Analysis of contact resistivity for multilevel transistors (Q1260888)

The objective of this paper is analysis of a weakly coupled system of elliptic equations; more precisely, the voltages \(u(x)\) and \(v(x)\) \((x=(x_ 1,x_ 2) \in \mathbb{R}^ 2)\) in the two layers satisfy the following system: \[ \Delta u-p \chi (S)(u-v)=0 \text{ in } \Omega_ 1, \quad \text{ and } \partial u/ \partial n=g \text{ on } \partial \Omega_ 1, \tag{1} \] \[ \Delta v-p \tau \chi (S) (u-v)=0 \text{ in } \Omega_ 2,\quad \text{ and } v=0 \text{ on } \partial \Omega_ 2. \tag{2} \] Here \(\Omega_ i \subset \mathbb{R}^ 2\), \(i=1,2\), represent the domains occupied by the two layers, respectively, and the contact area \(S\) is a subset of both \(\Omega_ 1\) and \(\Omega_ 2\) with \(\overline S \subset \Omega_ 1 \cap \Omega_ 2\). \(\chi (S)\) is the characteristic function of \(S\). The constants \[ p=R_{S1}/ \rho_ c \text{ and } \tau=R_{S2}/R_{S1} \] are positive, where \(R_{Si}\) is the sheet resistance of layer \(i\) \((i=1,2)\), and \(\rho_ c\) is the contact resistivity; \(g\) represents the density of the current applied through the boundary of one layer \((g \geq 0\) and \(g \not \equiv 0\) on \(\partial \Omega_ 1)\), and the voltage on the boundary of the other layer is assumed to be grounded. The author studies the dependence of \(u(x^ 0)\) \((x^ 0=(x^ 0_ 1,x^ 0_ 2) \in \partial \Omega_ 1)\) on the parameter \(p\), assuming the geometric setting and other parameters in this model are given. The existence, uniqueness, and positivity of the solutions to system (1)--(2) are established. Then the asymptotic properties of the boundary measurement \(u(x^ 0)\) as \(p\) goes to 0 and \(\infty\), respectively, and an example to illustrate these properties are given.