Nonlinear boundary value problems describing mobile carrier transport in semiconductor devices. (Q1771838)

The authors study the following boundary value problem which governs the steady state distributions of mobile carriers in a semiconductor device: \[ \begin{gathered} (D(| E| )(n'+nE))' = 0,\;E' = f-n \;\text{ in } (0,1),\\ E(0) = E(1) = E_0,\;D(| E(0)| )(n'(0)+n(0)E(0)) = j_0. \end{gathered} \] Here \(E(x)\) and \(n(x)\) describe the electric field and the density of mobile electrons, the constant \(f\) represents the homogeneous density of ionized impurities, \(D(| E| )\) is the diffusion coefficient depending on the electrical field, \(j_0\) is the electron current density at \(x=0\). It is assumed that \(E_0 > 0\), \(K(E_0) = 1+D'(E_0)E_0D^{-1}(E_0) <0\) which guaranties that the diffusion coefficient \(D\) as a function of the electric field has an \(N\)-shaped form and contains an interval \((E_1,E_2)\) in which this function has a negative derivative and \(D(E_0)+E_0D'(E_0) <0\) (the so called condition of negative differential conductivity). Under this assumption, there is a sequence of \(f_k(E_0)= -K(E_0)^{-1}(E_0^2/4+\pi ^2k^2)\), \(k=1, 2,\dots,\) at which nonconstant solutions bifurcate from the branch of constant solutions. A global behaviour of such bifurcation branches is shown and the character of solutions lying in the \(k\)-th branch is described. An asymptotic behaviour of solutions for large \(f\) is studied. Further, it is shown that the interior transition layer phenomena arises for a certain \(E_0^* \in (E_1,E_2)\). Existence and uniqueness of solutions to the corresponding nonstationary problem is also discussed and the stability of bifurcating solutions is studied.