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