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The quantum ring with spin–orbit interaction included is analyzed in a nonstandard way using Clifford or geometric algebra (GA). The solution of the Schrodinger–Pauli equation is presented in terms of rotors having clear classical mechanics interpretation, i.e., in GA the rotors act in 3D Euclidean space rather than as operators in an abstract Hilbert space. This classical-like property of spin control in GA provides a more transparent approach in designing and understanding spintronic devices. The aim of the paper is to attract readers attention to new possibilities in spin physics and to demonstrate how the quantum ring problem can be solved by GA methods.

Connection between optical Mueller matrices and geometrical (Clifford) algebra multivectors is established. It is shown that starting from 3-dimensional (3D) Cl 3,0 algebra and using isomorphism between Cl 3,0 and even Cl 3,1 + subalgebra one can generate canonical Mueller matrices and their combinations that describe an optical system. It appears that representation of polarization devices in terms of geometric algebra is very compact and, in contrast to Mueller matrix approach, there is no need for speculative physical restrictions. If needed, properties of media can be logically introduced into Maxwell equation in a form of Clifford algebra via constitutive relations. Since representation of polarization by Cl 3,1 algebra is Lorentz invariant it allows to include relativistic effects of moving bodies on light polarization as well. In this paper only simple examples of connection between Mueller matrices and geometric algebra multivectors is presented.

Electron reflection off and transmission through a stepped discontinuity in quantum well (QW), for example, due difference in spin–orbit (SO) interaction constants, effective masses, or potential step, is analyzed in a general case. Correct boundary conditions at the discontinuity are taken into account. It was found that spin flipping of the transmitted beam can be as high as 100%. A simple formula that relates the transmitted spin polarization and SO interaction constants on both sides of the discontinuity is presented and a relevant experimental set-up is proposed to measure SO interaction constants using a homobarrier in the QW.

The knowledge of exact wave functions is required in calculating physical parameters such as optical dipole moments, scattering matrix elements, or in wave function engineering. In this report we describe how a system of algebraic equations that follows from the Schrodinger equation can be reduced to a Grobner basis from which the exact wave function can be easily constructed. As an example, closed form solutions for a cylindrical electronic waveguide and double quantum well nanostructures are presented.

We apply Clifford algebra to investigate 2D electron spin reflection off and transmission through a stepped discontinuity of physical parameters in semiconducting quantum well. The discontinuity may be due to change of spin-orbit interaction constants, effective masses, or electrostatic potential. In the paper the posed problem has been solved exactly. It is shown that the reflected electronic beam has identical spin polarization as the incident one. However, the transmitted beam suffers spin flipping and in general case consists of a mixture of up and down spin states. Optimal conditions for total reversion of 2D electron spin polarization are found. Special attention is paid to correct boundary conditions in the presence of spin-orbit interaction. A simple formula that connects spin polarization of the transmitted beam and SO interaction constants is presented. Keywords: Clifford algebra, geometric algebra, semiconductors, spintronics, spin polarization, spin flipping, quantum well

The reflection of spin-polarized electrons from a potential barrier inside a quantum well (QW) is analyzed. Using Clifford (geometric) algebra it is shown that the spin–orbit interaction brings about a double reflection at an oblique incidence of an electronic beam. The reflected beam is in a superposition of two beams having different wavelengths and reflection angles which produces a two-period spatial beating pattern that can be observed experimentally in the electron spin polarization component which is normal to the QW.

Free electron and hole spin properties in Hg1–xCdx Te/CdTe quantum wells (QWs) are considered within eight-band k · p Hamiltonian, when the QW bands are inverted, or close to the inversion. The generic spin properties of 2D subbands are described by spin surface, an analogue of the Bloch sphere. In the inverted energy band arrangement, due to mutual interaction between conduction and valence subbands, strong deviation from the Bloch sphere is observed. However, at the normal band arrangement the spin surfaces of electrons are found to be close to spherical. The spin surfaces of the QW holes are strongly deformed in all cases, mainly due to strong spin–orbit interaction in valence bands of respective bulk semiconductors and their subsequent rearrangement due to spin–splitting. The corresponding graphs of the spin surfaces are presented. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Spin properties of electrons, the wave function of which is conned by cylindrical potential of a hollow semiconducting cylinder (quantum tube), and which simultaneously propagate along the cylinder axis are analysed. The spinnorbit interaction is included via Rashba and Dresselhaus Hamiltonians. The electron spin surface, which describes all possible spin eigenand superposition states, is shown to reduce to the Bloch sphere, independent of the electron energy and semiconductor band parameters. The electron dispersion can be tuned to a regime that is favourable for the operation of the spin-FET by trimming the diameter of the quantum tube.

Bloch-like surfaces that describe all possible spin states and, therefore, can be used to visualize spin precession and spin dynamics in external fields are presented for HgTe, a material with inverted energy bands. The spin and orbital surfaces for both free electrons and holes are shown, in general, to be spheroids, with the rotational axis parallel to a charge-carrier wave vector. The lengths of the principal axes of the spheroids depend on the energy band index and wave vector magnitude. At some wave vectors the surfaces may reduce to a line or a disc in the spin space. The corresponding graphs are presented.

The spin-FET (spin eld effect transistor) is the spintronics device the operation of which is based on unique properties of free electron or hole spin in semiconductors. In the spin-FET, the modulation of channel conductance is achieved by control of spin direction via the spinnorbit interaction. The present article, rstly , critically reviews the main difculties encountered in realizing the spin-FET. Secondly, using the concept of the spin surface, general spin properties of 2D holes in p-type channel of the FET are considered. The complex nature of the spin surfaces of the ballistic 2D holes is demonstrated. It is shown that in optimizing the spin-FET, in addition to the dispersion law of the channel one should pay a special attention on how the spin surface transforms between spin injector and collector.

To have a closed system, the Maxwell electromagnetic equations should be supplemented by constitutive relations which describe medium properties and connect primary fields (E, B) with secondary ones (D, H). J.W. Gibbs and O. Heaviside introduced the basis vectors {i, j, k} to represent the fields and constitutive relations in the three-dimensional vectorial space. In this paper the constitutive relations are presented in a form of Cl3,0 algebra which describes the vector space by three basis vectors {σ1, σ2, σ3} that satisfy Pauli commutation relations. It is shown that the classification of electromagnetic wave propagation phenomena with the help of constitutive relations in this case comes from the structure of Cl3,0 itself. Concrete expressions for classical constitutive relations are presented including electromagnetic wave propagation in a moving dielectric.

A practical computation method to find the eigenvalues and eigenspinors of quantum mechanical Hamiltonian is presented. The method is based on reduction of the eigenvalue equation to well known geometric algebra rotor equation and, therefore, allows to replace the usual det(H-E)=0 quantization condition by much simple vector norm preserving requirement. In order to show how it works in practice a number of examples are worked out in Cl_{3,0} (monolayer graphene and spin in the quantum well) and in Cl_{3,1} (two coupled two-level atoms and bilayer graphene) algebras., Comment: 12 pages, 3 figures in Advances in Applied Clifford Algebras, 2015

Spin properties of free charge carriers in PbTe and PbSe are analysed using for this purpose the concept of spin surface, which in the simplest case reduces to the Bloch sphere. It is shown that in lead chalcogenides the spin surfaces are ellipsoids, the principal axes of which depend on the charge carrier wave vector, spin–orbit interaction strength, and the basis functions used to construct the Hamiltonian. The present results may be useful in the study of dynamics of the polarized charge carriers injected electrically or optically into the volume of PbTe, PbSe or PbS.

The electron spin transport along a semiconducting quantum wire connected to spin-unpolarized electron reservoirs and the spin ltering properties of the wire are investigated. The wire is immersed in a magnetic eld that is perpendicular to the wire axis and substrate plane on which the wire is grown. The lateral conning potential is assumed to be parabolic. The spinnorbit interaction is included via Rashba Hamiltonian that in conjunction with the Zeeman Hamiltonian determines the spin conductance of the wire. Dependences of the spin current and conductance on the voltage applied over the ends of the wire as well as on the magnetic eld strength are analysed.

The scattering of hole spin during hole intra- and intervalence band transitions due to optical deformation potential interaction in tetrahedral semiconductors is considered. Matrix elements for transitions mediated by longitudinal and transverse optical phonons are presented. It is shown that for this type of scattering, the spin relaxation time can be found from hole momentum scattering time.

Two aspects related to valence-band hole spin are considered: spin surfaces in p-type semiconductors and flipping of the hole spin by an ultrashort electric field pulse. It is shown that heavy, light, and split-off holes have different spin surfaces. In general, the shape of the surface in real semiconductors may depend on the hole wave vector direction and magnitude. The concept of spin surface is used to explain very strong anisotropy of hole spin injection efficiency observed recently in ferromagnetic-semiconductor structures and in optimizing ultrafast spin switching. It was shown that, of all spin flipping mechanisms, the most effective one is associated with hole transfer between different spin surfaces in high electric fields. The less effective mechanisms are related to valence band warping and nonparabolicity. Examples of the hole spin flipping dynamics and the discussion on ultrafast control of spin in semiconductors by π-type electrical pulses are presented.

The scattering of valence band hole spin by acoustical and optical phonons due to deformation potential interaction is considered. Six-band deformation potential matrix, which takes into account doubly degenerate heavy-mass, light-mass, and split-off energy bands, is used to evaluate the matrix elements for spin conserving and flipping transitions. The concept of the spin surface was addressed to define the initial and final states of the hole spin. It was found that, in agreement with experiment, the spin conserving and flipping transitions are of comparable magnitude in both intravalence and intervalence band scattering mediated by acoustical as well as optical phonons.

Spin properties of holes in the wurtzite semiconductors, when the hole with a given wave vector ballistically propagates in either heavy-, light-mass or crystal-field split-off band, are considered. Analytical solutions for an averaged hole spin are found for two important cases, when the hole wave vector is parallel and perpendicular to the hexagonal axis. Shapes of the spin surfaces are presented for these cases. It is shown that the surfaces, in general, are spheroids. However, depending on the wave vector magnitude and valence band parameters, the surfaces may also be spheres or line-shaped. The properties of the spin surfaces are illustrated for parameters of the wurtzite GaN, where the crystal field dominates in the splitting of the valence band.

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. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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. (© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We present the results of time-resolved photoluminescence (TRPL) and optically detected microwave resonance (ODMR) spectroscopy investigations of semiconductor quantum dots and quantum wells. The ODMR spectra of InAs/GaAs QDs were detected via modulation of the total intensity of the QDs emission induced by 95 GHz microwave excitation and exciton fine structure was studied. Very long life times (up to 10 ns) of photoexcited carriers were observed in this system using TRPL at low temperatures and excitation intensities promising higher responsitivity of such QDs for quantum dot infrared photodetector development. The effects of proton and alpha particles irradiation on carrier dynamics were investigated on different InGaAs/GaAs, InAlAs/AlGaAs and GaAs/AlGaAs QD and QW systems. The obtained results demonstrated that carrier lifetimes in the QDs are much less affected by proton irradiation than that in QWs. A strong influence of irradiation on the PL intensity was observed in multiple QWs after high-energy alpha particles irradiation.

Ultrafast and coherent transitions induced by electric field between spin states of the valence band in semiconductors are considered. It is shown that using optimized femtosecond electric fields one can, in principle, control a selected component of the hole spin. The feasibility of hole spin switching is illustrated using the well-known Luttinger-Kohn Hamiltonian for valence bands of tetrahedral and zinc-blende semiconductors. The presented results may be useful in the development of the ultrafast semiconductor spintronics.

Intercollisional dynamics of hole spin, when free holes are subjected to an external electric field, is considered. Coherent evolution of spin projections and the population in valence bands is presented for dc and harmonic fields. Spin operators in the band representation are found and it is shown that there appears an uncertainty related with band degeneracy. The results obtained may be useful in the simulation of simultaneous spin and charge transport in p-type semiconductors, for example, by the Monte Carlo method.

Intervalence band dynamics of free holes in infrared fields is studied. A realistic valence band structure is used to simulate simultaneous excitation of split-off, heavy- and light-mass bands by femtosecond duration pulses. The decoherence due to phonon and ionized impurity scattering is taken into account in a quantum-mechanical density matrix. The excitation patterns in the Brillouin zone and hole distribution dynamics are presented. It is shown that it is possible to obtain high excitation levels in the valence band and dynamical inversion even in presence of dissipation, if femtosecond pulses are properly tailored.

Optimized intervalence band transitions of holes in p-type semiconductors subjected to femtosecond infrared optical fields are considered. The genetic algorithm is used to find minimal energy femtosecond pulses which maximize the transition probability of a heavy-mass hole to either light-mass or split-off band. Inversion of distribution of holes in wave vector k-space induced by such femtosecond pulses is studied. The distribution of coherently excited holes in both light-mass and split-off bands is shown to exhibit characteristic patterns in two-dimensional sections of k-space. The character of the patterns depends on the driving femtosecond pulse shape, energy and valence band properties. Examples of the optimized transitions and inversion patterns are presented for p-type GaAs.

Quantum control of coherent interband transitions is analyzed using a simple valence band as a model. A genetic algorithm is applied to optimize the shape of the pulsed driving radiation field. The fitness function used in this study maximizes the interband transition probability and simultaneously minimizes the energy of the driving field. It is shown that such optimized ultrashort pulses exist for holes that have a well-defined initial wave vector and band index. The effect of coherent excitation on inversion of a hole distribution was calculated, too. The inverted hole distribution is shown to exhibit a characteristic pattern which depends on the driving pulse shape and band structure.

Multiple quantum well (MQW) samples grown by the molecular beam epitaxy method were irradiated by alpha particles from isotope 239.Pu. The photoluminescence (PL) spectra and PL integrated intensity dependencies are presented at various alpha particle fluences, up to 10(exp 11) cm(exp -2). The experimental results are in agreement with a model which assumes that point centers (residual impurities and point defects introduced during irradiation) are responsible for PL intensity decrease with the alpha particle fluence. It was found that annealing of irradiated MQW samples at a temperature above 650 K nearly restores the PL intensity. An enhancement of PL by more than an order of magnitude was observed at annealing temperatures higher than 850 K, just before GaAs and Al(0.35)Ga(0.65)As interdiffusion begins.

Undoped multiple quantum well (MQW) samples grown by the molecular beam epitaxy method were irradiated by 5.15 MeV energy alpha particles. A strong influence of irradiation on the photoluminescence (PL) intensity was observed in these nearly intrinsic samples: the PL intensity at liquid nitrogen and room temperatures decreased by more than two orders of magnitude at alpha particle fluence of 10(exp 11) cm(exp -2). The dependence of the PL intensity on irradiation dose is described by a rate equation model that takes into account the generation of nonradiative point centers in barriers and wells of the MQW structure during irradiation.

A discussion is given of two types of ionizing particle detectors, which are based on radiation-generated charge carrier capture into quantum wells of semiconductors. The principle of operation of the two detector types depends on the strength of the coupling between the quantum wells. The first, isolated quantum well detector, is based on fast electron–hole pair recombination into 2D excitons and their decay into light quanta. The second, coupled quantum well detector, is based on electron and hole hopping between quantum wells. The advantages and limitations of the detectors are discussed.

We present a detailed experimental study on photoluminescence quenching due to exciton and donor impact ionization by accelerated electrons under an in-plane nanosecond duration electric field created in $\mathrm{GaAs}/{\mathrm{Al}}_{0.35}{\mathrm{Ga}}_{0.65}\mathrm{As}$ quantum wells. From the photoluminescence transients measured by the time-correlated single-photon counting technique, we have determined the experimental conditions under which donor impact ionization can have an influence on quenching of the excitonic photoluminescence. The coefficient of two-dimensional exciton impact ionization has been estimated; its dependences on the applied electric field, lattice temperature, and width of the quantum wells are given.

The shift of resonance energy in direct optical interband transitions is evaluated. The case of hole transitions between heavy and light mass valence bands is considered, using the full form of the Luttinger-Kohn Hamiltonian. In addition to anisotropy, very large Bloch-Siegert shift, of the order of 100%, was found at high IR light intensities in the teraherz frequency range. The shift is related to a periodic hole motion in the Brillouin zone.

A numerical method is described that allows to find differential interband absorption cross-section of infrared (IR) radiation in solids in the case of complex energy bands. The method is based on observation of carrier population oscillations between two selected energy bands. Absorption due to hole intervalence transitions in p-InP, when both the nonparabolicity and the warping of heavy, light and spin–orbit split-off valence bands is taken into account, is presented as an illustration of the method. The method allows us to calculate the resonance shift and change of the cross-section at high IR light intensities.

Hole de-trapping dynamics out of shallow acceptors subjected to high pulsed electric fields is investigated in pure p-GaAs used in radiation detectors. The characteristic de-trapping times are found from current transients due to impact and tunnel ionization of the acceptors. The de-trapping times are presented as a function of electric field strength.

The field ionization of beryllium acceptor in molecular-beam-epitaxy-grown GaAs was investigated at low temperatures, when acoustic lattice vibrations take part in the ionization process. It has been found that acoustic phonons enhance the field ionization of beryllium, the effect being stronger the higher the lattice temperature is. Experimental results are interpreted using a hydrogenic multiphonon field-ionization model. It is shown that at $Tl20 \mathrm{K}$ the acoustic-phonon-assisted acceptor\char21{}valence-band tunneling process is dominated by light mass holes and deformation-potential interaction.

Dynamical electrical breakdown in the depleted region of diodes has been investigated in the nanosecond timescale. A persistent conductivity due to Si-related DX centre impact ionization in high electric fields was observed after the onset of the breakdown. From the present investigation it follows that DX centre breakdown begins at electric fields of ( and is related to impact ionization of DX centres by free hot carriers.