Electrons and holes in InSb under crossed magnetic and stress fields. I. Theory

Abstract

In a series of four papers magnetooptical transitions are presented for InSb crystals, which are subjected to uniaxial stress perpendicular to the magnetic field. Here, in the first paper, we establish an 8 X 8 k·p Hamiltonian matrix for stress Tll[IOO) and field Bll[OOl) and diagonalize it exactly. The dependence of valence and conduction states on stress and longitudinal momentum is discussed and compared with the geometry of parallel fields TIIBII [OOl]. Characteristic features are extracted for inter-and intraband transitions. Under crossed fields, the levels are separated much stronger with stress, yielding more insight than in the parallel configuration. I. INTRODUCfiONThe electronic energy bands of zinc-blende structure semiconductors exhibit significant deviations from the simple isotropic and parabolic shape close to the gap. In both conduction 1 and valence bands 2 one finds strong terms of cubic and quartic order in k and warping. Uniaxial stress induces k-linear terms which shift the band extrema to nonzero wave vectors. 3 For the valence bands k-linear terms exist already in the unstressed state. 4 The band anomalies can be detected in Shubnikov-de Haas oscillations, 3 in spin-depolarization measurements, 5 or in far-infrared and Raman scattering experiments on doped semiconductors. 6 Under the influence of a magnetic field, the spindegenerate conduction bands form two Landau ladders, which, due to the nonparabolicities, are spaced irregularly. The differences in the spacings show up in splittings of the harmonic transition lines. The cyclotron mass and g factor are field dependent. 7 Higher harmonic and spin-flip transitions become allowed. 8 The uppermost valence band is fourfold degenerate at k=O. In a magnetic field, the four bands turn into four entangled Landau ladders with strongly varying separations of the rungs. Magnetooptical transitions can only be identified reliably if by application of uniaxial stress the band degeneracy is lifted and the ladders are pulled apart.Such a program was first pursued by Hensel and Suzuki 9 in the investigation of the valence bands of Ge. For most of their experiments, stress was applied parallel to the magnetic field along the crystal [001], [111], and [110] directions, and microwave absorption was regis-37 tered. The spectra were calculated from an effective-mass Hamiltonian which was formed in the space of the four degenerate valence-band functions of symmetry rt.Given the geometry of the experiments and the diamond lattice structure of Ge, the symmetry groups of this Hamiltonian are C 4 (consisting of rotations about the direction of the magnetic field by multiples of 27T/4), c3, and C 2 , respectively. The symmetries permit a simple labeling scheme for the magnetic states and simplify the numerical calculations.Similar experiments were performed soon afterwards by Ranvaud et al. 10 Since Ranvaud used far-infrared radiation, higher magnetic fields had to be applied than were necessary in the microwave experiments of Hensel and Suzuki. 9 The strong field...

Choosing a basis that eliminates spurious solutions ink∙ptheory

“…[70][71][72] The inequality ⌬Q 7 ⌬Q 8 is therefore nothing new from a physical standpoint. Nevertheless, it does serve to show that the standard experimental data-fitting procedure-namely, treating P as a fitting parameter and defining the modified Luttinger parameters for all of set A from the ⌫ 8v Luttinger parameters via Eq.…”

“…The latter result is hardly surprising, because the modified Luttinger parameters are known to have different values in each submatrix when spin-orbit coupling is treated exactly. 70,71,72 The inequality ∆Q 7 = ∆Q 8 is therefore nothing new from a physical standpoint. Nevertheless, it does serve to show that the standard experimental data-fitting procedure-namely, treating P as a fitting parameter and defining the modified Luttinger parameters for all of set A from the Γ 8v Luttinger parameters via Eq.…”

TRS: a program to calculate Landau levels and direct dipole transitions in uniaxially stressed semiconductors

“…Both KIKI* and X/' 1 are constructed with group theoretical methods (see refs. [2][3][4]). The parameters aK are the specific material constants (e.g.…”