Quasi-neutral limit to the drift-diffusion models for semiconductors with physical contact-insulating boundary conditions
Set \(\Omega=]0,1[\times ]0,\infty)\). This paper deals with problem \[ n_t^{\lambda}=(n_x^{\lambda}-n^{\lambda}\varphi_x^{\lambda})_x, \qquad p_t^{\lambda}=(p_x^{\lambda}+p^{\lambda}\varphi_x^{\lambda})_x, \qquad \lambda^2\varphi_{xx}^{\lambda}=n^{\lambda}-p^{\lambda}-D, \] \((x,t)\in\Omega\), associated to boundary conditions corresponding to the contact at \(x=0\) (\(n^{\lambda}(0,t) = \bar n(t)\), \(p^{\lambda}(0,t) = \bar p(t)\), \(\varphi^{\lambda}_x(0,t) = 0\)) and to the insulating at \(x=1\), and initial conditions \(n_o^{\lambda}(x),p_o^{\lambda}(x)\). Here the small parameter \(\lambda\) is the scaled Debye length of the semiconductor devices, the unknown functions \((x,t)\in\Omega \to n^{\lambda}(x,t) \) [resp. \(p^{\lambda}(x,t),\varphi^{\lambda}(x,t)\)] are the electron density [resp. the hole density, the electric potential]. The given function \(D(x)\) models the doping profile. The formal limit problem \(P_0\) of \(P_{\lambda}\), as \(\lambda\to 0\), consists of the system \(S_0\) (\(\lambda=0\) in \(S_{\lambda}\)) and suitable boundary and initial conditions. Under several supplementary conditions on the data, the authors study the asymptotic behavior of \(n^{\lambda},p^{\lambda},\varphi^{\lambda}\) in terms of a sufficiently smooth solution \(n,p,\varphi\) to \(P_0.\) In the case of well prepared boundary data, i.e., \(B(t)=\bar n(t)-\bar p(t) -D(0)=0\), the convergence is established. In the case of ill-prepared boundary data, i.e., \(B(t)\not=0\) the boundary layer which appears near \(x=0\) is investigated. The proofs are based on matched asymptotic analysis and energy estimates.
physical contact-insulating boundary conditions
scientific article; zbMATH DE number 5833536
scientific article