Effect of film thickness on the thermal resistance of confined semiconductor thin films

Abstract The thermal resistance of semiconductor thin films is predicted using lattice dynamics (LD) calculations and molecular dynamics (MD) simulations. We consider Si and Ge films with thicknesses, LF , between 0.2 nm and 30 nm that are confined between larger extents of the other species (i.e., Ge/Si/Ge and Si/Ge/Si structures). The LD predictions are made in the classical limit for comparison to the classical MD simulations, which are performed at a temperature of 500 K. For structures with LF < 2 nm, the thin film thermal resistance increases rapidly with increasing film thickness, a trend we attribute to changes in the allowed vibrational states in the film. These changes are found to affect the dependence of the phonon transmission coefficient on incidence angle for the Ge/Si/Ge structures and on frequency for the Si/Ge/Si structures. When LF > 2 nm, the MD-predicted thermal resistances are independent of the film thickness for the Ge/Si/Ge structures and increase with increasing film thickness for the Si/Ge/Si structures. We attribute these results to phonon transport that is ballistic in the Ge/Si/Ge structures and more diffusive in the Si/Ge/Si structures based on comparisons to the LD predictions, which assume ballistic phonon transport. We find that this difference between the structures cannot be predicted by comparing the mode-averaged phonon mean free path to the film thickness. It can be predicted, however, by considering the frequency-dependence of the phonon mean free paths.


I. INTRODUCTION
Semiconductor thin films are pervasive in advanced technological devices.For example, the active region in a quantum cascade laser or a light-emitting diode contains films of direct bandgap semiconductors such as GaAs, Al-GaAs, and GaN with thicknesses of 1-10 nm. 1,25][6] These film thicknesses are less than the mean free paths of many of the phonons ͑i.e., quantized lattice vibrations that dominate the thermal transport in semiconductors͒ in the corresponding bulk material.The thermal transport properties of semiconductor thin films are thus different than their corresponding bulk values. 7,8To aid in the design of devices employing thin films, accurate models for the film thermal transport properties are required.
0][11][12] In this approach, phonons are treated as particles and their wavelike nature is neglected.Closing the BTE requires specification of the phonon scattering rates ͑i.e., the phonon relaxation times or mean free paths͒ and boundary conditions.Typically, all of the phonons in the film are assumed to have a mode-independent mean free path. 9,104][15] The boundary conditions are often specified by assuming that a fraction of the phonons scatter specularly at the boundaries between the film and its adjoining medium, which may be a solid, fluid, or vacuum, while the remainder scatter diffusely. 10,13his fraction is often treated as a mode-independent fitting parameter, thereby neglecting the atomic-level detail of the boundaries.Even with more accurate methods to close the BTE ͑e.g., mode-dependent phonon properties obtained from ab initio calculations͒, the BTE-based approach will not be suitable for films with thicknesses on the same scale as the average wavelength of the thermally excited phonons, where the wavelike nature of the phonons is important. 16For Si at room temperature, this wavelength is estimated to be 1 nm. 168][19][20][21][22][23] In a MD simulation, the positions and momenta of a set of atoms evolve classically according to Newton's equations of motion.Such simulations are an ideal method for predicting thermal transport properties because no assumptions concerning the phonon scattering within the film or at the film boundaries are required.0][21][22][23] In these studies, the film thermal conductivity is predicted to increase toward the bulk value as the film thickness increases due to the decreasing effect of phonon scattering at the film/reservoir boundaries on the phonon mean free paths.5][26] These details can be more easily obtained using LD calculations, 27 in which the system dynamics are transformed from the real-space coordinates ͑the atomic positions͒ to the reciprocal-space coordinates ͑the phonon modes͒.By accessing the mode-dependent phonon properties ͑e.g., phonon frequencies, group velocities, and transmission coefficients͒, a more thorough understanding of thermal transport can be obtained.Because LD calculations are typically applied under the harmonic approximation, however, no information about inelastic phonon scattering ͑e.g., multiphonon scattering events͒ can be obtained because the phonon modes are decoupled. 17,28n this work, LD calculations and MD simulations are used to examine the dependence of thin film thermal resistance on film thickness.By comparing the predictions of the two methods, we examine how the thin film thermal resistance is affected by changes in the allowed vibrational states in the film and the presence of inelastic phonon-phonon scattering in the film.We consider two types of thin film structure ͑see Fig. 1͒.In one structure, a Ge film with thickness L F is confined between two Si leads ͑the Si/Ge/Si structure͒.In the other, a Si film is confined between two Ge leads ͑the Ge/Si/Ge structure͒.Similar configurations of a thin film confined by larger extents of a different species are often encountered in application, but have received minimal theoretical attention.In the Si/Ge/Si and Ge/Si/Ge structures, the Si/Ge interfaces are oriented along the ͑001͒ crystallographic plane and the film is symmetrically strained by setting the lattice constant in the x and y directions to the average of the bulk Si and bulk Ge lattice constants. 29The layer spacings in the z direction are chosen to give zero stress in that direction.These layer spacings are found using a steepest decent approach in the LD calculations and elasticity theory in the MD simulations, as described in Ref. 18.[32]

A. LD calculations
We predict the thermal resistance of the thin film region, which includes the film and the two lead/film interfaces ͑see Fig. 1͒.We refer to the thermal resistance of this region as the "thin film thermal resistance."It is defined as where q is the heat flux across the thin film region, and T L and T R are the lead temperatures at the lead/film boundaries.At the phonon-mode level, the steady-state heat flux across the thin film region is

͑2͒
where L and R denote the left and right leads, ប is the Planck constant divided by 2, denotes the phonon polarization, and , , and v z are the phonon wavevector, frequency, and z component of the group velocity.The first ͑second͒ integral is over the first Brillouin zone of the left ͑right͒ lead, and the first ͑second͒ summation is over phonons moving in the positive ͑negative͒ z direction.The mode-dependent phonon transmission coefficient, ␣ L→R , is defined as the fraction of the incident phonon energy that is transmitted from the left lead to the right lead ͑similar for ␣ R→L ͒.The variables f L and f R are the mode-dependent phonon distribution functions in the left and right leads at the lead/film boundaries.4][35][36][37][38] The equilibrium distribution function is where k B is the Boltzmann constant.Under this approximation, LD and MD predictions of the thermal resistance of an isolated Si/Ge interface ͑i.e., the Si/Ge thermal boundary re-sistance͒ are in agreement to within 5%. 18It is well known, however, that as the average phonon transmission coefficient approaches unity, the accuracy of this approximation decreases and the LD-predicted thermal resistances become too large. 18,39We therefore expect that approximating the phonon distributions using the equilibrium distribution will be suitable for modeling a thin film structure if the average phonon transmission coefficient across the film is less than or equal to that across the isolated Si/Ge interface.This condition is met when the thin film thermal resistance exceeds the Si/Ge thermal boundary resistance.Under this approximation, Eq. ͑2͒ can be substituted into Eq.͑1͒, yielding an expression for the thin film thermal resistance that can be simplified to ͑see Ref. 18 .

͑4͒
We evaluate R LD for the thin film structures using Monte Carlo integration with 10 5 random phonon wavevectors in the first Brillouin zone of the left lead, where the lead is defined using the two-atom diamond primitive cell.This level of Brillouin zone sampling results in a prediction uncertainty of Ϯ2%.For comparison to the classical MD predictions, Eq. ͑4͒ is evaluated in the classical limit by setting f BE = k B T / ប.At each wavevector, the frequencies and group velocities are obtained using LD calculations applied under the harmonic approximation.The phonon transmission coefficients are obtained using the harmonic LD-based scattering boundary method. 36,40,41This method assumes that ͑i͒ no inelastic phonon scattering occurs within the thin film region, ͑ii͒ the phonons scatter specularly at the lead/film boundaries, and ͑iii͒ the phonons are spatially delocalized.The assumption of no inelastic scattering is valid when the phonon scattering at the Si/Ge interfaces is elastic and the phonons travel ballistically across the film.In our previous work, we found that the Si/Ge thermal boundary resistance is temperature independent at temperatures less than or equal to 500 K, indicating that the first condition is met at these temperatures. 18The second condition is met when the film thickness is less than the phonon mean free paths in the corresponding bulk material.This point is discussed further in Sec.III C 2. The assumption of specular scattering is valid for our thin film structures because they contain no defects or roughness that would promote diffuse scattering. 13,42As discussed by Schelling and Phillpot, 43 in order to properly model the interference between phonons reflected from the lead/film interfaces when the phonon transport is ballistic, the extent of the phonon wave packet should be greater than the film thickness.The assumption that the phonons are spatially delocalized is thus appropriate.Full details related to our LD calculations can be found elsewhere. 18,27,44,45

B. MD simulations
To examine the effects of phonon-phonon scattering in the film, we compare the LD-predicted thermal resistances to values predicted using MD simulations, which require no assumptions about the nature of the phonon scattering.The MD predictions are made at a temperature of 500 K, where the phonon scattering at the Si/Ge interfaces is elastic. 18The velocity Verlet algorithm with a time step of 0.55 fs is used to numerically integrate the Newtonian equations of motion.The thin film thermal resistance is predicted using the direct method, in which a known heat flux is applied across the sample, the resulting steady-state temperature drop across the film is specified, and the thermal resistance is determined from Eq. ͑1͒.
A schematic diagram of our direct method simulation cell is shown in Fig. 2͑a͒.The system consists of either a Si/Ge/Si or Ge/Si/Ge structure that is bordered by hot and cold reservoirs of length L res = 50 monolayers and fixed boundaries in the z direction.The fixed boundary regions each contain four monolayers of fixed atoms, and the left and right leads have lengths L L = L R = 400 monolayers.The cross section of the simulation cell is square and has area A c equal to four unit cells by four unit cells.Based on the results of our previous examination into the effect of simulation cell size on the Si/Ge thermal boundary resistance, 18 we expect that these dimensions will generate size-independent predictions for the thin film thermal resistance.Periodic boundary conditions are imposed in the x and y directions.
The sample and reservoirs are initially set to a uniform temperature by scaling the atomic velocities for 0.55 ns ͑1 ϫ 10 6 time steps͒.A heat flux of q = 7.23 GW/ m 2 is then applied by adding a constant amount of kinetic energy to the hot reservoir and removing the same amount of kinetic energy from the cold reservoir at every time step using the method described by Ikeshoji and Hafskjold. 46This value of the heat flux does not influence the predicted thermal resistance. 18From this point, a period of 3.3 ns is allowed to reach steady-state conditions.By examining the evolution of the temperature profile during this period, we confirmed that steady-state conditions are met for each thin film structure.After steady-state conditions are reached, the steady-state temperature profile is obtained by averaging the temperature of each monolayer over an additional 2.75 ns.An example temperature profile is shown in Fig. 2͑b͒ for the Si/Ge/Si structure with L F = 10.6 nm.To minimize the uncertainty in specifying the temperature drop across the film, we apply a least-squares linear regression analysis to the temperature profile in each lead and evaluate the linear fits at the lead/ film boundaries.The nonlinear regions in the temperature profile found in the 100 monolayers closest to each reservoir/ lead boundary are neglected when performing the regression analysis, as shown in Fig. 2͑b͒.

A. Thin film thermal resistance
The LD-and MD-predicted thin film thermal resistances are provided in Fig. 3. Error bars representing the 95% confidence interval based on five independent simulations are  provided for four of the MD-predicted data points.The MD predictions are made for films with thicknesses between 0.2 and 30 nm, while the LD predictions are made for thicknesses between 0.2 and 4 nm.Because the LD-predicted thin film thermal resistances are independent of the film thickness for L F Ͼ 2 nm, however, we extrapolate the values at L F = 4 nm to allow for comparison to the MD predictions.The values of the thin film thermal resistance in the diffusive limit, R DL , reached when the film thickness is much greater than the phonon mean free paths in the corresponding bulk material, are also provided in Fig. 3 for comparison.The thin film thermal resistance in the diffusive limit is where R Si/Ge is the Si/Ge thermal boundary resistance and k is the bulk thermal conductivity of the film species.The L F / k term is the thermal conduction resistance often used in continuum-level heat transfer analyses. 47For our calculation of the diffusive limit, R Si/Ge is predicted using direct method MD simulations and k Si and k Ge are predicted using MD simulations and the Green-Kubo method.The predicted values of R Si/Ge , k Si , and k Ge at a temperature of 500 K are 2.93Ϯ 0.29ϫ 10 −9 m 2 K / W, 230Ϯ 47 W / m K, and 132Ϯ 34 W / m K. 18,44 For the Ge/Si/Ge structures, the MD-predicted thin film thermal resistances increase rapidly with increasing thickness for L F Ͻ 2 nm and are independent of thickness for 2 nm Ͻ L F Ͻ 30 nm.This trend is in qualitative agreement with that predicted from LD.The LD-predicted thin film thermal resistances, however, are greater than the MD-predicted values.For example, the LD-predicted thermal resistance of the Ge/Si/Ge structure with L F = 0.27 nm is two times greater than the MD-predicted value.We believe that this discrepancy is due to inaccuracy in the phonon distribution functions used in the LD calculations because the MD-predicted thin film thermal resistance is less than the Si/Ge thermal boundary resistance ͓see discussion preceding Eq. ͑4͔͒.For the structures with L F Ͼ 2 nm, the MD-predicted thin film thermal resistances are greater than the Si/Ge thermal boundary resistance, and the LD-and MD-predicted thermal resistances agree to within 15%.
For the Si/Ge/Si structures with L F Ͻ 2 nm, the MDpredicted thin film thermal resistances increase rapidly with increasing thickness.This trend is in agreement with that predicted from the LD calculations.Differences are observed between the MD and LD predictions, however, for structures with L F Ͼ 2 nm.For these structures, the MD-predicted thin film thermal resistances continue to increase toward the diffusive limit with increasing film thickness, while the LDpredicted values are independent of the film thickness.
We propose two mechanisms for understanding the thickness dependence of the thin film thermal resistances.The first mechanism is changes in the allowed vibrational states in the film, which can influence both the phonon transmission coefficients and the nature of the phonon-phonon scattering in the film ͑through the selection rules for multiphonon scattering events͒.Such changes arise due to the decreasing fraction of film atoms that are bonded to lead atoms and the increased resolution of the Brillouin zone of the film species ͑defined using the bulk primitive lattice vec-tors͒ as the film thickness increases.The second mechanism is increased phonon-phonon scattering in the film as the film thickness approaches or exceeds the mean free path of the phonons in the film.Such inelastic scattering leads to reduced coupling of phonons on either side of the film, which may alter the phonon transmission coefficients, and increased thermal conduction resistance of the film.In Secs.III B and III C, we will discuss how these two mechanisms affect the thermal resistance of the Si/Ge/Si and Ge/Si/Ge structures.

B. Allowed vibrational state effect
Because the LD calculations neglect inelastic phonon scattering, the thickness dependence of the LD-predicted thin film thermal resistances must be due to changes in the allowed vibrational states in the film.The MD-predicted thickness dependence for the Si/Ge/Si structures with L F Ͻ 2 nm and all of the Ge/Si/Ge structures is also likely to be a result of these changes due to the qualitative agreement between the LD-and MD-predicted trends.To quantify the thickness dependence of the allowed vibrational states, we use LD calculations to obtain the phonon density of states ͑PDOS͒, ͑͒, in the film.These calculations are performed using a unit cell that contains one atom for each monolayer in the film and one atom for each of 12 monolayers in each lead.This unit cell is periodically replicated in the x-y plane.Increasing the length of each lead has no discernible effect on the PDOS in the film.Extending on the concept of a local PDOS ͑i.e., the PDOS of a single atom in a supercell͒ used by others, 48 we define the PDOS in the film to be

͑6͒
where the j-summation is over the film atoms in the unit cell, the ␤-summation is over the three Cartesian coordinates, FIG. 3. ͑Color online͒ Thin film thermal resistances predicted from LDbased calculations and MD simulations.The LD-based calculations are performed in the classical limit for comparison to the classical MD simulations, which are performed at a temperature of 500 K.The thin film thermal resistance in the diffusive limit is also provided for comparison.
e j,␤ ͑ , ͒ is the j , ␤-component of the phonon polarization vector for mode ͑ , ͒, and When the PDOS is defined using Eq.͑6͒, its integral over frequency is equal to 3n, where n is the number density of atoms in the film.We evaluate Eq. ͑6͒ using Monte Carlo integration with 10 4 random phonon wavevectors in the twodimensional Brillouin zone for the thin film structure and a frequency bin width, ⌬, of 0.5 rad/ps.The PDOS in the films of three Ge/Si/Ge and three Si/ Ge/Si structures are provided in Figs.4͑a͒ and 4͑b͒.The PDOS for bulk Si and Ge, which are obtained under the same strain conditions that exist in the thin film structures, are also provided for comparison ͑denoted by L F = ϱ͒.For films with L F Ͻ 2 nm, the film PDOS is different than that of the corresponding bulk PDOS.Notable is the nonzero PDOS in the Ge film of the Si/Ge/Si structure between the maximum frequency in bulk Ge, max Ge , of 69 rad/ps and a frequency of ϳ90 rad/ ps ͓Fig.4͑b͔͒.As the film thickness increases, the PDOS in the film becomes increasingly bulklike, with minimal differences between the film and bulk PDOS observed for films with thickness L F Ͼ 2 nm.This transition to a bulklike PDOS coincides with the thickness where the thin film thermal resistances stop increasing rapidly with increasing film thickness.We take this agreement as evidence that the thickness dependence of the thin film thermal resistance for films with L F Ͻ 2 nm is due to changes in the allowed vibrational states in the film.
We further examine how the changes in the allowed vibrational states in the film affect the thermal transport by calculating the contributions to the LD-predicted thin film thermal conductance, G LD , as a function of either phonon frequency or incidence angle, .These functions, g and g , are defined such that In Figs.5͑a͒-5͑d͒, g and g are provided for the thin film ω (rad/ps) FIG. 5. ͑Color online͒ Dependence of the LD-predicted thin film thermal conductance on ͑a͒ phonon frequency and ͑b͒ phonon incidence angle for the Ge/Si/Ge structures and ͑c͒ phonon frequency and ͑d͒ phonon incidence angle for the Si/Ge/Si structures.For thin film structures with L F Ͼ 2 nm, g and g are independent of film thickness.structures for which we provided the PDOS in Figs.4͑a͒ and  4͑b͒.As with the film PDOS, g and g are independent of film thickness for L F Ͼ 2 nm.Vertical lines are provided in Figs.5͑a͒ and 5͑c͒ for the maximum phonon frequencies in bulk Si and Ge.We also provide a vertical line for the Si/Ge critical angle, crit , in Figs.5͑b͒ and 5͑d͒.By Snell's law, the phonon transmission coefficient is zero for a phonon incident on an interface from a material with a low group velocity to a material with higher group velocity ͑e.g., Ge to Si͒ if its incidence angle is greater than crit . 10,36,42We estimate crit to be 0.73 rad under the approximation that the phonon group velocities in each material are equal to the speed of sound, c, which we take to be the average of the three ͓001͔ acoustic phonon group velocities in the → 0 limit.Due to the realistic phonon dispersion used in our LD calculations, g is nonzero for incidence angles greater than this estimated crit .
We thus use crit only as a point of reference.
While the LD-predicted thermal resistances of the Ge/ Si/Ge and Si/Ge/Si structures are comparable in terms of magnitude and thickness dependence, the thickness dependence is manifested differently in each type of structure.As the film thickness in the Ge/Si/Ge structures decreases, g significantly increases only for greater than ϳ crit ͓see Fig. 5͑b͔͒, and g increases for all frequencies by an approximately frequency-independent factor ͓Fig. 5͑a͔͒.As the film thickness in the Si/Ge/Si structures decreases, g increases only for greater than ϳ30 rad/ ps ͓Fig.5͑c͔͒, and g increases for all incident angles by a factor that is independent of incidence angle ͓Fig.5͑d͔͒.For these structures, g is even nonzero for frequencies of max Ge Ͻ Ͻ 90 rad/ ps due to the vibrational states that exist in this frequency range when L F Ͻ 2 nm ͓see Fig. 4͑b͔͒.These trends are due solely to changes in the phonon transmission coefficients because the other phonon properties relevant to the thermal conductance ͓see Eq. ͑4͔͒ are independent of the film thickness.Furthermore, the frequency and angular dependences of the phonon transmission coefficients can be separated because there is little correlation between phonon direction and frequency in bulk Si and bulk Ge, as evidenced by their near-isotropic phonon dispersion curves. 49The g and g trends therefore suggest that the changes in the allowed vibrational states affect mostly the transmission coefficients of ͑i͒ phonons with Ͼ crit in the Ge/Si/Ge structures ͑independent of fre-quency͒ and ͑ii͒ phonons with Ͼ 30 rad/ ps in the Si/Ge/Si structures ͑independent of incidence angle͒.

Temperature profiles
As demonstrated in Sec.III B, thickness-dependent changes in the allowed vibrational states have negligible effect on the thin film thermal resistance for films with L F Ͼ 2 nm.Any thickness dependence of the thin film thermal resistance for these films must therefore be due to the effects of inelastic phonon-phonon scattering in the film.Because the MD-predicted thermal resistances of the Ge/Si/Ge structures with 2 Ͻ L F Ͻ 30 nm are thickness independent, we believe that the phonon transport is ballistic in the Si film.For the Si/Ge/Si structures with L F Ͼ 2 nm, however, the MD-predicted thin film thermal resistances increase with increasing film thickness, suggesting that the phonon-phonon scattering in the Ge film is non-negligible ͑i.e., the phonon transport is partially diffusive͒.Because the LD calculations assume ballistic phonon transport, this argument also explains why the LD-and MD-predicted thin film thermal resistances are in good agreement for the Ge/Si/Ge structures but are in poor agreement for the Si/Ge/Si structures with L F Ͼ 2 nm.
Additional evidence for ballistic phonon transport in the Ge/Si/Ge structures and more diffusive transport in the Si/ Ge/Si structures is found in the MD-predicted steady-state temperature profiles.The temperature profiles across a Ge/ Si/Ge structure with L F = 26.6 nm and a Si/Ge/Si structure with L F = 28.7 nm ͑the films in both structures have a thickness of 200 monolayers͒ are shown in Figs.6͑a͒ and 6͑b͒ as an example.The same heat flux of q = 7.23 GW/ m 2 was applied across both structures.To reduce statistical fluctuations, each temperature profile is averaged over five independent simulations.Using a least-squares linear regression analysis, we find that the temperature gradient in the Si film is −0.005Ϯ 0.007 K / nm ͑95% confidence͒.This value is a factor of 6 less than the value of −0.031K / nm that would develop in Si if the phonon scattering was entirely diffusive ͓calculated from the Fourier law, q =−k͑‫ץ‬T / ‫ץ‬z͔͒.Because phonon scattering is required to establish a temperature gradient, 11,[49][50][51] we take the small temperature gradient in the Si film relative to the value in the diffusive limit as evidence of ballistic phonon transport.In the Ge film, the temperature gradient is −0.165Ϯ 0.009 K / nm ͑95% confidence͒.This temperature gradient is a factor of 3 greater than the value of Ϫ0.060 K/nm that would develop if the phonon scattering was entirely diffusive, indicating the presence of phononphonon scattering in the film.
The MD-predicted thickness dependence of the film temperature gradient for a fixed value of the heat flux is also different between the Ge/Si/Ge and Si/Ge/Si structures.For all of the Ge/Si/Ge structures ͑L F Ͻ 30 nm͒, the temperature gradient is nearly zero and has no thickness dependence because the phonon transport is ballistic ͓see Fig. 6͑a͔͒.Flat temperature profiles have also been predicted by others using the lattice Boltzmann method for films in which the phonon transport is ballistic. 49,50As the film thickness increases, the lattice Boltzmann method predicts that the magnitude of the film temperature gradient increases toward the value in the diffusive limit as the phonon transport becomes increasingly diffusive.We expect that a similar trend would be observed for Ge/Si/Ge structures with L F Ͼ 30 nm.For the Si/Ge/Si structures, however, the opposite trend is observed, and the magnitude of the temperature gradient in the Ge film decreases toward the value in the diffusive limit as the film thickness increases.0][21][22][23] This difference suggests that the thermal transport in thin films depends not only on the species and thickness of the film but also on the materials that confine it.

Transition between ballistic and diffusive phonon transport
We now seek to determine why ballistic phonon transport is observed in the Ge/Si/Ge structures while more diffusive transport is observed in the Si/Ge/Si structures.The standard technique for determining whether the phonon transport is ballistic, diffusive, or partially diffusive is to compare the average bulk phonon mean free path, ⌳, to the film thickness.An estimate for ⌳ is obtained from the kinetic theory expression for thermal conductivity, 52 where C v is the specific heat per unit volume.Using thermal conductivities predicted from Green-Kubo MD simulations, classical specific heats, and our approximate sound speeds ͑see Sec.III B͒, we estimate ⌳ for both bulk Si and Ge to be 50 nm at a temperature of 500 K. Based on this result, the transition from ballistic to diffusive transport might be expected to occur for similar values of the film thickness in the Ge/Si/Ge and Si/Ge/Si structures, which is contrary to our MD predictions ͑see Fig. 3͒.A higher-level analysis of the phonon-phonon scattering in the bulk species is needed.We examine the details of the phonon-phonon scattering in bulk Si and Ge by calculating the phonon mean free path as a function of phonon frequency ¯z͑͒ at a temperature of 500 K.We define ¯z͑͒ to be ¯z͑Ј͒ Here, z is the mode-dependent phonon mean free path in the z-direction, which is equal to v z , where is the modedependent phonon relaxation time.We evaluate Eq. ͑10͒ using a frequency bin width of 1 rad/ps and phonon relaxation times predicted using anharmonic LD calculations for 32 3 evenly spaced wavevectors in the first Brillouin zone for the conventional diamond unit cell. 15,53The results are plotted in Fig. 7.
Consider Si/Ge/Si and Ge/Si/Ge structures with film thicknesses of L F = 1 nm.As shown in Fig. 7, ¯z is greater than this film thickness for all phonon frequencies in bulk Ge and bulk Si.The phonon transport in both films is thus ballistic.Now consider what happens as the film thickness increases.For the Si/Ge/Si structure, the film thickness will immediately begin to exceed ¯z in bulk Ge for frequencies near max Ge , leading to partially diffusive transport.When the film thickness reaches L F = 28.7 nm ͑our largest value for the Si/Ge/Si structure͒, the film thickness exceeds ¯z for Ͼ 31 rad/ ps.According to Fig. 5͑c͒, phonons in this frequency range contribute 51% to the thermal conductance when the phonon transport is ballistic ͓the LD calculations used to generate Figs.5͑a͒-5͑d͒ assume ballistic transport͔.For the Ge/Si/Ge structure, the phonon transport is entirely ballistic until the film thickness exceeds ¯z͑ max Ge ͒ = 5 nm because phonons with Ͼ max Ge do not contribute to the thermal transport ͓see Fig. 5͑a͔͒.When the film thickness reaches L F = 26.6 nm ͑our largest value for the Ge/Si/Ge structure͒, the film thickness exceeds ¯z for Ͼ 59 rad/ ps.According to Fig. 5͑a͒, phonons in this frequency range contribute only 7% to the thermal conductance when the phonon transport is ballistic.These results suggest that the phonon transport across the Si/Ge/Si structures should be more diffusive than that across the Ge/Si/Ge structures, which is consistent with our MD predictions of the thin film thermal resistance ͑see Sec.III A͒.We note that for both types of thin film structure, fully diffusive phonon transport is not expected until the film thickness is ϳ10 m due to the large phonon mean free paths at low frequencies and the lack of other scattering mechanisms in our models. 49

IV. SUMMARY AND CONCLUSIONS
We used LD calculations and MD simulations to predict the thin film thermal resistances of Ge/Si/Ge and Si/Ge/Si structures with film thicknesses, L F , between 0.2 and 30 nm.The LD calculations were performed in the classical limit for comparison to the classical MD simulations, which were performed at a temperature of 500 K.
For structures with L F Ͻ 2 nm, the thin film thermal resistances increase rapidly with increasing film thickness ͑see Fig. 3͒.Because the PDOS in the film is thickness dependent only in this range ͑Fig.4͒, we attributed this trend to changes in the allowed vibrational states in the film.We found that these changes affect the dependence of the phonon transmission coefficient on incidence angle for the Ge/Si/Ge structures and on frequency for the Si/Ge/Si structures ͓Figs.5͑a͒-5͑d͔͒.For Ge/Si/Ge structures with L F Ͼ 2 nm, the MDpredicted thermal resistances are thickness independent and agree with the LD-predicted values to within 15%.For Si/ Ge/Si structures with L F Ͼ 2 nm, the MD-predicted thermal resistances increase with increasing film thickness, while the LD-predicted values are thickness independent.Because the LD calculations assume that the phonon transport is ballistic, these results indicate that the phonon transport is ballistic in the Ge/Si/Ge structures and more diffusive in the Si/Ge/Si structures.We found that the standard technique of comparing the mode-averaged phonon mean free path to the film thickness is insufficient in predicting whether the phonon transport would be ballistic or diffusive in the film.The correct prediction can be made when the frequency dependence of the phonon mean free paths is considered ͑see Fig. 7͒.
The manner by which the temperature gradient in the film converges to the value in the diffusive for a fixed value of the heat flux was also found to be different between the two types of structure.For Si/Ge/Si structures, the magnitude of the temperature gradient decreases toward the value in the diffusive limit as the film thickness increases, while the opposite trend is expected for Ge/Si/Ge structures.We take this difference, along with the difference in the thickness dependence of the phonon transmission coefficients, as evidence that the thermal transport properties of a thin film depend on the materials that confines it in addition to the species and thickness of the film.This finding has implications in the interpretation of experimental data because it indicates that measured thermal conductivities may depend on the substrate material.

FIG. 4 .
FIG.4.͑Color online͒ PDOS in the films of ͑a͒ three Ge/Si/Ge and ͑b͒ three Si/Ge/Si structures.The PDOS for bulk Si and Ge ͑denoted by L F = ϱ͒ are also provided for comparison.
. Landry and A. J. H. McGaughey J. Appl.Phys.107, 013521 ͑2010͒ Author complimentary copy.Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp FIG.6.͑Color online͒ Steady-state temperature profiles near and across the thin film region of the Si/Ge/Si and Ge/Si/Ge structures with a film thickness of 200 monolayers ͑L F = 28.7 nm for the Ge film and L F = 26.6 nm for the Si film͒ predicted from MD simulation.The same heat flux of q = 7.23 GW/ m 2 was applied across both structures.

FIG. 7 .
FIG.7.͑Color online͒ Average phonon mean free path in bulk Si and Ge as a function of phonon frequency at a temperature of 500 K.
Author complimentary copy.Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp