Spin scattering in nonmagnetic semiconductors

Spin state

( ) ↓+↑= ↓↑ ,,, kkk ϕϕψ sWave function:

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛==

↓↓↓↑

↑↓↑↑

ρρρρ

ψψρ ss ,, kk

1112

z x ys

x y z

P P iPP iP P

ρ+ −⎛ ⎞

= ⎜ ⎟+ −⎝ ⎠

Spin density matrix:

In general, it should be

Spin polarization vector:

OR

( )ss ′′ ,;,kkρUse Wigner’s functionto get

( )ss ′,;,RKρ

( )zyx PPPP ,,=r

Local spin polarization

ΔV

X

Y

Z

x0

y0

z0

Number of electrons: N=n · ΔVMagnetic moment: M=N·μB·P

X

Y

Z

PZ(t)

P1

P2

P3

P4

P5

PSpin polarization |P|≤1

X

Y

Z ω

∆P

PZ(t)

Z

Evolution of spin polarization

Spin polarization vector

( ) ⊥−−−×= PT

PPT

BPdtPd rrrrrr

20||

1

11γBloch equation:

Spin scattering is here

Today we talk about T1 processes

Interaction with magnetic fieldInteraction with magnetic field

Hz

BH gμ=− =μH SH24

B 9.27 10μ −= × J/T

Px

t

Py

t

Pz

t

tϕ ω=

x

k

k’

No scattering!Pure dephasing.

Evolution of spin polarization vector

T2* instead of T2

T1 remains the same

Spin-orbit interactionElliot-Yafet mechanism

D’yakonov-Perel mechanism

Hyperfine interaction with nuclear spins

Exchange with holes (Bir-Aronov-Pikus)

Spin scattering

No direct interaction with phonons & non-magnetic impurities.

T1 processes We have to define basis for spin ↑ and ↓.Use small magnetic field.

Spin flip vs. spin precession

↓,k

↑,'k

νhE =

Phonon

↓,k

Impurity

↑,'k

OR

Type I

Sr

kr

phononωh

Impurity

Initialstate

Finalstate

k ′r

S ′r

Type II

Electron-hole exchange

Hyperfine interaction

)(rJSδAHexch =Exchange Hamiltonian:

Important in p-type semiconductors

)()(Hf rSIr δAH =Hyperfine Hamiltonian:

Suppressed in strong magnetic fields

Hole

Electron

momentum and J can relax

Nucleus

Electron

Elliott-Yafet mechanism

Spin states: ↑,k↓,k

k

E

( )( )

3/ 2

3/ 2

," " ( ) ( )

," " ' ( ) ' ( )

i

i

k L e u v

k L e u v

−

−

↑ = ↑ + ↓

↓ = ↓ + ↑

krk k

krk k

r r

r r

mixes pure spin up and down states

" "↑

" "↓

↓,k

↑,'k

νhE =

Phonon

↓,k

Impurity

↑,'k

Any momentum scattering can result in a spin flip.

D’yakonov-Perel’ mechanism

Sr

kr

phononωh

Impurity

Initialstate

Finalstate

k ′r

S′r

)()()( 222222yxzzxzyyzyxxSO kkkkkkkkkH −+−+−= Γ σσσα

Spin-orbit interaction:

Crystals without a center of inversion

GaAs

Fe

GaAs

S

V

GaAsFe n+

n+

Α=Α=

Α=

=

Γ

eVeV

eV

eVq

X

L

B

087.027.0

28

72.03

ααα

φ

1 μm

Region is studied

D-P scattering in a device structure, example

0.0 0.2 0.4 0.6 0.8 1.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

Pote

ntia

l, V

X, μm

Potential Profile, Vds = 2 V

Model for Monte-Carlo simulation

• Spin scattering mechanism:

• Charge transport (BTE):

• Spin density matrix evolution:

CtffVqf

tf

⎟⎟⎠

⎞⎜⎜⎝

⎛=∇⋅∇−∇⋅+

∂∂

∂∂

kv )(h

( )ds

22 )( NneV −−=∇ r

ε

hh // SOSO )()( dtiHi

dtiHi etedtt ρρ −=+

)(R yxxy kkH σση −=2

D ( )z y y x xH k k kβ σ σ= < > −

ρρ

φ

=′=′

=′

−=′

zz

yy

Bxx

EE

EE

qEE

h

h

h

/2

/2

/2

*

*

*

zz

yy

xx

Emk

Emk

Emk

′±=′

′±=′

′=′

Injection Mechanisms

Bq φ

fmE

cE

fsE

xE

xE′

eVbias

• Thermionic Emission: Bx qE φ> & spin is conserved

ΦΦ−=′

=′

=′

=′

iizz

yy

x

ee

EE

EE

E

ρρ

0

• Tunneling through the Schottky barrier: Bx qE φ<

h

h

/2

/2

0

*

*

zz

yy

x

Emk

Emk

k

′±=′

′±=′

=′Bq φ

fmE

cE

fsE

xE

tpx

)( tpc xE

eVbias

)])([22exp()(0

*tp dxExEmET

tpx

xc∫ −−=h

Tunneling probability:(WKB approximation)

Injection Mechanisms

TkEETkEE

Bfm

Bfme

eEf /)(

/)( ~1

1)( −−−+

=

• Electron distribution function:

• Equal average kinetic energy in x, y and z directions

Electrons at the Ferromagnetic Contact

• Probabilities of spin states are based on the densities of states

↑↓↓↑

↑↑ −=

+= PP

EDEDED

EP 1and)()(

)()(

Spin-orbit couplings in 3 valleys of bulk GaAs

1. Γ valley (000)

2. L valleys

3. X valleys

)()()( 222222yxzzxzyyzyxxSO kkkkkkkkkH −+−+−= Γ σσσα

⎪

⎪⎩

⎪⎪⎨

⎧

+−++−++−+−−

−++−+−+−+−

=

)()()(3/)()()(3/

)()()(3/)()()(3/

yxzzxyyzxL

yxzzxyzyxL

xyzzxyzyxL

yxzxzyzyxL

SO

kkkkkkkkkkkk

kkkkkkkkkkkk

H

σσσασσσα

σσσασσσα ( 1 1 1)

(-1-1 1)

(-1 1-1)

(1-1 -1)

⎪⎩

⎪⎨

⎧

−−−

=)()()(

yyxxX

zzxxX

yyzzX

SO

kkkkkk

Hσσασσασσα (±1 1 1)

( 1±1 1)( 1 1±1)

Dresselhaus Mechanism

Carrier ConcentrationT = 300 K, V = 2 V

The extremely high electric field at the Schottky barrier pumps the electrons onto up valleys, especially the X valley

0.00 0.05 0.10 0.151E18

1E19

1E20

1E21

1E22

1E23

conc

entra

tion,

m-3

X, um

Γ valley L valley X valley

Spin dephasing is much stronger in upper valleys due stronger SO coupling

Current Spin Polarization

T = 300 K, V = 2 V

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

Cur

rent

pol

ariz

atio

n

X, μm

Γ valley L valley X valley

0.00 0.05 0.10 0.15

1E14

1E15

1E16

Con

cent

rato

in m

-3

X, um

Gamma L X valley

T = 300 K, V = 0.5 VCarrier ConcentrationLow bias

T = 300 K, V = 0.5 V

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

Cur

rent

spi

n po

lariz

atoi

n

X, um

Gamma L X

Current Spin Polarization

T = 80 K, V = 2 VLow temperature

0.00 0.05 0.10 0.151E10

1E11

1E12

1E13

1E14

1E15

co

ncen

tratio

n, m

-3

X Axis Title

Γ valley L valley X valley

Carrier Concentration

Current Spin Polarization

T = 80 K, V = 2 VLow temperature

0.00 0.05 0.10 0.150.0

0.2

0.4

0.6

0.8

1.0

Cur

rent

pol

ariz

atio

n

X, μm

Γ valley L valley X valley

Spin-LED

GaAs (001)

p-GaAs

hν

Spin Polarizing contact

n-AlGaAsGaAs QW

p-AlGaAs

M. Yasar, PhD Thesis

A0

Do

e1

h1ħω

AlGaAs(n)

GaAs

AlGaAs (p)

ħΩ

A-

++++ +

- - - - -CBVB

X/K

Spin-LED

11600 11800 12000 12200 12400 126000

1x105

2x105

3x105

4x105

5x105

6x105

7x105

Sample 1T = 6 KCB A

EL Spectra X a XBulk

EL In

tens

ity (a

rb.u

.)

Energy (cm-1)

Electroluminescence (1,0,0)

12200 12300 12400 12500 12600

0.0

5.0x104

1.0x105

1.5x105

2.0x105

Sample 1T = 6 K

EL Spectra X a XBulk

EL In

tens

ity (a

rb.u

.)

Energy (cm-1)

ΔE~84 cm-1 = TA phonon at X

R. Mallory, et. al., Phys. Rev. B 73, 115308 (2006)

-6 -4 -2 0 2 4 6-30

-20

-10

0

10

20

30

Sample 1T = 6 K

Feature X Feature a

Pola

rizat

ion

(%)

Magnetic Field (T)

11600 11800 12000 12200 12400 12600

0.0

5.0x103

1.0x104

1.5x104

2.0x104

2.5x104

Sample 3T = 6 K

EL Spectrum X a'

XBulk Bulk feature

EL In

tens

ity (a

rb.u

.)

Energy (cm-1)

Spin-LED

Electroluminescence (1,1,0)

ΔE~105 cm-1 ~ phonon at K

Questions?