A Notation and Symbols

a lattice constant, inter-Landau-level ladder operatorA vector potentialb intra-Landau-level ladder operatorB magnetic fieldB effective magnetic fieldc index for conduction band state (Γ c

6 ), speed of lightD density of statese > 0 elementary chargee unit vector, polarization vectorE energyE electric fieldf frequency, occupation factorg group elementg0 = 2 g factor of free electronsg∗ effective g factorG = eB/(2π�) areal degeneracy of spin-split Landau levelsG (point) grouph index for heavy-hole state (m = ±3/2, Γ v

8 )� Planck’s constantH Hamiltonian, index for “heavy-hole state” (m = ±3/2, Γ c

8 )H multiband k · p Hamiltoniani imaginary unitj total angular momentumk (kinetic) wave vectork canonical wave vectorkB Boltzmann’s constantK tensor operatorl orbital angular momentum,

index for light-hole state (m = ±1/2, Γ v8 )

L Landau-oscillator index,index for “light-hole state” (m = ±1/2, Γ c

8 )

Roland Winkler: Spin–Orbit Coupling Effectsin Two-Dimensional Electron and Hole Systems, STMP 191, 197–199 (2003)c© Springer-Verlag Berlin Heidelberg 2003

198 A Notation and Symbols

m z component of angular momentumm0 free-electron massm∗ effective massn bulk band index (including spin), index of refractionN number of bands in multiband Hamiltonian,

Landau-level index (axial approximation)N Landau-level index (beyond axial approximation)NA acceptor charge densityND donator charge densityNd depletion charge densityNs 2D charge densityN± spin subband densityp = �k kinetic momentump = �k canonical momentumrs density parameter (6.53)r = (x, y, z) position vectorR full rotation groups index for spin split-off state (Γ v

7 )S index for spin split-off state (Γ c

7 )S generalized spin operator (6.65)T temperatureunk lattice-periodic part of Bloch functionV confining potentialv velocity operatorw quantum well widthα absorption coefficientα, β subband index, index for irreducible representationsΓ irreducible representationε dielectric constantε strain tensorθ, φ (see Fig. C.1)λc =

√�/(eB) magnetic length (cyclotron radius)

µB = e�/(2m0) Bohr magnetonν bulk band index (no spin)ξ z component of subband wave functionσ spin index, Pauli spin matricesΣ band offsetϕ direction of in-plane vectorψ 3D wave functionω excitation frequency

References 199

ωc = eB/m∗ cyclotron frequency[A,B] = AB −BA commutator{A,B} = 1

2 (AB +BA) symmetrized product

We denote the irreducible representations of the crystallographic point groupsin the same way as Koster et al. [1]. The band parameters and basis matricescharacterizing the extended Kane model are defined in Appendix C. Through-out this work, we use a coordinate system for quasi-2D systems where thex, y components correspond to the in-plane motion (represented by an index“‖”) and the z component is perpendicular to the 2D plane. We use SI unitsfor electromagnetic quantities.

Abbreviations

2D two-dimensionalBIA bulk inversion asymmetrycp cyclic permutation of the preceding term (in formulas)DOS density of statesEFA envelope function approximationEMA effective-mass approximationFIR far infraredHH heavy holeLH light holeMBE molecular-beam epitaxyMOS metal-oxide-semiconductorQW quantum wellSC semiclassicalSdH Shubnikov–de HaasSIA structure inversion asymmetrySO spin–orbit

References

1. G.F. Koster, J.O. Dimmock, R.G. Wheeler, H. Statz: Properties of the Thirty-Two Point Groups (MIT, Cambridge, MA, 1963) 6, 21, 23, 47, 72, 166, 199