Removing all periodic boundary conditions: Efficient nonequilibrium Green's function calculations

We describe a method and its implementation for calculating electronic structure and electron transport without approximating the structure using periodic supercells. This effectively removes spurious periodic images and interference effects. Our method is based on already established methods readily available in the nonequilibrium Green's function formalism and allows for nonequilibrium transport. We present examples of a nitrogen defect in graphene, finite voltage bias transport in a point contact to graphene, and a graphene-nanoribbon junction. This method is less costly, in terms of CPU hours, than the supercell approximation.

Figure 1

Defected structures using various PBC simulations; neighboring images are shown with transparency. (a) shows a typical DFT calculation with a single defect (marked region) and PBCs in all directions. (b) Using the current state-of-the-art NEGF, one can remove images in two directions, reducing the PBCs to one direction. Finally, in (c) our method allows removing all PBCs using an enclosing self-energy.

Figure 2

Green's function matrix elements of the graphene tight-binding model ($t=-2.7\mathrm{eV}$) for $z=(0.5+i{10}^{-4})\mathrm{eV}$ (${\mathbf{G}}_{\mathbf{k}}^{-1}=\mathbf{I}z-{\mathbf{H}}_{\mathbf{k}}-{\mathit{\Sigma }}_{\mathbf{k}}^L-{\mathit{\Sigma }}_{\mathbf{k}}^R$) for transverse $\mathbf{k}$ points. Several $\delta$ peaks are seen, which makes an integral extremely difficult.

Figure 3

The left system is completely bulk and coupled to an internal, also bulk, region (hatched) via $\mathbf{V}$. The right system is a single defect highlighted in color coupled to an infinite bulk region via the same $\mathbf{V}$. Calculation of the real-space self-energy in the colored region can be performed by calculating the real-space Green's function in the left bulk system, inverting, and subtracting the real-space matrices $\mathbf{H}$ and $\mathbf{S}$ to retrieve the self-energy describing the external bulk part. The resulting ${\mathit{\Sigma }}^{\mathcal{R}}$ may be used in any defected system as long as $\mathbf{V}$ and the potential in the surrounding region is not changed.

Figure 4

Projected DOS on a single carbon atom in the pristine graphene lattice. Three methods are compared (see the inset geometry), with the atom highlighted: siesta projected density of states (PDOS), two-terminal (bottom/top) transiesta, and, finally, ${\mathit{\Sigma }}^{\mathcal{R}}$. All methods yield exactly the same DOS.

Figure 5

Projected DOS on a single nitrogen defect in the graphene lattice. Three methods are compared (see inset geometry), with the nitrogen atom highlighted: siesta PDOS, two-terminal (bottom and top) transiesta, and, finally, ${\mathit{\Sigma }}^{\mathcal{R}}$. The unoccupied states have very different characters in the three cases.

Figure 6

Nonequilibrium transport calculation of tip-graphene contact (tip at $+0.5\mathrm{V}$). Top: Spectral DOS from the tip on the first carbon atom in the graphene layer. Bottom: Transmission from the tip into graphene comparing three-terminal (left and right graphene and tip) and two-terminal (graphene and tip) setups. A large difference in the unoccupied energy range is seen both in the spectral DOS and in the transmission.

Figure 7

Nonequilibrium transport calculation of the graphene-ribbon junction (ribbon at $+0.5\mathrm{V}$). Top: Spectral DOS from graphene for the four atoms starting in the ZGNR ribbon. Bottom: Transmission between graphene and ZGNR. A noticeable shift appears in the localized peaks above the graphene chemical potential, and there are some differences in the transmissions. The small differences between the real-space method and the standard way are because the system is already relatively wide and thus the interference effects are already minor.