Length-dependent thermal transport and ballistic thermal conduction

Length-dependent thermal transport and ballistic thermal conduction Bor-Woei Huang,1 Tzu-Kan Hsiao,1,2 Kung-Hsuan Lin,3 Dah-Wei Chiou,1,4 and Chih-Wei Chang1,a 1Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan 2Department of Physics, National Taiwan University, Taipei 10617, Taiwan 3Institute of Physics, Academia Sinica, Taipei 11529, Taiwan 4Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan

where is k B Boltzmann constant, T m is the transmission coefficient of the mth phonon mode.At low temperatures, Landauer's formula can be reduced to a simple form expressed by fundamental constants, as shown in Eq. (1).According to Eq. ( 1), no dissipation will occur inside a ballistic conductor and thermal resistance only happens at the contacts, resulting in a length-independent thermal resistance (1/K, where K is thermal conductance) effect for L < l.Equivalently, the behavior will result in a linear L-dependence of thermal conductivity (κ) for L < l, too.Furthermore, the length scale at which the thermal transport transits from ballistic to diffusive is generally denoted as the experimentally determined l.
Many experiments have been carried out to determine the l of Si.For example, Goodson's group studied thickness-dependent, in-plane κ of silicon-on-insulator wafer by repeated thermal oxidation and wet etching. 2,3The l was estimated to be 300nm instead of 43nm predicted by a dispersionless theory. 3Liu and Asheghi further studied κ of doped Si films with thinner thickness. 4i et al. studied diameter-dependent κ of Si nanowires and found that the κ decreases with decreasing diameter. 5Alvarez and Jou did theoretical analyses on the size dependent thermal transport phenomena of Si. 6 It should be noted that the above measurements were based on samples with different lateral dimensions and thus did not rigorously obey the operational definition of ballistic thermal conduction described above.Because investigating l by varying the lateral dimensions of a sample would need to adapt additional assumptions on the specular/diffusive boundary scatterings or the validity of Casimir limit, the estimation of l is indirect.In fact, the experimental verification of quantum thermal conduction of Eq. (1) provides a good example that the lateral confinement of the SiN x (200nm wide, 60nm thick) beam does not necessarily affect the ballistic thermal conduction along the beam (∼4µm). 7Rigorously speaking, the experimental data collected from varying lateral dimensions of various Si samples should not be regarded as evidence for ballistic thermal conduction below some characteristic size l.
On the other hand, there have been interests in utilizing optical techniques to probe l's of materials.Here heat is generated after the material absorbs the laser light.Then the temperature variation can be probed by another laser beam based on the temperature-dependence of the refractive index of the material.Recently, various techniques including time-domain thermoreflectance (TDTR), 8,9 frequency-domain thermoreflectance (FDTR) 10 have been employed to study l's.For these optical techniques, heat generation in materials requires absorption of laser power.But the investigated materials may not efficiently absorb light and convert it into heat.Besides, the modulation of refractive index of semiconductors could be mixed with phononic and electronic responses.To solve the problem, Al thin film is commonly employed as a thermal transducer in the experiments.However, it introduces another problem of thermal contact resistances in these optical measurements. 11The thermal transport at the interface is difficult to model, but it certainly plays an important role in enhancing the anisotropy in an otherwise isotropic material. 11The interpretation of optical measurement results without considering the role of interface may also lead to inconsistent conclusions.In TDTR or FDTR, changes of κ's were observed by changing spot sizes or the modulation frequency of the pump laser.The variation of spot sizes primarily reveals the in-plane heat conduction where the modulation frequency indicates the cross-plane phonon transport.Wilson and Cahill recently pointed out the discrepancy of previous results originated from anisotropic thermal transport in-plane and cross-plane. 11Yet deducing the l's is not at all straightforward, as experimental complexities associated with laser spot sizes, laser modulation frequencies, temperature profiles, and interface phonon scatterings are quite involved. 11Moreover, theoretical interpretations of the optical measurement results remain controversial.While it is commonly assumed that the observed deviation to be due to ballistic phonons, ab-initio calculations suggest that agreements on Si are better if simply considering harmonic and anharmonic phonon channels but without directly incorporating ballistic phonons. 12Thus the interpretation of the experimental data is model-dependent and nontrivial.
One should be reminded that the original definition of ballistic thermal conduction is based on the length-dependent thermal transport.Experimentally, the measured system should be kept as simple as possible to minimize unwanted complexities.4][15][16] Alvarez-Quintana et al. studied thickness-dependent cross-plane κ of germanium-on-insulator wafer and estimated the l ∼ 20nm. 13Johnson et al. employed laserinduced transient grating method to study in-plane κ of Si membranes and found the κ deviates from diffusive behavior at several micrometers, suggesting l > 1µm. 15In transient grating, changes of κ's were studied by varying the one-dimensional grating periods in silicon membrane and the experiment was designed to avoid problems of interfaces introduced by metal tranducers. 15The experiment thus reduced the complexities associated with TDTR or FDTR.Recently, we have also recognized the issues mentioned above and directly measured the length-dependent κ of homogeneouslyalloyed SiGe nanowires and heterogeneously-interfaced Si-Ge core-shell nanowires. 14,16Surprisingly, the l of these SiGe nanowires was found to be 5∼8µm, which is nearly 1000 times longer than that predicted by some theories. 17Interestingly, the diameter-dependent of κ of the SiGe nanowires displays a much weaker dependence than those observed in Si or Ge nanowires. 5,14,18The study on the SiGe nanowires also raises a subtle yet important issue on the validity of probing l by changing the lateral dimensions of a sample, as discussed above.
However, carrying out the desired length-dependent thermal transport measurement to determine l is not a simple task in practice.Because sometimes the data were collected from different samples of various sizes, they may inevitably exhibit large uncertainties due to variations of sample quality.Furthermore, some previous works had focused on the discussions on the length-dependent of κ alone, 13 rendering the observed behavior likely to be confused with effects of contact resistance.
Physically, thermal transport measurements usually require connecting leads to a sample, which would often result in finite contact thermal resistances.Thus the experimentally measured total thermal resistance (1/K total ) can be expressed as: where 1/K contact is contact thermal resistance, which can be reasonably modeled as a constant, and 1/K sample is sample resistance, which can be described as by assuming that ballistic behavior takes place and 1/K sample becomes constant when L < l, whereas diffusive behavior dominates and is proportional to L when L > l.One can refine the simple model of Eq. ( 3) and smooth out the kink at L = l by introducing a small but finite characteristic length ∆ for the ballistic-diffusive transition.By using the integral of the logistic function (viewed as a smoothed step function), a smooth function that provides a phenomenological model of 1/K sample can be constructed as If ballistic thermal conductance does not take place at all, we simply have the diffusive behavior 1/K sample = aL for all L > 0, and consequently the thermal conductivity κ ≡ K L/A yields a constant regardless of L. For this reason, the observation that κ decreases with decreasing L is often regarded as a signal of the occurrence of ballistic thermal conductance.This signal, however, could be fallacious, because the decrease of κ with decreasing L may be explained by the presence of a nonzero contact resistance.For finite contact resistance without any ballistic behavior, Eq. ( 2) reads as and consequently we have which leads to κ → 0 as L → 0. To illustrate the fallacy, we plot κ vs. L and 1/K total vs. L using Eq. ( 5) & (6) in Fig. 1(a) & 1(b), with added noises to simulate the experimental data.If one merely focuses on the decrease of κ, the phonon mean free path l could be erroneously identified to be about 7, significantly deviated from the actual value (l = 0).To determine the value of l rigorously, one has to perform regression analysis on the experimental data (of κ or 1/K total ) in response to all range of L, and it turns out Eq. ( 4) serves as a sound regression function.A reliable result of regression analysis not only gives the best-fitted values of l, 1/K Q , and 1/K contact but should also yield stringent standard errors for them.If regression analyses yields l > 0 with a small standard error, ballistic thermal conduction can be claimed.To easily visualize the ballistic thermal conduction, a convenient method is to plot the 1/K total vs. L relation.The 1/K total vs. L relation would allow us to separate the ballistic thermal conduction from the unwanted effect of contact thermal resistance by reading the offset at L → 0, as shown in Fig. 1(b).The ballistic thermal conduction can be seen by observing the deviation from diffusive behavior, extrapolating from the data of large L's (the dashed line of Fig. 1(b)).Although the method may give a conservative estimate on l, it is undoubtedly a more reliable method.
We now reexamine the previous data of Ge and Si, 13,15 whose κ vs. L relation is shown in Fig. 2  do not deviation from the diffusive behavior, there is no observable ballistic thermal conduction of Si or Ge at room temperature.
The reanalysis shown in Fig. 2(b) & 2(d) certainly raises concerns on the validity of our recent discovery of ballistic thermal conduction over 8.3µm for homogeneously-alloyed SiGe nanowires. 14specially, could the data be interpreted as ordinary diffusive thermal conductors with large contact thermal resistance?To answer the question, we plot the κ vs. L and A/K total vs. L on Fig. 3(a) & 3(b).Figure 3(a) shows that the κ increases linearly with L, followed by a slope change at L = 8.3µm, then the κ saturates at 8.2 W/m-K, agreeing with the bulk value of SiGe.Although Fig. 3(a) alone seems to suggest l = 8.3µm for SiGe nanowires, it is necessary to reexamine the data by plotting A/K total vs. L in Fig. 3(b).From Fig. 3(b), A/K total decreases linearly with L for L > 8µm, indicating the expected diffusive transport behavior.Interestingly, the A/K total vs. L changes the slope at L = 8µm and it significantly deviates from the diffusive behavior for L < 8µm.Applying the regression analysis mentioned above gives l = 8.05µm with standard deviation =0.31µm.Moreover, extrapolating the diffusive behavior to L → 0 suggests that the (classical) contact thermal resistance is nearly zero, which indicates that the observed ballistic thermal conduction shown in Fig. 3(a) cannot be attributed to effects of finite contact thermal resistance.We emphasize that the x-axes of Fig. 3(a) & 3(b) are "length" rather than "effective length" and thus the experimental investigations provide the direct probing of l.In addition, experimental demonstrations of quantum contact resistances and non-additive thermal resistance in series have provided independent strong supports to the claimed ballistic thermal conduction. 14Therefore, reexamining the data reaffirms the ballistic thermal conduction for at least 8µm in homogeneously-alloyed SiGe nanowires.][21][22] Similar analyses can be applied to data of heterogeneously-interfaced Si-Ge core-shell nanowires, too. 16However, unlike the homogeneously-alloyed SiGe nanowires discussed above, different heterogeneously-interfaced Si-Ge core-shell nanowires always exhibit distinct thickness of Si-core or Ge-shell.Besides, structural variations, changes of l's, or uncertainties in the contacts in different samples may further complicate the analyses.To minimize the unwanted effects associated with different samples, we thus focus our analyses on a sample investigated by an electron-beam heating technique. 16,23  Extrapolating the data of L > 5µm to L → 0 indicates that the contact thermal resistance is less than 1.6 × 10 6 K/W.Interestingly, 1/K total deviates from the diffusive behavior at L ∼ 2.5µm, indicating l ∼ 2.5µm for the sample (the regression analysis gives l = 2.61µm with standard deviation = 0.25µm).
In summary, we point out that directly investigating one-dimensional length-dependent (normalized) thermal resistance is a better experimental method to determine l.It directly follows the definition of ballistic thermal conduction and avoids many model-dependent complexities commonly exist in previous thermal transport or optical measurements.The 1/K total vs. L (or A/K total vs. L) relation also allows us to separate unwanted effect of contact thermal resistance.In this regard, we have discussed the problems in previous measurements and found that no clear experimental data strongly support ballistic thermal conduction of Si or Ge at room temperature.On the other hand, applying regression analyses to the data of both homogeneously-alloyed SiGe nanowires and heterogeneously-interfaced Si-Ge core-shell nanowires provide undisputed evidence for ballistic thermal conduction over several micrometers at room temperature.

FIG. 1 .
FIG.1.Simulated (a) κ vs. L and (b) 1/K total vs. L relations for 1/K contact = 1.Here some noises are added to the data to simulate the experimental observation.It can be seen that finite contact thermal resistance will create a pseudo ballistic thermal conduction effect in the κ vs. L relation even if the sample is a diffusive conductor.
FIG. 2. (a) κ vs. L and (b) A/K total vs. L relations for Ge film.The A/K total vs. L relation follows the diffusive behavior within the error bars for L > 42nm, as indicated by the gray dashed line.The data are reproduced from Ref. 13.The open symbols denote the bulk values of Ge.(c) κ vs. L and (d) A/K total vs. L relations for Si membranes.The A/K total vs. L relation follows the diffusive behavior for L > 1µm, as indicated by the gray dashed line.The data are reproduced from Ref. 15.Here the lengths are half of the grating period shown in Ref. 15 and the y-axes are normalized by the bulk values.

FIG. 3 .
FIG. 3. (a) κ vs. L and (b) A/K total vs. L relations for homogeneously-alloyed SiGe nanowires.Both κ vs. L and A/K total vs. L relations deviate from the diffusive behavior for L < 8µm, indicating the ballistic thermal conduction.The gray dashed line is a fitted line extrapolating from data of L > 8µm.The data are reproduced from Ref. 14.
Figure 4(a) displays κ vs. L for a Si-Ge core-shell nanowire of diameter ∼150nm.The slope of the κ vs. L changes at L = 5µm, which seems to indicate l = 5µm.Because the measurement was conducted on the same sample, we can plot 1/K total vs. L (instead of A/K total vs. L) in Fig. 4(b).

FIG. 4 .
FIG. 4. (a) κ vs. L and (b) 1/K total vs. L relations for a heterogeneously-interfaced Si-Ge core-shell nanowire (of diameter ∼150nm) investigated by an electron-beam heating method.It can be seen that both κ vs. L and 1/K total vs. L relations deviate from the diffusive behavior for L < 2.5µm, indicating the ballistic thermal conduction.The gray belt is a fitted line extrapolating from data of L > 5µm.The data are reproduced from Ref. 16.