Scholarly article on topic 'Thermal transport in 2- and 3-dimensional periodic “holey” nanostructures'

Thermal transport in 2- and 3-dimensional periodic “holey” nanostructures Academic research paper on "Nano-technology"

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Academic research paper on topic "Thermal transport in 2- and 3-dimensional periodic “holey” nanostructures"

Thermal transport in 2- and 3-dimensional periodic "holey" nanostructures

J. Ma, J. S. Sadhu, D. Ganta, H. Tian, and S. Sinha

Citation: AIP Advances 4, 124502 (2014); doi: 10.1063/1.4904073 View online: http://dx.doi.Org/10.1063/1.4904073

View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/4/12?ver=pdfcov Published by the AIP Publishing

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Thermal transport in 2- and 3-dimensional periodic "holey" nanostructures

J. Ma, J. S. Sadhu, D. Ganta, H. Tian, and S. Sinhaa

Department of Mechanical Science and Engineering, University of Illinois, Urbana-Champaign, Urbana, IL, USA

(Received 15 October 2014; accepted 2 December 2014; published online 9 December 2014)

Understanding thermal transport in two- and three-dimensional periodic "holey" nanostructures is important for realizing applications of these structures in thermo-electrics, photonics and batteries. In terms of continuum heat diffusion physics, the effective medium theory provides the framework for obtaining the effective thermal conductivity of such structures. However, recently measured nanostructures possess thermal conductivities well below these continuum predictions. In some cases, their thermal conductivities are even lower than predictions that account for sub-continuum phonon transport. We analyze current understanding of thermal transport in such structures, discussing the various theories, the measurements and the insights gained from comparing the two.© 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/L4904073]

I. INTRODUCTION

When compared against charge transport, thermal transport occurs over a limited range of conductivity across materials. Whereas electrical conductivity at room temperature varies over ~30 orders of magnitude between materials, thermal conductivity at room temperature varies over only ~5. Expanding the range of thermal conductivity can dramatically improve myriad engineering applications, especially if this is combined with desirable mechanical, optical and/or electrical properties. Nanostructuring is an example of a strategy that can engineer the thermal conductivity of a material over orders of magnitude.1 Within nanostructuring strategies, a recent and intriguing concept is that of periodically "holey" structures.2-5 Such a material is composed of a host material from which material is removed in the form of spatially periodic holes to create macroscopic arrays of microscale or nanoscale holes in two- and three-dimensions. The array of holes may be compared to the lattice of a crystal. The concept originates in optics where the term photonic crystal6 describes such structures with engineered optical properties. Modification of optical properties in these materials can generally be understood by applying the framework for the band theory of electrons in a crystal to light propagation under spatially periodic variation of optical properties. Success with modifying optical and later acoustic7 properties in this manner has prompted the question whether such periodic structures can be used to engineer thermal transport as well. In this paper, we analyze experimental and theoretical work that shed light on this question and discuss future directions. Key differences in the nature of light and sound propagation on one hand and heat flow on the other raise the obvious question whether heat flow can in fact be engineered using an idea borrowed from light propagation. A fundamental issue related to this question and much discussed in the recent literature is whether phonon transport at room temperature possesses any coherent wave-like characteristics that can be engineered. We discuss the issue of coherent phonon transport in detail.

A term "phononic crystal"5,7 is prevalent in the literature and is sometimes considered an umbrella term for describing both acoustic and thermal behavior. In our opinion, there are significant differences between sound propagation and heat flow that limits the utility of such an umbrella term.

Corresponding author: sanjiv@illinois.edu

2158-3226/2014/4(12)/!24502/17 4,124502-1 ~ " iilli if J nil \

In a crystalline semiconductor, sound and heat are both due to vibrations of the lattice; however, they are distinct in terms of the frequencies involved. Sound involves vibrations at low frequencies (typically kHz-MHz) whereas heat flow at room temperature primarily involves THz frequency vibrations. For the purpose of this paper, we consider the term phononics to refer to the engineering of materials into domains of dissimilar acoustic properties (density p and sound speed c) in order to control the propagation of sound in these materials. Sound propagation in 'phononic crystals' occurs in the audible range (20 Hz-20 KHz) as well as up to hypersonic frequencies (> GHz).7 The periodic modulation of acoustic properties in these structures prevents the propagation of specific phonon modes of wavelengths A that are comparable to the period of the structure, a. Prohibited modes with A~ 2a undergo coherent Bragg reflection and fall in the 'phononic band gap (PBG)'. Creating periodic structures with length scales from centimeters to sub-micrometers controls sound propagation from kHz to GHz respectively.7 Such "phononics" is excluded in our consideration.

Here, our focus is entirely on thermal transport where the primary physics is the cumulative, typically incoherent, transport of energy over a broadband range of phonon frequencies up to several THz. The search for improved thermoelectric materials primarily motivated initial investigations on heat flow in periodic "holey" structures. Most of the theory8-11 and experiments2-5,12 target silicon in particular. Hence, silicon is the material of focus for this paper. Figure 1(a) shows the schematic of a two-dimensional periodic holey nanostructure. Phonons conduct heat in such a structure. As an illustration, Figure 1(b) shows the computed atomic displacements corresponding to a specific phonon mode in such a periodic nanostructure. Three-dimensional periodically holey nanostructures form another interesting class of materials. Figure 1(c) shows the structure of an "inverse opal". Such three-dimensional periodic materials are currently under investigation for applications in optics as well as battery electrodes. They also represent an interesting direction for simultaneously engineering thermal with mechanical (elastic) properties.

In this paper, we first discuss fundamental thermal transport in terms of the physics of phonon propagation. We next examine theory for describing thermal transport in two- and three-dimensional periodically "holey" nano- and microstructures. We compare these against experimental data from the literature and attempt to summarize the understanding so far as well as provide additional insight. While phonon coherence is in general not a useful concept in describing thermal transport at room temperature, much of the attention on thermal transport in periodic "holey" structures centers round the issue of coherence. We thus pay special attention to the debate on coherent versus incoherent transport. Fabrication of periodic structures can in practice introduce aperiodicity into the structures. Thus, we discuss implications of aperiodic structures on transport. Finally, we discuss future directions for research on "holey" periodic structures for engineering thermal transport.

II. SCALES AND FIGURES FOR FUNDAMENTAL THERMAL TRANSPORT IN SILICON

In this section, we present, in brief, scales and figures associated with fundamental thermal transport in bulk silicon. These numbers are particularly relevant for understanding transport in the

(a) (b) (c)

(->100 nm)

FIG. 1. (a) Schematic of a two-dimensional periodic holey nanostructure with subcontinuum phonon transport. (b) Atomic displacements corresponding to a phonon mode in a 2-D holey silicon. (c) Schematic of a three-dimensional periodic silicon inverse opal structure.

more complex holey nanostructures. Silicon at room temperature, has a thermal conductivity of 148 W/mK and a thermal diffusivity of 0.89 cm2/s. Phonons are the dominant carriers of heat in a dielectric. Emerging work suggests that phonon mean free paths in silicon at room temperature span a broad range from ~10 nm to ~10 ^m1315 Dominantly contributing wavelengths span the range ~0.5-5 nm.13 A majority of acoustic phonon modes possess lifetimes in the range ~10-500 ps. Acoustic phonons are the dominant carriers at room temperature, conducting ~95% of heat. While the relative contribution amongst different phonon polarizations has been much debated in the literature, recent ab initio molecular dynamics calculations suggest that this is unimportant with each of the 3 polarizations contributing approximately equally at room temperature. In almost all experiments on periodic silicon nanostructures reported till date, feature sizes are chosen to be less than 1 ^m to alter the transport of long mean free path phonons. Phonons with mean free paths >1 ^m may contribute as much as 50% to the thermal conductivity of silicon at room temperature.13

III. MODELING APPROACHES FOR PERIODIC HOLEY STRUCTURES

In this section, we discuss different levels of theory for considering thermal transport in periodic holey structures, starting from continuum heat diffusion and culminating in sub-continuum incoherent and coherent transport of phonons respectively.

A. Continuum considerations

Transport properties in a multiphase system can be very different from that of each of the constituents. There are several models to describe the relationship between effective properties and intrinsic properties of each material, among which the simplest are the parallel and series resistance network models.16 These yield the theoretical upper and lower bounds on effective transport properties. They apply to the ideal situation where the two materials are stacked on each other and the direction of transport is either parallel or perpendicular to the material. If the conductivities of the two materials are k1 and k2(k1 > k2), the effective conductivities given by each model are

= (1 - ф)Ь + #2 (1)

1 - Ф + Ф I-1

respectively, where 0 is the volume fraction of the second material. In practice, the structure is not physically realistic and the bounds are quite far apart to make reasonable estimations.

A more realistic and narrower bound approach is the geometry-independent Hashin and Shtrik-man model,17 which applies to one material embedded into a continuous host material. The upper and lower bounds of the effective conductivity are

ku - k1 +_30k1(k2 - k1)__(3>

keff = k1 + 3k1 + (k2 - k1)(1 - 0) (3)

kff-k2+ <4>

This model works well if the contrast between k1 and k2 is not large. For instance, if the ratio r-k2/k1 is 0.2, taking the geometric mean of the upper and lower bounds gives a reasonable estimation with maximum uncertainty of ±11%. However, for a porous system where the second phase is air or vacuum, the lower bound becomes zero and remains far apart from the upper bound. The geometric mean is therefore not an accurate estimation of the effective conductivity in this case.

The well-known effective medium theory (EMT), originally proposed by Maxwell,18 takes better account of geometric effects. In the model, the temperature field perturbation due to many small inclusions is thought to be the same as one big inclusion, for which the effective conductivity is then calculated. The shape of the imbedded material is generalized as a spheroid and a is the ratio between the length of unequal axis and that of one of the equal axes. It is found that a system with sphere inclusions (a-1) represents the upper limit of the H-S model. If these inclusions are voids,

the H-S model reduces to the well-known Maxwell-Garnett (MG) model

(1 - 0)ko

11 1 + 0/2

where k0 is the conductivity of the host medium. When the inclusions are thin-disk like (a^0), the EMT predicts that the effective conductivity is the lowest and coincides with the lower bound of the H-S model. Another special case applies to a 2D system with infinitely long cylindrical voids where

k (1 - 0)ko keff = • (6)

which lies in between aforementioned bounds. Therefore EMT does not narrow the bounds by H-S model, but provides a more accurate estimation if the shape of the inclusions is available. Predictions from the linear model and the two MG models for a porous structure are shown in Figure 2.

In general, EMT provides a good estimate at small 0 or when the contrast between the two materials is small. However, the prediction for a system with large porosity and/or big contrast between materials is not reliable. A more intensive but accurate approach is to solve the heat diffusion equation to obtain the effective conductivity for a specific periodic structure. Albrecht et al.19 have formulated a boundary-integral method for calculating the conductivity of structures with two- and three-dimensional periodic lattices.

Figure 2 also shows the effective conductivities of a 2D square lattice with cylindrical voids, an opal20 and an inverse opal21 structure respectively. Unsurprisingly, the curve for the 2D square lattice agrees well with the MG model for 0 as large as 0.5. Beyond this, it deviates sharply until it finally tends to 0 at 0=0.785, at which porosity the system becomes discontinuous. The behavior is different in an opal structure, which is an FCC array of overlapping spheres. When 0 is smaller than 0.1, the conductivity approximately follows the MG model. However, for increasing 0, the

FIG. 2. The effect of porosity on thermal conductivity from different continuum models including the notable effective medium models. The calculations by boundary-integral method for FCC opal, 2D square lattice (cylindrical voids) and inverse opal are shown for comparison.

overlapping volume between adjacent spheres becomes smaller, creating a bottleneck effect. The conductivity goes to zero when 0=0.26, at which the spheres disconnect from each other.

Another interesting structure is an inverse opal, which can be made by filling the pores inside an opal structure followed by removal of the original opal template. It is a highly porous structure as the to-be-removed opal template occupies 74% of the volume. Surprisingly, the trend for the conductivity of an inverse opal is very similar to the predictions of the MG model. In fact, the prediction by the MG model overestimates the exact solution by only ~10% for the realistic porosity range. The accuracy results from the fact that an inverse opal is effectively a thin film network, where all of the material comprising the inverse opal contributes similarly to the resistance. In contrast, in an opal structure, the dominating resistance arises in the overlapping portions of spheres and the remainder of the material has markedly lesser contribution to the resistance.22

B. Random walk in periodic structures

While an effective thermal conductivity can be readily estimated from measurements through use of results from continuum theory as discussed in Sec. III A, it is not clear at what feature size this approach is no longer justified. Sofo and Mahan22 have considered the problem of a classical particle diffusing inside a periodic structure when the mean free path is still smaller than the period but within an order of magnitude. By integrating out random walks inside individual blocks and instead focusing on diffusion from block to block through the interconnecting necks, they elegantly reduced the problem to diffusion in a lattice but with residence time at each site. In this manner, they derived diffusion coefficients for 1-D, 2-D and 3-D periodic structures. For a random walk in 3-D, it is well known that the diffusion coefficient, Do in the bulk is related to average speed of the particle, v and its mean free path, A as Do = 1/6vA. Sofo and Mahan show that the coefficient 1/6 appearing in this relation is reduced by >50% in a face-centered cubic opal when the diameter of the characteristic sphere is a few times (< 10) the mean free path. This implies that even for phonon modes with mean free paths smaller than the feature size, we may expect a reduction in effective transport. When the mean free path is comparable to the feature size, sub-continuum rather than continuum transport is expected to play a significant role. This is discussed in the next section.

C. Sub-continuum phonon transport

The discussion so far has implicitly assumed that feature sizes are much larger than the phonon mean free paths. As mentioned in Sec. II, phonon mean free paths larger than 1 ^m contribute significantly to heat conduction.1315 In structures with sub-micrometer feature sizes (film thickness and/or pitch of holes), phonon transport will be modified from the bulk. Clearly, at a small enough feature size, the distribution function for phonons transporting heat in the structure would depend on the specific geometry of the structure and would not be the same as in the bulk. The most detailed consideration of transport in periodic structures till date involves modeling phonons through either atomistic simulations23-25 or tracking their trajectories through Monte Carlo simulations.9,10 We review basic phonon scattering physics here and discuss specific calculations later in Sec. VI.

Phonons in a periodic structure will be scattered at the boundaries of the crystal and if present, at grain boundaries, by other phonons and from impurities. For bulk silicon at room temperature, the phonon MFP is dominated by three-phonon Umklapp scattering. The surface boundary scattering is relatively more influential at low temperature, where three-phonon processes are substantially suppressed. The Casimir model26 for boundary scattering assumes complete thermalization of incident phonons at the boundaries of a crystal to predict a phonon mean free path (MFP) comparable to the crystal dimensions. It is illustrative to consider rates associated with various phonon scattering mechanisms. Scattering rates available in the literature27,28 for mass difference (isotope) and Umklapp scattering are of the form r,m1= Am4 and T~1=BTu2e~C/T. The value of A depends on the isotope concentration and we assume the commonly used value,29 A=1.32x10-45 s3. The values of B, C can be determined by recursively fitting the high temperature thermal conductivities of bulk silicon and nanostructures, and are found to be B=1.6x10-19 s/K, C=152 K. Using Casimir's

approach, the boundary scattering rate is T-1=v/(Fd), where v is the phonon speed, F is a geometric factor, and d is the characteristic dimension.

In polycrystalline silicon, grain boundary scattering is usually treated as a frequency independent process, in which the grain size is the limiting dimension.30 However, recent measurements show that the scattering process due to grain boundary is a rather complicated mechanism.31,32 Different approaches have been proposed to fit the corresponding measurement values. Here we adopt a scattering rate that explains the low temperature behavior of thermal conductivity of silicon inverse opals:32

-1 = 4 ^ (Av/v )2 (7)

Tg 3 v l

where t is the thickness of intergrain region, l is the average grain size, Av is the change in velocity at the boundary. The origin of this scattering process will be explained in section VII.

In Figure 3 we plot all the empirical scattering rates as a function of frequency. To address the effect of temperature, Umklapp scattering is plotted at 30 K and 300 K. We use d=l=100 nm for both surface and grain boundary scattering. Overall, surface scattering dominates at low frequencies and grain boundary scattering dominates at higher frequencies. The contribution from grain boundary scattering is always significant except for very low frequencies. We note that Umklapp scattering rates are always smaller than the dominant rates for feature size at 100 nm, even at 300 K.

Consideration of phonon transport in periodically porous nanostructures is complicated due to geometry. First attempts in this direction11,33,34 simplified the treatment of phonon scattering while focusing on solving the Boltzmann equation in the challenging geometry. The main conclusion from these studies was that sub-continuum transport is indeed an important factor to consider in evaluating thermal conductivity of these structures. Atomistic simulations23-25 provided more insight into the role of various phonon scattering mechanisms, particularly surface scattering, and

cr) <u

-I—>

CC 108

A—' -1—>

1 1 1 Surface 1 i | i | i | i | i Grain (f1~a?)

^____-" "■* Umklapp (300 K)____,____-

if Mass Difference

l.i. Umklapp (30 K) i . i . i . i . i

0 1 2 3 4 5 6 Phonon Frequency, co/27i (THz)

FIG. 3. Frequency dependent phonon scattering rates calculated for silicon. Rates for Umklapp scattering are shown at 30 and 300 K. The characteristic dimension for both surface and grain boundary scattering is 100 nm. Adapted with permission. Copyright 2013 American Chemical Society.32

suggested that it was possible to obtain thermal conductivity close to the amorphous limit with truly nanoscale features. More recent theory,9,10 following the publication of data on nanostructures, has revisited the Boltzmann transport simulations while considering frequency dependent scattering rates. These are discussed further in Sec. VI.

D. Coherent phonon transport

Coherent phonon transport refers to the condition where the relative phase of phonons plays a role in the transport process typically introducing wave effects such as interference or localization.35 While similar transport has been investigated extensively in the case of electrons36 and photons,37 only a relatively small amount of work has focused on phonons, notably in core-shell nanowires.38-40 An important concept in describing coherent transport of energy is the coherence length, Lc. This concept has origins in optics where the coherence of a light beam is related to the coherence in the source. In the framework of quantum mechanics, the simplest interpretation of coherence length for a quantum particle is that Lc is the spatial spread of the wave packet representative of the particle; its estimation, however, differs between different particles and between different prevalent scattering mechanisms. For example for electrons in a metal, Lc is typically the product of the Fermi velocity and the phase relaxation time. The phase relaxation time is typically shorter than the momentum relaxation time that defines the electron mean free path. However, it is possible to have situations where the phase relaxation time exceeds the momentum relaxation time i.e. the phase is conserved over several collisions even as the direction of momentum is randomized. In this case the coherence length is related to the diffusivity of the electron.

In the case of photons, one interpretation is that Lc~c/Aw where c is the speed of light and Am is the bandwidth of waves. A ray of electromagnetic waves comprising a stream of photons is a series of wave packets emitted from individual emitters at the source. Each wave packet then has a coherence length and can interfere with itself. For (incoherent) thermal radiation, the effective bandwidth arises from the Heisenberg uncertainty relation as the energy spread kBT divided by the Planck constant, h. This bandwidth is ~6 THz at room temperature. Replacing the speed of light with the speed of sound, vsound, the coherence length for thermal phonons is then Lc~ vsound h/kBT. When estimated in this approximate manner, Lc <10 A at room temperature. Recent calculations41 of the frequency dependent coherence length for phonons in silicon consider coherence to be the spatial correlation of the atomic displacement fluctuations at equilibrium. Using the Stillinger-Weber potential, the calculated coherence lengths are larger than the figure for Lc from above and can be as high as several tens of nanometers for THz frequency phonons.

There is considerable debate on the role played by coherence. The effect may be any combination of opening of band gaps, alteration in the density of states, reduction of group velocities or localization of modes. Significant band gaps are necessary in the THz frequencies to affect thermal transport and it is unlikely that current feature size of ~100 nm or larger are sufficient at creating such gaps. Change in density of states and a reduction in phonon velocities is also much debated. We further discuss transport theory related to coherence in Sec. VI.

IV. EXPERIMENTAL RESULTS FOR 2- AND 3-D PERIODIC STRUCTURES

In this section, we discuss experimental results for thermal transport in 2-D and 3-D periodic holey nanostructures. We briefly discuss the methods for fabrication as well as the measurement techniques and finally summarize key data reported till date.

A. Two-dimensional structures

In this section we present an overview of the methods reported for fabricating 2-D periodic holey silicon2-5,12 and discuss techniques used in measuring thermal conductivity. In the first experimental work on thermal transport in periodic silicon structures, Song et al.2 used a silicon-on-insulator (SOI) wafer to fabricate a silicon membrane resting on buried silicon oxide. The top surface was coated

with Si3 N4 prior to membrane preparation and the buried oxide served as the etch stop. Finally, micrometer scale pores were patterned on the silicon membrane using photolithography followed by reactive-ion-etching (RIE) on the nitride film and deep RIE to create micrometer scale pores on the silicon membrane. In more recent work that has received much attention, Tang et al.3 used two methods to fabricate holey silicon from an SOI wafer. The first method is nanosphere lithography using polystyrene spheres followed by DRIE etching of silicon using a chromium mask. The second method is based on block copolymer assembly to define features. Nanomesh films fabricated by Yu et al.,4 also from SOI wafers, used a well-known superlattice nanowire pattern transfer (SNAP) technique. Periodic structures fabricated at Sandia used a focused ion beam to create features.42,43 The SOI wafer was first patterned and then plasma etched to release trenches in silicon. The buried oxide was removed using HF vapor. Finally, Marconnet et al.12 used electron-beam lithography to fabricate a one-dimensional periodic holey structure.

Both steady-state and transient heating techniques have been used to measure temperature dependent thermal conductivity. Song et al.2 used the steady-state Volklein technique44 to measure in-plane thermal conductivity of thin films. Here a metal heater and thermometer are placed at each end of the suspended silicon thin film. They calculated the film thermal conductivity by measuring the temperature difference and estimating the heat flux along the film. Accurate estimation of heat flux requires knowledge of thermal conductivities of other materials in the device. More recent in-plane measure-ments3,4 employed suspended heating and sensing platforms exploiting a technique developed by Li Shi et al.45 Marconnet et al.12 used a metal line as both heater and thermometer to measure the thermal conductivity of suspended silicon membrane beneath it. The device was constructed such that a simple 1D conduction model is valid. One obvious advantage is the precise determination of heat flux. For cross plane thermal conductivity determination, Hopkins et al.5 used the time-domain thermoreflectance technique, in which a pump laser introduces a temperature perturbation and a probe laser detects the change in thermoreflectance.

We now summarize data on the thermal conductivity of 2D periodically holey silicon. Sec. VI presents a detailed discussion of the data and discusses various interpretations. The data cover porosities up to ~40% with the limiting feature size spread over a broad range from ~15 nm to 10 ^m. Measured in-plane thermal conductivities at room temperature range from ~100 W/mK at 10 ^m feature size to ~ 2 W/mK at ~15 nm limiting features. Cross-plane thermal conductivities are ~ 10 W/mK for ~500 nm limiting feature size. The temperature trend shows distinct Umklapp peaks when limiting features > 1 ^m. These peaks are noticeably absent in the data for smaller limiting features (< 100 nm).

B. Three-dimensional structures

Three-dimensional periodic dielectric structures have distinctive optical properties46-48 and are of great interest in designing new functionalities in optoelectronics. They may also possess promising thermoelectric properties.49 In comparison to the top-down fabrication of 2D periodic structures, the fabrication of an ordered 3D structure is not as straightforward. Usually the fabrication uses a bottom-up approach that incorporates a self-assembled 3D structure. The opal structure is one example but has rather limited material options. It has also been shown that the negative replica of an opal give rise to better photonic band gap properties. Using opal as a template, one can deposit a variety of materials into the interstices to form the inverse structure. This enables applications beyond optics. For instance, carbon inverse opal, as a battery electrode, shows superior transport properties and mechanical integrity.50

Ma et al.32 fabricated 3D periodic silicon nanostructure in the form of inverse opals.21 The opal template is comprised of silica spheres made from the Stober method.51 Amorphous silicon is deposited by CVD and recrystallized to produce polysilicon. Finally, BOE etches away the silica template, releasing the inverse structure. The periodicities and shell thicknesses are in the range 420-900 nm and 18-38 nm respectively in their measurement. Ma et al. employed the 3m method52 to measure the thermal conductivity as a function of temperature from 15-400 K. During the measurement, a sinusoidal current (at 1m frequency) through a metallic heater sets up a 2m temperature oscillation, which in turn creates a 3m voltage across the metal line. By measuring the 3m voltage,

the thermal property of the sample material underneath can be extracted. The results show low effective thermal conductivity of the inverse opal structure, ~0.6-1.4 W/mK at 300 K which arise due to macroscopic bending of heat flow lines in the structure. After accounting for porosity as discussed in Section III A, the intrinsic material thermal conductivity is ~5-12 W/mK. This data is discussed in depth later in Sec. VI.

V. TRANSPORT IN NON-PERIODIC MESO- AND NANO-POROUS STRUCTURES

Porous silicon is one of the most studied meso-porous nanoporous structure. It is usually fabricated by electrochemical etching in aqueous or ethanoic HF solution.53,54 The morphology of porous silicon is generally classified by its pore size as nanoporous (<2 nm), meso-porous (2-50 nm) and macro-porous (>50 nm).55 In this section, we present an overview of thermal conductivity in porous silicon. The data are important in that they represent the limiting case for periodically holey structures that are essentially porous structures with ordered pores.

The first reported value of thermal conductivity for porous silicon at room temperature is 1.2 W/mK56 for nano-porous silicon (40% porosity). In the same work, the thermal conductivity of as-prepared meso-porous silicon (45% porosity) was measured to be 80 W/m K which dropped to 2.7 W/m K upon oxidation at 300°C. Temperature dependent measurements in the range of 35 K-320 K using the 3ro method were first reported by Gesele et al.57 The thermal conductivities of all the investigated samples increased with increasing temperature, and were less than 1W/ m K at room temperature. A minimum value of 0.03 W/m K was reported for p-type porous silicon with 89% porosity and 4.5±0.6 nm crystallite size. The ultra-low thermal conductivity in nano/meso-porous silicon is mainly attributed to strong phonon confinement and scattering at the crystallite boundary. Further, porosity also plays an important role in reducing the effective thermal conductivity, which can be explained by various effective medium theories mentioned in previous section. Theoretical approaches to model phonon transport in porous silicon include, for example, solution of the BTE by the discrete ordinate method,33 3D Monte-Carlo simulations,58 molecular dynamic simulation23,24 and a combination of analytical and phonon-tracking methods.59 It remains a challenge to clearly understand thermal transport in a non-periodic porous structure.

VI. DISCUSSION ON 2-D PERIODIC STRUCTURES

Figure 4 plots data for thermal conductivity of 2-D periodically holey silicon at room temperature. It is illustrative to first compare against predictions from continuum heat diffusion discussed in Sec. III A. Figure 4(a) plots the data as a function of the porosity and compares it against the Maxwell-Garnett model with cylindrical voids. The Maxwell-Garnet formula predictions exceed the measured thermal conductivities in all cases. Since this holds across different samples, feature sizes and measurement techniques, it is reasonable to assume that the explanation does not lie in measurement error. Rather the comparison suggests that sub-continuum phonon transport is indeed a factor in all samples.

As discussed in Sec. III B, the main difference between bulk and periodically holey silicon in terms of incoherent phonon transport, is in the relative importance of surface scattering. In the Casimir limit, the mean free path associated with surface scattering is the limiting dimension of the crystal. Figure 4(b) plots the intrinsic material thermal conductivity against the limiting dimension. A prediction of thermal conductivity assuming boundary scattering at the Casimir limit of completely diffuse scattering is also plotted. We obtain the Casimir limit by following Mingo's modified Callaway model.60 In assessing boundary scattering, we choose the geometric factor F to be two limiting cases: F=1 for cylindrical wire, and F=4 for thin film geometry.61 For large dimensions (> 200 nm) the data are close to the Casimir limit. However, for limiting dimensions below ~200 nm, the data fall significantly below the Casimir limit. It is worthwhile to note that a similar trend exists for nanostructured silicon in general and is not just special to periodic nanostructures. For example, a similar plot of thermal conductivity versus limiting size for silicon thin films and/ornanowires would reveal the same trend. As feature sizes shrink below 100 nm, there is greater discrepancy between Casimir-limited

-t—» 4—•

ZJ T3 £Z

l| 0) >

140 120 100 80 60 40 20 0

K 1 1 \ MG (cylindrical voids) A 1 Song and Chen

► Hopkins ef al.

Marconnet etal. -

■ Yu etal.

• Tang et al.

♦ Kim et al.

v O Jain etal.

♦ ♦

■ . « * , •

Porosity

4—»

Z3 "O £Z

......... ....... A Song and Chen ► Hopkins et at. « Marconnet et al. . ■ Yu ef al. - • Tang ef al. _ - ♦ Kimefai'' U 1 i i i i i i i Casimir Limit ▲

■ '' ►►

100 1000 10000 Limiting Dimension (nm)

FIG. 4. (a) The room temperature effective thermal conductivity of various 2-D periodic holey silicon nanostructures as a function of porosity. Measurement values reported as intrinsic conductivity are converted to effective conductivity using Eq. (6). Open symbols correspond to theory and closed symbols to experimental data. The MG model is shown for comparison. (b) Summary of room temperature intrinsic material thermal conductivity data of 2-D periodic structures plotted against the limiting dimension for surface scattering. Measurement values reported as effective conductivity are converted to intrinsic conductivity using Eq. (6), except for the cross-plane measurement (Hopkins et al.), which is converted using solid fraction. The corresponding Casimir limit is shown for comparison.

and the measured thermal conductivities. However, it is not clear at present whether this trend has origins in the measurements themselves. Fabrication of features at the smaller end is clearly challenging and the presence of subtle defects such as at the boundaries is difficult to detect as well as rule out. Characterizations of the sample other than thermal conductivity measurements, such as Raman scattering and high-resolution transmission electron microscopy, are not widely reported.

In the absence of detailed characterization beyond thermal measurements, various mechanisms have been suggested to explain the anomalously low thermal conductivities. These can be categorized into two: coherent and incoherent transport. A key hypothesis in coherent transport is that the periodic structure induces Bragg diffraction of phonons, opening up band gaps.4,5 While this may be expected at low temperature where the coherence length associated with long wavelength phonons can be large, this is inconsistent with the expected thermal coherence length of Lc~1 nm at room temperature. We note that very recent calculations of the frequency dependent coherence lengths

yield figures an order of magnitude larger. Partially coherent transport is another possibility where a part of the phonon population with coherence lengths comparable to or larger than the feature size exhibits coherent transport and phonons with shorter coherence lengths undergo incoherent diffusive transport. The arguments supporting incoherent transport rely on surface scattering with or without disorder as the mechanism responsible for reduced thermal transport. We first discuss the possibility of coherent transport in detail and then discuss incoherent transport theories from the literature.

Since phonon transport at room temperature is broadband, only modes whose coherence length is comparable to the feature size may possibly exhibit coherent effects. A simplistic estimate of the coherence length for thermal phonons was discussed in Sec. III whereby Lc~ vsound h/kBT. We note however, that the coherence lengths of individual modes with long wavelengths could be much larger than this figure. The key question is whether such modes have an appreciable impact on thermal transport. Recent work points both ways, further confusing the issue. Marconnet et a/.12 have argued that even in the extreme case that the coherent part of the spectrum corresponding to wavelengths comparable to the spacing does not contribute to thermal conductivity (i.e. coherent modes have zero transmission), the reduction in thermal conductivity at room temperature is negligible. They suggest that coherent transport may play a role only at much lower temperatures and would require periods and features on the order of tens of nanometers. On the other hand, Dechaumphai and Chen62 have reported simulations of partially coherent transport to show good agreement with the data of Yu et a/. even for the very low thermal conductivities (~ 2 W/mK). A key difference between the two is the criterion to decide coherent character. In the first case, coherence is assumed for modes with wavelengths comparable to or longer than the feature size. In the second case, coherence is assumed for modes whose Umklapp scattering limited mean free paths are larger than the feature size. The separation of modes into coherent and incoherent is arbitrary in both cases but critically affects the answer.

On the incoherent side, one explanation is the "necking" effect8 where phonons with mean free paths larger than the size of the "necks" connecting the pores cannot transport heat as effectively as in the bulk. This is, in principle similar to the physical picture of particle diffusion in a lattice with residence time, discussed in Sec. III B. However, sub-continuum phonon transport rather than diffusive random walk is considered. The predictions from the model are larger than the measured conductivities. However, effects such as disordered pores, variance in pore size, surface disorder and roughness have not been considered and may serve to further reduce thermal conductivity to the observed values. Recent Monte Carlo simulations of the phonon Boltzmann transport equation using a mean free path sampling technique9 yield an excellent match with the measured conductivities for limiting dimensions > 100 nm. Results from the simulations are included in Figure 4(a). This provides strong support to the "necking effect" in incoherent transport as an important factor behind the reduced thermal conductivities.

However, the results for the smaller features still remain puzzling. It is illustrative to consider the effect of feature size in the extreme limit of nanometer features. Figure 5 shows lattice dynamics calculations of "nanomesh" silicon with a square mesh of periodicity ~4.3 nm. The geometry is shown in Figure 5(a) and the phonon dispersion along two high-symmetry directions is shown in Figure 5(b). The interatomic potential used in the calculation includes two- and three-body terms and has been used previously60 in modeling the thermal conductivity of silicon nanowires. When comparing between different mesh sizes, these computations suggest that features smaller than ~20 nm are necessary to see any significant changes in both the density of states and the speed of sound. Recent measurements63 on periodic holey structures in silicon nitride membranes with a limiting feature size of ~60 nm, suggest that phonon transport is suppressed through coherent effects at temperatures < 1K. Interestingly, corresponding simulations attributed the reduced transport not to the opening of band gaps but rather to reductions in group velocities. Similar reduction in group velocities may possibly be a factor in the nanomesh. Figure 5(c) compares the reduction in group velocity of long wavelength transverse acoustic phonons from bulk silicon to a ~4.3 nm diameter nanowire to a nanomesh with ~4.3 nm periodicity and 2.2 by 2.2 nm voids. In the nanomesh structure, there is indeed an absolute band gap but at a very high frequency between ~10.5 and 11 THz. This is unlikely to affect thermal transport at room temperature and below.

FIG. 5. (a). Schematic of a two-dimensional nanomesh silicon with square holes. The dimensions are shown in the figure, where a=0.543 nm is the lattice constant of Si. (b) Phonon dispersion along two high-symmetry directions, a band gap appears above 10 THz. (c) Reduction in group velocity of long wavelength transverse acoustic phonons in nanowire and nanomesh silicon. The diameter of nanowire and the periodicity of nanomesh are both ~4.3 nm.

Further, significant reductions in group velocity occur only at extremely small features that are not relevant to the experimental structures under discussion.

While the effect of coherence at room temperature may be difficult to resolve without additional data and characterization, the temperature trend of existing data may provide clues on whether coherence is indeed a factor at low temperatures. The frequency dependence in the mean free paths should be different between coherent and incoherent transport leading to different temperature dependencies at low temperatures. Figure 6 plots representative thermal conductivity data versus temperature. For large features such as in the work of Song and Chen,2 a modified effective medium model that combines an analytical solution to the Boltzmann transport equation under the gray medium approxima-tion,11 yields an excellent match with data over the entire temperature range. The MFP used depends on the choice of dispersion and is somewhat of a fitting parameter in these calculations. At the other end of feature size, Ravichandran and Minnich10 have reported Monte Carlo simulations of the Boltzmann equation over the entire temperature range using an efficient variance reduction technique. They assumed the presence of a disordered surface, which effectively increases the pore size. Using the thickness of the disordered surface, they are able to fit the thermal conductivity against the data of Yu et al. That surface disorder may play a significant role in reducing thermal transport in periodic structures has been previously suggested in the work of Lee et al.24 though conclusive proof is missing. In their atomistic simulations of nanoscale pores on the order of a few nanometers spaced apart again by a few nanometers, Lee et al.24 find that the thermal conductivity is a strong function of the pore spacing. Further, not only porosity but also the ratio of the surface area of the pore to the volume of the crystal strongly affects thermal conductivity. However, in an overall sense, the emphasis on the magnitude of thermal conductivity rather than the slope of conductivity with temperature renders it difficult to provide a definitive answer on the issue of coherence.

FIG. 6. Thermal conductivity measurements of 2-D periodic structures versus temperature. Corresponding fitting curves are also shown. We note that the data from Yu et al. is rescaled to account for a disordered layer assumed in the Monte Carlo simulation.

One aspect that has not been studied in depth is the effect of disorder in the period of the structure. In atomistic simulations23 that are limited in system size to a few nanometers and therefore cut off larger phonon wavelengths, thermal conductivity does not appear to be sensitive to such disorder. This is expected in incoherent scattering. In coherent transport, however, we may expect a significant impact of disorder. To understand this, we can use the Helmholtz wave equation in the acoustic limit and consider multiple scattering of waves to show the impact of disorder. We solve the Helmoltz wave equation in the domain defined by silicon. Assuming the continuity of the displacement and stress at the silicon-hole interfaces, we formulate the scattered field as

^scat(r) = 2 2 AmHm(k\r - Tt\)eim*r'ri (8)

i=1 m=-TO

where Aim is the scattering strength of mth harmonic from the ith cylinder (centered at ri)at a point r={x, y}, H is the Hankel function of the first kind, 0 is a polar co-ordinate and N is the number of cylinders. The coefficient Aim is calculated by considering the acoustic impedance at hole-silicon interface and the multiply scattered wavefronts from all the other cylinders j+i. Specifically, in order to obtain Aim, we solve a hierarchy of equations64

Aim = iCmFim

Fjn = (-1)n exp(ikrj cos j + ^ ^ iCm+nFm+nHm(krij)eim^i' (9)

i=1, i+j m=-TO

where the coefficient Cm is obtained from the boundary condition at the hole-silicon interface and involves Bessel and Hankel functions. The treatment enables consideration of randomly arranged holes. The total field is the sum of incident and scattered field: = fine + $ scat.

Ordered Array Disordered Array Highly Disordered Array

FIG. 7. The scattered field intensity increases and becomes randomized as disorder in the array increases. The x and y axis in each plot denote the dimension of the simulated region in nm.

A simulation for scattering of a unit amplitude plane wave from an array of 25 holes of diameter 80 nm and pitch 150 nm for periodic and disordered configurations is shown in Figure 7. Figure 7(a) and 7(b) plot the intensities of the scattered field at ~0.3 and 3.0 THz respectively with respective wavelengths of ~ 18 nm and 1.8 nm. The phonon field is incident from the top. When the array of holes is ordered, the field is either backward or forward scattered. At 0.3 THz the Rayleigh parameter has the same order of magnitude with unity, corresponding to Mie scattering. Interference effects are evident. At 3 THz, Rayleigh parameter is much larger than 1 and geometric scattering occurs. Here, geometric shadowing effects are evident. As the array becomes more random, the field is scattered in other directions. Localized fields are evident at 0.3 THz where phase effects are appreciable since the period is comparable to the wavelength. The scattered field is "trapped" through multiple scattering events resulting in reduced transport along the direction of propagation. In localization theory, this attenuation of the field in the forward direction can be related to an effective mean free path though a formal theory for phonons is still in development. However, for the higher frequency, phase effects are less evident with increasing disorder since scattering is approximately in the geometric limit. We note that the experimental conditions for smaller features (<100 nm) correspond to the disordered case rather than the highly ordered case. It is not clear whether these effects are indeed present in the structures measured till date but such effects should be increasingly observable as feature sizes are further reduced.

In summarizing the data and theory for 2-D periodic holey silicon, we observe that for features larger than 100 nm, incoherent transport appears a plausible explanation for the data at room temperature. For structures with smaller features, it is not yet clear whether coherence can be ruled out at room temperature. In theory, incoherent scattering may reduce thermal conductivity to the observed values due to surface disorder and partially coherent transport may also result in the same without consideration of disorder. A key aspect of the difference between coherent and incoherent transport is that the resulting thermal conductivity should have different temperature trends at low temperatures.

VII. DISCUSSION ON 3-D PERIODIC STRUCTURE

We finally discuss data for three-dimensional periodic nanostructures. Due to the nature of the bottom-up fabrication, the inverse opal structure is rarely single crystal. Even after post annealing, the grain size is limited by shell thickness,65 and much smaller than their bulk counterparts under same annealing condition. Therefore, grain boundary scattering could be the dominant scattering mechanism at or below room temperature in inverse opal structures.

20 40 100 200 400

Temperature (K)

FIG. 8. Material thermal conductivity data for silicon inverse opals as a function of temperature and for various shell thicknesses. The numbers in brackets indicate the (shell thickness, grain size) of each set of data. The effect of structure is already accounted for and thus the y-axis represents the intrinsic thermal conductivity of the material of the inverse opal. The thermal conductivity varies as ~T18 at low temperatures in all samples. Coherent scattering of phonons at grain boundaries is depicted schematically in the inset and explains this low temperature behavior. Adapted with permission. Copyright 2013 American Chemical Society.32

Phonons scatter with a grain boundary due to change in orientation of crystal planes as well as disorder in the inter-grain region. In order to explain this scattering process, Klemens66 adopted a perturbative approach, which introduces a perturbation in wave velocity Av, caused by either a tilt boundary or an inter-grain region. The scattering rate from mode q to q' has the form

1 V -2 -2

Tg (q) (2n)3

P-2^-2 X/ v-12 IM(q,q')|2dS' (10)

where V is the volume of a crystal containing one boundary, p is the density, vg is the phonon velocity, dS is the surface element, j is polarization. The perturbation element M (q,q') is

M(q, q') = V-12pio2 J dróv (r)eiQr(e • e') (11)

where e, e' are unit vectors along the polarization directions, and Q=q'-q. Klemens showed that randomized scattering at a tilt boundary gives rise to a frequency independent scattering rate

- = - (v/l )(Av/v )2 (12)

where l is average grain size, Av is the change in velocity at the tilt boundary. Since typically Av/v ~ 0.1, the magnitude of this rate is very small. In fact it is approximately two orders of magnitude smaller than the gray model, which uses l as the limiting dimension.

Phonons also scatter with the region between grains. This scattering is much stronger due to the greater disorder in the inter-grain region, which yields a stronger perturbation in velocity: Av is comparable to v in this case. Since the thickness of the inter-grain region t is typically ~ A scale, which is smaller than the wavelength of most conducting phonons,13 the scattering process is coherent. This

suggests a critical frequency, wcr « v/t, above which phonons would scatter incoherently with the inter-grain region. However, those phonons contribute little to conduction due to low group velocities. The scattering rate for phonons below the critical frequency has a quadratic frequency dependence, which recovers to Eq. (7), r-1=(4w2t2/3vl)(Av/v)2.

Figure 8 plots the temperature dependent data for the thermal conductivity of silicon inverse opals. The fitting parameter here is the thickness of intergrain region. They are between 1.8 to 2.2 A, consistent with the molecular dynamics calculation of bulk silicon for grain boundaries of different energies.67,68 At low temperatures, the thermal conductivity has an anomalous ~T18 dependence, distinct from the typical ~T3 behavior of bulk polycrystalline silicon.30 This low temperature behavior suggests a strong dependence of the scattering rate on frequency. A model of thermal conductivity of inverse opals that incorporates both diffusive thermal transport as well as microscopic phonon transport through coherent phonon scattering at grain boundaries agrees well with the data. The predictions are depicted as solid lines in Figure 8.

VIII. CONCLUSION

In conclusion, the reported room temperature thermal conductivities in periodically holey silicon are well below the Casimir limit corresponding to the limiting dimension. Current understanding of the physics responsible for such reduction points toward incoherent scattering of phonons at the surfaces of the pores and in the neck region connecting the poresfor features >100 nm. Coherent effects may also play a role at lower temperatures and in smaller features. However, this is not clear at present and more work is necessary to first clearly define coherence for thermal phonons and then construct a theory for partially coherent phonon transport. From the perspective of applications, the thermal conductivities are attractive only at the very low end of the reported values. Thus, it is critical that these low values are carefully verified. Retaining the power factor close to the bulk in these structures remains challenging and has not been confirmed beyond the first report. A second interesting category of holey silicon material is the three-dimensional periodic structure. Inverse opals in particular are attractive for future optics and battery applications. Thermal transport in these structures is well explained by the conventional incoherent diffusion of phonons. At low temperatures, however, phonons appear to scatter coherently with the thin grain boundaries in these structures. These materials can be promising for thermal applications provided their thermal insulating properties can be combined with improved mechanical properties at high porosity to create for example, lightweight and rigid thermal insulators.

ACKNOWLEDGMENT

This work was supported in part by the U.S. Navy through Grant No. N66001-11-1-4154, and in part by the AFOSR under Contract No. AF FA9550-12-1-0073.

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