Ab initio tight-binding LMTO method for nonequilibrium electron transport in nanosystems

We present an ab initio method for calculation of the electronic structure and electronic transport of nanoscale systems coupled to electrodes with applied voltage bias. The method is based on the local density approximation of density functional theory and implemented in the framework of the tight-binding linear muffin-tin orbital approach in its atomic sphere approximation. A fully atomistic description of the electrodes and the nanosystem is used, and the self-consistent charge and electrostatic potential for the system under applied bias is calculated using the nonequilibrium Green’s function (NEGF) approach. General expressions for the lesser Green’s function and transmission coefficient obtained within NEGF theory are rewritten using auxiliary Green’s functions that are defined by the inverse of the short-ranged structural constants. This reformulation of the theory with auxiliary Green’s functions allows the use of very effective and well-developed tight-binding techniques. The method is applied to three systems: a single benzene di-thiol molecule coupled to (111) gold electrodes, a single gold atom coupled to (100) gold electrodes, and a single platinum atom coupled to (100) platinum electrodes.

Figure 1

(Color online) The difference between the electrostatic potential $\Delta V$ in the center of the spheres (top panel) and charges $\Delta Q$ on the spheres (bottom panel) for the “capacitor” with applied bias $V=-1.36\mathrm{V}$ and with zero bias as a function of the sphere’s $z$ coordinate (in units of Bohr radius $a_0$) calculated by the NEGF approach (circles) and by the equilibrium approach (squares). For the sake of comparison the potential of the equilibrium calculation is shown shifted by $+V∕2$ for $z>0$ and by $-V∕2$ for $z<0$. The circles are connected by solid lines and squares are connected by dashed lines. Vertical dotted lines show the positions of the left-most and right-most gold layers of the capacitor surfaces.

Figure 2

(Color online) The computational cell used for simulation of transport properties of the BDT molecule coupled to two gold (111) surfaces. The cell repeats itself in the $xy$ plane. Left and right semi-infinite gold leads are attached to the cell. The atoms between the two vertical lines constitute the central region, while all atoms in the figure form a supercell, periodic in three dimensions, used to calculate the electrostatic potential.

Figure 3

(Color online) The density of states projected on the BDT molecule atomic spheres obtained by the LMTO-ASA method (solid line) for an equilibrium system in the multilayer geometry and by full potential supercell calculations (dotted line).

Figure 4

(Color online) DOS projected on the left (top panel) and right (bottom panel) sulfur atomic sphere for four different bias voltages.

Figure 5

(Color online) DOS projected on the left-most (top panel) and right-most (bottom panel) carbon atomic sphere for four different bias voltages.

Figure 6

(Color online) DOS projected on the ${\mathrm{C}}_4{\mathrm{H}}_4$ inner atoms of the BDT molecule for four different bias voltages.

Figure 7

The Induced potential at the sphere centers as a function of the sphere’s $z$ coordinate for the system of BDT molecules attached to two (111) gold electrodes with an applied bias of 2.72 V. The induced potential for 241 spheres is shown, including 139 empty spheres, 12 BDT spheres, 54 Au spheres in the central region, and an additional 36 Au spheres in the two left lead layers and two right lead layers that are included in the construction of the supercell for electrostatic potential calculation. The atomic spheres that have the same values of the $z$-coordinate and induced potential value are shown in the figure as a single point. Vertical dotted lines show the positions of the left-most and right-most gold layers of the right and left electrode, respectively.

Figure 8

Current as a function of bias. Solid line with dots represents current obtained by the NEGF approach. Dashed line represents current calculated from zero-bias transmission coefficient (see text for details).

Figure 9

Transmission coefficient calculated by the NEGF approach for four different bias voltages. The bias window is shown with the vertical dotted lines.

Figure 10

(Color online) The computational cell used for simulation of transport properties of a single Au (Pt) atom coupled to two Au (Pt) surfaces. The cell repeats itself in the $xy$ plane. Left and right semi-infinite leads are attached to the cell. The atoms between the two vertical lines constitute the central region, while all atoms in the figure form a supercell, periodic in three dimensions, used to calculate the electrostatic potential.

Figure 11

(Color online) Zero-bias transmission spectrum for a single Au atom coupled to semi-infinite Au electrodes for different empty spheres packing structures, with 83 (solid line), 87 (dashed line), and 95 (dotted line) empty spheres.

Figure 12

(Color online) The $spd$ angular momentum resolved DOS projected on the central gold atomic sphere for the single gold atom system.

Figure 13

(Color online) Transmission spectrum for a single Au atom coupled to semi-infinite Au electrodes calculated at zero bias and at a bias of 1.09 V.

Figure 14

(Color online) Current as a function of applied voltage for the gold system (solid line with filled circles) and platinum system (solid line with open squares). The currents for these two systems calculated from the zero bias transmission spectrum are also shown by dotted lines that coincide at small voltage with appropriate solid lines.

Figure 15

(Color online) Zero-bias transmission spectrum for a single Pt atom coupled to semi-infinite Pt electrodes for different empty spheres packing structures, with 83 (solid line), 91 (dashed line), and 107 (dotted line) empty spheres.

Figure 16

(Color online) The $spd$ angular momentum resolved DOS projected on the central platinum atomic sphere for the single platinum atom system.

Figure 17

(Color online) Transmission spectrum for a single Pt atom coupled to semi-infinite Pt electrodes calculated at zero bias and at a bias of 0.88 V.

Figure 18

(Color online) The induced potential in the center of spheres as a function of the sphere’s $z$ coordinate for the gold system (solid line with filled circles) and the platinum system (solid line with open squares). In both cases the bias voltage equals 0.68 V. The induced potential is shown for all atomic and empty spheres of the central region and for the additional 36 spheres of two left lead layers and two right lead layers that are included in the construction of the supercell for the electrostatic potential calculation. Vertical dotted lines show the positions of the left-most and right-most Au(Pt) layers of the right and left electrode, respectively.