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 of non-interacting spins 2

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 ∗ • , , and • Bloch equation The Zeeman Hamiltonian 3

Hamiltonian for an spin in a

of an electron spin : magnetic moment : magnetic field : Landé g-factor (=2 for free ) : Bohr magneton 58 eV/T , ) : electronic charge : Planck constant : free electron mass

: spin operator : Pauli operator The Zeeman Hamiltonian 4

Hamiltonian for an electron spin in a magnetic field

magnetic 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 : spin operator 2 : Pauli operator The energy eigenstates 5

Without loss of generality, we can set the -axis to be the direction of , i.e., 0,0,

1 Spin “up” |↑ |↑ 2 1 2 2

1 Spin “down” |↓ |↓ 2

1 2 2 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 are

1 ex. |↑ |↑ is equivalent to 2

1 10 1 1 1 1 |↑ |↑ |↑ 2 2 01 0 2 0 2 Expectation values of the Pauli vector 7

Expectation values for the spin up state

01 1 0 ↑| |↑ 10 10 0 10 0 1

0 1 0 ↑| |↑ 10 10 0 00

10 1 1 ↑| |↑ 10 10 1 01 0 0

0,0,1 spin is pointing in the +z direction Expectation values of the Pauli vector 8

One more example: Expectation values for a coherent superposition 1 1 1 | |↑ |↓ 2 2 1

1 01 1 1 1 | | 11 11 1 2 10 1 2 1 1 0 1 1 | | 11 11 0 2 01 2 1 10 1 1 1 | | 11 11 0 2 01 1 2 1

1,0,0 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” The Bloch sphere 9

In general, | |↑ |↓ where and are complex numbers.

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

one-to-one correspondence 1 |↑ |↑ |↓ 2 | cos |↑ sin |↓ 2 2 θ 1 |↑ |↓ This is the most general state. 2 complex 2 coefficients have 4 degrees of freedom, 1 minus normalization condition and ϕ |↑ |↓ arbitrariness of the global phase. 2

1 |↑ |↓ |↓ 2 Time evolution of spin states 10

How does a spin state that is perpendicular to magnetic field evolve with time? 1 Initial state: | 0 |↑ |↓ 2

1. Obtain the time-evolution operator exp

2. Compute the state at time

| | 0

3. Calculate the expectation value of

Time-evolution operator 11

1. Time-evolution operator is given by

10 exp where 2 01 so

Ω exp 0 exp 0 2 2 Ω 0exp 0exp 2 2

where Ω is the Larmor frequency 14 GHz/T 2 2. The state at time is obtained by applying on the initial state 1 1 1 | 0 |↑ |↓ 2 2 1 1 exp Ω⁄ 2 0 1 1 exp Ω⁄ 2 | | 0 2 0expΩ⁄ 2 1 2 exp Ω⁄ 2 Larmor precession 12 3. Expectation value of 1 01 || 2 10 1 2 cosΩ 1 0 || 2 0 2 spin precession in the plane at sinΩ an angular frequency Ω⁄

1 10 || 110 2 01 Heisenberg picture of Larmor precession 13

More intuitive semi-classical picture can be obtained

1 , , ,

using commutation relations: , , , , ,

combined into one vector equation

where Ω Semi-classical equation of motion 14

in terms of magnetization and spin (agrees with classical result!) Ω rate of change for exerted by angular momentum magnetic field • Torque is perpendicular to both magnetic field Ω and spin • Parallel component of Ω spin is conserved • Perpendicular component precesses

Spin precesses in a cone at frequency Ω

Spin relaxation and the Bloch equation 15

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

∗ Larmor precession revisited 16

For spins initially pointing along in a magnetic field along , the solution for spin component along is given by Ω 1 exp ∗ cos Ω 2

Summary: physics of non-interacting spins 17

• Zeeman Hamiltonian

• Bloch sphere Ω • Larmor precession ∗ • , , and • Bloch equation

Optical spin injection and detection 18

Time-resolved Kerr rotation data In direct gap nonmagnetic semiconductors, T = 5 K Larmor precession can be directly observed B = 0 T in the time domain 0 0

B = 1 T • Optical selection rules 0 0 (a.u.) x

• Time-resolved S B0 = 3 T Kerr/Faraday rotation 0 AlGaAs quantum well • Hanle effect 0 1 2 Time (ns)

Electron 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 band

E c 6 cb

3113

    + +   v 8 hh lh mj = 3/2 mj = 1/2 mj = +1/2 mj = +3/2 0 k valence band

Zinc-blende: 50% polarization Wurtzite and quantum wells: 100% polarization Time-resolved Faraday (Kerr) rotation 20

1. Circularly polarized pump creates spin population B S

(1) ~100fs pulses at 76MHz pump Ti:Sapphire

sx = -1/2 sx = 1/2 Conduction electrons

sx=±1/2

+  Heavy and j = -3/2 j = 3/2 light holes x x jx = -1/2 jx = 1/2 31: : 13: 1. Circularly polarized pump creates spin population

2. Linearly polarized probe measures Sx

B S (2) probe  F  Sx

 + K Sx absorption sx = -1/2 sx = 1/2  +

+   +  index of refraction j = -3/2 j = 3/2 x x photon energy 1. Circularly polarized pump creates spin population

2. Linearly polarized probe measures Sx 3. t is scanned and spin precession is mapped out

B S (3) t

(2) probe  F  Sx (1) pump  K  Sx

* t /T2 F F  Ae cos(t)

  gB B /  t S. Crooker et al., PRB 56, 7574 (1997) TRFR in bulk semiconductors 23

GaN GaAs PRB 63, 121202 (2001) PRL 80, 4313 (1998) TRFR in quantum wells and quantum dots 24

ZnCdSe quantum well Science 277, 1284 (1997) CdSe quantum dots PRB 59, 10421 (1999) Hanle effect 25

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

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

S0

“Optical Orientation” (Elsevier, 1984) Imaging the spin accumulation due to spin Hall effect 26

Science 306, 1910 (2004) ns (a.u.) Reflectivity (a.u.) -2-1 021 1 2 3 4 5

Spin accumulation at the edges are imaged 150 by modulating the externally applied magnetic field and measuring the signal at the second harmonic frequency 100

B ext 50 m)

2  T = 30 K 0 1 rad)  0 x = - 35 m Position ( -50 x = + 35 m 0

-1 unstrained -100 Kerr rotation ( GaAs -2 -40 -20 02040 -150 Magnetic field (mT) -40-20 02040-40-20 02040 Position (m) Position (m) Spin manipulation in nonmagnetic semiconductors 27

There are many interactions in nonmagnetic semiconductors that allow spin manipulation

• g-factor engineering • spin-orbit interaction • hyperfine interaction • AC Stark effect g-factor engineering 28

H = gB B·S Control the g-factor through material composition in g: material dependent effective g-factor semiconductor heterostructures B: magnetic field

g= -0.44 in GaAs g-factor in Al Ga As g= -14.8 in InAs x 1-x 0.6 g= 1.94 in GaN 0.4 0.2

Change g in a static, global B! 0.0 g-factor -0.2 -0.4 0.0 0.1 0.2 0.3 0.4 Al concentration x move electrons into different materials using electric fields Weisbuch et al., PRB 15, 816 (1977) Quasi-static electrical tuning of g-factor 29

Nature 414, 619 (2001)

T=5K, B=6T 100 nm g-factor 0.0V E=0 negative 0

-0.8V 0

-1.4V

0

-2.0V Kerr Rotation (a.u.) Kerr Rotation 0

-3.0V 0 E>0 positive

0 500 1000 Time Delay (ps)

g-factor is electrically tuned Frequency modulated spin precession 30

Time dependent voltage

V(t)=V0+V1 sin(2t)

S B

laser

Blue: Kerr Rotation Red: microwave voltage

Fits well to equation  : t Science 299, 1201 (2003)      Ae 1 cos t  1 cost   Spin precession is frequency modulated  0     g-factor tuning at GHz frequency range Spin-orbit interaction 31

H = ħ/(4m2c2) (V(r) × p)·

lab frame electron’s frame “E” “B”

e- e- relativistic transformation k

Allows zero-magnetic field spin manipulation by electric field through control of k Rashba and Dresselhaus effects 32 time-reversal symmetry Ek, E k, Kramers degeneracy (i.e., at zero magnetic field) inversion symmetry Ek, E k, Zero-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)

Ω 2 , , In a (001) quantum well Ω 2 , Strain-induced terms

Ω ∝ , ,

Ω ∝ , , 33 Spatiotemporal evolution of spin packet in strained GaAs

Nature 427, 50 (2004) -20 E = 0 V cm-1 E = 66 V cm-1 m) V=0  0 0

pump m) 20 FR (a.u.) 20  -1 0 132 E = 33 V cm-1 40 e- 0 E = 100 V cm-1 d 0 Pump-probe distance ( z 20 E 20 3 probe 2 z E Pump-probe distance ( 1 33 V cm-1 40 T=5K -1 0 -1 66 V cm B=0.0000T V=V FR (a.u.) 100 V cm 0 -1 0 1 2435 0 13524 Time (ns) Time (ns) Spin precession at zero magnetic field! z (“E”) k Bint

0 V cm-1

50 V cm-1

100 V cm-1 Coherent spin population excited with electrical pulses 35

Phys. Rev. Lett. 93, 176601 (2004) 10 m -2 -1 0 12 F (au) InGaAs 60 m T = 5 K Vpp = 20 V 10 100 m 90 m   0 +44 mT

[110] probe pump -44 mT (ns) background t 5  (arb. units) 0 F

contact to contact to  0 InGaAs substrate 0 0510 -40 -20 02040 t (ns) semi-insulating GaAs substrate B (mT) Precession at electron Larmor frequency of InGaAs Beff S0 B Faraday rotation signal due to electrons The signal shows sign change with magnetic field E spins excited in the plane of the sample (along the effective magnetic field) Electron spins are generated using picosecond electrical pulses Hyperfine interaction 36

Nuclear spin polarization acts as an effective magnetic field for electron spins : nuclear spin : electron spin Dynamic nuclear polarization: spin injection causes nuclear spin polarization Lampel, PRL 20, 491 (1968)

PRL 56, 2677 (2001) 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 Kerr/Faraday rotation • Bloch sphere • Hanle effect • Larmor precession ∗ • g-factor engineering • , , and • Bloch equation • spin-orbit interaction • hyperfine interaction • AC Stark effect