Geometric phases and spin-orbit effects

Alexander Shnirman (KIT, Karlsruhe)

Lecture 2

Outline

• Geometric phases (Abelian and non-Abelian)

• Spin manipulation through non-Abelian phases a) Toy model; b) “Moving” quantum dots

• Spin decay due to random geometric phase

• Spin pumping

Geometric spin manipulations

We look for alternative ways to manipulate spin

Question:

Can one manipulate spin with electric fields only, at B=0?

Answer:

Yes, provided strong spin-orbit coupling

Motivation

Spin-orbit interaction in a 2DEG

A: Spin-orbit interaction ↔ momentum dependent ‘magnetic

field’ (Bext=0)

B: Semiclassical picture: electron moves a distance dr

in time dt the spin is rotated by U[dr], independent of dt (‘geometric’)

W. A. Coish, V. N. Golovach, J. C. Egues, D. Loss.Physica Status Solidi (b) 243, 3658 (2006)

Rashba Dresselhaus

Semiclassical description of geometric spin drift

Spin-orbit interaction in a quantum dot

H =p2

2m+ V (r) + HSO

Effect of SO on quantum dot orbitals:

spin textureEffective spin-orbit strength:

Spin-orbit interaction in a quantum dot

H =p2

2m+ V (r) + HSO

Spin-orbit coupling

• Eigenstates are spin-textures• For B=0 the basis is two-fold

degenerate (Kramer’s theorem)• The lowest doublet will be labeled

by τ

Spin-orbit interaction in a quantum dot

H =p2

2m+ V (r) + HSO

H =p2

2m+ V (r) + HSO + e r · E(t)

Parabolic dot in a 2DEG subject to electric field

Effect of electric field in parabolic dot

H =p2

2m+ V (r) + HSO + e r · E(t)

Parabolic dot in a 2DEG subject to electric field

Effect of electric field in parabolic dot

H =p2

2m+ V (r) + HSO + e r · E(t)

Parabolic dot in a 2DEG subject to electric field

Effect of electric field in parabolic dot

H =p2

2m+ V (r) + HSO + e r · E(t)

Parabolic dot in a 2DEG subject to electric field

Position shift

Effect of electric field in parabolic dot

Evolution in the instantaneous basis

1. Choose at each time the basis that instantaneously diagonalizes H(t)

Dot displacement

Evolution in the instantaneous basis

1. Choose at each time the basis that instantaneously diagonalizes H(t)

Dot displacement

This sets a ‘reference frame’ for the description of the electron state at each moment/position

Evolution in the instantaneous basis

1. Choose at each time the basis that instantaneously diagonalizes H(t)

Dot displacement

This sets a ‘reference frame’ for the description of the electron state at each moment/position

2. Compute Heff(t) that governs the dynamics in the instantaneous basis

Evolution in the instantaneous basis

1. Choose at each time the basis that instantaneously diagonalizes H(t)

Dot displacement

This sets a ‘reference frame’ for the description of the electron state at each moment/position

2. Compute Heff(t) that governs the dynamics in the instantaneous basis

Evolution in the instantaneous basis

1. Choose at each time the basis that instantaneously diagonalizes H(t)

Dot displacement

This sets a ‘reference frame’ for the description of the electron state at each moment/position

2. Compute Heff(t) that governs the dynamics in the instantaneous basis

Exact evolution in the instantaneous basis

Evolution in the instantaneous basis

Adiabatic theory

Adiabatic theorem: A system prepared initially in a degenerate subspace τ(0) of energy Eτ(0) and driven infinitely slowly will remain within the subspace τ(t) of instantaneous

eigenenergy Eτ(t).

Adiabatic theory

Adiabatic theorem: A system prepared initially in a degenerate subspace τ(0) of energy Eτ(0) and driven infinitely slowly will remain within the subspace τ(t) of instantaneous

eigenenergy Eτ(t).

Adiabatic theory

Adiabatic theorem: A system prepared initially in a degenerate subspace τ(0) of energy Eτ(0) and driven infinitely slowly will remain within the subspace τ(t) of instantaneous

eigenenergy Eτ(t).

Adiabatic evolution within subspace τ

Adiabatic theory

Adiabatic theorem: A system prepared initially in a degenerate subspace τ(0) of energy Eτ(0) and driven infinitely slowly will remain within the subspace τ(t) of instantaneous

eigenenergy Eτ(t).

Adiabatic evolution within subspace τ

Adiabatic theory

Spin dressing

What is the explicit dependence on the spin-orbit coupling strength λso?

Spin dressing

What is the explicit dependence on the spin-orbit coupling strength λso?

Spin dressing

What is the explicit dependence on the spin-orbit coupling strength λso?

Spin dressing

P. San-Jose, B. Scharfenberger, G. Schön, A.S., G. Zarand, PRB 77, 045305 (2008)

Spin-orbit coupling is modified due to spin dressing

What is the explicit dependence on the spin-orbit coupling strength λso?

Spin dressing

P. San-Jose, B. Scharfenberger, G. Schön, A.S., G. Zarand, PRB 77, 045305 (2008)

Spin-orbit coupling is modified due to spin dressing

(dressed spin)

What is the explicit dependence on the spin-orbit coupling strength λso?

Spin dressing

P. San-Jose, B. Scharfenberger, G. Schön, A.S., G. Zarand, PRB 77, 045305 (2008)

Spin-orbit coupling is modified due to spin dressing

(dressed spin)

What is the explicit dependence on the spin-orbit coupling strength λso?

Spin dressing

P. San-Jose, B. Scharfenberger, G. Schön, A.S., G. Zarand, PRB 77, 045305 (2008)

What is the explicit dependence on the spin-orbit coupling strength λso?

Final result

Spin dressing

P. San-Jose, B. Scharfenberger, G. Schön, A.S., G. Zarand, PRB 77, 045305 (2008)

O(3)

Rotation of a sphere of radius R0 rolling on a surface

Rotation of electron spin due to spin-orbit interaction

SU(2)isomorphism

(double covering)

Geometrical interpretation

Can one use this effect to manipulate spin effectively?

Can one use this effect to manipulate spin effectively?

Problem: displacements should be comparable to spin-orbit length

Can one use this effect to manipulate spin effectively?

Problem: displacements should be comparable to spin-orbit length

Miller, Zumbhül, Marcus, et al. Phys. Rev. Lett 90, 076807

Different measurements agree onin GaAs/AlGaAs heterostructres

Can one use this effect to manipulate spin effectively?

Problem: displacements should be comparable to spin-orbit length

Miller, Zumbhül, Marcus, et al. Phys. Rev. Lett 90, 076807

Different measurements agree onin GaAs/AlGaAs heterostructres

Other materials? Recent measurements suggest that other semiconductors such as InAs could have

Fasth, Fuhrer, Samuelson, et al., cond-mat/0701161

Can one use this effect to manipulate spin effectively?

Problem: displacements should be comparable to spin-orbit length

Miller, Zumbhül, Marcus, et al. Phys. Rev. Lett 90, 076807

Different measurements agree onin GaAs/AlGaAs heterostructres

Is it possible to perform arbitrary manipulations with realistic (small) displacements?

Other materials? Recent measurements suggest that other semiconductors such as InAs could have

Fasth, Fuhrer, Samuelson, et al., cond-mat/0701161

Can one use this effect to manipulate spin effectively?

Problem: displacements should be comparable to spin-orbit length

Miller, Zumbhül, Marcus, et al. Phys. Rev. Lett 90, 076807

Different measurements agree onin GaAs/AlGaAs heterostructres

Is it possible to perform arbitrary manipulations with realistic (small) displacements?

Other materials? Recent measurements suggest that other semiconductors such as InAs could have

Fasth, Fuhrer, Samuelson, et al., cond-mat/0701161

Yes.

Purely electrical spin control in GaAs/AlGaAs dots

Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation

Purely electrical spin control in GaAs/AlGaAs dots

Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation

Purely electrical spin control in GaAs/AlGaAs dots

Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation

Many repetitions = spinning motion!

Purely electrical spin control in GaAs/AlGaAs dots

Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation

Observation 2: Optimal path for a spin flip without spinning = straight line Optimal path for a spin flip with uniform spinning = curved line!

Many repetitions = spinning motion!

Purely electrical spin control in GaAs/AlGaAs dots

Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation

Observation 2: Optimal path for a spin flip without spinning = straight line Optimal path for a spin flip with uniform spinning = curved line!

Many repetitions = spinning motion!

Observation 3: A spin flip is possible without straying far from the origin if there is a constant spinning component

Purely electrical spin control in GaAs/AlGaAs dots

Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation

Observation 2: Optimal path for a spin flip without spinning = straight line Optimal path for a spin flip with uniform spinning = curved line!

Many repetitions = spinning motion!

Observation 3: A spin flip is possible without straying far from the origin if there is a constant spinning component

Optimal path: Two-component path. Fast component = spinning motion. Slow component = spin flip Optimal relative frequency given by size

Purely electrical spin control in GaAs/AlGaAs dots

Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation

Observation 2: Optimal path for a spin flip without spinning = straight line Optimal path for a spin flip with uniform spinning = curved line!

Many repetitions = spinning motion!

Observation 3: A spin flip is possible without straying far from the origin if there is a constant spinning component

Optimal path: Two-component path. Fast component = spinning motion. Slow component = spin flip Optimal relative frequency given by size Spyrograph

Purely electrical spin control in GaAs/AlGaAs dots

Electrical manipulation: large displacements

Another possibility: a multiple dot pump in GaAs/AlGaAs

Electrical manipulation: large displacements

Another possibility: a multiple dot pump in GaAs/AlGaAs

Transporting a single electron around the ring can result in a more general rotation depending on the tunneling amplitudes

Electrical manipulation: large displacements

Experimental realization: electronic conveyor belt

Stotz, Hey, Santos, Ploog. Nature Materials 4, 585 (2004)

Experimental realization: electronic conveyor belt

Many electron moving quantum dots are created by the piezoelectric potential of interfering surface sound waves

Stotz, Hey, Santos, Ploog. Nature Materials 4, 585 (2004)

Experimental realization: electronic conveyor belt

Many electron moving quantum dots are created by the piezoelectric potential of interfering surface sound waves

A spin polarization is created in each dot by exciting with circularly polarized laser

Stotz, Hey, Santos, Ploog. Nature Materials 4, 585 (2004)

Experimental realization: electronic conveyor belt

Many electron moving quantum dots are created by the piezoelectric potential of interfering surface sound waves

A spin polarization is created in each dot by exciting with circularly polarized laser

The total spin polarization at each position/time is indirectly measured by pholuminiscence (recombination rate)

Stotz, Hey, Santos, Ploog. Nature Materials 4, 585 (2004)

Experimental realization: electronic conveyor belt

Many electron moving quantum dots are created by the piezoelectric potential of interfering surface sound waves

A spin polarization is created in each dot by exciting with circularly polarized laser

The total spin polarization at each position/time is indirectly measured by pholuminiscence (recombination rate)

Stotz, Hey, Santos, Ploog. Nature Materials 4, 585 (2004)

Experimental realization: electronic conveyor belt

• Usual spin decay theory through electric fields:

• Spin decay rate (piezoelectric ph.)

Spin decay through noisy electric fields

�B ωB

Relaxation mechanism:phonons + spin-orbit + magnetic field

T−1 ∝ ω2Bρph(ωB) ∝ B5

Khaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005)

Vanishing decay rates at

B → 0

Geometric spin dephasing

Multiple (N)circles with random direction

Moving a dot around a ‘small’ closed path results in a z-axis rotation

Many repetitions = spinning motion!

“Area diffusion”

The dominant sources to electric noise:

• Piezo-electric longitudinal phonons:

• Ohmic charge fluctuations:

Weak electric fields: (x0=dot size)

(Dominant at high fields)

(Dominant at low fields)

These vanish at B=0

Geometric contribution

Coupling to electric field

Derivation

Needed: Time evolution operator projected onto lowest spin doublet subspace (n=0)

perturbation

P. San-Jose et al. Phys. Rev. Lett. 97 , 076803 (2006)

Time evolution operator projected onto lowest spin doublet subspace (n=0)

Adiabatic expansionexpansion in ω/ε

Integrating out higher Zeeman doublets (poor man)

Full evolution operator

Evolution operator projected on the lowest (ground state) doublet

Look for effective coupling that would give such evolution

General theory (beyond Born-Oppenheimer)

- Hamiltonian of slow orbital env. (phonons)

Adiabatic expansion

one ‘phonon’

two ‘phonon’, dynamic, survives at B=0

two ‘phonons’, ‘static’, Van Vleck cancellation

Spin decay results

Phys. Rev. B 77, 045305 (2008)Physica E 40, pp. 76-83 (2007)Phys. Rev. Lett. 97 , 076803 (2006)

Spin pumping at B=0

QL = − e

2π

� T

0Im

�Tr

�(ΛL ⊗ σ0)

dSdtS†

��dt

�SL = − �2π

� T

0Im

�Tr

�(ΛL ⊗ �σ)

dSdtS†

��dt

Brower’s formulae

Hd =

�ε1 σ0 −i�α · �σi�α · �σ ε2 σ0

�

Pumping via:�1(t), �2(t)

v1,L(t), v2,L(t)v1,R(t), v2,R(t)

Minimal model: two orbital levels + SO coupling

S(t)Scattering

matrix

Previous works: Sharma, Brouwer 2003Governale, Taddei, Fazio 2003

Scattering matrix

Uo =�

eiφL 00 eiφR

�⊗ σ0 Us =

�UL 00 UR

�T =

�−√

1− T0√

T0√T0

√1− T0

�⊗ σ0 Transmission

Charge phases Spin rotations

S = UoUs T U†s U†

o convenient representation

QL =e

2π

� T

0

�(1− T0)

�φ̇R − φ̇L

��dt

�SL =i�2π

� T

0T0 Tr

��U†

L �σ UL

� �U†

LU̇L − U†RU̇R

��dt

J. Avron et al., 2000

“peristaltic” pumpingT0 → 0

[UL,UR] �= 0→ �SL + �SR �= 0 non-conservation of spin

Minimal model

Sσ =�−eiφ

√1− T0 eisσ θ

√T0

e−isσ θ√

T0 e−iφ√

1− T0

�S = ei(φL+φR)S↑ ⊕ S↓

Bs = ∂r1T0∂r2θ − ∂r2T0∂r1θ

SL = −SR =�4π

�d2r Bs

Pumped spin = flux of effective “magnetic field”

In eigenbasis of �αSO�σ

tan(θ) =|�αSO| (v1Lv2R − v2Lv1R)ε1 v2Lv2R + ε2 v1Lv1R

Geometric effect

φ = φL − φR

Effective magnetic field SL = −SR =

�4π

�d2r Bs(r1, r2)

r1 = ε1 + ε2 r1 = ε1 + ε2

r2 = ε1 − ε2 r2 = ΓL = 2π|vL|2ρL

• Non-Abelian phases -> robust, timing-independent, spin manipulations at B=0 (strong spin-orbit interaction needed)

• Spin decay at low magnetic fields (saturation at B=0)

• Spin pumping at B=0

Summary