Self-energy matrices for electron transport calculations within the real-space finite-difference formalism

Shigeru Tsukamoto, Tomoya Ono, Kikuji Hirose, and Stefan Blügel

Phys. Rev. E 95, 033309 – Published 20 March 2017

Abstract

The self-energy term used in transport calculations, which describes the coupling between electrode and transition regions, is able to be evaluated only from a limited number of the propagating and evanescent waves of a bulk electrode. This obviously contributes toward the reduction of the computational expenses in transport calculations. In this paper, we present a mathematical formula for reducing the computational expenses further without using any approximation and without losing accuracy. So far, the self-energy term has been handled as a matrix with the same dimension as the Hamiltonian submatrix representing the interaction between an electrode and a transition region. In this work, through the singular-value decomposition of the submatrix, the self-energy matrix is handled as a smaller matrix, whose dimension is the rank number of the Hamiltonian submatrix. This procedure is practical in the case of using the pseudopotentials in a separable form, and the computational expenses for determining the self-energy matrix are reduced by 90% when employing a code based on the real-space finite-difference formalism and projector-augmented wave method. In addition, this technique is applicable to the transport calculations using atomic or localized basis sets. Adopting the self-energy matrices obtained from this procedure, we present the calculation of the electron transport properties of $C_{20}$ molecular junctions. The application demonstrates that the electron transmissions are sensitive to the orientation of the molecule with respect to the electrode surface. In addition, channel decomposition of the scattering wave functions reveals that some unoccupied $C_{20}$ molecular orbitals mainly contribute to the electron conduction through the molecular junction.

Received 31 March 2016
Revised 22 November 2016

DOI:https://doi.org/10.1103/PhysRevE.95.033309

Physics Subject Headings (PhySH)

Computational complexity Density functional theory

Atomic, Molecular & Optical

Authors & Affiliations

Shigeru Tsukamoto^1,*, Tomoya Ono², Kikuji Hirose³, and Stefan Blügel¹

¹Peter Grünberg Institut & Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, D-52425 Jülich, Germany
²Center for Computational Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan
³Graduate School of Engineering, Osaka University, Suita, Osaka 565-0871, Japan

^*Corresponding author: s.tsukamoto@fz-juelich.de

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Issue

Vol. 95, Iss. 3 — March 2017

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Images

Figure 1
Schematic representation of a junction system used in electron transport calculations typically. The junction system is partitioned into a transition region and left and right electrode regions by boundary planes represented by the thick lines. The dark gray areas represent the boundary regions, only in which the interaction matrices $B_{L}$ and $B_{R}$ act on the scattering wave function $ψ$ , as seen in Eq. (2). The parts of the scattering wave function $ψ$ , which are included in the boundary regions, are denoted by $ψ_{M_{L}}, ψ_{N_{L}}, ψ_{M_{R}}$ , and $ψ_{N_{R}}$ . The matrices $H_{0}, H_{L}$ , and $H_{R}$ represent the Hamiltonians truncated by the transition region, a unit cell of the left electrode, and a unit cell of the right electrode, respectively.
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Figure 2
Schematic representations of a $C_{20}$ fullerene molecular junction. Panel (a) shows the side view of the molecular junction before structural optimization. The brown and blue spheres represent C and Al atoms, respectively. A $C_{20}$ molecule is sandwiched between a couple of Al(001) bulk electrodes, so the C–C bonds close to the electrodes are parallel to the electrode surfaces. The orientation of the molecule with respect to the $z$ axis is varied, and the angle between the $x$ axis and the C–C bond attaching to the electrode surface is denoted by $θ$ .
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Figure 3
Optimized geometry and electron transmission spectra of a $C_{20}$ molecular junction after structural optimization. In (a), the blue spheres represent Al atoms, and brown ones C atoms. In (b), the total and channel-decomposed transmissions of the molecular junction are plotted by filled and open circles, respectively. $E_{F}$ denotes the Fermi energy.
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Figure 4
Spatial distributions of the transmission channels of a $C_{20}$ molecular junction and $C_{20}$ molecular orbital after deformation. Panels (a), (b), and (c) exhibit the first, second, and third transmission channels of the electrons injected from the left electrode at the Fermi energy, respectively. Panels (d)–(g) show the fourth-, third-, second-, and first-lowest unoccupied molecular orbitals of the free-standing molecule, which is taken from the optimized junction structure shown in Fig. 3.
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Figure 5
Schematic representation of a bulk electrode. The matrices $B_{L}$ and $H_{L}$ represent the interaction matrix between the neighboring unit cells and the Hamiltonian matrix truncated by a unit cell, respectively. The vectors $q_{M_{L}}^{i - 1}, q_{N_{L}}^{i}, q_{M_{L}}^{i}$ , and $q_{N_{L}}^{i + 1}$ denote the generalized Bloch wave functions in the respective boundary regions, which are represented by dark gray areas and are subject to the influence of the interaction matrix $B_{L}$ .
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