An analogy between the band structure of narrow gap semiconductors and the Dirac equation for relativistic electrons in vacuum is used to demonstrate that semiconductor electrons experience a Zitterbewegung (trembling motion). Its frequency is ${\ensuremath{\omega}}_{Z}\ensuremath{\approx}{\mathcal{E}}_{g}∕\ensuremath{\hbar}$ and its amplitude is ${\ensuremath{\lambda}}_{Z}$, where ${\ensuremath{\lambda}}_{Z}=\ensuremath{\hbar}∕{m}_{0}^{*}u$ corresponds to the Compton wavelength in vacuum (${\mathcal{E}}_{g}$ is the energy gap, ${m}_{0}^{*}$ is the effective mass, and $u\ensuremath{\approx}1.3\ifmmode\times\else\texttimes\fi{}{10}^{8}\phantom{\rule{0.3em}{0ex}}\mathrm{cm}∕\mathrm{s}$). Once the electrons are described by a two-component spinor for a specific energy band there is no Zitterbewegung but the electrons should be treated as extended objects of size ${\ensuremath{\lambda}}_{Z}$. The magnitude of ${\ensuremath{\lambda}}_{Z}$ in narrow gap semiconductors can be as large as $70\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$. Possible consequences of the above predictions are indicated.

Zitterbewegung and its effects on electrons in semiconductors

We show that free-carrier (Drude) absorption of both polarized and unpolarized terahertz radiation in quantum well (QW) structures causes an electric photocurrent in the presence of an in-plane magnetic field. Experimental and theoretical analysis evidences that the observed photocurrents are spin dependent and related to the gyrotropy of the QWs. Microscopic models for the photogalvanic effects in QWs based on asymmetry of photoexcitation and relaxation processes are proposed. In most of the investigated structures the observed magneto-induced photocurrents are caused by spin-dependent relaxation of non-equilibrium carriers.

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Magneto-gyrotropic photogalvanic effects in semiconductor quantum wells

Five-level $\mathbf{k}∙\mathbf{p}$ band model is used to describe the spin $g$ factor of conduction electrons in undoped $\mathrm{Ga}\mathrm{As}∕{\mathrm{Ga}}_{0.65}{\mathrm{Al}}_{0.35}\mathrm{As}$ quantum wells (QWs) with an external magnetic field parallel and transverse to the growth direction [001]. Far-band corrections to the model are also included in the theory. It is found that for QWs narrower than $10\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ the transverse ${g}_{\ensuremath{\perp}}$ factor is slightly higher than the parallel one ${g}_{\ensuremath{\Vert}}$. Our theory without any adjustable parameters describes very well the existing experimental data of various authors for both field configurations and well widths $3\phantom{\rule{0.3em}{0ex}}\mathrm{nm}\ensuremath{\leqslant}d\ensuremath{\leqslant}25\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. Comparison between the theory and experiment probes the validity of the $\mathbf{k}∙\mathbf{p}$ approach for lengths of few lattice periods.

Anisotropy of spin g factor in Ga As ∕ Ga 1 − x Al x As symmetric quantum wells

We have performed a numerical calculation of the energy dispersion relation of hybrid magneto‐electric states (both propagating and evanescent) in an electron waveguide subjected to a magnetic field. Our results are considerably different from those obtained through the Bohr–Sommerfield quantization condition. We have also calculated the density of the magneto‐electric states as a function of energy and the velocity versus energy relationships. Finally, we show how the wavefunctions of these states evolve with increasing magnetic field from particle in a box states to edge states. These results are useful in the analysis of numerous recent magnetotransport experiments performed in electron waveguides.

Numerical calculation of hybrid magneto‐electric states in an electron waveguide

An analogy between behavior of electrons in narrow-gap semiconductors (NGS) and relativistic electrons in vacuum is reviewed. Energy band structures [Formula: see text] are considered for various NGS materials and their correspondence to the energy-momentum relation in special relativity is emphasized. It is indicated that special relativity for vacuum is analogous to a two-band [Formula: see text] description for NGS. The maximum electron velocity in NGS is [Formula: see text], which corresponds to the light velocity in vacuum. An effective mass of charge carriers in semiconductors is introduced, relating their velocity to quasimomentum and it is shown that this mass depends on electron energy (or velocity) in a way similar to the mass of free relativistic electrons. In [Formula: see text] alloys one can reach vanishing energy gap at which electrons and light holes become three-dimensional massless Dirac fermions. A wavelength [Formula: see text] is defined for NGS, in analogy to the Compton wavelength in relativistic quantum mechanics. It is estimated that [Formula: see text] is on the order of tens of Angstroms in typical semiconducting materials which is experimentally confirmed in tunneling experiments on energy dispersion in the forbidden gap. Statistical properties of the electron gas in NGS are calculated and their similarity is demonstrated to those of the Juttner gas of relativistic particles. Interband electron tunneling in NGS is described and shown to be in close analogy to the predicted but unobserved tunneling between negative and positive energies resulting from the Dirac equation for free electrons. It is demonstrated that the relativistic analogy holds for orbital and spin properties of electrons in the presence of an external magnetic field. In particular, it is shown that the spin magnetic moment of both NGS electrons and relativistic electrons approaches zero with increasing energy. This conclusion is confirmed experimentally for NGS. Electrons in crossed electric and magnetic fields are described theoretically and experimentally. It is only the two-band description for NGS, equivalent to the Dirac or Klein-Gordon equations for free particles, that gives a correct account of experimental results in this situation. A transverse Doppler shift in the cyclotron resonance observed in crossed fields in InSb indicates that there exists a time dilatation between an oscillating electron and an observer. The phenomenon of Zitterbewegung (ZB, trembling motion) for electrons in NGS is considered theoretically, following the original proposition of Schrodinger for free relativistic electrons in vacuum. The two descriptions are in close analogy, but the frequency of ZB for electrons in NGS is orders of magnitude lower and its amplitude orders of magnitude higher making possible experimental observations in semiconductors considerably more favorable. Finally, graphene and carbon nanotubes, as well as topological insulators are considered in the framework of relativistic analogy. These systems, with their linear energy-quasimomentum dispersions, illustrate the extreme semirelativistic regime. Experimental results for the energy dispersions and the Landau quantizations in the presence of a magnetic field are quoted and their analogy to the behavior of free relativistic electrons is discussed. Approximations and restrictions of the relativistic analogy are emphasized. On the other hand, it is indicated that in various situations it is considerably easier to observe semirelativistic effects in semiconductors than the relativistic effects in vacuum.

http://iopscience.iop.org/article/10.1088/1361-648X/aa7932/pdf

Semirelativity in semiconductors: a review.

Semirelativistic Behavior of Electrons in InSb in Crossed Magnetic and Electric Fields

Electrons in inversion layers of narrow-band-gap semiconductors in the presence of an external magnetic field parallel to the interface (crossed-field configuration) are considered theoretically. A three-level model of the \ensuremath{\kappa}\ensuremath{\cdot}P theory is used to describe electrons in external electric and magnetic fields, taking into account the main features of the band structure in InSb-type semiconductors: a small energy gap and a strong spin-orbit interaction. The electric potential is taken to be in the form of a triangular well and the presence of the interface is accounted for by appropriate boundary conditions for the wave function. An analytic description of the eigenenergies is obtained for arbitrary intensities of the homogeneous electric and magnetic fields, from magnetic surface levels (vanishing electric field) to purely electric subbands (vanishing magnetic field). Experimental data on electron cyclotron resonance in InSb in the presence of crossed fields (the bulk limit) are theoretically described and interpreted using an analogy between electrons in narrow-gap semiconductors and relativistic electrons in vacuum. The influence of a transverse magnetic field on the energies of intersubband resonances in metal-oxide-semiconductor structures is discussed. The relation of the theoretical results obtained to existing and possible experiments is emphasized throughout.

W Zawadzki

Papers

Semirelativistic Behavior of Electrons in InSb in Crossed Magnetic and Electric Fields

Inversion electrons on narrow-band-gap semiconductors in crossed electric and magnetic fields