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Ballistic electron transport described by a generalized Schrödinger equation

Giulia Elena Aliffi, Giovanni Nastasi, Vittorio Romano

TL;DR

This work introduces a generalized, higher-order Schrödinger equation derived from the Kane non-parabolic dispersion to model ballistic charge transport in nanoscale devices. It establishes a finite-domain formulation with transparent boundary conditions and derives a universal current expression valid at any order, including explicit forms for $J_4$ and $J_6$. The authors prove self-adjointness of the Hamiltonians and well-posedness under realistic potentials, and demonstrate current conservation within the TBC framework. Numerical simulations of a resonant tunneling diode show that higher-order terms capture interference effects and can significantly alter the current relative to the second-order model, underscoring the importance of non-parabolic dispersion in device modeling.

Abstract

We propose a Schrödinger equation of arbitrary order for modeling charge transport in semiconductors operating in the ballistic regime. This formulation incorporates non-parabolic effects through the Kane dispersion relation, thereby extending beyond the conventional effective mass approximation. Building upon the framework introduced in G. E. Aliffi, G. Nastasi, V. Romano, {ZAMP} {76}, 155 (2025), we derive a hierarchy of models, each governed by a Schrödinger equation of increasing order. As in the standard second-order case, the problem is formulated on a finite spatial domain with suitable transparent boundary conditions. These conditions are designed to simulate charge transport in a quantum coupler where an active region -- representing the electron device -- is connected to leads acting as reservoirs. We investigate several analytical properties of the proposed models and derive a generalized expression for the current, valid for any order. This formula includes additional terms that account for interference effects arising from the richer wave structure inherent in higher-order Schrödinger equations, which are absent in the effective mass approximation. Numerical simulations of a resonant tunneling diode (RTD) illustrate the key features of the solutions and highlight the impact of the generalized formulation on device behavior.

Ballistic electron transport described by a generalized Schrödinger equation

TL;DR

This work introduces a generalized, higher-order Schrödinger equation derived from the Kane non-parabolic dispersion to model ballistic charge transport in nanoscale devices. It establishes a finite-domain formulation with transparent boundary conditions and derives a universal current expression valid at any order, including explicit forms for and . The authors prove self-adjointness of the Hamiltonians and well-posedness under realistic potentials, and demonstrate current conservation within the TBC framework. Numerical simulations of a resonant tunneling diode show that higher-order terms capture interference effects and can significantly alter the current relative to the second-order model, underscoring the importance of non-parabolic dispersion in device modeling.

Abstract

We propose a Schrödinger equation of arbitrary order for modeling charge transport in semiconductors operating in the ballistic regime. This formulation incorporates non-parabolic effects through the Kane dispersion relation, thereby extending beyond the conventional effective mass approximation. Building upon the framework introduced in G. E. Aliffi, G. Nastasi, V. Romano, {ZAMP} {76}, 155 (2025), we derive a hierarchy of models, each governed by a Schrödinger equation of increasing order. As in the standard second-order case, the problem is formulated on a finite spatial domain with suitable transparent boundary conditions. These conditions are designed to simulate charge transport in a quantum coupler where an active region -- representing the electron device -- is connected to leads acting as reservoirs. We investigate several analytical properties of the proposed models and derive a generalized expression for the current, valid for any order. This formula includes additional terms that account for interference effects arising from the richer wave structure inherent in higher-order Schrödinger equations, which are absent in the effective mass approximation. Numerical simulations of a resonant tunneling diode (RTD) illustrate the key features of the solutions and highlight the impact of the generalized formulation on device behavior.

Paper Structure

This paper contains 9 sections, 5 theorems, 89 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Dispersion relation and generalized Schrödinger equation
  3. Generalization of the Current
  4. Transparent boundary conditions for the generalized stationary Schrödinger equation
  5. Case $k_1>0$
  6. Case $k_1<0$
  7. Well posedness of the higher order SE with transparent boundary conditions
  8. Comparison of the current between the second and the fourth order SE
  9. Simulation of a resonant tunneling diode

Key Result

Proposition 1

If $V$ is real and $V \in L^{\infty}(\mathbb{R}^3)$, then for any $s \in \mathbb{N}$, the Hamiltonian $\mathcal{H}_{2s}$ is well-defined on the Schwartz space $\mathcal{S}(\mathbb{R}^3)$ as a symmetric operator with respect to the scalar product in $L^2(\mathbb{R}^3)$.

Figures (7)

  • Figure 1: Numerical evaluation of the ensemble incident, reflected and transmitted probability current density versus the wave-vector of the incident wave.
  • Figure 2: RTD type potential in the case $V_0=0$ V, $V_L= 0.1$ V, $V_b = -0.3$ V, $L=135$ nm.
  • Figure 3: Plot of the function $g (\sigma,k_x)$. One can note that numerically $g (\sigma,k_x)$ has compact support.
  • Figure 4: Electron density obtained with the fourth-order SE (left) and the second order SE (right) in the case $V_0=0$ V, $V_L= 0.1$ V, $V_b = -0.3$ V, $L=135$ nm by using the Kane dispersion relation.
  • Figure 5: Electron current density obtained with the fourth-order SE (left) and the second order SE (right) in the case $V_0=0$ V, $V_L= 0.1$ V, $V_b = -0.3$ V, $L=135$ nm by using the Kane dispersion relation.
  • ...and 2 more figures

Theorems & Definitions (12)

  • Proposition 1
  • Proposition 2
  • Lemma 1
  • proof
  • Remark 1
  • Proposition 3
  • Remark 2
  • Remark 3
  • Remark 4
  • Proposition 4
  • ...and 2 more