Fast algorithms in NEGF approach for steady-state quantum transport with its application to multi-terminal nano-device simulation

Abstract

With the miniaturization of emerging electronics, the charge carrier transport at the nano-scale has attracted great interest among researchers in the past decade. The non-equilibrium Green's function(NEGF) approach is a systematic method to handle non-equilibrium problems in open quantum system. The application of NEGF in quantum transport has led to tremendous success for the simulation of nano-devices. However, the intensive computational cost in NEGF prohibits its application to realistic systems at large scale, especially devices with a multi-terminal setup. The major part of the cost in self-consistent field (SCF) NEGF calculations comes from the evaluation of electron density, or Mulliken charge, which is obtained by a numerical integration of the Green's function, and can be divided into two parts : the equilibrium part involved involving complex contour integration of a retarded Green's function ($\mathbf{G^r}$), the non-equilibrium part involving real axis integration of a term similar to lesser Green's function ($\mathbf{G^r \Gamma G^a}$). To speed up the charge calculation in NEGF, several efficient and accurate methods are developed in this thesis from two aspects: the integral term: $\mathbf{G^r}$ and $\mathbf{G^r \Gamma G^a}$; the numerical integration involving these two terms.

The evaluation of charge only needs a subset of Green's function corresponding to sparsity pattern of transposed overlap matrix $\mathbf{S^{T}}$, while Selective Inversion algorithm provides an efficient way to compute certain elements of $\mathbf{G^r}$, the inversion of the sparse matrix. The computed elements, including the pattern of $\mathbf{S^{T}}$, are adequate for the equilibrium charge computation. Similarly, Selected Inversion algorithm is extended for non-equilibrium charge evaluation by constructing a new recursive relation for $\mathbf{G^r \Gamma G^a}$. Meanwhile, the X-formula algorithm exploits the low-rank property of $\Gamma$ to evaluate the non-equilibrium part. The inversion involved term $\mathbf{G^r \Gamma G^a}$ thus could be obtained from the solution of the sparse linear solver with mutiple right-hand-side(RHS) by X-formula($\mathbf{AX = Y}$). Furthermore, for larger number points of non-equilibrium integration, a projection-based Model Order Reduction(MOR) approach adaptively chooses solutions of X-formula at sample energy points to construct the solution subspace. The solution for other energy points could be obtained very efficiently within the solution subspace. Besides,the numerical integration technique, multi-contour and adaptive quadrature, are also implemented in my NEGF regime. Multi-contour technique systematically improves the NEGF charge integration accuracy by assigning and optimizing a customized weight combination for each orbital. Adaptive quadrature technique gives an acceptable tolerance for the non-equilibrium charge integration by adaptively choosing the integration points at contour. Finally, all the algorithms and techniques developed in the thesis are applied to the simulation of the junctionless nanowire transistor(JNT), which is a novel muti-terminal device. The Current-Voltage characteristics and electronic properties are systematically studied under the fast NEGF implementation.