Optical transient-grating measurement of spin propagation in a two-dimensional electron gas

Chris Weber, June 5, 2006 National High Magnetic Field Laboratory

Optical transient-grating measurement of spin propagation in a two-dimensional electron gas

Chris WeberUC Berkeley and Lawrence Berkeley National Lab


LBNL, UC Berkeley, Stanford, and UCSB collaboration

CW, Nuh Gedik, Joel Moore, Joe OrensteinUC Berkeley and LBNL

Jason Stephens and David AwschalomCenter for Spintronics and Quantum ComputationUCSB

Andrei Bernevig and Shouchang ZhangStanford


Outline Introduction to fast optics

Spin physics in GaAs 2DEGs

Measuring spin propagation: the transient spin grating

Observation of anomalous diffusion

Prediction of the persistent spin helix, and preliminary observations

Observation of spin Coulomb drag: e-e collisions suppress spin diffusion









Fast optics: time resolution and broad dynamic range

1 ns 10 ps 100 fs

1 ueV 0.1 meV 10 meV

e-e collisonsElectron-phonon int’n

Spin lifetimes

Spin-orbit splitting

Energy splittings or linewidths

Electron-hole pairs

Quasiparticles (high-Tc)Excited statesof biomolecules


Example of pump-probe: spin dynamics in a GaAs quantum well

ħ= 1.5 eV

Step 1:Optical orientation with a circular pump


Spin dynamics after circular excitation

0 10 20 30 40

Mag

netiz

atio

n (a

.u.)

Time (ps)

Pump ħ= 1.5 eV

GaAs QW(n-doped)


Measuring spin dynamics with a time-delayed probe

Detector

Probe (variable delay)

Pumpħ= 1.5 eV

GaAs QW(n-doped)

Anti-parallel circular polarizations

Parallel circular polarizations

0 10 20 30 40-1

0

1

2

3

4

5

T/T

(x10

4 )

Delay (ps)

spin(parallel) = -spin(antiparallel)


Pump-probe schematic

Detector

Center wavelength 800 nm ~ 1.5 eV

Pulse duration 100 fsRep rate 80 MHzAvg. power at sample 20 mW

Delay stage

Sample

Pump

Probe


Samples: 10-layer, modulation-doped quantum wells

GaAs (12nm)

Al0.3Ga0.7As

+ + + +

Si in barrier layer

n [1011 cm-2] TF [K] [cm2/Vs]

7.8 400 230,000

4.3 220 93,000

1.9 100 70,000


Spin dynamics at T = 50 K

s = 26 ps

Why do we care about spin dynamics, anyway?

You can learn most from pump-probe data when you have another “knob to turn”:

• B field• T temperature• n doping• q wavevector• l disorder

} In this talk









Spin dynamics: physics in, physics out

H = H0 + He-e + HSO + Hdis + …

Spin Coulomb drag

Spin helix

Spin Hall effect

Weak (anti-) localization

Spin Coulomb dragSpin helix

,qSpin dynamics of


Spin-orbit coupling creates an effective magnetic field

Rashba term (due to electric field)

Dresselhaus term (from crystal structure)

yk

xk kBeff

k

S

yk

xk

Typical field size ~ 2 T


Spin-orbit coupling: hero and villain of spintronics

Control over spin state via E field: good

Non-conservation of spin angular momentum: bad

…butTuning different contributions to spin-orbit interaction may

provide an elegant solution.yk

xk

yk

xk

yk

xk

yk

xk

yk

xk

yk

xk

Datta & Das Applied Physics Letters 56, 665 (1990).









How to measure spin dynamics & propagation?

Frequency shift [cm-1]

Energy [meV]

Juss

eran

d et

al.,

PR

L 69

, 848

-51

(199

2)

Spin-flip Raman (low T) -domain:

Time-domain:

Neutron scatteringSpin-flip Raman

Transient spin gratings

“motional narrowing” creates sharp peaks centered on zero frequency

1E 1E

1E 1E

Low q (where the action is!)


Transient spin gratings

Interference of two orthogonally polarized beams….

Creates a helicity wave…which generates a spin density wave.

Cameron et al., Phys. Rev. Lett. 76, 4793 (1996)


Detecting the transient grating

Pump beamsProbe beam

transmitteddif

fracte

dAmplitude of diffracted beam

Time delay




transmitteddif

fracte


Time delay




transmitteddif

fracte


Time delay


Ordinary diffusion: higher-q gratings decay faster

0 20 40 60 800.1

1

Spi

n po

lariz

atio

n

Time [ps]

14 m4.8 m3.5 m2.5 m

Low q

High q


Ordinary diffusion: higher-q gratings decay faster

Cameron et al., Phys. Rev. Lett. 76, 4793 (1996)

q2 [cm-2]0 20 40 60 80

0.1

1

Spi

n po

lariz

atio

n

Time [ps]

14 m4.8 m3.5 m2.5 m


Rapid acquisition of data: more is different

Points in (n,T,l)-space at which Ds has been measured.

Before this work:

In this work:

Technical innovations:

• Rapid-scanned heterodyne detection of diffracted beam

• Phase-mask for changing q

2

Hundreds









Anomalous diffusion: Decay faster for finite q than for q = 0 !

q=0

q=0.6 x 104 cm-1

0 20 40 60 800.1

1

Spi

n po

lariz

atio

n

Time [ps]

T = 50 K


Dispersion of double-exponential decay (50 K)

q=0.6 x 104 cm-1

Dec

ay ti

me

[ps]

Wavevector [104 cm-1]

Slow component

Fast component


Dec

ay ti

me

[ps]


Why the long lifetime?

Imagine that the sample was one-dimensional …


Spin-orbit precession: random walks in one-D

Motion along / sx L

These two paths have the same net precession.

Path (1)

Path (2)

x

z

Spin precesses in x-z plane

xy||Beff


In one-D, spin helix has infinite lifetime!

At the resonant q, spin precesses by 2 as it propagates one period of the helix

q1/Ls

Sz + iSx

Sz - iSx

q

2Ls


…back to two-dimensional reality.


For spin diffusion in 2-D,

q1/Ls

q

One-D

Two-D

Precession angle is path dependent…

leading to weaker, but nonzero, spin/space correlations at the same

critical wavevector.


Rashba coupling only:

Froltsov PRB (2001)Burkov, Nunez, MacDonald PRB (2004) Mishchenko, Shytov, Halperin PRL (2004)Bernevig, Zhang PRL (2006)

Theoretical description in 2D

z2s

2s

z SL2qD

tS

zSOx2s

2s

x qSiSL2qD

tS

xSOqSi


Coupling of Sz and Sx…

q

q1/Ls

Sz + iSx

Sz - iSx

… leads to normal modes that are linear combinations of the two spin-components.

At the resonant q, the normal modes are spin helices of opposite chirality


Simple theory predicts two exponentials of equal weight

One mode is fast, the other slow, depending on the sign of the internal field

Initial condition

Sz

= +


Our spin lifetime is even longer than simple theories predict

q1/Ls

q

One-D

Two-D

Dec

ay ti

me

[ps]



Why the very long lifetime?

Imagine that the Rashba and Dresselhaus couplings were equal …

Dec

ay ti

me

[ps]










Equal contributions to SO coupling

Rashba term (due to electric field)

Dresselhaus term (from crystal structure)

yk

xk

yk

xk

yk

xk+ =

Spin-orbit field at every k points in the same direction

0ˆ coseff y


Perfect correlation of precession with displacement along x

0ˆ coseff y

Precession in x-z plane:

yk

xk

All of these paths experience exactly the same net rotation!

F0eff v

cos|k|t

xvF

0


So if Rashba = Dresselhaus

• Persistent spin helix: in analogy with one-D, spin lifetime diverges at q = 1/Ls

• There is an exact SU(2) symmetry (Bernevig & Zhang)

Test this prediction: design QW samples with Rashba = Dreselhaus, measure transient spin grating at q = 1/Ls (future work)

Precession equal

Precession not equal

Also predicts anisotropic spin transport:

• Can have strong spin-orbit without dephasing spins!


…back to reality, where Rashba and Dressalhaus terms are unequal.


Predictions are surprisingly robustyk

xk

Dresselhaus = Rashba Dresselhaus = 3 x Rashba

• Spin-helix lifetime diverges • Spin-helix lifetime is long

• Anisotropic spin transport • Anisotropic spin transport


Anisotropic lifetimes at q = 1/Ls

t [ps]

Sz Precession equal

Precession not equal


Theory for arbitrary Rashba, Dresselhaus (Bernevig & Zhang)

Dispersion of double-exponential decay along the two directions:


Fits to Bernevig-Zhang theory

Fits give:

31

sDresselhauRashba

)T(Ds









Spin diffusion coefficient

0 50 100 150 200 250 3000

1

2

3

Ds (1

000

cm2 /s

)

T (K)

n-GaAs QWn=7.81011 cm-2

Nature 437, 1330-1333 (2005)


Compare Ds with charge diffusion coefficient, Dc0

Einstein relation (for charge)

020c e

D

n0where

(non-interacting susceptibility)

0 100 200 3000

200

400

600

D (c

m2/s

)T (K)

0 100 200 3000

500

1000

1500

D (c

m2/s

)

T (K)0 100 200 300

0

2000

4000

6000

D (c

m2/s

)

T (K)

7.8 E11 cm-2 4.3 E11 1.9 E11

Nature 437, 1330-1333 (2005)


e-e collisions affect spin current, not charge current

cJ

spinJ

spinJ

e-e collisions conserve total momentum, but exchange momentum between spin up and spin down populations.

Spin Coulomb drag (D’Amico &Vignale)


‘Drag’ damps diffusive spin current

Counter-propagation of spin populations

nnspin Coulomb drag resistance

/1DD 0c

s

0s

(sCd theory)

C(measured)

0 100 200 3000.0

0.5

1.0

1.5

k

T (K)

D’Amico & Vignale, PRB 68, 45307 (2001)


Comparison of diffusion coefficients & sCd theory

0 100 200 3000

200

400

600

D (c

m2/s

)

T (K)0 100 200 300

0

500

1000

1500

D (c

m2/s

)

T (K)0 100 200 300

0

2000

4000

6000

D (c

m2/s

)

T (K)

7.8 E11 cm-2 4.3 E11 1.9 E11

/1D

D 0c

s

0s

Photons

Hot electrons

Warm Fermi sea

Lattice

Nature 437, 1330-1333 (2005)


Advantage of spin Coulomb drag: how far can spin packet drift in E-field before spreading?

w

F

D eEwwL

DL

nn

Enhancement due to spin Coulomb drag

Diffusion (but not drift) involves counter-propagation of populations

s

0c

DD

and

Nature 437, 1330-1333 (2005)


Future directions: Tune a sample to

Dressalhaus = Rashba: divergent lifetime at qc?

Disordered sample: weak (anti-) localization?

yk

xk

yk

xk=


Uses for a (high) magnetic field

(Quasi-) infinite spin lifetimes (ordinary samples)

Destroy persistent spin-helix (Dresselhaus = Rashba sample)

Spin-polarized 2DEG / QH ferromagnet:

Spin-waves?

Skyrmions?

Spin transport w/o charge transport?


Conclusions

Transient grating technique successfully probes spin transport in ps time regime

Observed anomalous diffusion (fast & slow modes; maximum lifetime at nonzero q) due to spin-orbit coupling

Exact mixing of Rashab & Dresselhaus couplings should produce a persistent spin helix

Spin Coulomb drag: e-e collisions suppress spin diffusion to far below charge diffusion

Documents

Optical transient-grating measurement of spin propagation in a two-dimensional electron gas