Electron density and transport in top-gated graphene nanoribbon devices: First-principles Green function algorithms for systems containing a large number of atoms

Abstract

The recent fabrication of graphene nanoribbon (GNR) field-effect transistors poses a challenge for first-principles modeling of carbon nanoelectronics due to many thousand atoms present in the device. The state of the art quantum transport algorithms, based on the nonequilibrium Green function formalism combined with the density-functional theory (NEGF-DFT), were originally developed to calculate self-consistent electron density in equilibrium and at finite bias voltage (as a prerequisite to obtain conductance or current-voltage characteristics, respectively) for small molecules attached to metallic electrodes where only a few hundred atoms are typically simulated. Here we introduce combination of two numerically efficient algorithms which make it possible to extend the NEGF-DFT framework to device simulations involving large number of atoms. Our first algorithm offers an alternative to the usual evaluation of the equilibrium part of electron density via numerical contour integration of the retarded Green function in the upper complex half-plane. It is based on the replacement of the Fermi function $f(E)$ with an analytic function $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{f}(E)$ coinciding with $f(E)$ inside the integration range along the real axis, but decaying exponentially in the upper complex half-plane. Although $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{f}(E)$ has infinite number of poles, whose positions and residues are determined analytically, only a finite number of those poles have non-negligible residues. We also discuss how this algorithm can be extended to compute the nonequilibrium contribution to electron density, thereby evading cumbersome real-axis integration (within the bias voltage window) of NEGFs which is very difficult to converge for systems with large number of atoms while maintaining current conservation. Our second algorithm combines the recursive formulas with the geometrical partitioning of an arbitrary multiterminal device into nonuniform segments in order to reduce the computational complexity of the retarded Green function evaluation by extracting only its submatrices required for electron density and transmission function. We illustrate fusion of these two algorithms into the NEGF-DFT-type code by computing charge transfer, charge redistribution and conductance in zigzag-$\text{GNR}\ensuremath{\mid}\text{variable}$-width-armchair-$\text{GNR}\ensuremath{\mid}\text{zigzag}$-GNR two-terminal device covered with a gate electrode made of graphene layer as well. The total number of carbon and edge-passivating hydrogen atoms within the simulated central region of this device is $\ensuremath{\simeq}7000$. Our self-consistent modeling of the gate voltage effect suggests that rather large gate voltage $\ensuremath{\simeq}3\text{ }\text{eV}$ might be required to shift the band gap of the proposed AGNR interconnect and switch the transport from insulating into the regime of a single open conducting channel.