”

122

|↓

1
|↓
22

Spinor notation and the Pauli operators

6

Quantum states are represented by a normalized vector in a Hilbert space
Spin states are represented by 2D vectors

Spin operators are represented by 2x2 matrices. For example, the Pauli operators areex.

|↑|↑

1|↑2is equivalent to|↑1 12 0201

101 1
2 01
|↑2

Expectation values of the Pauli vector

Expectation values for the spin up state↑|↑||↑|↑↑| |↑0,0,1

1 0

1 0

1 00 1
1 0

1000

10011010

1 0

1 0

1 00
100

10

spin is pointing in the +z direction

017

8Expectation values of the Pauli vector
One more example: Expectation values for a coherent superposition

1|

2

|↑|↓

1

12 1|

1
1 1
20 1
1 0|

1
1 1
20| |

1
1 1
210|
|

1,0,0

111
1 1201111 120
11111 1211

10
1
10

spin is pointing in the +x direction

spin can point in any direction, but when measured along a particular axis,
it can only be “up” or “down”

9The Bloch sphere
In general, ||↑|↓ whereandare complex numbers.

Any spin state (and any qubit) can be represented by a point on a spheresin cos, sin sin, cos

one-to-one
correspondence1|↑|cos |↑2

sin |↓2|↓θ1This is the most general state. 2 complex
coefficients have 4 degrees of freedom,minus normalization condition andarbitrariness of the global phase.

2|↑|↑2

|↓

ϕ12

|↑

|↓|↓12|↑|↓

Time evolution of spin states
How does a spin state that is perpendicular to magnetic field evolve with time?
Initial state: |

102

|↑|↓

1. Obtain the time-evolution operatorexp
2. Compute the state at time||0

3. Calculate the expectation value of10

11Time-evolution operator
1. Time-evolution operator is given by
where

exp

2

1001

so
exp
0
where

exp

02
exp

2. The state at time is obtained by applying
||

012

|↑|

|↓0002is the Larmor frequency

ΩΩ2expΩ
214 GHz/T
2on the initial state

112 11 exp
2Ω ⁄20exp0Ω ⁄2111exp
2 exp

Ω ⁄2
Ω ⁄2

12Larmor precession
3. Expectation value of

|

|

|

12
12
cos Ω01120|

1002
sin Ω| |12

spin precession in theplane at⁄an angular frequency Ω100
1

1

10

Heisenberg picture of Larmor precession
More intuitive semi-classical picture can be obtained1,using commutation relations:,,,,,,,

combined into one vector equation

whereΩ

13

Semi-classical equation of motion

14

in terms of magnetizationand spin angular momentum(agrees with classical result!)

Ωrate of change forangular momentum

torque exerted bymagnetic field•

Torque is perpendicular
to both magnetic fieldand spin•

Parallel component ofspin is conserved•

Perpendicularcomponent precessesΩΩ

Spin precesses in acone at frequency ΩSpin relaxation and the Bloch equation
In a real world, interaction with the environment will eventually randomize spin
1. Longitudinal spin relaxation
Requires energy relaxation
2. Transverse spin relaxation
Phase relaxation is enough∗For an ensemble, Inhomogeneous dephasing∗
Bloch equation

∗

15

16

Larmor precession revisited
For spins initially pointing along in a magnetic
field along , the solution for spin component along
is given by1
exp
2

Ω
∗

cos Ω

Summary: physics of non-interacting spins• Zeeman Hamiltonian• Bloch sphere• Larmor precession•,, and∗• Bloch equation

Ω

17

Optical spin injection and detection

18

Time-resolved Kerr rotation data
In direct gap nonmagnetic semiconductors,Larmor precession can be directly observedin the time domain

T=5K

B0 = 0 T• Optical selection rules• Time-resolved
Kerr/Faraday rotation• Hanle effectSx (a.u.)0
B0 = 1 T
0
B0 = 3 T

0

AlGaAs quantum well01Time (ns)2Electron spins are optically injected at t=0

Optical selection rules

19

left circularly polarized photons carry angular momentum of +1
right circularly polarized photons carry angular momentum of -1(in units of ħ)
selection rules from angular momentum conservation
conduction bandE

6c3

+11

+cb3mj = 3/2 mj = 1/2 mj = +1/2 mj = +3/2

8v

lh0hhkvalence band
Zinc-blende: 50% polarization

Wurtzite and quantum wells: 100% polarization

Time-resolved Faraday (Kerr) rotation20

1. Circularly polarized pump creates spin population
BS(1)
pump

Conduction electrons
sx=±1/2

Heavy andlight holes~100fs pulses at 76MHz

Ti:Sapphiresx = -1/2

sx = 1/2

+

jx = -3/2

jx = 3/2

3

jx = -1/2 jx = 1/2

:

1:

1:

3

1. Circularly polarized pump creates spin population
2. Linearly polarized probe measures Sx
BS(2) probeF  SxK  Sx
sx = 1/2

sx = -1/2

+jx = -3/2

jx = 3/2

+absorption

++  index ofrefraction

photon energy

1. Circularly polarized pump creates spin population
2. Linearly polarized probe measures Sx
3. t is scanned and spin precession is mapped outBS(3) t

F  Sx(1)
pump(2) probeK  Sx F  AeFt t / T2*cos(t )  g B B / 
S. Crooker et al., PRB 56, 7574 (1997)

TRFR in bulk semiconductors
GaAs
PRL 80, 4313 (1998)GaN
PRB 63, 121202 (2001)

23

TRFR in quantum wells and quantum dots

ZnCdSe quantum well
Science 277, 1284 (1997)

CdSe quantum dots
PRB 59, 10421 (1999)

24

Hanle effect

25

In DC measurements, magnetic field dependenceof the spin polarization becomes Lorentzian.

 t el
expcostdtS
0  0  T2* * 2 T2  1S0

“Optical Orientation” (Elsevier, 1984)

Imaging the spin accumulation due to spin Hall effectScience 306, 1910 (2004)

ns (a.u.)
Reflectivity (a.u.)-2 -1 0 1 2 1
2 3 45Spin accumulation at the edges are imagedby modulating the externally applied magnetic
field and measuring the signal at the second
harmonic frequency

150100Bext

50

T = 30 K

100-1-2x = - 35 m
x = + 35 m
unstrained
GaAs-40 -20
020 40Magnetic field (mT)

Position (m)

Kerr rotation (rad)

226

0-50

-100

-150
-40 -20 0 20 40 -40 -20 0 20 40
Position (m) Position (m)

Spin manipulation in nonmagnetic semiconductors
There are many interactions in nonmagnetic semiconductors thatallow spin manipulation• g-factor engineering• spin-orbit interaction• hyperfine interaction• AC Stark effect

27

28

g-factor engineering

H = gB B·S
g: material dependent effective g-factor
B: magnetic field

Control the g-factor throughmaterial composition insemiconductor heterostructures

g= -0.44 in GaAs
g= -14.8 in InAs
g= 1.94 in GaN

0.6

g-factor

Change g in a static, global B!

g-factor in AlxGa1-xAs
0.40.20.0-0.2-0.4
0.0

move electrons into different materials
using electric fields

0.1

0.2

0.3

0.4

Al concentration xWeisbuch et al., PRB 15, 816 (1977)

29

Quasi-static electrical tuning of g-factor

Nature 414, 619 (2001)
T=5K, B=6T

100 nmg-factor
0.0V

Kerr Rotation (a.u.)

0E=0 negative-0.8V
0-1.4V0-2.0V0-3.0V

0E>0 positive
0500

1000

Time Delay (ps)g-factor is electrically tuned

Frequency modulated spin precession

30

Time dependent voltage

V(t)=V0+V1 sin(2t)SBlaser

Blue: Kerr Rotation
Red: microwave voltage
Fits well to equation :Ae



t

1
1
cos 0t cos  t   


Science 299, 1201 (2003)
Spin precession is frequency modulated
g-factor tuning at GHz frequency range

Spin-orbit interaction31

H = ħ/(4m2c2) (V(r) × p)·electron’s framelab frame
“E”

“B”
e-e-

relativistic transformationk

Allows zero-magnetic field spin manipulation
by electric field through control of kRashba and Dresselhaus effects
time-reversal symmetry(i.e., at zero magnetic field)

E k ,    E  k ,  Kramers degeneracy

E k ,    E  k ,  inversion symmetryZero-magnetic-field spin
splitting requires asymmetry

Rashba effect: structural inversion asymmetry (SIA) of quantum wells
Sov. Phys. Solid State 2, 1109 (1960)
Ω,Dresselhaus effect: bulk inversion asymmetry (BIA) of zinc-blende crystal
Phys. Rev. 100, 580 (1955)
Ω2,,In a (001) quantum wellΩ

2

Strain-induced termsΩ∝Ω∝,,,,32,

33

Spatiotemporal evolution of spin packet in strained GaAs

e-dEprobe

V=V0z-20

E = 0 V cm-1

E = 66 V cm-10020
E = 33 V cm-10203210-1z

E
33 V cm-1

100 V cm-1

01

Pump-probe distance (m)pump

FR (a.u.)

V=0

Pump-probe distance (m)Nature 427, 50 (2004)

20FR (a.u.)-1 0 1 2 340

E = 100 V cm-1

02040

T=5K
B=0.0000T

66 V cm-1

2 3 4Time (ns)501

2 3 4Time (ns)5Spin precession at zero magnetic field!

z (“E”)

Bint

0 V cm-150 V cm-1

100 V cm-1kCoherent spin population excited with electrical pulses

35Phys. Rev. Lett. 93, 176601 (2004)-2 -1 0 1 2F (au)

10 m
60 m

[110]

probe pumpcontact toInGaAs

contact tosubstratesemi-insulating GaAs substrate

T=5K
Vpp = 20 V
0+44 mT-44 mT
background000510t (ns)10t (ns)100 m 90 mF (arb. units)

InGaAs50-40 -20 0 20 40
B (mT)

Precession at electron Larmor frequency of InGaAs

Beff S0B

Faraday rotation signal due to electrons
The signal shows sign change with magnetic fieldE

spins excited in the plane of the sample(along the effective magnetic field)

Electron spins are generated using picosecond electrical pulses

Hyperfine interaction
Nuclear spin polarization acts as an effective
magnetic field for electron spins: nuclear spin
: electron spin

PRL 56, 2677 (2001)

Dynamic nuclear polarization:
spin injection causes nuclear spin polarization
Lampel, PRL 20, 491 (1968)36

AC Stark effect

37

Optical pulse below bandgap shifts the band gap (AC Stark effect)
Circularly polarized pulse shifts one of the spin subbands, causing spin splitting

+

mj = 3/2

mj = +3/2

Science 292, 2458 (2001)

Topics covered in this talk• Optical selection rules• Time-resolved• Zeeman Hamiltonian• Bloch sphere

Kerr/Faraday rotation• Hanle effect• Larmor precession•,, and∗• Bloch equation• g-factor engineering• spin-orbit interaction• hyperfine interaction• AC Stark effect