Spectral Properties of a 2D SpinOrbit Hamiltonian

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Spectral Properties ofa 2D Spin-Orbit Hamiltonian
Denis Bulaev Department of Physics University of
Basel, Switzerland
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Outline

• Motivation
• k.p method
• 2DEG
• Quantum Dots
• Summary

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Motivation

• ..
• Quantum Computing

Supercoducting A.Shnirman, G.Shön, Z.Herman,
PRL 79, 2371 (1997) Quantum-Dot-based D.Loss
and D.P.DiVincenzo, PRA 57, 120 (1998)
Nanoll make 1T/yr by 2015
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k.p method
Pauli Hamiltonian
Thomas term (s-o coupling)
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Inversion asymmetric strs. (Td)
CB l0 (s) jls1/2 VB l1 (p) j3/2 1/2
Bir and Pikus. Symmetry and Strain-Induced
Effects in Semiconductors (Wiley, New York, 1974).
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Inversion asymmetric strs. (Td)
Single group Double
group
E
E
j1/2 G6
G1 l0
D x G1 G6 D x G15 G7 G8
Eg
D
j3/2 G8
G15 l1
j1/2 G7
k
k
Optical Orientation, ed. by Zakharchenya and F.
Meier (North - Holland, Amsterdam, 1984) Bir and
Pikus. Symmetry and Strain-Induced Effects in
Semiconductors (Wiley, New York, 1974).
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Kane Hamiltonian
Folding down
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Electron effective Hamiltonian
Dresselhaus SO (DSO) coupling
Dresselhaus, Phys. Rev. 100, 580 (1955).
(GaAs, InAs, InSb, etc - inversion
asymmetry) For Ge, Si - inversion symmetric strs
(point group Oh Td x Ci ) DSO 0!
Remark No. 1 DSO is due to bulk inversion
asymmetry (BIA)
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2DEG
GaAs
GaAs
AlxGa1-xAs
AlyGa1-yAs
AlxGa1-xAs
AlxGa1-xAs
V(z)
V(z)
z
z
D2d (E C2 2C2 2sd 2S4)
C2v (E C2 2sv)
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Dresselhaus SO interaction
D'yakonov Kocharovskii, Sov. Phys. Semicond.
20, 110 (1986)
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Rashba SO interaction
After folding down
Bychkov Rashba, JETP Lett. 39, 78 (1984).
Remark No. 2 RSO is due to structure inversion
asymmetry (SIA)
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Spin degeneracy splitting
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Energy spectrum of 2DEG
Ganichev, et al., PRL 92, 256601 (2004).
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Spin decoherence anisotropy
Averkiev Golub PRB 60, 15582 (1999).
Remark No. 3 SO coupling leads to anisotropy in
dispersion and spin decoherence
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Effective Hamiltonian for a QD
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Canonical transformation
Geyler, Margulis, Shorokhov, PRB 63, 245316
(2001).
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Three lowest electron energy levels
Dresselhaus SO coupling
Rashba SO coupling
Anti-crossing (crossing) of the levels E2 and E3
at
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Anticrossing due to Rashba coupling
E3 E1
0.25
0.20
orbital
Energy meV
0.15
0.10
Zeeman
E2 E1
0.05
E1 E1
0
8
2
4
6
10
B T
Bulaev, Loss, PRB 71, 205324 (2005).
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Summary

• SO coupling is due to space inversion asymmetry
• Dispersion anisotropy in a 2DEG
• Anticrossing due to RSO in a QD