geometric phases and spin-orbit effects - uni-due.dekoenig/dpg_school_10/shnirman_2.pdfkhaetskii,...
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![Page 1: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/1.jpg)
Geometric phases and spin-orbit effects
Alexander Shnirman (KIT, Karlsruhe)
Lecture 2
![Page 2: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/2.jpg)
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
![Page 3: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/3.jpg)
Geometric spin manipulations
![Page 4: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/4.jpg)
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
![Page 5: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/5.jpg)
Spin-orbit interaction in a 2DEG
![Page 6: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/6.jpg)
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
![Page 7: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/7.jpg)
Spin-orbit interaction in a quantum dot
H =p2
2m+ V (r) + HSO
![Page 8: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/8.jpg)
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
![Page 9: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/9.jpg)
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
![Page 10: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/10.jpg)
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
![Page 11: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/11.jpg)
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
![Page 12: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/12.jpg)
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
![Page 13: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/13.jpg)
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
![Page 14: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/14.jpg)
Evolution in the instantaneous basis
![Page 15: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/15.jpg)
1. Choose at each time the basis that instantaneously diagonalizes H(t)
Dot displacement
Evolution in the instantaneous basis
![Page 16: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/16.jpg)
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
![Page 17: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/17.jpg)
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
![Page 18: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/18.jpg)
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
![Page 19: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/19.jpg)
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
![Page 20: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/20.jpg)
Adiabatic theory
![Page 21: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/21.jpg)
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
![Page 22: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/22.jpg)
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
![Page 23: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/23.jpg)
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
![Page 24: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/24.jpg)
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
![Page 25: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/25.jpg)
Spin dressing
![Page 26: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/26.jpg)
What is the explicit dependence on the spin-orbit coupling strength λso?
Spin dressing
![Page 27: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/27.jpg)
What is the explicit dependence on the spin-orbit coupling strength λso?
Spin dressing
![Page 28: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/28.jpg)
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)
![Page 29: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/29.jpg)
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)
![Page 30: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/30.jpg)
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)
![Page 31: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/31.jpg)
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)
![Page 32: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/32.jpg)
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)
![Page 33: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/33.jpg)
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
![Page 34: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/34.jpg)
Can one use this effect to manipulate spin effectively?
![Page 35: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/35.jpg)
Can one use this effect to manipulate spin effectively?
Problem: displacements should be comparable to spin-orbit length
![Page 36: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/36.jpg)
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
![Page 37: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/37.jpg)
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
![Page 38: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/38.jpg)
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
![Page 39: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/39.jpg)
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.
![Page 40: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/40.jpg)
Purely electrical spin control in GaAs/AlGaAs dots
![Page 41: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/41.jpg)
Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation
Purely electrical spin control in GaAs/AlGaAs dots
![Page 42: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/42.jpg)
Observation 1: moving a dot around a ‘small’ closed path results in a z-axis rotation
Purely electrical spin control in GaAs/AlGaAs dots
![Page 43: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/43.jpg)
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
![Page 44: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/44.jpg)
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
![Page 45: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/45.jpg)
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
![Page 46: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/46.jpg)
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
![Page 47: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/47.jpg)
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
![Page 48: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/48.jpg)
Electrical manipulation: large displacements
![Page 49: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/49.jpg)
Another possibility: a multiple dot pump in GaAs/AlGaAs
Electrical manipulation: large displacements
![Page 50: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/50.jpg)
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
![Page 51: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/51.jpg)
Experimental realization: electronic conveyor belt
![Page 52: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/52.jpg)
Stotz, Hey, Santos, Ploog. Nature Materials 4, 585 (2004)
Experimental realization: electronic conveyor belt
![Page 53: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/53.jpg)
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
![Page 54: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/54.jpg)
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
![Page 55: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/55.jpg)
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
![Page 56: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/56.jpg)
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
![Page 57: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/57.jpg)
• 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
![Page 58: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/58.jpg)
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”
![Page 59: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/59.jpg)
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
![Page 60: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/60.jpg)
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)
![Page 61: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/61.jpg)
Time evolution operator projected onto lowest spin doublet subspace (n=0)
Adiabatic expansionexpansion in ω/ε
![Page 62: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/62.jpg)
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
![Page 63: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/63.jpg)
General theory (beyond Born-Oppenheimer)
- Hamiltonian of slow orbital env. (phonons)
![Page 64: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/64.jpg)
Adiabatic expansion
one ‘phonon’
two ‘phonon’, dynamic, survives at B=0
two ‘phonons’, ‘static’, Van Vleck cancellation
![Page 65: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/65.jpg)
Spin decay results
Phys. Rev. B 77, 045305 (2008)Physica E 40, pp. 76-83 (2007)Phys. Rev. Lett. 97 , 076803 (2006)
![Page 66: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/66.jpg)
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
![Page 67: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/67.jpg)
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
![Page 68: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/68.jpg)
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
![Page 69: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/69.jpg)
Effective magnetic field SL = −SR =
�4π
�d2r Bs(r1, r2)
r1 = ε1 + ε2 r1 = ε1 + ε2
r2 = ε1 − ε2 r2 = ΓL = 2π|vL|2ρL
![Page 70: Geometric phases and spin-orbit effects - uni-due.dekoenig/DPG_School_10/Shnirman_2.pdfKhaetskii, Nazarov (2001). Golovach, Khaetskii, Loss (2004). Stano, Fabian (2005) Vanishing decay](https://reader034.vdocuments.site/reader034/viewer/2022042409/5f2582ad5cff2c19af0ae294/html5/thumbnails/70.jpg)
• 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