^{1,a)}, Wenchang Lu

^{1,2}, J. Bernholc

^{1,2}and Vincent Meunier

^{1,b)}

### Abstract

We present a generalized approach for computing electron conductance and characteristics in multiterminal junctions from first-principles. Within the framework of Keldysh theory, electron transmission is evaluated employing an O(N) method for electronic-structure calculations. The nonequilibrium Green function for the nonequilibrium electron density of the multiterminal junction is computed self-consistently by solving Poisson equation after applying a realistic bias. We illustrate the suitability of the method on two examples of four-terminal systems, a radialene molecule connected to carbon chains and two crossed-carbon chains brought together closer and closer. We describe charge density, potential profile, and transmission of electrons between any two terminals. Finally, we discuss the applicability of this technique to study complex electronic devices.

Portions of this research was sponsored by the Laboratory Directed Research and Development Program of Oak Ridge National Laboratory (ORNL), managed by UT-Battelle, LLC for the U.S. Department of Energy under Contract No. De-AC05-00OR22725 (KKS and VM), by DOE Grant Nos. DE-FG02-03ER46095 and DE-FG02-98ER45685, and by ONR Grant No. N000140610173 (WL and JB).

I. INTRODUCTION

II. THEORY

A. System setup

B. Density Matrix

C. Generalized density matrix for multiterminal junction

D. Computation of the conductance

E. Finite bias

F. Computational details

G. Parallelization on supercomputers

III. APPLICATIONS

A. Radialene molecule

B. Crossed carbon chains

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Lead
- 47.0
- Carbon
- 14.0
- Carrier density
- 12.0
- Green's function methods
- 10.0
- Density functional theory
- 9.0

## Figures

Schematic diagram of a multiterminal junction. The barrier region is divided into -number of blocks . The leads are connected to blocks , respectively, whereas the remaining leads are connected to block .

Schematic diagram of a multiterminal junction. The barrier region is divided into -number of blocks . The leads are connected to blocks , respectively, whereas the remaining leads are connected to block .

(Left) Schematic of a four-terminal system. The leads , , , and are connected to the molecular barrier via the subregions , , , and , respectively. The subregions and the molecular region are considered together as a single region shown by the blue-dotted box. The black-dotted box shows the extended-scattering region . (Right) Tridiagonal matrix representation of this system. Any matrix-element in the lower off-diagonal-blocks is the complex conjugate of the corresponding element in the upper off-diagonal-blocks and therefore one can avoid storing the lower off-diagonal-blocks. The shaded blocks of the matrix are directly connected to the leads and hence these contain the same matrix-elements as in the respective leads.

(Left) Schematic of a four-terminal system. The leads , , , and are connected to the molecular barrier via the subregions , , , and , respectively. The subregions and the molecular region are considered together as a single region shown by the blue-dotted box. The black-dotted box shows the extended-scattering region . (Right) Tridiagonal matrix representation of this system. Any matrix-element in the lower off-diagonal-blocks is the complex conjugate of the corresponding element in the upper off-diagonal-blocks and therefore one can avoid storing the lower off-diagonal-blocks. The shaded blocks of the matrix are directly connected to the leads and hence these contain the same matrix-elements as in the respective leads.

An initial bias profile (size ) for a four-terminal junction to be applied to the system at the beginning of a nonequilibrium calculation. For the planar molecules considered here, it has been generated by solving a 2D Laplace’s equation with appropriate boundary conditions (see text for details). An identical bias voltage 0.8 V is applied through all four leads , , ,and . The leads are denoted by the numbers 1–4, respectively, and the scale bar of the potential is in electron volt.

An initial bias profile (size ) for a four-terminal junction to be applied to the system at the beginning of a nonequilibrium calculation. For the planar molecules considered here, it has been generated by solving a 2D Laplace’s equation with appropriate boundary conditions (see text for details). An identical bias voltage 0.8 V is applied through all four leads , , ,and . The leads are denoted by the numbers 1–4, respectively, and the scale bar of the potential is in electron volt.

Flowchart of the SC loop used to calculate characteristics. The terminology is explained in the text.

Flowchart of the SC loop used to calculate characteristics. The terminology is explained in the text.

Charge convergence of the radialene system with SC steps at zero bias. In the inset, a schematic view of the central region of the four-terminal radialene junction is shown, with the number 1–4 marking the positions of the leads. The size of the system is .

Charge convergence of the radialene system with SC steps at zero bias. In the inset, a schematic view of the central region of the four-terminal radialene junction is shown, with the number 1–4 marking the positions of the leads. The size of the system is .

(Left) SC converged potential profile (same as system size, i.e., ) at zero bias of the four-terminal radialene system, plotted 4.88 Bohr above the atomic plane. (Right) The converged potential profile after applying an identical bias at 0.8 V through all the leads. Both the images are shown in same color scale (in electron volt), to compare the relative heights of the potentials.

(Left) SC converged potential profile (same as system size, i.e., ) at zero bias of the four-terminal radialene system, plotted 4.88 Bohr above the atomic plane. (Right) The converged potential profile after applying an identical bias at 0.8 V through all the leads. Both the images are shown in same color scale (in electron volt), to compare the relative heights of the potentials.

(Left) Converged potential profile of the radialene system with a nonuniform bias voltage. (The color scale is in electron volt.) An identical bias of 0.8 V is applied through three of the leads (denoted by 1, 2, 3) and no bias is applied through the fourth lead. (Right) Potential drop along the central line between the leads 3 and 4.

(Left) Converged potential profile of the radialene system with a nonuniform bias voltage. (The color scale is in electron volt.) An identical bias of 0.8 V is applied through three of the leads (denoted by 1, 2, 3) and no bias is applied through the fourth lead. (Right) Potential drop along the central line between the leads 3 and 4.

Transmission curves of the four-terminal radialene system with different bias voltages. The is the Fermi energy of the lead 4. The bias geometry is the same as in Fig. 7. The left and right panels show transmission through the leads and , respectively.

Transmission curves of the four-terminal radialene system with different bias voltages. The is the Fermi energy of the lead 4. The bias geometry is the same as in Fig. 7. The left and right panels show transmission through the leads and , respectively.

Current-voltage characteristics of the four-terminal radialene junction. The bias geometry is as shown in Fig. 7. The current contributions through the leads , are displayed.

Current-voltage characteristics of the four-terminal radialene junction. The bias geometry is as shown in Fig. 7. The current contributions through the leads , are displayed.

A schematic diagram of the crossed-carbon-chains system. Three cases are considered, with distances between the chains of 7.5, 5.0 and 2.5 Bohr.

A schematic diagram of the crossed-carbon-chains system. Three cases are considered, with distances between the chains of 7.5, 5.0 and 2.5 Bohr.

(a) and (c) Comparison of the converged potential profiles of the crossed-carbon-chains system when the distances between the two carbon chains are 7.5 and 5.0 Bohr. In both the cases, we apply same bias 0.5 V through the leads , and and bias −0.5 V through the fourth lead . The plotting passing through one of the chains. (b) and (d) Potential drops between the leads to for (a) and (c) case.

(a) and (c) Comparison of the converged potential profiles of the crossed-carbon-chains system when the distances between the two carbon chains are 7.5 and 5.0 Bohr. In both the cases, we apply same bias 0.5 V through the leads , and and bias −0.5 V through the fourth lead . The plotting passing through one of the chains. (b) and (d) Potential drops between the leads to for (a) and (c) case.

Comparison of transmission curves in all three cases, where the distances between the chains are 7.5, 5.0 and 2.5 Bohr, of the crossed-carbon-chain system with zero and nonzero biases (as in Fig. 11). The left and right panels show transmission through leads and , respectively.

Comparison of transmission curves in all three cases, where the distances between the chains are 7.5, 5.0 and 2.5 Bohr, of the crossed-carbon-chain system with zero and nonzero biases (as in Fig. 11). The left and right panels show transmission through leads and , respectively.

Comparison of curves with varying distances between the chains of the crossed-carbon-chain system. The left and right panels show the current contributions through leads and , respectively, with the bias voltage.

Comparison of curves with varying distances between the chains of the crossed-carbon-chain system. The left and right panels show the current contributions through leads and , respectively, with the bias voltage.

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