Transmission eigenchannels for coherent phonon transport

J. C. Klöckner, J. C. Cuevas, and F. Pauly

Phys. Rev. B 97, 155432 – Published 26 April 2018

Abstract

We present a procedure to determine transmission eigenchannels for coherent phonon transport in nanoscale devices using the framework of nonequilibrium Green's functions. We illustrate our procedure by analyzing a one-dimensional chain, where all steps can be carried out analytically. More importantly, we show how the procedure can be combined with ab initio calculations to provide a better understanding of phonon heat transport in realistic atomic-scale junctions. In particular, we study the phonon eigenchannels in a gold metallic atomic-size contact and different single-molecule junctions based on molecules such as an alkane chain, a brominated benzene-diamine, where destructive phonon interference effects take place, and a $C_{60}$ junction.

Received 30 January 2018
Revised 11 April 2018

DOI:https://doi.org/10.1103/PhysRevB.97.155432

Physics Subject Headings (PhySH)

Ballistic transport Density functional theory First-principles calculations Lattice dynamics Phonons Scattering theory

Molecular junctions

Landauer formula Nonequilibrium Green's function

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Authors & Affiliations

J. C. Klöckner^1,2,*, J. C. Cuevas^1,3, and F. Pauly^1,2

¹Department of Physics, University of Konstanz, D-78457 Konstanz, Germany
²Okinawa Institute of Science and Technology Graduate University, Onna-son, Okinawa 904-0395, Japan
³Departamento de Física Teórica de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, E-28049 Madrid, Spain

^*jan.kloeckner@uni-konstanz.de

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Issue

Vol. 97, Iss. 15 — 15 April 2018

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Images

Figure 1
(a) Sketch of the 1D junction. Two semi-infinite leads with nearest-neighbor coupling constants $k_{l}$ are connected at sites $- 1$ and 0 with the coupling constant $k_{c}$ . The region $(- \infty, - 2]$ is considered as the L part, the region $[- 1, 0]$ as the C part, and $[1, \infty)$ as the R part. (b) Transmission eigenchannel ${\tilde{Q}}_{1} (E, t = 0)$ for the 1D chain represented in the complex plane for energies of $E = 1$ , 10, and 19 meV. The red arrow shows the complex number ${\tilde{Q}}_{1, - 1} (E, t = 0)$ for the atom $- 1$ , the blue one ${\tilde{Q}}_{1, 0} (E, t = 0)$ for atom 0. (c) The same as in (b), but now we display only the real part of the solution $Re {\tilde{Q}}_{1} (E, t = 0)$ . (d) The corresponding transmission eigenvalue $τ_{1} (E)$ as a function of energy together with the LDOS of one of the atoms ( $- 1$ or 0) in the central part. The energies of those eigenchannels, which are studied in (b) and (c), are indicated with arrows. For (b)–(d), we assumed spring constants of $k_{l} = 100 {meV}^{2}$ and $k_{c} = 20 {meV}^{2}$ .
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Figure 2
(a) Transmission eigenchannels for a gold dimer contact at an energy of 1.5 meV. We show the three eigenchannels with the highest transmissions, which are equal to $τ_{1} = 0.943$ , $τ_{2} = 0.238$ and $τ_{3} = 0.002$ . To better visualize the perpendicular polarizations of eigenchannels 1 and 2, we have inserted an inset between these two channels, showing the top atom of the left electrode in transport direction. It shows both $Re {\tilde{Q}}_{1} (E)$ and $Re {\tilde{Q}}_{2} (E)$ on the dimer atoms. Red are the displacements for eigenchannel 1, blue for eigenchannel 2, and the geometry has been rotated such that the displacement vectors of channel 1 point out of plane. (b) The same as in (a), but for an energy of 10.5 meV. The channel transmissions are $τ_{1} = 0.968$ , $τ_{2} = 0.122$ , and $τ_{3} = 0.098$ . (c) The three highest transmission coefficients as a function of energy for the gold atomic contact shown in (a) and (b). The energies of the eigenchannels considered in these two panels are indicated by arrows, and colored frames around the channel representations serve to identify the corresponding transmission values in (c).
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Figure 3
(a) Transmission eigenchannels for an Au-decane-dithiol-Au junction at an energy of 7.61 meV. We show the three eigenchannels with the highest transmissions of $τ_{1} = 1.00$ , $τ_{2} = 0.84$ , and $τ_{3} = 0.13$ as well as two vibrational modes of the free molecule at energies of 6.4 and 9.8 meV. (b) The most transmissive eigenchannel at an energy of 19.23 meV and a vibrational mode of the free molecule at energy 18.76 meV. (c) Three highest transmission eigenvalues as a function of energy for the Au-decane-dithiol-Au junction. The energies of the eigenchannels shown in (a) and (b) are indicated with arrows. In these two panels the angle of view differs for modes with out-of-plane as compared to in-plane character.
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Figure 4
The three highest eigenchannel transmissions as a function of energy for a Au-2-bromo-1,4-diaminobenzene-Au junction. Above the graph, one can see the eigenchannels with the highest transmission for the three different energies of 17.03, 18.85, and 19.37 meV. The energies are selected by peaks of $τ_{1}$ in the transmission plot and are indicated by corresponding arrows. Above the eigenchannels we show the vibrational modes of the free molecule that are responsible for the destructive interference.
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Figure 5
The three largest transmission coefficients for a Au- $C_{60}$ -Au junction as a function of energy. The most transmissive eigenchannels at energies of 5.4, 6.0, and 7.8 meV, as indicated by arrows in the transmission plot, are shown in the upper part of the figure.
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