Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research

Abstract

The Green's function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form $X+(C^*+{\rm i} \eta D^*)X^{-1}(C+{\rm i} \eta D)=R+{\rm i}\eta P$, where $R$ and $P$ are Hermitian, $P+\lambda D^*+\lambda^{-1} D$ is positive definite for all $\lambda$ on the unit circle, and $\eta \to 0^+$. For each fixed $\eta>0$, we show that the required solution is the unique stabilizing solution $X_{\eta}$. Then $X_*=\lim_{\eta\to 0^+} X_{\eta}$ is a particular weakly stabilizing solution of the matrix equation $X+C^*X^{-1}C=R$. In nano applications, the matrices $R$ and $C$ are dependent on a parameter, which is the system energy $\mathcal E$. In practice one is mainly interested in those values of $\mathcal E$ for which the equation $X+C^*X^{-1}C=R$ has no stabilizing solutions or, equivalently, the quadratic matrix polynomial $P(\lambda)=\lambda^2 C^*-\lambda R+ C$ has eigenvalues on the unit circle. We point out that a doubling algorithm can be used to compute $X_{\eta}$ efficiently even for very small values of $\eta$, thus providing good approximations to $X_*$. We also explain how the solution $X_*$ can be computed directly using subspace methods such as the QZ algorithm by determining which unimodular eigenvalues of $P(\lambda)$ should be included in the computation. In some applications the matrices $C, D, R, P$ have very special sparsity structures. We show how these special structures can be expoited to drastically reduce the complexity of the doubling algorithm for computing $X_{\eta}$.