Quantized thermal conductance via phononic heat transport in nanoscale devices at low temperatures

M. Käso and U. Wulf

Phys. Rev. B 89, 134309 – Published 28 April 2014

Abstract

We study phononic heat transport in nanoscale devices. In the nonequilibrium Green's function formalism, an analytical small-frequency expansion of the phonon current transmission is derived for an arbitrary oscillator chain with typical contact-device-contact structure. Applying this expansion in a Landauer formula, it is possible to construct a systematic low-temperature expansion of the thermal conductance. It follows that quantized thermal conductance occurs as a plateau of the thermal conductance divided by the temperature within second order of the temperature expansion for completely heterogeneous systems as long as the product of force constant and oscillator mass is identical in both contacts, independent of the scattering area. Beyond this plateau, the higher-order terms of the low-temperature expansion yield a finite-temperature correction exhibiting the form of a cubic power law depending on the details of the scattering area. These findings are in agreement with experiments and numerical calculations. Our general results are applied to a double junction chain, where we find as the first phenomenon beyond our low-temperature expansion a second plateau. This plateau is associated with a thermal phase averaging of the phonon transmission, which leads for increasing temperatures to an independence of the thermal conductance from the device length.

Received 6 May 2013
Revised 19 September 2013

DOI:https://doi.org/10.1103/PhysRevB.89.134309

Authors & Affiliations

M. Käso^* and U. Wulf^†

Department of Theoretical Physics, Brandenburg University of Technology Cottbus, Konrad-Wachsmann-Allee 1, 03046 Cottbus, Germany

^*matkaeso@physik.tu-cottbus.de
^†wulf@physik.tu-cottbus.de

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Vol. 89, Iss. 13 — 1 April 2014

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Images

Figure 1
Schematic representation of a one-dimensional contact-device-contact structure with a arbitrary scattering area (device) coupled to two homogeneous, not necessarily identical contacts.
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Figure 2
Current transmission according to Eq. (17) for DJ structures with $N = 5$ . (Left) Si-Ge- ${Si}_{X}$ with $M_{X}$ equal to $M_{Si}$ (solid), $2 M_{Si}$ (dashed), and $3 M_{Si}$ (dotted). (Right) Ge-Si- ${Ge}_{X}$ with $M_{X}$ equal to $M_{Ge}$ (solid), $2 M_{Ge}$ (dashed), and $3 M_{Ge}$ (dotted). Further parameters are taken from Ref. [10]: $M_{Si} = 4.7 \times 10^{- 26}$ kg, $f_{Si} = 16.9$ N/m, $d_{Si} = 5.43 \times 10^{- 10}$ , $M_{Ge} = 1.2 \times 10^{- 25}$ kg, $f_{Ge} = 13.7$ N/m, and $d_{Ge} = 5.66 \times 10^{- 10}$ m. In addition, $f_{0} = (f_{C 1} + f_{D}) / 2$ , and $f_{N} = (f_{D} + f_{C 2}) / 2$ .
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Figure 3
Representation of $Ξ (ω \to 0) = Ξ_{0}$ as a function of the ratio of the contact masses $y = M_{X} / M_{C 1}$ . Points: for a Si-Ge- ${Si}_{X}$ structure with a device region consisting of five Ge atoms and a Si contact. Circle: for a Ge-Si- ${Ge}_{X}$ structure with a device region consisting of five Si atoms and a Ge contact. The points and circles are calculated from (17) and the solid line is calculated from the universal result (7).
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Figure 4
(Left) Transmission for a Si-Ge-Si DJ structure with a device region consisting of five Ge atoms (solid) as well as the averaged transmission according to Eq. (19) (solid line with points). The peaks resulting from multiple reflections between the two interfaces of the DJ system. The minima of the transmission are enveloped by the square of the averaged transmission function (dotted). The arrow marked the maximum device frequency $2 {\tilde{ω}}_{0} = 21.4$ THz. In the “forbidden” gray region above $2 {\tilde{ω}}_{0}$ , we observe phonon tunneling as the small tail. (Right) Thermal conductance $Λ (T)$ divided by $T$ for a Si-Ge-Si DJ structure with a device region consisting of 5 (solid), 10 (dashed dotted), 25 (dashed), and 50 (long dashed) Ge--atoms. The upper plateau corresponds to the quantum of the thermal conduction. We see also the forming of a second plateau for increasing the number of atoms in the device with the value (20). The dotted line represents the expansion corresponding to Eq. (6) for a device consisting of five atoms. The arrow indicates the plateau width $T_{ε} = 1.6$ K according to a deviation of the thermal conductance from the quantum value of about 1% ( $ε = 0.01$ ). All curves for different $N$ merge with a asymptote (solid line with points), which is given by $Λ_{av} (T) / T$ .
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Figure 5
The plateau width $T_{ε}$ of the Si-Ge-Si DJ structure in Fig. 4 as a function of the device length $N$ . We choose $ε = 0.01$ corresponding to a maximum deviation of the thermal conductance from its quantized value by 1%.
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