Effect of crystalline/amorphous interfaces on thermal transport across confined thinfilms and superlatticesAshutosh Giri, Jeffrey L. Braun, and Patrick E. Hopkins Citation: Journal of Applied Physics 119, 235305 (2016); doi: 10.1063/1.4953683 View online: http://dx.doi.org/10.1063/1.4953683 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/119/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Crystalline-amorphous silicon nano-composites: Nano-pores and nano-inclusions impact on the thermalconductivity J. Appl. Phys. 119, 175104 (2016); 10.1063/1.4948337 Optical bandgap of single- and multi-layered amorphous germanium ultra-thin films J. Appl. Phys. 119, 014304 (2016); 10.1063/1.4939296 Lattice thermal conductivity of crystalline and amorphous silicon with and without isotopic effects from the ballisticto diffusive thermal transport regime J. Appl. Phys. 116, 043514 (2014); 10.1063/1.4891500 Effect of film thickness on the thermal resistance of confined semiconductor thin films J. Appl. Phys. 107, 013521 (2010); 10.1063/1.3275506 Interfacial effects on the thermal conductivity of a -Ge thin films grown on Si substrates J. Appl. Phys. 104, 074903 (2008); 10.1063/1.2986443

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07

Effect of crystalline/amorphous interfaces on thermal transport acrossconfined thin films and superlattices

Ashutosh Giri, Jeffrey L. Braun, and Patrick E. Hopkinsa)

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville,Virginia 22904, USA

(Received 29 March 2016; accepted 30 May 2016; published online 16 June 2016)

We report on the thermal boundary resistances across crystalline and amorphous confined thin

films and the thermal conductivities of amorphous/crystalline superlattices for Si/Ge systems

as determined via non-equilibrium molecular dynamics simulations. Thermal resistances across

disordered Si or Ge thin films increase with increasing length of the interfacial thin films and in

general demonstrate higher thermal boundary resistances in comparison to ordered films. However,

for films �3 nm, the resistances are highly dependent on the spectral overlap of the density of states

between the film and leads. Furthermore, the resistances at a single amorphous/crystalline interface

in these structures are much lower than those at interfaces between the corresponding crystalline

materials, suggesting that diffusive scattering at an interface could result in higher energy transmis-

sions in these systems. We use these findings, together with the fact that high mass ratios between

amorphous and crystalline materials can lead to higher thermal resistances across thin films, to

design amorphous/crystalline superlattices with very low thermal conductivities. In this regard, we

study the thermal conductivities of amorphous/crystalline superlattices and show that the thermal

conductivities decrease monotonically with increasing interface densities above 0.1 nm�1. These

thermal conductivities are lower than that of the homogeneous amorphous counterparts, which

alludes to the fact that interfaces non-negligibly contribute to thermal resistance in these superlatti-

ces. Our results suggest that the thermal conductivity of superlattices can be reduced below the

amorphous limit of its material constituent even when one of the materials remains crystalline.

Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4953683]

I. INTRODUCTION

The resistance to thermal transport across an interface

between two solids (thermal boundary resistance; TBR) can

severely restrict the dissipation of heat in systems reliant on

interfaces and junctions.1 This resistance can limit thermal

transport in nanosystems and microelectronic devices, and in

many cases ultimately leads to device failure.2–4 The knowl-

edge of phonon driven TBR is particularly crucial where sev-

eral layers of semiconductor thin films are stacked together

to form the active region in devices such as quantum cascade

lasers and light-emitting diodes5 as well as superlattices

(SLs) for thermoelectric applications.6,7

While the magnitude of the TBR is dependent upon

interface roughness, disorder, dislocations, and bonding,8

even interfaces on epitaxially grown crystalline films have

been demonstrated to have significant thermal resistance.9–12

Moreover, in practical applications that utilize sputtering or

evaporation techniques resulting in non-epitaxial film depo-

sition, disorder and film oxidation at the interfacial layer

between two solids (e.g., metal-oxide-semiconductor (MOS)

structures containing silicon include a non-stoichiometric ox-

ide layer present at the Si/SiO2 interface13) can introduce

additional resistance to thermal transport. Therefore, the dis-

ruption of crystallinity in thin films can influence the materi-

al’s vibrational characteristics, which in turn influences the

TBR across the thin films. It should be noted that throughout

this work, we refer to the TBR across thin interfacial films as

a lumped resistance that takes into account Kapitza resistan-

ces at both film/lead boundaries and the resistance due to the

interfacial film layer.

Computationally, the study of thin films acting as interfa-

cial layers between two solids has demonstrated the impor-

tance of film thickness and vibrational bridging (between the

leads) on TBR. English et al.14 showed that tuning the vibra-

tional properties of an interfacial layer between two dissimilar

materials can decrease the TBR. Landry and McGaughey15

showed that when the length scales of the interfacial thin film

are on the order of the intrinsic phonon mean free paths, bal-

listic phonon transport can reduce the resistance across these

crystalline interfacial layers such that the often employed ther-

mal circuit model fails to capture this phenomenon. Taken

together, the role of the interfacial layer’s length, intermixing

between the leads, and mass-mismatch with the leads can

greatly influence the total TBR across confined thin films;

manipulating these properties can provide useful tunability in

the overall thermal transport across devices with ordered or

disordered interfacial layers.

The aforementioned computational studies have mainly

focused on confined crystalline films with atomically smooth

interfaces. These “perfect” interfaces are generally not pres-

ent in real world applications (e.g., MOS structures, which

inherently include an amorphous native oxide interfacial

layer). Therefore, the main goal of this study is to understanda)Electronic mail: phopkins@virginia.edu

0021-8979/2016/119(23)/235305/8/$30.00 Published by AIP Publishing.119, 235305-1

JOURNAL OF APPLIED PHYSICS 119, 235305 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07

the role of amorphous interfacial layers and the effect of

mass and thickness of these layers on the TBR.

Generally speaking, pristine interfaces are considered

ideal for interfacial heat flow, and diffusive scattering caused

by disruption of crystallinity in the vicinity of interfaces is

usually thought to impede thermal transport. In this regard,

we also seek to understand the contribution of the individual

lead/film resistances on the total TBR across amorphous

interfacial layers. Finally, we also seek to design amorphous/

crystalline superlattices (SLs) with high interface densities

where the resistances across the amorphous/crystalline inter-

faces non-negligibly contribute to lowering the thermal

conductivities of these structures (even beyond that of the

amorphous counterpart). For these purposes, in this work, we

systematically investigate the combined role of disorder,

mass-mismatch, and layer length on thermal transport across

thin confined films and amorphous/crystalline SLs using mo-

lecular dynamics (MD) simulations.

For lattice-mismatched crystalline films confined

between two crystalline leads, we find that ballistic trans-

port across Ge/Si/Ge structures and diffusive transport

across Si/Ge/Si structures affect the TBR across the respec-

tive interfacial regions, similar to the findings of Landry

and McGaughey16 for lattice-matched Si/Ge structures. In

contrast, the TBR across amorphous confined films is

shown to monotonically increase with film thickness,

regardless of the leads or the material comprising the amor-

phous film. Furthermore, for amorphous films with thick-

nesses �3 nm, we demonstrate that the spectral overlap in

the vibrational density of states between the leads and the

thin films can significantly affect the scattering of the pho-

non flux impinging upon the interfacial layer from the crys-

talline leads. Interestingly, we also find that the resistance

across a single amorphous/crystalline interface is lower

than that across the crystalline/crystalline counterpart.

These findings suggest that careful manipulation of the

vibrational properties of the disordered confined thin film

could lead to reduced TBR in comparison to the case where

the interface is comprised of two crystalline materials.

The results from our original goal mentioned in the

previous paragraph led us to design amorphous/crystalline

SLs that demonstrate very low thermal conductivities in the

second part of our study. More specifically, we utilize the

fact that crystalline/crystalline interfaces can demonstrate

higher TBRs compared to amorphous/crystalline interfaces

in designing SL structures with very low overall thermal

conductivities. In this regard, the main motivation in

designing these SLs is driven by the push towards achiev-

ing ultralow thermal conductivities (beyond the amorphous

limit to thermal conductivity of a material17) for applica-

tions such as thermal barrier coatings and thermoelectric

power generation.3 Several groups have overcome the

amorphous thermal conductivity limit by nanostructuring

and utilizing multilayered systems with high density of

interfaces.18–22 Consequently, it has been shown that varia-

tion in length scale in multilayered systems such as semi-

conductor superlattices (SLs) can shift phonon behavior

from a coherent, wave-like nature in which phonons tra-

verse the structure without scattering at internal interfaces

to an incoherent, particle-like nature in which scattering of

vibrational energy at interfaces can severely limit thermal

transport in SLs.23–26 As a result, computational studies

have predicted that crystalline superlattices can possess

lower thermal conductivities compared to that of their con-

stituent components.27–29 This is particularly significant in

Si-based SLs, where a significant amount of heat is carried

by phonons with long mean-free-paths.30–33

Along with the motivation of designing SLs with very

low thermal conductivities, our objective for the second part

of this study is also driven by the lack of attention given

towards understanding thermal transport across SLs compris-

ing of alternating amorphous/crystalline layers (that result

due to large lattice mismatch between constituent materi-

als)34 due to the generalization that in amorphous systems,

amorphicity is expected to dominate thermal transport while

interfaces do not pose as significant resistors.35,36 While

the understanding of thermal transport across amorphous/

crystalline heterojunctions has been limited, there has been

realization of such heterojunctions as potential materials for

solar cell and various photo-electronic device applica-

tions.37–39 Therefore, in the second part of our study, we

investigate the effect of layer thickness and mass-mismatch

on thermal conductivity in Si/Ge-based amorphous/crystal-

line superlattices. For these superlattices, we show that the

introduction of interfacial densities �0.1 nm�1 can severely

hinder thermal transport to result in thermal conductivities

even below that of the amorphous constituent comprising the

SL. Moreover, we do not observe size effects in the thermal

conductivities of these amorphous/crystalline superlattices,

which is in contrast to the size dependent thermal conductiv-

ities of crystalline/crystalline superlattices.40 This implies

that at interface densities �0.1 nm�1, vibrational transport in

these structures is dictated by incoherent boundary scattering

effects. We also show that by making use of the fact that the

resistance across a crystalline/crystalline interface can be

greater than at an amorphous/amorphous interface, the ther-

mal conductivities of these superlattices can be lowered by

as much as 10% by creating superlattice periods that consist

of two adjacent crystalline layers followed by two adjacent

amorphous layers. Finally, when comparing our results of

mass-mismatched crystalline/amorphous SLs to that of

amorphous/amorphous SLs, we find that at very high mass

ratios between the adjacent layers in the SLs, the thermal

conductivities are similar for the two different structures

(regardless of crystallinity in the amorphous/crystalline SL),

indicating the strong role of interfacial resistances in SLs

with large mismatch in the vibrational density of states

between the layers.

II. METHODOLOGY

The TBRs across confined thin films and the thermal

conductivities of amorphous/crystalline SLs are calculated

via non-equilibrium MD (NEMD) simulations using the

LAMMPS package.41 A schematic of the computational do-

main for confined thin films between crystalline leads is

shown in the top panel of Fig. 1. These systems are either

composed of (i) crystalline Si leads with an amorphous or

235305-2 Giri, Braun, and Hopkins J. Appl. Phys. 119, 235305 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07

crystalline layer of Ge in-between the leads or (ii) crystalline

Ge leads with an amorphous or crystalline layer of Si in-

between the leads. The Si-Si, Ge-Ge, and Si-Ge interatomic

potentials are described by the Tersoff potential,42,43 which

accounts for the strain associated with the lattice mismatch

at a Si/Ge interface.43 All simulations for these systems are

performed at an average temperature of 500 K, where elastic

scattering has been shown to dominate the TBR.16 We apply

fixed boundary conditions on the z-direction by fixing 4

monolayers of atoms that border the hot and cold reservoirs

as shown in the top panel of Fig. 1. Periodic boundary condi-

tions are applied in the x- and y-directions. The leads are

each 400 monolayers, and the cross section area Ac of the

simulation domain is set to five unit cells by five unit cells.

Based on previous MD simulations on similar Si/Ge systems,

we do not expect size effects to affect our MD-predicted

TBRs.15,16 A time step 0.5 fs is applied throughout the simu-

lations, and the structures are initially equilibrated with the

NPT integration (which is the isothermal-isobaric ensemble

with the number of particles, pressure and temperature of the

system held constant)44 at zero-pressure and 500 K for a total

of 2.5 � 106 time steps.

To determine the TBR across the confined amorphous

and crystalline films, we implemented the NEMD method

where a thermal flux, q, is applied across a computational do-

main to establish a steady-state temperature gradient, @T/@z.

During the implementation of a heat flux, we remove the

thermostat and perform NVE integration (which is the

microcanonical ensemble with the number of particles, vol-

ume, and energy of the system held constant) while adding

and removing a fixed amount of energy per time step from

warm and cool baths, respectively. In doing so, we apply a

thermal flux, q¼ 3.0 GW m�2, across the computational

domains; note, increasing q by 50% produced statistically

invariant TBRs for the Si/Ge-based structures. After the ini-

tial application of the heat flux, we let the system reach a

steady-state for 3 ns. The steady-state temperature profile is

determined by averaging the temperature of the computa-

tional domain (by dividing it into equally spaced bins) for an

additional 3 ns. From the temperature profiles, the TBRs

across the interfacial films are determined by the relation,

q ¼ R�1K DT, as shown in the bottom panel of Fig. 1. To

determine the temperature drop at the film boundaries, we

apply a linear regression analysis to the temperature profiles

of the leads and calculate the exact temperature at the bounda-

ries between the leads and the film from the linear fit to the

MD-data (Fig. 1). The linear fits to the temperature profiles

reduce the uncertainty associated with determining the exact

temperature at the boundaries; this approach has been used

previously to determine interfacial resistances across solid-

solid,14–16,45,46 solid-liquid,47,48 and solid-gas49–51 systems.

The NEMD approach implemented in this study does

not directly shed light on the modal contributions to inter-

face conductance, which would substantially improve the

understanding of the NEMD results presented in this work.

In this regard, we note that recently there has been consid-

erable improvement over the analytical models (such as the

acoustic mismatch model and diffuse mismatch model1)

that consider modal contributions to TBR by the more rig-

orous approaches that implement atomistic simulations to

spectrally decompose the dynamics of individual phonon

eigen modes and how they interact both elastically and

inelastically.52–57 Although implementation of such meth-

ods to understand the modal contributions to TBR across

confined thin films is beyond the scope of this work, it is

worth mentioning that these in-depth analyses could poten-

tially enhance the significance of the results presented in

this work.

For our amorphous/crystalline SLs, the domains are

setup by first creating a homogeneous amorphous slab. The

crystalline layers of the SL are prepared by deleting atoms in

the amorphous domain according to the position of the atoms

and then substituting the deleted atoms with atoms arranged

along a diamond cubic lattice with an average value of the

lattice constants of Si and Ge. A similar method has also

been implemented in Refs. 34 and 58 to produce Si-based

amorphous/crystalline structures. Finally, energy minimiza-

tion is performed to set up the computational cells for the

amorphous/crystalline SLs. We separated the species of

atoms between the amorphous and crystalline layers depend-

ing on the position of the atoms in the computational cell,

and the final structures are relaxed under the Nose-Hoover

thermostat and barostat for a total of 2 � 106 time steps.59

The radial distribution function and atomic coordination

numbers were calculated for each amorphous and crystalline

layer in the SL structures to ensure no voids are formed dur-

ing and after energy minimization. It is well known that for

amorphous solids, the thermal conductivity is highly depend-

ent on the density of the solid.17,60,61 Therefore, we deter-

mine the densities of each a-Si and c-Ge layers (based on the

local volume and the number of atoms present in the respec-

tive layers in the relaxed SL structures). We confirm that the

calculated densities match with their bulk counterparts for

all of our SL structures with different interface densities.

To check for finite size effects, two domain lengths with

d� 21 nm and 43 nm were created for the SLs. The cross

section area for the SLs is set to five by five unit cells with

periodic boundary conditions in the x- and y-directions. The

FIG. 1. (Top panel) Schematic of the simulation cell for thin films confined

between crystalline leads. Thermal flux is applied across in the simulation

cell in the z-direction. (Bottom panel) Temperature profile of Si/Ge/Si sys-

tems with disordered and crystalline Ge films.

235305-3 Giri, Braun, and Hopkins J. Appl. Phys. 119, 235305 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07

top panel of Fig. 2 shows an example of the computational

domain created for the cystalline/amorphous superlattices,

and the bottom panel shows the temperature gradient

induced by the steady-state heat flux applied across the com-

putational domain (with a similar procedure as explained

above for the systems with confined thin films between two

leads). From the temperature gradient, the thermal conduc-

tivity is calculated by invoking the Fourier law as shown in

Fig. 2. The local temperature profiles (as shown in the inset

of Fig. 2) for the amorphous and crystalline layers are similar

due to the high interface densities in these structures; how-

ever, we note that the radial distribution functions calculated

for the crystalline layers show proof of crystallinity in those

layers.

III. RESULTS AND DISCUSSIONS

A. Thermal boundary resistance across the interfacialthin film

Figure 3 shows the MD predicted Kapitza resistances

across the amorphous and crystalline confined films (for the

Si/Ge/Si and Ge/Si/Ge systems) as a function of the film

thickness, Lf. The MD simulations are carried out for films

with thicknesses between 2 and 25 nm. We do not consider

films with thicknesses <2 nm due to the fact that the allowed

vibrational states highly depend on the length of the films

below 2 nm; in these cases, the TBRs increase rapidly with

increasing thickness of the confined films until 2 nm for both

Si and Ge films.15,62 We only study the thickness regime

where the material’s intrinsic vibrational density of states

does not depend on the thickness of the film.

For the Si/c-Ge/Si structures, the TBR increases monot-

onically with Lf (hollow triangles in Fig. 3). In contrast, the

resistances across the Ge/c-Si/Ge structures are independent

of the film thicknesses studied in this work (hollow squares).

Often, the resistance across interfacial thin films is described

by a thermal circuit model, which inherently represents the

resistance for the diffusive limit where the film thickness is

much greater than the mean-free-path of the phonons in the

film. The total resistance calculated by this model is the sum

of the two resistances associated with the film boundaries

and the resistance due to the finite thickness of the film.

However, the validity of this model is limited to situations

where non-diffusive transport is negligible. When comparing

the MD-predicted resistances across the Ge/c-Si/Ge films for

all thicknesses and for films <20 nm for the Si/c-Ge/Si struc-

tures, the thermal circuit model grossly overpredicts MD

simulated resistances.15 This is due to the fact that the heat

carrying phonons have mean-free-paths greater than or on

the order of the thickness of the thin films, and therefore, the

diffusive transport description fails to replicate the MD pre-

dictions due to considerable size effects. This deviation from

the diffusive limit is more apparent in the Ge/c-Si/Ge struc-

tures where we do not observe a change in the TBR with

increasing film thicknesses from 2 nm to 20 nm due to the

fact that the phonon transport across the thin films is mostly

ballistic.15 This is also evident from the increase in the ther-

mal conductivities (�6-folds) predicted from the temperature

profiles of the thin films compared to the bulk value of Si. In

contrast, for the Si/c-Ge/Si structures, the thermal resistance

for thicker films increases monotonically as shown in Fig. 3,

suggesting that phonon-phonon coupling becomes prominent

for these films. These results are consistent with the NEMD

predictions by Landry and McGaughey15 on lattice matched

Stillinger Weber (SW)-based Si/Ge structures where they

compare their MD-predictions to that of lattice dynamics cal-

culations and conclude that phonon transport is mostly bal-

listic for the Ge/c-Si/Ge and more diffusive for the Si/c-Ge/

Si structures.

In comparison to the crystalline films, the TBR for the

structures with the amorphous thin films is significantly

FIG. 2. (Top panel) Schematic of a 27� 27� 212 A3 simulation cell with

N¼ 0.57 nm�1 for an amorphous/crystalline superlattice. (Bottom panel)

Temperature gradient induced due to the applied flux across the computa-

tional domain. (Inset) Local temperature profiles for the amorphous/crystal-

line layers are similar due to high interface densities.

FIG. 3. Thermal boundary resistances across the confined thin films pre-

dicted via NEMD simulations performed at 500 K temperature. The TBRs

across Ge/c-Si/Ge are represented as hollow square symbols, solid square

symbols represent Ge/a-Si/Ge, hollow green triangles represent Si/c-Ge/Si,

and solid green triangles represent Si/a-Ge/Si. For the amorphous thin films,

the TBRs for interfacial layers with twice the mass of the layers are also

shown.

235305-4 Giri, Braun, and Hopkins J. Appl. Phys. 119, 235305 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07

greater for thicker films as shown in Fig. 3. The linear

increase in resistance is suggestive of diffusive transport

across the amorphous interfacial regions. Moreover, the

thickness trend in resistances predicted from the thermal

circuit model, which only considers the resistance due to

the amorphous layer (Lf/k), agrees well with the MD predic-

tions due to diffusive nature of heat propagation in the amor-

phous interfacial regions. Owing to the very low thermal

conductivities of the amorphous layers, it is very difficult to

accurately determine the resistance across a single amor-

phous/crystalline interface from the MD predicted tempera-

ture profiles for these structures. Typically, for crystalline

systems, the distinction between an interfacial resistance and

film thermal conductivity is well defined under the NEMD

framework due to the distinct temperature discontinuity at

interfaces. This distinction across the crystalline/amorphous

interfaces in our simulations is not clear, as the temperature

discontinuities at these interfaces are minute. However, the

accurate prediction of interfacial resistance at these bounda-

ries could be achieved via equilibrium MD simulations

where the resistance is calculated based on tracking the equi-

librium fluctuations and not dependent on the temperature dis-

continuity.63,64 Previously, this method has been used to

predict thermal interface conductance in crystalline/crystalline

Si/Ge superlattices63 and the resistance across a Lennard-Jones

based solid-gas interface.65 In fact, in Ref. 64, it is shown that

conductances calculated based on the equilibrium MD

approach are consistent with transmission describing diffusive

scattering, while the NEMD framework consistently describes

specular phonon scattering processes as determined for interfa-

ces consisting of atomically perfect Lennard-Jones solids.

Even though we cannot quantitatively prescribe an

accurate finite interface resistance, the NEMD-predicted

temperature profiles across our crystalline and amorphous

films suggest that the resistance across a single crystalline/

crystalline interface is greater than that at a single cystal-

line/amorphous interface. This result is consistent with

recent findings of interfacial resistance being higher across

a graphene/crystalline-SiC interface compared to a gra-

phene/amorphous-SiC interface, which was mainly attrib-

uted to a better overlap of density of states between the

graphene and amorphous-SiC compared to crystalline-

SiC.66 The lower resistance associated with amorphous sys-

tems is also similar to the findings reported in our previous

work on amorphous SLs where we show that the Kapitza

resistance across a single a-Si/a-Ge interface is �6 times

lower than between the corresponding crystalline materi-

als.67 Furthermore, Beechem and Hopkins68 have shown

that interfacial resistances predicted under the diffuse mis-

match model approach demonstrate lower resistances for an

amorphous/crystalline interface as compared to the predic-

tions for the crystalline counterparts. They attribute the

lower resistance at amorphous/crystalline interface to the

lack of abruptness of the system’s change from one material

to another, which ultimately augments heat flow across the

disordered interface. These results allude to the fact that

nanoscale imperfections around an interface could result in

higher energy transmission across the interface.

For the Si/a-Ge/Si structures, the values of TBR do not

agree with those of the Si/c-Ge/Si for all film thicknesses

considered. This can be mainly attributed to the lower ther-

mal conductivities of the amorphous Ge compared to its

crystalline counterpart. Increasing the mass of the “Ge” layer

by a factor of two also increases the TBR across the thin

films (red triangles in Fig. 3). In contrast, for the Ge/Si/Ge

structures with amorphous and crystalline films, the TBRs

for the thinner films (with Lf< 5 nm) are comparable in val-

ues with each other, regardless of the order and mass of the

interfacial region. Although the bandwidths in the density of

states for amorphous and crystalline Si are very similar to

each other (Fig. 4(b)),67 the heat carrying vibrations in the

two phases are considerably different due to the fact that the

mean-free-path of vibrations in the crystal is much longer

than those in amorphous Si. This suggests that for thinner Si

films (with Lf< 5 nm) confined between Ge leads, the order

and thus the “mean-free-path” of the vibrations in the film do

not significantly influence the TBR as much as the relative

frequencies of the film and the leads (i.e., the spectral pho-

non “mismatch”). Therefore, to evaluate the spectral band-

width effect on the TBR across Ge/a-Si/Ge structures, we

perform additional NEMD simulations by varying the mass

of the a-Si, which effectively alters the available frequency

modes in the thin films.

The TBRs across Ge/a-Si/Ge with mfilm¼ 14 g mol�1 to

112 g mol�1 are plotted as a function of mass ratio between

the film and the leads in Fig. 4(a). Within uncertainties, the

MD predicted TBRs for 28.09 g mol�1<mfilm< 84.27 g mol�1

are similar. However, for relatively higher or lower masses

of the thin film, the TBRs increase considerably. As evident

from the MD predicted temperature profiles, part of the reason

FIG. 4. (a) Kapitza resistance of Ge/a-Si/Ge structures with Lf¼ 3 nm as a

function of mass-mismatch between the leads and the film. (b) Vibrational

density of states of crystalline Si and Ge (shaded region), and Si structures

with different mass ratios compared to the mass of the Ge lead.

235305-5 Giri, Braun, and Hopkins J. Appl. Phys. 119, 235305 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07

for the increase in TBRs for the heavier masses is due to the

reduction in thermal conductivities with increasing mass

(mfilm), while part of it is also due to the mismatch in the over-

lap of density of states between the film and the leads (calcu-

lated from the method described in Ref. 69) as shown in Fig.

4(b). The cutoff frequencies for the vibrational spectra shown

in Fig. 4(b) decrease with increasing mass due to the relation,

x / 1=ffiffiffiffiffiffiffiffiffiffi

mfilmp

. The vibrational spectra for 28.09 g mol�1

<mfilm < 84.27 g mol�1 fall within the cutoff frequency of

the crystalline Ge leads. Therefore, we do not observe a sig-

nificant difference in the TBR across films within this mass

range. However, for the film with relatively lighter mass

(mfilm¼ 14 g mol�1), even though the thermal conductivity

increases due to reduced mass, the drastic mismatch in the

overlap of density of states with that of the Ge leads (as shown

in Fig. 4(b)) significantly increases the TBR across the Ge/

light-a-Si/Ge structure. This suggests that the Kapitza resist-

ance across an individual amorphous/crystalline interface can

be increased by significantly enhancing the mismatch between

the density of the states of the two materials by effectively

creating a high mass ratio between the materials. This could

be particularly beneficial in designing SLs with small period

thicknesses (and high interface densities) where low thermal

conductivities are desired.

B. Thermal conductivity of amorphous/crystallinesuperlattices

The thermal conductivities of amorphous/crystalline Si/

Ge SLs were predicted for two different domain lengths,

d¼ 21 nm and 43 nm. In NEMD simulations, domain lengths

have been shown to significantly influence the thermal con-

ductivities of the structures due to the fact that the fixed

domains (in the direction of the applied flux) serve as phonon

scattering sites by effectively shortening the mean-free-paths

of the energy carriers. This is exemplified by the thermal

conductivities plotted as a function of interface density in

Fig. 5 for SW-based crystalline/crystalline Si/Ge SLs that

are taken from Ref. 40 (solid triangle and diamond symbols).

As is clear, the two domain lengths produce completely dif-

ferent thermal conductivities for the c-Si/c-Ge SLs, suggest-

ing that size effects can greatly influence the thermal

conductivities (by as much as four-fold) in these superlattice

structures. In contrast to these cystalline/crystalline SLs, we

do not observe any statistically significant change in the pre-

dicted thermal conductivities of our amorphous/crystalline

SLs as evident from the overlap of the predicted thermal

conductivities for the two domain lengths as shown in Fig. 5

(hollow and solid squares).

The MD-predicted thermal conductivities for our amor-

phous/crystalline SLs decrease linearly with increasing N,

implying that interfaces contribute non-negligibly to thermal

resistance in these structures. The trend observed in Fig. 5

also suggests that vibrations are mostly particle-like in nature

and incoherent scattering at the internal boundaries of the

SLs leads to a monotonic reduction in thermal conductivity

with increasing N. In contrast, for the crystalline/crystalline

SLs, the thermal conductivity increases with increasing N,

which can be attributed to the wave nature of phonon

transport in structures where the phonons do not scatter at

the internal boundaries of the SLs or to ballistic transport

across relatively thicker periods;23–25 note this coherence

effect in the calculated thermal conductivity of SW Si/Ge

crystalline SLs with “perfect” interfaces has been previously

observed via MD simulations.70

Another aspect of the results shown in Fig. 5 worth not-

ing is the fact that the thermal conductivities of our amor-

phous/crystalline SLs are below the thermal conductivity of

amorphous Si as predicted by additional MD simulations

(dotted line in Fig. 5). Therefore, by creating interfaces, we

can effectively lower the thermal conductivity even below

that of the amorphous material, which is often thought to be

the minimum limit to thermal conductivity in materials. We

have previously shown this for Lennard-Jones and Stillinger

Weber-based amorphous/amorphous SLs; however, we note

that the thermal conductivities of our amorphous/crystalline

structures are slightly higher than those of the a-Si/a-Ge

SLs.67 In Ref. 67, we have shown that the Kapitza resistance

across a single a-Si/a-Ge interface can be six times lower as

compared to that in their crystalline counterparts. This

implies that creating SL structures with two amorphous

layers and two crystalline layers forming a period in the SL

would lead to lower thermal conductivities due to the higher

Kapitza resistances across c-Si/c-Ge interfaces relative to a-

Si/a-Ge interfaces. To validate this hypothesis, we run addi-

tional NEMD simulations on SLs structures with a-Si/a-Ge/

c-Si/c-Ge and find that the thermal conductivities of these

structures are systematically �10% lower than the a-Si/c-Ge

structures with the same layer thicknesses (red squares in

Fig. 5). Thus, strategic placement of layers and control over

the thicknesses of the individual layers in these SLs can pro-

duce structures with thermal conductivities that are below

the minimum limit of their amorphous counterparts.

FIG. 5. Thermal conductivities of Si/Ge SLs as a function of interface

density N. Solid square symbols are MD predictions on a-Si/c-Ge SLs with

domain lengths of 21 nm, while hollow square symbols are predictions for

a-Si/c-Ge SLs with domain length of 43 nm. For comparison, MD predic-

tions taken from Ref. 40 on crystalline Si/Ge superlattices with two domain

lengths are also plotted.

235305-6 Giri, Braun, and Hopkins J. Appl. Phys. 119, 235305 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07

The thermal conductivities of our amorphous/crystalline

SLs can be further lowered by increasing the mass-mismatch

of the layers in the SL. Figure 6(a) shows the thermal con-

ductivities of a-Si/c-Ge with d¼ 21 nm, N¼ 0.57 nm�1, and

with the mass of the Ge layer ranging from 56 g mol�1 to

145 g mol�1 (solid circles). As is evident in the plot, the ther-

mal conductivities of these SLs decrease with increasing

mass-mismatch. The reduction in thermal conductivity is

due to the combined effect of increase in the Kapitza resist-

ance and decrease in the thermal conductivity of the layers

from the increase in mass, as mentioned above for our con-

fined thin films. For comparison, we have also included our

MD predicted thermal conductivities of mass-mismatched

amorphous SLs from Ref. 67. The thermal conductivities for

the two different SLs converge as the mismatch between the

masses in the layers increases. This demonstrates the ability

of vibrational mismatch and resulting Kapitza resistance at

SL interfaces to dominate the thermal transport mechanisms,

regardless of the degree of crystallinity of the materials.

The thermal conductivity of the a-Si/c-Ge with

N¼ 0.57 nm�1 is plotted as a function of temperature in Fig.

6(b) (solid circles). Along with c-Ge/a-Si systems, c-Si/a-Ge

SLs have a slightly lower thermal conductivities (as shown

in Fig. 6(b)), which is mainly due to the lower thermal con-

ductivity of a-Ge compared to a-Si. The thermal conductiv-

ities do not show a temperature dependency, contrary to

purely crystalline SLs70,71 but consistent with results for

amorphous SLs.67 In Ref. 34, similar results are obtained for

a-Si/mass-heavy c-Si SLs where they show a very slight

increase in thermal conductivity with temperature. We note

that from the analyses presented in this work, it is not possi-

ble to separate the mode dependent harmonic and anhar-

monic contributions to the lack of temperature dependencies

observed in Fig. 6(b). However, more rigorous analyses

methods as presented in Refs. 56, 72, and 73 can be used to

quantitatively give insight into the mode level details to fully

understand these temperature dependencies.

IV. CONCLUSIONS

We have used molecular dynamics simulations to inves-

tigate the thermal boundary resistance across amorphous and

crystalline confined films of Si/Ge/Si and Ge/Si/Ge struc-

tures with film thicknesses ranging from 2 nm to 25 nm. We

find that for crystalline thin films, depending on the film and

the leads, the combined ballistic and diffusive behavior of

phonon transport can influence the thermal boundary resist-

ance. Conversely, for amorphous films, the resistance

increases monotonically regardless of the leads or the amor-

phous material comprising the thin film. From the tempera-

ture profiles obtained from the NEMD simulations, the

predicted resistances across single amorphous/crystalline

interfaces are lower than that across the crystalline counter-

parts. This suggests that diffusive scattering at amorphous/

crystalline interfaces can lead to higher energy transmissions

as compared to crystalline/crystalline interfaces for these

systems. Finally, for structures with film thicknesses �3 nm,

we find that the thermal boundary resistance across the inter-

facial layer is highly dependent on the overlap of density of

states between the leads and the confined films.

The fact that high resistances arise due to high mass-

mismatch between the �3 nm confined films and the leads

and the fact that crystalline/crystalline interfaces demon-

strate higher resistances compared to crystalline/amorphous

interfaces are used to design SL structures with very low

thermal conductivities. In this regard, NEMD simulations

are performed to predict the thermal conductivities of amor-

phous/crystalline Si/Ge SLs with varying interface densities

and mass ratios between the layers. In contrast to crystalline/

crystalline SLs, the thermal conductivities of our amorphous/

crystalline SLs do not demonstrate size effects for interface

densities above 0.1 nm�1. This suggests that for these inter-

face densities, the wave-nature of phonon transport observed

for crystalline Si/Ge SLs is absent in our amorphous/crystal-

line SLs. We also find that the thermal conductivities

decrease monotonically with increasing interface densities,

which further supports the claim that incoherent scattering of

vibrations limits thermal transport across these SLs. We

design the period of the SLs to consist of two crystalline

layers followed by two amorphous layers to show �10%

reduction in the thermal conductivity, which is mainly attrib-

uted to the higher resistance at the crystalline/crystalline

interface. Furthermore, we have shown that at high mass

ratios between the layers in the amorphous/crystalline SLs,

the thermal conductivities are comparable to those of amor-

phous/amorphous SLs.

FIG. 6. (a) Thermal conductivities of SW-based a-Si/c-Ge SLs with

N¼ 0.57 nm�1 plotted as a function of mass-mismatch between the layers.

For comparison we also include thermal conductivities of SW-based a-Si/

a-Ge from Ref. 67. Note, for fair comparison, we show the thermal conduc-

tivities of SW-based SLs as a function of mass ratios. However, SLs defined

by the Tersoff potential demonstrate statistically similar thermal conductiv-

ities. (b) Thermal conductivity of amorphous/crystalline Si/Ge SLs

described by the Tersoff potential (layers differentiated by mass and bond).

235305-7 Giri, Braun, and Hopkins J. Appl. Phys. 119, 235305 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07

ACKNOWLEDGMENTS

We would like to thank the support from the Office of

Naval research (N00014-15-12769).

1E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989).2D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar,

H. J. Maris, R. Merlin, and S. R. Phillpot, J. Appl. Phys. 93, 793 (2003).3D. G. Cahill, P. V. Braun, G. Chen, D. R. Clarke, S. Fan, K. E. Goodson,

P. Keblinski, W. P. King, G. D. Mahan, A. Majumdar, H. J. Maris, S. R.

Phillpot, E. Pop, and L. Shi, Appl. Phys. Rev. 1, 011305 (2014).4E. Pop, Nano Res. 3, 147 (2010).5Y.-L. Li, Y.-R. Huang, and Y.-H. Lai, Appl. Phys. Lett. 91, 181113

(2007).6G. Chen, M. S. Dresselhaus, G. Dresselhaus, J.-P. Fleurial, and T. Caillat,

Int. Mater. Rev. 48, 45 (2003).7M. Dresselhaus, G. Chen, M. Tang, R. Yang, H. Lee, D. Wang, Z. Ren,

J.-P. Fleurial, and P. Gogna, Adv. Mater. 19, 1043 (2007).8P. E. Hopkins, ISRN Mech. Eng. 2013, 682586.9R. M. Costescu, M. A. Wall, and D. G. Cahill, Phys. Rev. B 67, 054302

(2003).10B. Krenzer, A. Hanisch-Blicharski, P. Schneider, T. Payer, S. M€ollenbeck,

O. Osmani, M. Kammler, R. Meyer, and M. Horn-von Hoegen, Phys. Rev.

B 80, 024307 (2009).11A. Hanisch, B. Krenzer, T. Pelka, S. M€ollenbeck, and M. Horn-von

Hoegen, Phys. Rev. B 77, 125410 (2008).12J. Wang and J.-S. Wang, J. Phys.: Condens. Matter 19, 236211 (2007).13F. Jolly, F. Rochet, G. Dufour, C. Grupp, and A. Taleb-Ibrahimi, J. Non-

Cryst. Solids 280, 150 (2001).14T. S. English, J. C. Duda, J. L. Smoyer, D. A. Jordan, P. M. Norris, and L.

V. Zhigilei, Phys. Rev. B 85, 035438 (2012).15E. S. Landry and A. J. H. McGaughey, J. Appl. Phys. 107, 013521 (2010).16E. S. Landry and A. J. H. McGaughey, Phys. Rev. B 80, 165304 (2009).17D. G. Cahill, S. K. Watson, and R. O. Pohl, Phys. Rev. B 46, 6131 (1992).18H. Mizuno, S. Mossa, and J.-L. Barrat, Sci. Rep. 5, 14116 EP (2015).19E. Dechaumphai, D. Lu, J. J. Kan, J. Moon, E. E. Fullerton, Z. Liu, and R.

Chen, Nano Lett. 14, 2448 (2014).20X. Mu, X. Wu, T. Zhang, D. B. Go, and T. Luo, Sci. Rep. 4, 3909 EP

(2014).21C. Chiritescu, D. G. Cahill, N. Nguyen, D. Johnson, A. Bodapati, P.

Keblinski, and P. Zschack, Science 315, 351 (2007).22G. Pernot, M. Stoffel, I. Savic, F. Pezzoli, P. Chen, G. Savelli, A. Jacquot,

J. Schumann, U. Denker, I. M€onch, C. Deneke, O. G. Schmidt, J. M.

Rampnoux, S. Wang, M. Plissonnier, A. Rastelli, S. Dilhaire, and N.

Mingo, Nat. Mater. 9, 491 (2010).23M. V. Simkin and G. D. Mahan, Phys. Rev. Lett. 84, 927 (2000).24M. N. Luckyanova, J. Garg, K. Esfarjani, A. Jandl, M. T. Bulsara, A. J.

Schmidt, A. J. Minnich, S. Chen, M. S. Dresselhaus, Z. Ren, E. A.

Fitzgerald, and G. Chen, Science 338, 936 (2012).25J. Ravichandran, A. K. Yadav, R. Cheaito, P. B. Rossen, A. Soukiassian,

S. J. Suresha, J. C. Duda, B. M. Foley, C.-H. Lee, Y. Zhu, A. W.

Lichtenberger, J. E. Moore, D. A. Muller, D. G. Schlom, P. E. Hopkins, A.

Majumdar, R. Ramesh, and M. A. Zurbuchen, Nat. Mater. 13, 168 (2014).26Y. Wang, H. Huang, and X. Ruan, Phys. Rev. B 90, 165406 (2014).27Y. Chen, D. Li, J. R. Lukes, Z. Ni, and M. Chen, Phys. Rev. B 72, 174302

(2005).28G. Chen, J. Heat Transfer 119, 220 (1997).29E. S. Landry, M. I. Hussein, and A. J. H. McGaughey, Phys. Rev. B 77,

184302 (2008).30Z. Aksamija and I. Knezevic, Phys. Rev. B 88, 155318 (2013).31M. Hu and D. Poulikakos, Nano Lett. 12, 5487 (2012).32S. T. Huxtable, A. R. Abramson, C.-L. Tien, A. Majumdar, C. LaBounty,

X. Fan, G. Zeng, J. E. Bowers, A. Shakouri, and E. T. Croke, Appl. Phys.

Lett. 80, 1737 (2002).

33D. Li, Y. Wu, R. Fan, P. Yang, and A. Majumdar, Appl. Phys. Lett. 83,

3186 (2003).34A. France-Lanord, S. Merabia, T. Albaret, D. Lacroix, and K.

Termentzidis, J. Phys.: Condens. Matter 26, 355801 (2014).35Z. Zhang, J. Roger, D. Fournier, A. Boccara, and J. Wang, Thin Solid

Films 186, 361 (1990).36B. Shen, Z. Zeng, C. Lin, and Z. Hu, Int. J. Therm. Sci. 66, 19 (2013).37A. Descoeudres, L. Barraud, S. De Wolf, B. Strahm, D. Lachenal, C.

Gucrin, Z. C. Holman, F. Zicarelli, B. Demaurex, J. Seif, J. Holovsky, and

C. Ballif, Appl. Phys. Lett. 99, 123506 (2011).38H. Ohta, R. Huang, and Y. Ikuhara, Phys. Status Solidi RRL 2, 105

(2008).39R. W. Fathauer, T. George, E. W. Jones, W. T. Pike, A. Ksendzov, and R.

P. Vasquez, Appl. Phys. Lett. 61, 2350 (1992).40K.-H. Lin and A. Strachan, Phys. Rev. B 87, 115302 (2013).41S. Plimpton, J. Comput. Phys. 117, 1 (1995).42J. Tersoff, Phys. Rev. B 37, 6991 (1988).43J. Tersoff, Phys. Rev. B 39, 5566 (1989).44W. G. Hoover, Phys. Rev. A 31, 1695 (1985).45Z. Liang and P. Keblinski, Phys. Rev. B 90, 075411 (2014).46S. Majumdar, J. A. Sierra-Suarez, S. N. Schiffres, W.-L. Ong, I. C. Fred

Higgs, A. J. H. McGaughey, and J. A. Malen, Nano Lett. 15, 2985

(2015).47H. Acharya, N. J. Mozdzierz, P. Keblinski, and S. Garde, Ind. Eng. Chem.

Res. 51, 1767 (2012).48B. H. Kim, A. Beskok, and T. Cagin, J. Chem. Phys. 129, 174701

(2008).49Z. Liang, W. Evans, and P. Keblinski, Phys. Rev. E 87, 022119 (2013).50Z. Liang, W. Evans, T. Desai, and P. Keblinski, Appl. Phys. Lett. 102,

061907 (2013).51Z. Liang and P. Keblinski, Int. J. Heat Mass Transfer 78, 161 (2014).52K. Gordiz and A. Henry, New J. Phys. 17, 103002 (2015).53K. Gordiz and A. Henry, Sci. Rep. 6, 23139 EP (2016).54Y. Chalopin, N. Mingo, J. Diao, D. Srivastava, and S. Volz, Appl. Phys.

Lett. 101, 221903 (2012).55Y. Chalopin and S. Volz, Appl. Phys. Lett. 103, 051602 (2013).56K. S€a€askilahti, J. Oksanen, J. Tulkki, and S. Volz, Phys. Rev. B 90,

134312 (2014).57T. Murakami, T. Hori, T. Shiga, and J. Shiomi, Appl. Phys. Express 7,

121801 (2014).58A. France-Lanord, E. Blandre, T. Albaret, S. Merabia, D. Lacroix, and K.

Termentzidis, J. Phys.: Condens. Matter 26, 055011 (2014).59W. Shinoda, M. Shiga, and M. Mikami, Phys. Rev. B 69, 134103

(2004).60D. Hondongwa, L. Olasov, B. Daly, S. King, and J. Bielefeld, Thin Solid

Films 519, 7895 (2011).61S.-M. Lee, D. G. Cahill, and T. H. Allen, Phys. Rev. B 52, 253 (1995).62C. Shao and H. Bao, Int. J. Heat Mass Transfer 85, 33 (2015).63Y. Chalopin, K. Esfarjani, A. Henry, S. Volz, and G. Chen, Phys. Rev. B

85, 195302 (2012).64S. Merabia and K. Termentzidis, Phys. Rev. B 86, 094303 (2012).65B. Peng, W. He, X. Hao, Y. Chen, and Y. Liu, Comput. Mater. Sci. 87,

260 (2014).66M. Li, J. Zhang, X. Hu, and Y. Yue, Appl. Phys. A 119, 415 (2015).67A. Giri, P. E. Hopkins, J. G. Wessel, and J. C. Duda, J. Appl. Phys. 118,

165303 (2015).68T. Beechem and P. E. Hopkins, J. Appl. Phys. 106, 124301 (2009).69J. C. Duda, T. S. English, E. S. Piekos, W. A. Soffa, L. V. Zhigilei, and P.

E. Hopkins, Phys. Rev. B 84, 193301 (2011).70E. S. Landry and A. J. H. McGaughey, Phys. Rev. B 79, 075316 (2009).71Y. K. Koh, Y. Cao, D. G. Cahill, and D. Jena, Adv. Funct. Mater. 19, 610

(2009).72W. Lv and A. Henry, New J. Phys. 18, 013028 (2016).73M. Gill-Comeau and L. J. Lewis, Phys. Rev. B 92, 195404 (2015).

235305-8 Giri, Braun, and Hopkins J. Appl. Phys. 119, 235305 (2016)

Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Download to IP: 137.54.2.234 On: Thu, 16 Jun 2016

15:32:07