Theory of Tunneling, Including Photon-Assisted Tunneling, in Semiconductors in Crossed and Parallel Electric and Magnetic Fields

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Tunneling phenomena in crossed electric and magnetic fields cannot be properly described using the one-band effective-mass approximation. For this purpose, the two-band Hamiltonian is solved in the presence of crossed electric and magnetic fields and also in parallel fields. For crossed fields, two types of solutions are obtained. The first, for $E<H{(\frac{{\mathcal{E}}_{g}}{2{m}^{*}{c}^{2}})}^{\frac{1}{2}}$, are of the harmonic-oscillator type, with quantized energy levels. In this region there is no interband tunneling in a pure material. In the region of high electric field, where $E>H{(\frac{{\mathcal{E}}_{g}}{2{m}^{*}{c}^{2}})}^{\frac{1}{2}}$, the solutions are of the electric-field type with a continuous energy spectrum. The analogy of this model to the motion of free classical relativistic electrons in crossed fields is discussed. WKB solutions in the region of high electric field are used to calculate tunneling (Zener) current and photon-assisted tunneling [Franz-Keldysh (FK) effect]. The Hamiltonian is solved to obtain quasistationary solutions, neglecting a term which acts as the perturbation causing the Zener tunneling. The tunneling integrals are computed by the method of steepest descent. The results are nearly identical to those of Aronov and Pikus, obtained by a different method. In general, the magnetic field decreases both Zener and FK tunneling. The result for Zener tunneling predicts that the current will depend on $E$ and $H$ approximately as $\mathrm{exp}(\frac{{\ensuremath{-}H}^{2}}{{E}^{3}})$, in good agreement with experiment. In the FK effect, for photon energies close to that of the gap, the ratio of the absorption in crossed fields to that at $H=0$ varies with frequency and field approximately as $\mathrm{exp}[\frac{\ensuremath{-}{({\mathcal{E}}_{g}\ensuremath{-}\ensuremath{\hbar}\ensuremath{\omega})}^{\frac{5}{2}}{H}^{2}}{{E}^{3}}]$. This is confirmed experimentally, both in the frequency and field dependence, by Reine, Vrehen, and Lax. Thus the main features of electron tunneling in crossed fields can be explained by a WKB treatment which in addition provides a good physical picture of the tunneling process. In order to provide a unified picture of tunneling in crossed and parallel fields, we also obtain expressions for Zener and FK tunneling in parallel fields using WKB solutions to the two-band model. The results for FK tunneling are very similar to those of the preceding paper, in that the electron motion separates into quantized motion transverse to the magnetic field and nonquantized motion parallel to both fields. The magnetic field reduces the tunneling by increasing the effective energy gap by the energy of the transverse motion. Our expressions reduce to those of the preceding paper in the limit of large ${\mathcal{E}}_{g}$. The ratio of the absorption in parallel fields to that at $H=0$ varies approximately as $\mathrm{exp}[\frac{\ensuremath{-}{({\mathcal{E}}_{g}\ensuremath{-}\ensuremath{\hbar}\ensuremath{\omega})}^{\frac{1}{2}}H}{E}]$. The result for Zener tunneling predicts $\mathrm{exp}(\frac{\ensuremath{-}H}{E})$ behavior, in contrast to $\mathrm{exp}(\frac{\ensuremath{-}{H}^{2}}{{E}^{3}})$ for crossed fields. The model in the preceding paper does not predict any Zener tunneling.