PACS 71.18.+y, 71.20.Nr Properties of spin surfaces of heavy-mass, light-mass and split-off bands of GaAs, InAs, GaP, InP, GaSb, and InSb are presented under various approximations. A simple two-band, analytically solvable model is used as a starting point and the reference. It is shown that the strongest deviation from the simple spherical and parabolic two-band model comes from the valence band nonparabolicity and nonshpericity. The spin-splitting in all three valence bands was found to have negligible influence on shape of the spin surfaces at thermal hole energies.

Surfaces and trajectories traced out by spin vectors in spin space are calculated and systematized for holes in tetrahedral semiconductors. It is shown how various trajectories are related to the wave-function phases and amplitudes of the doubly degenerated heavy-mass, light-mass, or split-off bands. Relations between hole wave vectors and spin surface shapes are examined for high symmetry directions. The presented results may be useful in ultrafast spintronics, where both the spin state and the wave vector of a hole should be taken into account simultaneously in the description of the hole spin dynamics.

The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl 3,0, Cl 2,1, Cl 1,2 and Cl 0,3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.The Clifford algebra Cl p,q is an associative noncommutative algebra, where (p, q) indicates vector space metric. In 3D case, the MV consists of the following elements (basis blades) {1, e 1 , e 2 , e 3 , e 12 , e 13 , e 23 , e 123 ≡ I}, where e i are the orthogonal basis vectors, and e ij are the bivectors (oriented planes). The last term is called the pseudoscalar. The shorthand notation, for example, e ij means e ij = e i • e j , where the circle indicates the geometric or Clifford product. Usually, the multiplication symbol is omitted, and one writes e ij = e i e j . The number of subscripts indicates the grade of basis element, so that the scalar is a grade-0 element, the vectors are the grade-1 elements etc. In the orthonormalized basis, the products of vectors satisfies e i e j + e j e i = ±2δ ij .

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

Geometric algebra is a relatively new and powerful tool in mathematical physics that allows one to solve physical problems in a very elegant way. In the present paper, which also bears a pedagogical flavor, geometric algebra is applied to a simple problem of electron spin dynamics in the conduction bands of cubic semiconductors. It is shown that the spin-related multivector, which is a basic object of geometric algebra, is a more transparent and flexible physical object as compared with spinor eigenvectors in the standard formulation of the problem in Hilbert space. Using geometric algebra the general equation of motion for electron spin in A 3 B 5 and A 2 B 6 semiconductors in the presence of spin-orbit interaction is solved exactly for the first time.

Recently the Clifford (geometrical) algebra was addressed to describe dynamical properties of electron spin in semiconductors (Dargys 2009 Phys. Scr. 79 055702). In this paper, the Clifford algebra is used to investigate heavy and light hole spins in valence bands of cubic semiconductors. Owing to strong spin–orbit interaction in the valence band, the precession trajectories of hole spin polarization are ellipses, or even lines, rather than circles as usually found in the case of electrons. The paper shows how one can investigate the valence band spectrum and free-hole spin precession trajectories within a framework of the Clifford algebra Cl 4, 1. General formulae that describe free heavy- and light-mass hole spin precession are presented.

Field ionization properties of shallow carbon acceptors in MBE GaAs, where thermal lattice vibrations have an effect on the hole emission rate, are presented. The excited energy levels of the carbon atom are relatively far from the ground level; therefore, it appeared possible to observe pure phonon-assisted tunnelling of a hole from a single (ground) level over a wide temperature range, from 4.2 K to 22 K. Experimental results are interpreted using a multiphonon field ionization model that takes into account the interaction of acoustic phonons with a localized centre via deformation and piezoelectric potentials. In the temperature range considered, the influence of the deformation potential is found to predominate over the piezoelectric one.

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