Non‐equilibrium Green function implementation of boundary conditions for full band simulations of substrate‐nanowire structures

Abstract

The density of states of a nanowire exhibits peaks at energies in which the individual transverse modes begin to propagate. Under ideal conditions these modes are eigenstates solely determined by the cross sectional size and shape of the nanowire. However, for realistic nanowires, which are grown on semiconductor substrates, the density of states of the subsrate-nanowire structure is dependent on the distance from the nanowire endpoints. Near the substrate, the density of states is is nonzero far below the energy corresponding to the first eigenstate of the ideal nanowire. This initial increase in the density of states occurs at energies near the conduction and valence band edges of the semiconductor substrate on which the nanowire is grown. Away from the substrate, the density of states begins to acquire ideal nanowire characteristics. In the present work this effect is captured by imposing boundary conditions, with properties of bulk material, at the nanowire base. The calculations utilize a first neighbor sp 3 s*d 5 orbital basis within the non-equilibrium Green function formalism.

Electronic properties of silicon nanowires

“…The algorithms are described in detail in [39]. Transmission coefficients that we plot are summed over all spin and orbital channels at a given energy.…”

Contact Block Reduction method for ballistic quantum transport with semi-empirical sp³d⁵s* tight binding band models

“…RGF significantly reduces this numerical load, 9,10 however, it is still expensive and more efficient alternatives need to be considered, 11 one of which is CBR since it reduces the matrix inversions to find G R . …”

“…Then the simplification of (6) to (8) becomes difficult, since for square matrices K and W, K -1 WK cannot be reduced to W and if W is singular. Therefore, one needs to evaluate the self-energy term Σ with the solution in (6) with alternatives for X to make the CBR method still practical with TB models, one of which we suggested in (9), where ε n is an eigenvalue at the n th sub-band minima in the conduction band (maxima for the valence band) of the semi-infinite contact.…”

“…To measure the numerical efficiency of the suggested method with the RGF, we assume two contacts for the carrier transport in open system, which is the usual case in the modeling of quantum transport. 5,9 As a detailed example, we assume a [100] Si quantum wire with 2 contacts in Fig. 3 and divide the device space into boundary region c=c 1 +c 2 and region d for the rest, where Σ c x in (3) is written as the expression in (10) since the unit block in the boundary region c 1 is not coupled with the one in the region c 2 .…”

A multi-level parallel simulation approach to electron transport in nano-scale transistors

“…Special attention has been paid to the calculation of the open-boundary conditions (OBCs) which may severely limit the size of the simulation domain if it is not optimized. The usual iterative solutions [10] and generalized eigenvalue problems [11] have been abandoned in favor of a shift-and-invert procedure [12]. The Non-equilibrium Green's Function (NEGF) [13] and the Wave Function (WF) [14] formalisms have been implemented to solve the transport problem and to obtain density-of-states, transmission coefficients, carrier, and current densities.…”