Analysis of thermal conductance of ballistic point contacts

Substantial reduction of thermal conductance (Kph) was recently reported for air gap heterostructures (AGHs) in which two bulk layers were connected by low-density nanopillars. We analyze Kph using a full phonon dispersion and including important phonon scattering. We find a transition from ballistic at low temperatures to quasi-ballistic transport near room temperature and explain the slow roll-off in Kph that occurs near room temperature. We show that the density of nanopillars deduced from the analysis depends strongly on the phonon dispersion assumed. Our model provides a good agreement with experiment that will be necessary to design AGHs for thermoelectric applications.

Substantial reduction of thermal conductance (K ph ) was recently reported for air gap heterostructures (AGHs) in which two bulk layers were connected by low-density nanopillars. We analyze K ph using a full phonon dispersion and including important phonon scattering. We find a transition from ballistic at low temperatures to quasi-ballistic transport near room temperature and explain the slow roll-off in K ph that occurs near room temperature. We show that the density of nanopillars deduced from the analysis depends strongly on the phonon dispersion assumed. Our model provides a good agreement with experiment that will be necessary to design AGHs for thermoelectric applications. Recently, Bartsch et al. 1 demonstrated the fabrication of so-called air-gap heterostructures (AGHs) for which GaAs pillars with controlled length of a few nanometers and diameters of a hundred nanometer are sandwiched between GaAs substrate and capping layers. The thermal conductance of these structures was dramatically reduced in comparison to bulk structures. Bartsch et al. 1 explained the measured results with a simple model that assumed pure ballistic transport from low temperatures to room temperature. Since the measurement of thermal conductance (K ph ) was done for an ensemble of nanopillars, the thermal conductance through a single pillar was not independently measured. Instead, the K ph for a single pillar was estimated by comparing the measurement to a model calculation for which the transmission was one (i.e., ballistic transport) and a sine-type phonon dispersion was assumed. The pillar density was taken as a fitting parameter, and good agreement was found with the experiment-if the pillar density was assumed to be somewhat larger than expected. In this letter, we report an analysis of the AGH experiment that includes a full description of the phonon dispersion and includes realistic scattering processes. Our analysis supports the simpler analysis by Bartsch et al., 1 but we find that at room temperature, phonon transport through the nanopillars is quasi-ballistic, and the inferred density of nanopillars is significantly lower than that from the simpler analysis. This analysis also explains the slow roll-off in conductance observed experimentally. The results illustrate the importance of including realistic phonon dispersions and scattering when optimizing AGHs for applications such as thermoelectrics.
Heat conduction at length scales comparable to or smaller than phonon mean free paths (MFPs), which can be hundreds of nanometers in crystalline materials at room temperature, has received great attention with the increasing interest in nanostructures. Ballistic transport has been studied in nanostructures such as thin films, 2 nanowires, 3 and superlattices, 4 and the ballistic thermal conductance of an ideal one-dimensional channel has been predicted to have a quantum of thermal conductance, k 2 B T L p 2 =3h. 5,6 Direct evidence of ballistic transport, however, has not been demonstrated in such nanostructures due to non-ideal coupling between thermal reservoirs and thermal conductors, which prevent trans-mission across two reservoirs from being close to one. Schwab et al. 7 addressed this issue by fabricating both reservoirs and conductor from a homogeneous suspended silicon nitride layers and observed the quantum of thermal conductance. Due to the use of a micron-meter long conductor, however, the ballistic thermal conductance was observed only at low temperatures. The work of Bartsch et al. 1 shows that ballistic phonon transport (or at least quasi-ballistic, as we shall argue here) can be observed at room temperature.
In this study, we show that: (1) transport should not be considered to be pure ballistic even when the lengths of pillars are well below the average phonon MFP in the bulk material, (2) considering the full phonon dispersion and phonon scattering provides a clear understanding of measured K ph , which saturates at around 150 K and then roll-offs at room temperature, and (3) the analysis yields a fitted pillar density closer to the measured one. The paper is organized as follows. To calibrate the scattering parameters, we first compare thermal conductivity of bulk GaAs samples to the computed results by using a Landauer approach with a full dispersion description of phonons. Next, we examine quantitatively the effect of sine-type dispersions to demonstrate the importance of the full phonon dispersions-especially in near ballistic transport. Then, we compare thermal conductance without scattering to thermal conductance with scattering to show that pure ballistic transport model holds only at low temperatures. Finally, we show that the computed results match well the experimental data given in Ref. 1. The use of a full dispersion and inclusion of scattering yields a nanopillar density that is significantly lower than the simple model, and it also captures the slow roll-off in conductance for temperatures above 150 K. Finally, we summarize our conclusions.
To determine how close a nanopillar operates to its ballistic limit, the Landauer approach is used because it provides a simple, physically insightful description of ballistic transport and has been widely used to describe quantized electrical and thermal transport in nanostructures. Although not as widely appreciated, the Landauer approach describes diffusive transport as well and provides a simple way to treat the ballistic to diffusive transition. The Landauer formula for thermal conductance (K ph ) is expressed as 8 where k 2 B T L p 2 =3h is the quantum of thermal conductance, T ph is the transmission at a given energy hx, M ph is the number of conducting channels at a given energy, and W ph is a "window function," which determines the occupation of the channels and is given by 8 with n 0 being Bose-Einstein distributions. The transmission T ph is given as 9 where k ph ðxÞ is the mean-free-path for backscattering and L is the length of the conductor, i.e., the length of the pillar in this study. Equation (1a) applies to ballistic limit (L ( k ph ðxÞ) for which T ph ¼ 1; quasi-ballistic regime (L $ k ph ðxÞ) for which T ph ¼ k ph ðxÞ=ðL þ k ph ðxÞÞ as well as to diffusive limit (L ) k ph ðxÞ) for which T ph ¼ k ph ðxÞ=L. In the diffusive limit, the Landauer expression for lattice thermal conductivity is equivalent to the conventional expression from BTE. 8 Given an accurate phonon dispersion, M ph ðxÞ can be readily computed by a simple numerical technique-the "band counting" method. 3,8 To evaluate M ph ðxÞ in this work, a full band description of phonon dispersion of GaAs was obtained from the Tersoff interatomic pair potential model 10 within General Utility Lattice Program (GULP). 11 It is straightforward to compute K ph per cross-sectional area (A) with T ph ¼ 1 in Eq. (1a), i.e., ballistic thermal conductance per area K ph;BAL =A. A sine-type approximation for phonon dispersion is often used because it provides a compromise between rigorous full phonon dispersions and a simple Debye approximation and is given by 12 where x 0 is the maximum phonon frequency and k 0 is the Debye cutoff wave vector, for which the values are computed as defined in Ref. 12. Figure 1(a) compares M ph for a full phonon dispersion to M ph for a sine-type dispersion. For a sine-type dispersion, M ph is found to be where the factor of 3 comes from the three phonon polarizations. 13 It is clearly shown in Fig. 1(a) 15 and crystalline boundary scattering rates (s b À1 ¼ constant), where B, C, D, and s b À1 are fitted to match the measured data for bulk samples in Ref. 1. The inset of Fig. 1(b) shows that the resulting fit is excellent, regardless of phonon dispersion model. However, it can be seen that a simple sine model overestimates average phonon MFP by a factor of 2, where the average phonon MFP can be obtained by taking the ratio of thermal conductivity (j ph ) to ballistic thermal conductance per area (K ph;BAL =A), as presented in Ref. 8. For example, the average MFP for full phonon dispersion is 72 nm, compared to 166 nm for sine-type dispersion at room temperature. The difference in average MFP can be understood from Fig. 2(a), which compares ballistic thermal conductance per area, K ph;BAL =A, for a full phonon dispersion to that from a sine-type dispersion. At temperatures higher than 150 K where almost all conducting channels contribute to thermal conduction, neglecting optical phonon modes within the sine dispersion model leads to $2 times smaller ballistic conductance than that from full dispersion. The 2Â smaller K ph;BAL =A forces the average phonon MFP to be 2Â larger to match the measured thermal conductivity.
To examine the thermal conductance of the nanopillars, the surface roughness of pillar boundaries should be considered via a frequency-dependent specularity parameter. The phonon MFP by the boundary scattering is given by 16 where the diameter of a pillar is D pillar ¼ 100 nm as given in Ref. 1, and the specularity parameter p is given by 17 p ¼ expðÀ16p 3 g 2 =K 2 ph Þ; with K ph being the phonon wavelength and g the surface roughness assumed to be 0.2 nm. 18 For pillars with lengths of 4 nm and 6 nm, thermal conductance with scattering and full phonon dispersions is evaluated, as plotted in Fig. 2(a). It is found that the computed thermal conductance per area, K ph =A, for 4 nm and 6 nm long pillars is smaller compared to the pure ballistic thermal conductance, which demonstrates that the nanopillars operate below a fully ballistic limit.
To estimate how close a pillar operates to the ballistic limit, the average transmission is evaluated and presented in Fig. 2(b). The average transmission is defined as the ratio of thermal conductance with scattering being accounted for to ballistic thermal conductance. Figure 2(b) shows that although the transmission at low temperature is indeed close to one, it is reduced to 0.6-0.7 at 300 K, which clearly illustrates the importance of treating quasi-ballistic transport for the analysis of such nanostructures. Note that the effect of boundary scattering (Eq. (4)) and the length of pillars are negligible; Including boundary scattering reduces K ph =A only by $5% (Fig. 2(a)) and the transmission for 4 nm pillar is on average about 7% higher than that for 6 nm pillar ( Fig. 2(b)).
Additional insights can be obtained from Fig. 3, where M ph ; W ph , and T ph in Eq. (1a) vs phonon energy are plotted for three temperatures of 50, 150, and 300 K. As the temperature increases, the width of window function (W ph ) broadens and more conducting channels participate in conduction until the entire spectrum of channels contributes, which occurs at around 150 K. This results in an increase of thermal conductance with temperature as well as the saturation of thermal conductance for temperatures higher than 150 K, as shown in Fig. 2(a). In contrast, Fig. 3(b) shows that transmission (Eq. (1c)) keeps decreasing with increasing temperatures mainly due to umklapp scattering. Therefore, thermal conductance with scattering ( Fig. 2(a)) roll-offs at 300 K as the decreasing transmission term takes over the saturated effective number of conducting channels. This effect is not captured in the simple model. Note that as shown in Fig.  3(b), transmission for low energy phonons is close to 1, but high energy phonons have small transmission.
As shown in Fig. 3(a), the increasing importance of optical phonons is clearly visible as temperature increases. To quantitatively examine the contribution of optical phonon modes, Figure 4 shows the cumulative distribution function of thermal conductance as a function of energy with and without scattering for three temperatures of 50, 150, and 300 K. At 50 K, low energy channels (mostly acoustic phonon modes) contribute to more than 80% of the thermal conductance and two curves obtained with scattering and without scattering almost overlap, which indicates the pillars operates at the near-ballistic limit. At 150 and 300 K, it can be seen that all energy channels contribute to the ballistic thermal conductance and about 40-50% of ballistic conduction is carried by optical phonon modes. So a sine dispersion model, which considers only acoustic phonons, should be used with caution for quasi-ballistic transport. It is also  2)), and shown in Fig. 1(a). Ball: without scattering, scatt: with scattering.
observed that with scattering being included, high energy optical phonon modes contribute less to K ph at high temperatures because optical phonons have short MFPs. Figure 5 compares computed thermal conductance of a single pillar and the experimental data. The experimental data are calculated by adjusting the pillar densities to match computed K ph . For the best fit, a sine-type dispersion and pure ballistic model produces a density of 6.4 lm À2 for the 4 nm long and 3.75 lm À2 for the 6 nm long pillars, as presented in Ref. 1. Using the full phonon dispersion and quasiballistic model, best agreement is obtained using a density of 3.7 lm À2 for the 4 nm long and 2.5 lm À2 for the 6 nm long pillars. Note that these values compare better to the measured density (of a different sample) of about 2-3 lm À2 . 19 The important point is that the pillar density extracted from the analysis depends on the dispersion and scattering models used. Also, it can be seen that calculations with full phonon dispersion and scattering exhibits an excellent agreement with the experimental data at around 300 K, for which a good fit is no longer possible with the pure ballistic model. The results, shown in Fig. 5, demonstrate that considering full phonon dispersion and quasi-ballistic transport is important for the quantitative modeling of thermal transport in nanostructures.
In this paper, we showed that a Landauer approach with full phonon dispersion provides excellent fit to measured thermal conductance of the nanopillar over a wide temperature range with reasonable fitting parameters, i.e., pillar density. We demonstrate that although a sine-type dispersion approximation provides a compromise between rigorous full band and simple Debye models, a full band description of phonon dispersion should be considered in the quasi-ballistic regime because optical phonons significantly contribute to thermal conduction. Our analysis shows that nanometer long pillars operate below the ballistic limit except for very low temperatures; they deliver about 60% of the ballistic limit at 300 K. Our analysis shows that the measured thermal conductance of pillars in a wide temperature range up to 300 K provides direct evidence of a transition from ballistic to quasi-ballistic transport. An increase of thermal conductance for temperatures lower than $150 K is a signature of ballistic conductance, while a decrease of thermal conductance at $300 K is attributed to reduction in transmission due to strong diffusive scattering. The theoretical framework presented here is important for a clear understanding of fundamental thermal transport in nanostructures and can be useful as a tool for designing AGHs to enhance performance of thermoelectric devices and integrated circuits where the increasing importance of power dissipation, self-heating, and the management of hot spots necessitates rational thermal design.