Size-dependent model for thin film and nanowire thermal conductivity

We present an analytical model for the size-dependence of thin film and nanowire thermal conductivity and compare the predictions to experimental measurements on silicon nanostructures. The model contains no fitting parameters and only requires the bulk lattice constant, bulk thermal conductivity, and an acoustic phonon speed as inputs. By including the mode-dependence of the phonon lifetimes resulting from phonon-phonon and phonon-boundary scattering, the model captures the approach to the bulk thermal conductivity of the experimental data better than gray models based on a single lifetime. VC 2011 American Institute of Physics. [doi:10.1063/1.3644163]

Modeling work at different levels of sophistication has attempted to predict the thermal conductivity reduction in films and wires.Flik and Tien 1 and Majumdar 2 proposed film models based on a single phonon group velocity and lifetime (i.e., the gray approximation).More detailed calculations that include quantum statistics and the mode dependence of the phonon properties have proven successful at predicting experimental data for silicon nanostructures with minimal fitting parameters. 8,10,16Lattice dynamics calculations 17,18 and Monte Carlo-based methods 11 have also been used to elucidate the underlying phonon physics.
We recently proposed a closed-from high-temperature model for the cross-plane thermal conductivity of a film and used it to assess size effects in molecular dynamics thermal conductivity predictions. 21Here, we extend this model, which contains no fitting parameters, to the in-plane direction and to wires.The predictions are then compared to experimental data for silicon nanostructures.Our model does not seek to supplant more detailed calculations but to provide a straightforward method for predicting thermal conductivity reduction in nanostructures.
The film and wire geometries are shown in Figs.1(a) and 1(b).The film has thickness L, and the wire has diameter D. The derivation begins with an expression for the thermal conductivity in the i direction that is obtained by solving the Boltzmann transport equation under the relaxation time approximation and using the Fourier law 22 The summation is over all phonon modes with wave vector j and dispersion branch m, c ph is the mode volumetric specific heat, and v g,i (j, m) is the i component of the group velocity vector.The phonon transport is described using a set of mode-and system-size-dependent lifetimes, s(j, m, L or D), defined as the average time between successive scattering events.
Converting the summation over wave vector in Eq. ( 1) to an integral in spherical coordinates over the first Brillouin zone (BZ) and assuming that optical phonons do not contribute to thermal conductivity gives where V is the sample volume, j BZ (/, /) is the magnitude of the wave vector at the first BZ boundary along the h, u direction, and the summation is restricted to acoustic modes (ac).
While optical phonons can contribute to thermal conductivity in nanostructures, 18,20 we ignore them here to preserve the simplicity of the model.We now make the Debye approximation for phonon dispersion, which assumes a single acoustic branch such that x ¼ v ac j.Under this approximation, for a classical (i.e., high temperature) system, Eq. ( 2) simplifies to where the integral over wave vector has been changed to an integral over frequency and c ph has been replaced with k B /V, where k B is the Boltzmann constant.The Debye frequency, 22 where X is the primitive cell volume.For silicon, which has a diamond structure, the primitive cell volume is a 3 /4, where a is the lattice constant.The expressions for v g,i are provided in Table I.
The Matthiessen rule is used to combine the effects of phonon-phonon (s 1 ) and phonon-boundary (s b ) scattering so that 22 1 The phonon-phonon lifetimes are modeled using a relationship proposed by Callaway 23 for low frequencies, s 1 ¼ A/x 2 .This form agrees with lattice dynamics predicted phononphonon lifetimes times for SW silicon at a temperature of 300 K for frequencies below 3 THz. 21The constant A is calculated in the bulk limit and is 2p where k 1 is the bulk thermal conductivity.Phonon-boundary scattering at high temperatures is expected to be diffuse. 24The expressions for s b are provided in Table I.Following substitution of Eq. ( 4) into Eq.( 3) and using the expressions from Table I, the integrals can be performed analytically for the film, leading to Here, L is a non-dimensional length defined as 3 2 . For the wire, the angular integrals can be carried out analytically, but the frequency integral must then be performed numerically.While there is no closed-form solution for the thermal conductivity, the ratio k Nanowire ðDÞ=k 1 can be plotted as a function of As shown by the solid lines in Figs.2(a)-2(c), the thermal conductivity models approach k 1 as the limiting dimension (L or D) increases.The in-plane thermal conductivity of  the film is always higher than the value.This result can be understood a cubically isotropic material by noting that phonons scatter more frequently with boundaries when traveling in the cross-plane direction. 17,18vailable experimental thermal conductivity data for silicon films 3,5,9,12,19 and wires 7,14 at a temperature of 300 K are plotted in Figs.2(a)-2(c).Because the thermal conductivity of silicon is dominated by low-frequency phonons, 17,18 our high-temperature models should give reasonable predictions even though the Debye temperature of silicon is 645 K. 25 In converting the experimental data to dimensionless units, the lattice constant is 5.43 A ˚, the velocity is the average of the one longitudinal and two transverse acoustic phonon branches in the low-frequency limit in the [001] direction and is 6733 m/s, and the bulk thermal conductivity is 149 W/m-K. 25he in-plane thermal conductivity model captures the trend measured by four separate studies.The model predictions are generally higher than the experimental data.The result may be due to the flatness of the transverse acoustic phonon branch in silicon at mid-to high-frequencies, 25 which a single phonon velocity cannot capture.Making the boundary scattering in the model somewhat specular increases the predicted thermal conductivity away from the experimental data.While the agreement of the cross-plane thermal conductivity model with the one available experimental measurement is poor, more data are needed to assess its performance.
For the wire, the model overpredicts the experimental data of Li et al. 7 by $0.1k/k 1 .The similar slopes of the experimental data and the model suggest that the boundary scattering is properly handled.The wire model may suffer due to the single group velocity 8 and/or the possibility that the phonon-phonon scattering is not bulk-like due to confinement in two dimensions.In comparison, the model does not predict either the magnitude or diameter dependence of the rough silicon nanowires of Hochbaum et al., 14 pointing to the different nature of the phonon-boundary scattering in those wires. 15he Flik 1 and Majumdar 2 film thermal conductivity models are derived under the gray approximation (i.e., a single, mode-independent velocity, v ac , and lifetime, s G ) for L=ðs G v ac Þ ) 1 and use the Matthiessen rule for diffuse boundary scattering.They take the form The values of a are 3p/2 (Flik, In-Plane), 8/3 (Majumdar, In-Plane), and 3/4 (Majumdar, Cross-Plane).For the wire, with D ¼ L in Eq. ( 7), we set a ¼ 4/3. 22y using the kinetic theory expression for the bulk thermal conductivity, k 1 ¼ 1 3 Cv 2 ac s G , where C is the volumetric specific heat (1.66 Â 10 6 J/m 3 -K for silicon at a temperature of 300 K), L and D can be expressed as 3 where n pc is the number of atoms in the primitive cell (two for silicon) and C 1 is the high-temperature harmonic specific heat (3n pc k B /X).The gray models are plotted as dashed lines in Figs.2(a)-2(c) for all values of L or D. In all cases, the gray models predict a much faster approach to the bulk thermal conductivity than the experimental data and our models.In full Brillouin-zone lattice dynamics calculations on SW silicon thin films, Sellan et al. 18 and Turney et al. 17 also found that gray models predict a too-fast approach to the bulk thermal conductivity.This faster approach to bulk occurs because gray models tend to underpredict the lifetime of the majority of the phonons, which can span many orders of magnitude. 18n summary, we presented a high-temperature analytical model for the size-dependence of film and wire thermal conductivity that requires no fitting parameters.For the film, the model provides convenient algebraic expressions.We tested the model by comparing its predictions against experimental data for silicon nanostructures.Good agreement was found for the in-plane direction for the film.When compared to gray models, our model better captures the approach to the bulk thermal conductivity for both films and wires.Extending this approach to lower temperatures, where phonondefect scattering and quantum statistics must be considered, can be done in this framework but will require fitting parameters and numerical integration.

FIG. 1 .
FIG. 1.(a) Film and (b) wire models.The film is infinite in the in-plane directions, and the wire is infinite along its axis.

FIG. 2 .
FIG. 2. (Color online) Comparison of thermal conductivity models and experimental data for silicon nanostructures at a temperature of 300 K: (a) film in-plane, (b) film cross-plane, (c) wire.

TABLE I .
Group velocities and diffuse boundary scattering lifetimes.For the film, the average distance traveled ballistically by a phonon before hitting the top or bottom surface is 1=ðLjcos hjÞ Ð L 0 zdz ¼ L=ð2jcos hjÞ.For the wire (with D ¼ 2R), the average distance traveled ballistically by a 2 À r 2 cos 2 / p À rsin/Þrdrd/ ¼ 4D=ð3pjsin hjÞ.h and / are the spherical angles.