On the time-dependent drift-diffusion model for semiconductors (Q1890530)

The paper is concerned with the nonstationary drift-diffusion equations of semiconductor theory in a cylinder \(Q= \Omega\times (0, T)\) (\(\Omega\subset \mathbb{R}^d\) a bounded domain, \(d= 1,2,3\)): \[ {\partial n\over \partial t}- \text{div } J_n= R(n, p)(1- np)+ q,\tag{1} \] \[ {\partial p\over \partial t}- \text{div } J_p= R(n, p)(1- np)+ q,\tag{2} \] \[ -\varepsilon \Delta u= p- n+ N,\tag{3} \] where \(J_n= D_n \nabla n- \mu_n n\nabla u\), \(J_p= -(D_p \nabla p+ \mu_p p\nabla u)\) and \(n=\) electron density, \(p=\) hole density, \(u=\) electrostatic potential, \(R=\) given non-negative, bounded, locally Lipschitz function, \(g\) and \(N\) given functions of position \(x\in \Omega\), \(\varepsilon=\text{const}> 0\). System (1)--(3) is studied under the usual mixed boundary conditions on \(n\), \(p\) and \(u\), and initial conditions on \(n\) and \(p\). The conditions on the diffusivities \(D_n\), \(D_p\) and mobilities \(\mu_n\), \(\mu_p\) are as follows: \(D_n= \mu_n\), \(D_p= \mu_p\) and \[ \mu_n= \mu_1+ \nu_1(x, \xi),\;\mu_p= \mu_2+ \nu_2(x, \xi),\quad x\in \Omega,\quad \xi\in \mathbb{R}^d\tag{4} \] (\(\mu_k= \text{const.}> 0\), \(|\nu_k(x, \xi)\xi\mid\leq\text{const } \forall x\in \Omega\), \(\xi\in \mathbb{R}^d\), \(k= 1,2\)). If \(\nu_1= \nu_2\equiv 0\), the authors prove the existence of a weak solution to the above problem for any \(T> 0\), where \(n\), \(p\) are bounded from above and below by a positive constant. In the case of general mobilities (4) an analogous result is established for sufficiently small \(T> 0\). The proofs are based on a time discretization of (1), (2), combined with a truncation of the nonlinear terms \(n\nabla u\) and \(p\nabla u\).