Ballistic electron transport described by a generalized Schrödinger equation
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arXiv
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| Format: | Preprint |
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2025
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| author | Aliffi, Giulia Elena Nastasi, Giovanni Romano, Vittorio |
| author_facet | Aliffi, Giulia Elena Nastasi, Giovanni Romano, Vittorio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_03765 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ballistic electron transport described by a generalized Schrödinger equation Aliffi, Giulia Elena Nastasi, Giovanni Romano, Vittorio Mathematical Physics 81Q05, 35J10, 34L40 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. |
| title | Ballistic electron transport described by a generalized Schrödinger equation |
| topic | Mathematical Physics 81Q05, 35J10, 34L40 |
| url | https://arxiv.org/abs/2510.03765 |