Spin-Orbit Interactions in Semiconductor Nanostructures

Branislav K. NikolićDepartment of Physics and Astronomy, University of Delaware, U.S.A.

http://www.physics.udel.edu/~bnikolic

Jyväskylä Summer School 2004, Finland

Spin-Orbit Hamiltonians from Dirac Equation

� Expand Dirac equation for relativistic electron in the low energy limit and second order in :

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Example: Rashba SO Term

� Rashba Hamiltonian:

� is the expectation value over the lowest subband with energy

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Spectral Problem of Rashba Hamiltonian

� Rashba Hamiltonian of infinite 2DEG: Kinetic energy + SO coupling

� Since Hamiltonian commutes with the 2D momentum operator, we can classify its eigenvectors and eigenvalues with wave numbers :

� Spin degeneracy on the Fermi surface is lifted, but Rashba term is not able to produce a spontaneous spin polarization of electron quantum states:

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Eigenvectors and Eigenvalues in Pictures

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Symmetries and Conservation Laws in QM

� Wigner: Symmetries in Quantum Mechanics are represented by unitary or antiunitary operators in the Hilbert space (or, more properly, space of rays).

� Generators of symmetries represented by a unitary operator and …

� … constants of motion

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Time-Reversal Symmetry for Spin-½

� Time reversal (i.e., “motion reversal”) operator reverses the linear and angular momentum, while leaving the position unchanged:

� Time reversal operator has to be antilinear:

� Time reversal for spin:

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Kramers Degeneracy

� Invariance under antiunitary time reversal operator does not produce a conserved quantity, but it sometimes increases the degree of degeneracy of the energy eigenstates:

are linearly independent.

� In many cases the degeneracy implied by Kramers theorem is merely the degeneracy between spin-up and spin-down states, or something equally obvious.

�The theorem is nontrivial for systems with SO coupling in unsymmetrical electric field, where neither spin nor angular momentum are conserved. Kramerstheorem implies that no such field can split the degenerate pairs of energy levels.

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Kramers Degeneracy of Rashba Hamiltonian

� Since SO coupling terms are time-reversal invariant , the two Rashba eigenstates are connected by time reversal operator, while their spinor factor states are orthogonal to each other.

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Spectrum of Rashba Hamiltonian in 1D

� Rashba Hamiltonian of infinite 1DEG: Kinetic energy + SO coupling

� 1D Eigenenergies:

� 1D Eigenstates:

� The general spin evolution is precession about the y-axis (in the x-zplanne) with an angular frequency angular modulation of the spin orientation for a channel of length L (c.f. Datta-Das spin-FET).

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Rashba + Dresselhaus SO couplings in 2DEG

� Rashba + Dresselhaus Hamiltonian of infinite 2DEG: Kinetic energy + two (apparently) different SO couplings

� 2D Eigenenergies:

� 2D Eigenstates:

� If we tune SO couplings to , spinors cease to depend on momentum:

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Rashba Hamiltonian on 2D Lattice

� Rashba Hamiltonian of infinite 2DEG on periodic tight-binding lattice:

� 2D Eigenenergies:

� 2D Eigenstates:

� For small this reduces to continuum parabolic dispersion:

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Rashba SO Hamiltonian vs. 2D Ferromagnet

E

k

E

k

2 ∆

k R

a )

b )

� Velocity operator for the Rashba Hamiltonian:

� Velocity operator for the 2D Ferromagnet:

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Eigenvalues and Eigenvectors of Rashba Wires

� In wires of finite width complete translation invariance, which exists in homogeneous 1D or 2D systems, is lost!

�For a wire of finite width along the y-axis :

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Rashba Eigenvalues and Eigenvectors in Wires

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–20 –10 0 10 20

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–20 –10 0 10 20

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Rashba Spin-Splitting in Wires in Pictures