Generalized phonon-assisted Zener tunneling in indirect semiconductors with non-uniform electric fields: A rigorous approach

Abstract

A general framework to calculate the Zener current in an indirect semiconductor with an externally applied potential is provided. Assuming a parabolic valence and conduction band dispersion, the semiconductor is in equilibrium in the presence of the external field as long as the electronphonon interaction is absent. The linear response to the electron-phonon interaction results in a non-equilibrium system. The Zener tunneling current is calculated from the number of electrons making the transition from valence to conduction band per unit time. A convenient expression based on the single particle spectral functions is provided, enabling the evaluation of the Zener tunneling current under any three-dimensional potential profile. For a one dimensional potential profile an analytical expression is obtained for the current in a bulk semiconductor, a semiconductor under uniform field and a semiconductor under a non-uniform field using the WKB (WentzelKramersBrillouin) approximation. The obtained results agree with the Kane result in the low field limit.A numerical example for abrupt p − n diodes with different doping concentrations is given, from which it can be seen that the uniform field model is a better approximation than the WKB model but a direct numerical treatment is required for low bias conditions.

A review of selected topics in physics based modeling for tunnel field-effect transistors

“…First the 1D Schrödinger equation for electrons and holes is solved in each slice along the channel, then the conduction (E C ) and valence (E V ) band profiles are modified so as to suppress E C values below the energy of the lowest electron subband E e0 and the E V values above the highest hole subband E h0 . The non-local model with the effective gap correction was found to match quite well the full-quantum non-self-consistent model described in section 2.3 [43]. More recently, the authors of [47] proposed a methodology based on the rejection of those tunneling paths that involve states in the CB below E e0 or states in the valence band above E v0 .…”

“…A recent model that is applicable to non-uniform potential profiles and to a carrier gas with different dimensionality has been proposed by Vandenberghe et al [43]. The approach makes use of the diagonal elements of the spectral functions that, for a 2D gas described by the parabolic effective mass approximation (EMA), can be written as…”

“…In [43] it is shown that, if the modeling framework described above is applied to the bulk crystal (using the corresponding spectral functions) and for a uniform electric field, then one obtains the same expression for the tunneling generation rate as in [42]. For a non uniform field, instead, it can be shown that, by introducing suitabe approximations for the envelope wave-functions, the formalism in equation (20) provides an expression for non-local tunneling in bulk systems similar to the dynamic-path, non-local-phonon-assisted model used in TCAD [25], and similar also to the model derived in [24].…”

First-principles electron transport with phonon coupling: Large scale at low cost

“…In Fig. 2 we consider transport in a short (6.5 nm) and a long (19.6 nm) silicon p-n junction [7,47,48] with transport in the [100] crystal direction.…”

“…Reducing the computational cost of inelastic, compared to elastic, device simulations has therefore been an important and unsolved challenge for decades since the first ultrascaled transistors emerged. In the extreme limit of molecular-scale devices there are accurate first-principles methods for inelastic processes available [2][3][4][5][6][7][8][9][10][11][12][13], while in the opposite bulk continuum limit, deformation potentials (DPs) are extracted for Boltzmann transport equations (BTEs) that accurately describe low bias transport [14][15][16][17]. However, in between these two regimes efficient computational methods are missing.…”