Anisotropic Heat Conduction in Two-Dimensional Periodic Silicon

Article pubs.acs.org/JPCC

Anisotropic Heat Conduction in Two-Dimensional Periodic Silicon Nanoporous Films Yu-Chao Hua and Bing-Yang Cao* Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, P. R. China ABSTRACT: Research on heat conduction in periodic nanoporous silicon films has drawn much attention due to its importance for developing highly efficient thermoelectric devices. Here, the thermal transport in two-dimensional (2D) periodic silicon nanoporous films is studied by a phonon Monte Carlo (MC) method and the theoretical analyses based on the Boltzmann transport equation (BTE). It is found that both the cross-plane and the in-plane effective thermal conductivities are significantly reduced when compared to those in the diffusive limit, and decrease with the increasing porosity or the decreasing period length. Importantly, our work reveals a strong anisotropy of the effective thermal conductivity of the 2D periodic nanoporous films; that is, the effective thermal conductivity in the in-plane direction is significantly less than that in the crossplane direction, due to the anisotropic effects of material removal and pore boundary scattering. Interestingly, even the influence of the specular parameter that depends on the pore boundary roughness is anisotropic in this case, which could provide an anisotropic tuning method on the effective thermal conductivities of the periodic nanoporous films. In addition, the effective thermal conductivity models, which well concern the anisotropy and the specular parameter dependence, are derived on the basis of Matthiessen’s rule by introducing the geometrical factors that can be obtained from the MC simulations. The good agreements have been achieved between the present models and the MC simulations, verifying the validity of the models.

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2D periodic silicon nanoporous films is still necessarily essential for the further development of thermoelectric devices. The effective thermal conductivity of a specific porous structure at the macroscale is reduced due to the material removal, which has been fully investigated by using the effective medium theory (EMT) in the diffusive limit.9 However, the material removal effect cannot be utilized to enhance the thermoelectric performance, since the electrical conductivity is also reduced by the same factor. For nanoporous films whose characteristic lengths are comparable to the phonon mean free path (MFP), boundary scattering will predominate the phonon transport, leading to an excess reduction of the effective thermal conductivity.10,11 Importantly, since phonons (the dominant heat carriers in silicon) are much more sensitive to boundary scattering than charge carriers, we could utilize boundary scattering to tune thermal conductivity without adversely influencing electrical conductivity.12 Generally, phonon transport can be characterized in the frame of particle transport at room temperature, and thus, Boltzmann transport equation (BTE) is used13

INTRODUCTION Experiments have demonstrated that by etching periodic nanoscale holes in silicon films the effective thermal conductivity can be greatly reduced without adversely influencing the electrical transport ability, which provides a promising way to develop highly efficient thermoelectric devices.1−6 For instance, Yu et al.7 found that the in-plane effective thermal conductivity for a 22 nm thick periodic nanoporous film with a pore diameter of 11 nm and period of 34 nm could only be 2 W/(m K) which is about 2 orders of magnitude smaller than the silicon bulk value (148 W/m K) at room temperature. Hopkins et al.8 investigated four 500 nm thick films with pore diameters and periods between 300 and 800 nm, and found that the cross-plane effective thermal conductivity was between 5 and 7 W/(m K) at room temperature. For a two-dimensional (2D) periodic silicon nanoporous film, its strong columnar microscopic structure indicates the high anisotropy of the effective thermal conductivity that determines which direction (in-plane or cross-plane) is more efficient for thermoelectric applications. However, there have been no systematical investigations on the anisotropy of the effective thermal conductivity of the 2D periodic silicon nanoporous films, and the effective thermal conductivity models that can well account for the anisotropy and the specular parameter dependence are also highly desired. Therefore, the study on the anisotropic thermal transport in the © XXXX American Chemical Society

Received: November 25, 2016 Revised: February 14, 2017 Published: February 15, 2017 A

DOI: 10.1021/acs.jpcc.6b11855 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 1. (a) Two-dimensional periodic silicon nanoporous film: the period is denoted by Lp and the pore radius is Rp. (b) Cross-plane heat conduction: the heat flow is along the pore axis (along the x-direction). (c) In-plane heat conduction: the heat flow is perpendicular to the pore axis (along the y-direction).

vg⃗ ωj ·∇fωj =

fω0j − fωj τj(ω)

neously at each time step, to simulate the in-plane phonon transport in the 2D porous silicon films with aligned square pores, and gave an explanation about the mechanism for the reduction of effective thermal conductivity by boundary scattering; that is, phonons with MFPs longer than the distance between two pores become “trapped” behind pores. Wolf et al. 22 demonstrated that the in-plane effective thermal conductivity slightly decreases with the increasing boundary roughness by using the ensemble MC simulations. Besides, a phonon MC method,23 which simulates the trajectories of individual phonons independently, has been used to investigate thermal transport in periodic silicon nanoporous films. Péraud et al.23 and Ravichandran et al.24 investigated the in-plane effective thermal conductivity of the periodic silicon nanoporous films by using the phonon MC simulations. Hua et al.25 also employed this method to study the cross-plane effective thermal conductivity of the periodic silicon nanoporous films. Additionally, the modification of phonon dispersion induced by the artificial secondary periodicity can further reduce the effective thermal conductivity, called the coherent effect.26,27 The experiments of Yu et al.7 showed that the periodic silicon nanoporous film exhibits a considerably lower in-plane thermal conductivity than an array of silicon nanowires, even though the nanowire array has a higher surface-to-volume ratio, and they attributed this phenomenon to the coherent effect. Hopkins et al.8 and Alaie et al.28 measured the effective thermal

(1)

where j represents the polarization, ω is the angular frequency of phonons, vg⃗ ω is the group velocity, fω is the phonon distribution function, fω0 is the equilibrium distribution function, and τj(ω) is the relaxation time. The phonon BTE incorporated with proper boundary conditions can well characterize the effect of boundary scattering on phonon transport.14,15 Yang et al.16 studied the cross-plane effective thermal conductivity of the periodic nanoporous films by numerically solving the phonon BTE with the Debye approximation. Hsieh et al.17 used a frequency-dependent phonon BTE solver to investigate the influence of pore geometries on the in-plane effective thermal conductivity of 2D periodic silicon nanoporous films. Tang et al.18 studied the inplane periodic nanoporous films with square pores by numerically solving the phonon BTE, and emphasized the influence of pore structure and placement on the reduction of effective thermal conductivity. The Monte Carlo (MC) method, which well handles the problems with complicated geometries and multiple scattering events, is a favorable technique to simulate phonon transport in nanostructures, equivalent to directly solving the phonon BTE.19,20 Hao et al.21 used the ensemble MC technique, in which the trajectories of all phonons are simulated simultaB


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The Journal of Physical Chemistry C conductivity of periodic silicon nanoporous films at room temperature, respectively, and both found the phenomenological thermal conductivity models considering the material removal and the boundary scattering could overpredict the experimental results, deducing the existence of the coherent effect. However, the attribution from the coherent effect for reducing the effective thermal conductivity of periodic silicon nanoporous films at room temperature is still on debate. Lee et al.3 measured the cross-plane thermal conductivity of periodic nanoporous silicon films, and found that the model not considering the coherent effect can also well reproduce the experimental data at room temperature. Besides, Jain et al.29 calculated the in-plane and cross-plane effective thermal conductivities using the MC simulations, and by comparing them with some relevant experimental data, they found that the coherent effect does not contribute to thermal transport in porous films with feature sizes greater than 100 nm. Ravichandran et al.24 also demonstrated that the incoherent boundary scattering dominates thermal transport in periodic silicon nanoporous films at room temperature by using the MC simulations. In fact, if one wants to evaluate the coherent effect for reducing effective thermal conductivity of periodic nanoporous films, it is necessary to accurately characterize the effect of boundary scattering in quantity first. However, in the papers of Hopkins et al.8 and Alaie et al.,28 the semiempirical models were employed to calculate the boundary scattering effect, while the accuracy of them is lacking of validation. The accurate predictive models for both the in-plane as well as the crossplane effective thermal conductivities of the periodic nanoporous films are needed for clarifying the influence of the coherent effect. In the present work, we focus on the influence of boundary scattering and systematically investigate the anisotropic heat conduction in the 2D periodic silicon nanoporous films by the phonon MC simulations and the theoretical analyses based on the phonon BTE. Moreover, the effective thermal conductivity models for both the in-plane and cross-plane directions, which concern the specular parameter dependence, are derived on the basis of Matthiessen’s rule by introducing the geometrical factors that can be obtained from the MC simulations.

P is equal to 1, the phonon scattering is completely specular, while P = 0 corresponds to the diffusive scattering. For silicon at room temperature, the dominant phonon wavelength is approximately less than 1 nm,30 and thus any realistic surface roughness will make P equal to 0, corresponding to the completely diffusive boundary scattering. Here, only the acoustic phonon contributions to thermal transport are taken into account, and thus, the optical phonon contributions that could be less than 5% for bulk silicon31 are neglected. Actually, the optical phonon contributions can increase with the reducing characteristic lengths and the increasing temperature.31,32 Nevertheless, particularly in the present work, since the simulation temperature is 300 K and the minimum characteristic length (i.e., LP − 2RP) is about 40 nm (LP = 300 nm, porosity = 0.6), referring to the works of Sellan et al.31 and Tian et al.,32 the optical phonon contributions could be roughly less than 10% even in the case with the minimum characteristic length. Therefore, the neglect of the optical phonon contributions should be reasonable. The Brillouin zone boundary condition (BZBC) model is employed to characterize the silicon dispersion curves of acoustic phonons in the [001] crystal direction.33 The BZBC model, which well reproduces the experimental phonon dispersion by using a low order polynomial, is given by34 ω = v0,Lκ mκ * + (ωL − v0,Lκ mκ *)2

for longitudinal phonon (L) and ω = v0,Tκ mκ * + (3ωT − 2v0,Tκ m)κ *2 + (v0,T − 2ωT)κ *3 (4)

for transverse phonon (T), where κm is the wavenumber at the edge of the first Brillouin zone, κ* = κ/κm is the dimensionless wavenumber, ωm,L = 570kB/ℏ rad/s, ωm,L = 210kB/ℏ rad/s, ν0,L = 8480 m/s, and ν0,T = 5860 m/s. Once the dispersion relation is given, the group velocity can be calculated vg,L(T) =

∂ωL(T) (5)

∂κ

In addition, the intrinsic phonon relaxation time is calculated from Matthiessen’s rule

I. SIMULATION DETAILS I.A. Problem Formulation. Figure 1a illustrates the structure of the 2D periodic nanoporous films. Pores are squarely arrayed with a period length of Lp, and the pore radius is denoted by Rp. The heat flow is along the x-direction (along the pore axis) in the cross-plane heat conduction, as shown in Figure 1b, while the heat flow is along the y-direction (perpendicular to the pore axis) in the in-plane heat conduction, as shown in Figure 1c. In these two cases, the phonon transport in the z-direction is regarded as periodic. Besides, the pore boundary is thermally adiabatic, that is, all phonons that strike it will be reflected back to the domain. Generally, a specular parameter, P, is introduced to describe the characterization of phonon scattering at such boundaries.13 The specular parameter is given by ⎛ 16π 3Δ2 ⎞ P = exp⎜ − ⎟ ⎝ λ2 ⎠

(3)

τj−1(ω) = τIj−1(ω) + τ3phj−1(ω)

(6)

for j = L or T, where τIj and τ3phj are the relaxation time of impurity and three phonon scatterings, respectively. Then, the intrinsic MFP is lint j = νgωjτj(ω). According to ref 33, the relaxation time can be readily calculated using a set of semiempirical formulations. The impurity relaxation time is given by τIj−1(ω) = BIj ω4 ,

j = L or T

(7)

The relaxation time of three phonon scatterings for the L mode is τ3phL−1(ω) = τL,U−1 + τL,N−1 = BL ω 2T 3

τL,U−1

(8)

τL,N−1

in which and are the relaxation times for U and N processes, respectively. The relaxation time of three phonon scatterings for the T mode is

(2)

τ3phT−1(ω) = τT,U−1 + τT,N−1

in which Δ is the root-mean-square value of the roughness fluctuations and λ is the dominant phonon wavelength.13 When

(9)

with C


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The Journal of Physical Chemistry C 4 ⎧ ω < ω1/2,T ⎪ B NT ωT τT,N−1 = ⎨ ⎪ ω > ω1/2,T ⎩0

τT,U

−1

⎧0 ω < ω1/2,T ⎪ =⎨ ω2 ω > ω1/2,T ⎪BUT sinh(ℏω/kBT ) ⎩

(10)

(11)

in which ω1/2,T = 2.47 × 1013 rad/s corresponds to κ* = 0.5. The relaxation time parameters in eqs 7−11, i.e., BIj, BL, BNT, and BUT, have been determined from a fit between theoretical and experimental thermal conductivities of bulk silicon over a wide temperature range.33 I.B. Phonon Monte Carlo Simulation. A phonon MC method20,23 is applied to simulate the phonon transport in the 2D periodic nanoporous films. The MC technique simulates phonon transport processes by random number samplings, equivalent to directly solving the phonon BTE. In our simulations, a temperature difference is imposed in the structure to induce a heat flux, as shown in Figure 1a and b, and then, the effective thermal conductivity can be obtained by using Fourier’s law

keff =

qL ΔT

(12)

where q is the heat flux, L is the distance between these two phonon baths, and ΔT is the temperature difference. Isothermal boundary conditions35 are usually used to establish the temperature difference in the standard MC simulations, which requires a computational domain consisting of many periods to eliminate the end effects.21 In this case, the exact effective thermal conductivity can be obtained only if the simulation results will no longer vary with further increasing number of periods, resulting in a considerable large computational expense. Jeng et al.36 proposed a periodic boundary condition specially for the ensemble MC simulation. Following the method proposed by Jeng et al.,36 Hao et al.21 and Péraud et al.37 studied the thermal transport in periodic nanostructures. In the present work, we use a newly proposed two-step algorithm,38 which can significantly reduce the computational expense, to eliminate the end effects.

Figure 2. In-plane and cross-plane effective thermal conductivities of 2D periodic nanoporous silicon films with various periods (Lp = 300, 1100, and 3000 nm) at room temperature as a function of porosity.

which the heat flow is perpendicular to the pore axis, is given by9 1−ε keff_in,diff = k bulk (14) 1+ε

II. RESULTS AND DISCUSSION II.A. Anisotropy of Effective Thermal Conductivity for 2D Periodic Nanoporous Films. Figure 2 shows the in-plane and cross-plane effective thermal conductivities of the 2D periodic nanoporous silicon films at room temperature. The period lengths (Lp) are set as 300, 1100, and 3000 nm, respectively, and the specular parameter is equal to 0 at room temperature. We first calculate the in-plane and cross-plane effective thermal conductivities in the diffusive limit (that is to say, Lp is much larger than the average MFP) via the MC simulations. As for a 2D periodic porous film, the influence of material removal on the effective thermal conductivity is anisotropic. When the heat flow is along the pore axis, the cross-plane effective thermal conductivity is given by9 keff_cr,diff = (1 − ε)k bulk

As shown in Figure 2, the good agreements between the simulations and the EMT models can be regarded as a verification of our MC simulations. By contrast, when the pore boundary-scattering is concerned (Lp = 300, 1100, and 3000 nm), both the in-plane and crossplane effective thermal conductivities obtained by the MC simulations are reduced as compared to the diffusive-limit results, and they both decrease with the increasing porosity or the decreasing period length. Besides, we calculate the effective ratio, i.e., the ratio of the normalized effective thermal conductivity to the corresponding material removal factor ((1 − ε)/(1 + ε) for in-plane and 1 − ε for cross-plane). Since the effective ratio reflects the effect of pore boundary scattering alone, it can be used to evaluate the improvement of thermoelectric performance; that is to say, the smaller the

(13)

where ε is the porosity and kbulk is the bulk thermal conductivity. The in-plane effective thermal conductivity, for D


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The Journal of Physical Chemistry C effective ratio is, the more significant the improvement of thermoelectric performance is. As shown in Figure 3, the

keff_in /

γex =

( 11 −+ εε )

keff_cr /(1 − ε)

(15)

In the case excluding the material removal effect, the anisotropy ratio increases but still less than 1 (ranging from about 0.7 to 0.9), which indicates the pore boundary scattering effect is anisotropic for the 2D periodic silicon nanoporous films. Besides, the anisotropy ratio excluding the material removal effect also decreases with the increasing porosity or the decreasing period length. The boundary scattering effect plays an important role for the silicon nanoporous films utilized as a thermoelectric material, and thus, its anisotropy indicates that the in-plane direction is more efficient for the enhancement of thermoelectric performance. II.B. Effective Thermal Conductivity Models for 2D Periodic Nanoporous Films. The phonon boundary scattering causes the size effect of the effective thermal conductivity of nanoporous films. As for the in-plane direction, Prasher30 proposed an approximate ballistic-diffusive effective medium model with the gray media approximation by adding the ballistic and diffusive resistances

Figure 3. Effective ratio for in-plane and cross-plane thermal conductivities of 2D periodic nanoporous silicon films at room temperature as a function of porosity.

effective ratios for both the in-plane and cross-plane effective thermal conductivities also decrease with the increasing porosity or the decreasing period length, indicating that the improvement of thermoelectric performance is enhanced with the increasing porosity or the decreasing period length. Importantly, according to Figures 2 and 3, our simulation results reveal a strong anisotropy of the effective thermal conductivity of the 2D periodic nanoporous silicon films; that is, the in-plane effective thermal conductivity is significantly less than the cross-plane effective thermal conductivity. The experiments by Kim and Murphy39 demonstrated the same phenomenon, but the authors did not give enough explanations about the underlying mechanism. Actually, the strong anisotropy should be attributed to both the material removal and the pore boundary scattering. As stated above, the anisotropy of the material removal effect on the heat conduction in the 2D periodic nanoporous films has been well clarified on the basis of the EMT. What is still ambiguous is the anisotropy of the pore boundary scattering effect. Here, an anisotropy ratio, γ = keff_in/keff_cr, is then calculated, as shown in Figure 4. In the case including the material removal effect, the anisotropy ratio ranges from about 0.4 to 0.8, and it decreases with the increasing porosity or the decreasing period length. Furthermore, we can exclude the material removal effect in the anisotropy ratio by eliminating the material removal factors, that is,

keff l = b = km lave

1 1+ε 1−ε

+

4 lave 1 3 Lp F

(16)

with F = 1 − 2rp/Lp[π/2 − arcsin[Lp/(2rp)] − ((2rp/Lp)2 − 1)1/2 − Lp/(2rp)], in which lave is the average intrinsic MFP of silicon and lb is the effective MFP accounting for the boundary scattering. It could be straightforward to take the phonon dispersion into account in the model above. At room temperature, the contribution of momentum-conserving collisions (normal scattering) can be negligible for silicon, so the standard relaxation time approximation40 is employed to calculate thermal conductivity k=

1 3

∑∫ j

ωmj

ℏω

0

∂f0 ∂T

vgωjlbωj DOSj (ω) dω (17)

where the frequency-dependent effective MFP, lbωj, can be derived on the basis of eq 16 as follows l bω j =

l int j(ω) 1+ε 1−ε

+

4 l int j(ω) 1 3 Lp F

(18)

41

Besides, Alvarez et al. utilized phonon hydrodynamics to analyze the influence of porosity and pore size on reduction of effective thermal conductivity in porous silicon. As for the cross-plane effective thermal conductivity of the 2D periodic nanoporous films, Prasher42 derived an analytical model on the basis of the phonon BTE with the gray media approximation. Then, Hua and Cao25 extended this model by considering the phonon dispersion and the size dependence due to film thickness. However, Bera et al.43 found that the cylindrical geometry approximation used in the analytical model above could lead to an overestimation in the small pore limit, and then they proposed a semiempirical model with an adjustable parameter. It should be noted that these previous models mostly hold a complicated expression, and cannot well account for the anisotropy and the influence of the specular parameter. In practice, the semiempirical models based on Matthiessen’s rule have been widely utilized to predict the effective thermal conductivity of various nanostructures. By selecting a proper

Figure 4. Anisotropy ratio (γ) for the effective thermal conductivity of 2D periodic nanoporous silicon films at room temperature. E


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about 2.25 (αin = 2.55) for the in-plane and about 4.65 (αcr = 4.65) for the cross-plane. As shown in Figure 2, the predictions by our models well agree with those by the MC simulations, and the maximum deviation between them is less than 10% for both the in-plane and cross-plane directions. II.C. Anisotropic Influence of the Specular Parameter on the Effective Thermal Conductivity. In the above sections, we mainly investigate the anisotropic heat conduction in the 2D periodic nanoporous silicon films with the completely diffusive scattering pore boundaries at room temperature. However, at low temperature, when the dominant phonon wavelength of silicon is larger than the boundary roughness, the specular scattering can occur at the pore boundaries.22,45 Besides, for the material whose dominant phonon wavelength of heat conduction is large even at room temperature, such as silicon germanium alloy46 or graphene,47 the specular phonon boundary scattering plays an important role in thermal transport. Figure 5a shows the cross-plane effective thermal conductivity of the 2D periodic nanoporous silicon films with the

characteristic length and introducing a geometrical factor, the model can calculate the effective thermal conductivity of diverse nanostructures.44 In this way, the in-plane and cross-plane effective thermal conductivities of the 2D periodic nanoporous films can hold a unified form that is given by keff_in(cr) = Gin(cr)k m_in(cr)

(19)

where the material removal factor Gin(cr) is (1 − ε)/(1 + ε) for the in-plane and 1 − ε for the cross-plane and km_in(cr) is the thermal conductivity of the host material which can be calculated by using eq 17 incorporated in Matthiessen’s rule. Here, Matthiessen’s rule is utilized to calculate the frequencydependent effective MFP 1 l bω j

=

1 l int j

+

1 αLch

(20)

where Lch is the characteristic length that depends on the period length Lp and the pore radius Rp and α is the geometrical factor. In eq 20, the term αLch refers to the MFP if there was boundary scattering but no other source of scattering. We note that, for a structure only with boundary scattering, this MFP should merely depend on the characteristic length but not always directly be equal to the characteristic length; thus, a geometrical factor is frequently introduced for better estimating this MFP.44 For the in-plane effective thermal conductivity of the 2D periodic nanoporous films, Alaie et al.28 chose

Lp2 − πR p2 as the characteristic length with a geometrical factor equal to 1. For the cross-plane direction, Hopkins et al.8 chose the neck length Lp − 2Rp as the characteristic length with a geometrical factor equal to 1. By combining eqs 17 and 20, the thermal conductivity of the host material is calculated, and then, the effective thermal conductivities can be obtained according to eq 19. Figure 2 compares the results predicted by some of the existing models we mentioned above with those calculated by the MC simulations. For the in-plane direction, the model by Prasher30 significantly underpredicts the effective thermal conductivities by the simulations and the maximum deviation is about 40%. This is because the end effect is not eliminated in th is model. The m odel by Alaie et al. 2 8 w it h Lch_in = Lp2 − πR p2 and α in = 1 underpredicts the simulation results as the porosity is less than 0.3, whereas it overpredicts the simulation results. Its maximum deviation is about 30%, indicating that the choice of Lch_in or αin could be inapplicable in this case. Besides, for the cross-plane direction, the model by Hopkins et al.8 with Lch_cr = Lp − 2Rp and αcr = 1 underpredicts the simulation results with a maximum deviation equal to about 40%, and thus, the choice of Lch_cr or αcr could also be inapplicable. Therefore, in order to improve the prediction accuracy of the effective thermal conductivity models, we need to explore the proper characteristic lengths with the corresponding geometrical factors for both the in-plane and cross-plane directions. As a plausible assumption, we think a specific nanostructure should (or had better) just hold a specific characteristic length, and the anisotropy of the effective thermal conductivity could be characterized by introducing diverse geometrical factors. Here, we select the neck length (Lp − 2Rp) as the characteristic length of the 2D periodic nanoporous films for both the inplane and cross-plane directions. Then, according to the MC simulation results, we find the geometrical factor should be

Figure 5. (a) Cross-plane effective thermal conductivity of 2D periodic nanoporous silicon films with various specular parameters (P = 0, 0.5, and 1). (b) In-plane effective thermal conductivity of 2D periodic nanoporous silicon films with various specular parameters (P = 0, 0.5, and 1).

various specular parameters (P = 0, 0.5, and 1). The cross-plane effective thermal conductivity significantly increases with the increasing P. In the case of P = 1, the completely specular boundary even does not cause the excess reduction of the crossplane effective thermal conductivity, and thus, the simulation results agree with the predictions by the EMT model that only concerns the material removal effect, despite the diverse period lengths. The same phenomenon has been found for the effective thermal conductivity of nanowires. Ziman13 studied F


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in-plane effective thermal conductivity with P = 0 and that with P = 1 is less than 1.05, despite the diverse period lengths. Hence, if ε is small, the influence of the specular parameter could even be neglected. Wolf et al.22 calculated the in-plane thermal conductivity of the random-arrayed nanoporous films, and obtained a similar conclusion. Besides, much different from the cross-plane direction, as P = 1, the completely specular boundary scattering also significantly reduces the in-plane effective thermal conductivity. As illustrated in Figure 6, although the specular boundary scattering in the in-plane 2D periodic nanoporous films does not randomize the phonon transport direction, it does change the vector component of the phonon transport direction along the heat flow, and thus leads to the reduction of the heat flow even at P = 1. Therefore, we can conclude that the specular parameter dependence of the effective thermal conductivity is anisotropic for the 2D periodic nanoporous films. Here, we modified eq 22 for the in-plane direction by introducing two adjustable parameters (B1 and B2) to characterize the anisotropy of the specular parameter dependence, that is,

the specular parameter dependence of the effective thermal conductivity of nanowires in analogy to the discussion of the flow of a very rarefied gas in a tube, and he found 1−P [αwireLch_wire]P = [αwireLch_wire]P = 0 (21) 1+P where [αwireLch_wire]P=0 is the value for completely rough boundaries (P = 0). By combining eqs 17, 20, and 21, the effective thermal conductivity model concerning the specular parameter for nanowires is then obtained. As shown in Figure 6a and b, both in the nanowires and in the cross-plane 2D

[αinLch_in]P =

1 − B1P B2 1 + B1P B2

[αinLch_in]P = 0

(23)

where [αcrLch_cr]P=0 = 2.25(Lp − 2Rp). When B1 = 1 and B2 = 1, the form of eq 23 is reduced to that of eq 22. By combining eqs 17, 19, 20, and 23, an in-plane effective thermal conductivity model for the 2D periodic nanoporous films, concerning the specular parameter dependence, is obtained. Then, according to the MC simulation results, we find B1 = 0.15 and B2 = 0.2. As shown in Figure 5b, the predictions by our model well agree with those by the MC simulations, and the maximum deviation is about 10%. II.D. Analyses of Experimental Results. In this section, we compare the MC simulation results with some available experimental data. Alaie et al.28 measured the in-plane effective thermal conductivities for 336 nm thick periodic silicon nanoporous films with a period of 1100 nm at room temperature. Hopkins et al.8 measured the cross-plane effective thermal conductivity of four 500 nm thick periodic silicon nanoporous films at room temperature. In Figure 7, the measured effective thermal conductivity is normalized to the thermal conductivity of the corresponding film without pores. We assume that the size effects due to the film thickness and the nanopores could be decoupled in the case of thick films,

Figure 6. Schematics for phonon boundary scatterings in 2D periodic nanoporous films and nanowires: (a) cross-plane heat conduction in periodic nanoporous film; (b) heat conduction in nanowire; (c) inplane heat conduction in periodic nanoporous film.

periodic nanoporous films, the diffusive boundary scattering randomizes the phonon transport direction, leading to the reduction of the heat flow, while the specular boundary scattering will not change the vector component of the phonon transport direction along the heat flow. Besides, a cross-plane 2D periodic nanoporous film can be roughly regarded as an array of tubular nanowires. Therefore, it is reasonable that a cross-plane 2D periodic nanoporous film holds the same specular parameter dependence of the effective thermal conductivity as a nanowire, that is, 1−P [αcrLch_cr ]P = [αcrLch_cr ]P = 0 (22) 1+P with [αcrLch_cr]P=0 = 4.65(Lp − 2Rp). Then, we obtain a crossplane effective thermal conductivity model for the 2D periodic nanoporous films, which takes the specular parameter dependence into account, by combining eqs 17, 19, 20, and 22. As shown in Figure 5a, our model well predicts the cross-plane effective thermal conductivities by the MC simulations despite the diverse period lengths and specular parameters, and the maximum deviation is only about 10%. Figure 5b shows the in-plane effective thermal conductivity of the 2D periodic nanoporous silicon films with various specular parameters (P = 0, 0.5, and 1). The in-plane effective thermal conductivity also increases with increasing P, but the increment is not significant when compared to that of the crossplane direction. For instance, as ε = 0.1, the ratio between the

Figure 7. In-plane and cross-plane effective thermal conductivities of 2D periodic nanoporous silicon films in comparison with experimental data at room temperature. G


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(3) By comparing the experimental data and the simulation results, we find that the material removal and the boundary scattering could in principle explain the reduction of the in-plane effective thermal conductivity of the 2D periodic silicon nanoporous films at room temperature, and thus, the coherent effect may not be significant in this case. (4) The effective thermal conductivity models for both the in-plane and cross-plane directions, which concern the anisotropic specular parameter dependence, are derived on the basis of Matthiessen’s rule by introducing the geometrical factors that are obtained from the MC simulations, and the relevant parameters of the models have been summarized in Table 1. The good agreements

and the normalized effective thermal conductivity mainly reflects the influence of nanopores on the phonon transport. As shown in Figure 7, the results by our model and MC simulations for the in-plane effective thermal conductivity with a period length of 1100 nm and P = 0 are slightly less than the measured values of Alaie et al.,28 and the deviation may result from the phonon dispersion approximation.33 Importantly, by comparing the experiment data and the simulation results, it is found that the material removal and the pore boundary scattering could in principle explain the reduction of the inplane effective thermal conductivity of the periodic nanoporous films. Hence, we may obtain a conclusion similar to that by Ravichandran et al.24 that the coherent effect is not significant for the heat conduction in the periodic silicon nanoporous films at room temperature. In fact, the recent experiments by Lee et al.48 further confirmed this conclusion. They measured the inplane thermal conductivities of the films with the periodic and aperiodic square holes, and concluded that the phonon coherence should be unimportant for thermal transport in silicon nanomeshes with periodicities of 100 nm and higher and temperatures above 14 K. This conclusion could be easily understood in physics. The frequency of the phonons that could be tuned by the periodic nanomeshes is in the gigahertz regime,4 but the thermal transport in silicon at room temperature is governed by the terahertz frequency phonons,27 so it is not surprising that the coherence has a minor effect on the thermal conductivity of silicon nanomeshes at room temperature. As for the cross-plane effective thermal conductivity, our simulations as well as the present model both overpredict the measured values by Hopkins et al.8 Actually, Jain et al.29 also found the modeling results by their MC simulations are significantly larger than the experimental data by Hopkins et al.8 However, there has been no reasonable explanation about this issue yet.

Table 1. Material Removal Factors, Characteristic Lengths, Geometrical Factors, and Specular Parameter Dependence for Cross-Plane and In-Plane Effective Thermal Conductivities of 2D Periodic Nanoporous Films material removal factor

geometrical factor

in-plane

1−ε 1+ε

Lp − 2Rp

2.25

cross-plane

1−ε

Lp − 2Rp

4.65

■

III. CONCLUSIONS

characteristic length

specular parameter dependence 1 + 0.15P 0.2 1 − 0.15P 0.2 1+P 1−P

have been achieved between the present models and the MC simulations, verifying the validity of the models. The results and models in our work can provide a more indepth understanding about the anisotropic heat conduction in the 2D periodic nanoporous films including but not limited to silicon, and be useful for developing the highly efficient thermoelectric devices.

AUTHOR INFORMATION

Corresponding Author

*Phone/Fax: +86-10-6279-4531. E-mail: [email protected]. cn.

(1) The heat conduction in the 2D periodic silicon nanoporous films is studied by the phonon MC simulations. Both the in-plane and the cross-plane effective thermal conductivities are significantly reduced by the pore boundary scattering and decrease with the increasing porosity or the decreasing period length. Their effective ratios also decrease with the increasing porosity or the decreasing period length, indicating the improvement of thermoelectric performance could be enhanced with the increasing porosity or the decreasing period length. (2) The strong anisotropy of the heat conduction in the 2D periodic silicon nanoporous films is revealed. The effective thermal conductivity in the in-plane direction is significantly less than that in the cross-plane direction, due to the anisotropic effects of material removal, pore boundary scattering, and specular parameter dependence. The anisotropy becomes enhanced with the increasing porosity or the decreasing period length. We thus suggest that the in-plane direction of the 2D periodic silicon nanoporous films is more efficient for thermoelectric applications. Moreover, the anisotropic specular parameter dependence also provides an anisotropic tuning method on the effective thermal conductivity of the periodic nanoporous films by boundary surface modification.

ORCID

Yu-Chao Hua: 0000-0002-1351-6474 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by National Natural Science Foundation of China (No. 51676108, 51356001) and Science Fund for Creative Research Group (No. 51321002), the Tsinghua National Laboratory for Information Science and Technology of China (TNList).

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Anisotropic Heat Conduction in Two-Dimensional Periodic Silicon

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