Abstract
We consider a one-dimensional, isentropic, hydrodynamical model for a unipolar semiconductor, with the mobility depending on the electric field. The mobility is related to the momentum relaxation time, and field-dependent mobility models are commonly used to describe the occurrence of saturation velocity, that is, a limit value for the electron mean velocity as the electric field increases. For the steady state system, we prove the existence of smooth solutions in the subsonic case, with a suitable assumption on the mobility function. Furthermore, we prove uniqueness of subsonic solutions for sufficiently small currents.
Highlights
The hydrodynamic model for semiconductors was first introduced by Bløtekjær in1970 [1]
The hydrodynamic model for semiconductors describing the dynamics of charged fluid particles, such as electron and holes in semiconductor devices and ions in plasmas, has attracted the attention of many scholars
We have considered the effect of an electric field dependent mobility
Summary
The steady-state version of this one-dimensional model has been extensively studied, starting from the existence result, and uniqueness for small currents, in the subsonic case [5]. We prove the existence of subsonic solutions to the semiconductor hydrodynamic model with field-dependent mobility, for sufficiently small currents. We prove the uniqueness of solutions for small currents Both proofs are inspired by the classical results in [5], with some substantial changes due to the particular nature of the source term considered in our case. Mathematics 2021, 9, 2152 with μ0 low-field mobility, Ec critical field value, related to the saturation velocity, and β ≥ 1 a real exponent After scaling, for this specific model we get q(E) = (1 + (cq|E|)β) β , q (E) = q(E)(cq|E|)β , E(1 + (cq|E|)β). The resulting solution will still depend on the boundary value φ1, which will be adjusted so that (14) is satisfied
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