1

Spectral Control of Thermal Boundary Conductance

between Copper and Carbon Crystals by Self-

Assembled Monolayers

Shih-Wei Hung†, ‡, Shiqian Hu†, Junichiro Shiomi†*

†Department of Mechanical Engineering, The University of Tokyo, Tokyo, Japan

‡Department of Materials Science and Engineering, City University of Hong Kong, Hong Kong,

China

2

ABSTRACT

Controlling the thermal boundary conductance (TBC) between copper and carbon crystals is

important since it can bottleneck the thermal conductivity when reinforcing copper with carbon-

crystals fillers, namely diamond or graphite, to develop heat sinks and spreaders needed for the

thermal management. In this work, by using the non-equilibrium molecular dynamics simulation,

we show how the TBC of copper/diamond is smaller than that of copper/graphite by an order of

magnitude due to poorer overlap of the lattice vibrational spectra. To improve the TBC at the

copper/diamond interface, the covalently bonded self-assembled monolayers (SAMs) with

different chain lengths are installed at the interface. The TBC is significantly improved, and

increases with the chain length to approach the value of copper/graphite interface. The spectral

analysis further identifies that this is because the low frequency vibrational modes of SAMs

becomes softer with increasing chain length due to disordering of the collective SAMs structure,

enhancing the spectral overlap with copper. The obtained results are useful to improve the thermal-

conductivity of metal/carbon-crystal composite materials.

3

KEYWORDS

thermal boundary conductance, metal matrix composite, molecular dynamics simulation, self-

assembled monolayer, chain-length dependence, vibrational spectral analysis, frequency-

dependent transmissivity

4

INTRODUCTION

In recent years, the increasing need for rapid dissipation of heat in electronics demands the

pursuit of materials with high thermal conductivity. One excellent candidate for thermal

management application is the metal matrix composite.1 Owing to the exceptionally high thermal

conductivity (up to 2200 W/mK), diamond reinforced metal matrix composites have gained

considerable attention to achieve further excellent thermal properties.2 Because copper can

provide an acceptable combination of high thermal conductivity and low coefficient of thermal

expansion, the copper/diamond composite is one of the most promising materials for high-

efficiency thermal management.3 According to the effective medium theory,4 the thermal

boundary conductance (TBC) plays a critical role in determining the effective thermal

conductivity of the composite. In fact, the thermal conductivity of the copper/diamond composite

has been limited mainly due to the poor TBC,5 which has been thought as a result of the weak

interfacial bonding between copper and diamond, and thus, one way to improve the TBC is to

strengthen the interfacial bonding chemistry by introducing the covalent-bonded molecules at the

interface.6,7

Self-assembled monolayers (SAMs),8 two-dimensional periodic arrays of organic molecules

that form spontaneously from solution on solid surfaces, have been widely investigated to

improve the TBC between two solid surfaces,7,9–19 because they are stable, highly ordered, easy

to prepare, and provide a wide range of organic functionality. Because the structural and

vibrational information at atomic scale is difficult to be achieved experimentally, molecular

dynamics (MD) simulations serve as a powerful tool to complement the experimental studies of

TBC at interfaces between materials. Previous MD studies20,21 have provided evidences that the

TBC between carbon nanotube and surrounding matrix is correlated to the spectral coupling of

5

atoms. In order to further tailor the thermal transport across two solid surfaces, the dependence

of TBC between two solid surfaces on the chain length of SAMs was recently discussed

extensively based on the MD simulations.9–11,13,16

Luo and Lloyd performed both equilibrium and non-equilibrium MD simulations10,16 to study

the thermal transport across gold/SAM/gold interface. It was observed that the chain-length

dependence of TBC is weak; the TBC decreases slightly with the increasing chain length.

Sasikumar et al.11 studied the dependence of TBC on both the chain length and conformation of

one single-molecule covalent junction with two crystalline silicon surfaces using the non-

equilibrium molecular dynamics (NEMD) simulations. They concluded that the TBC does not

depend significantly on the chain length but on the conformation instead. The values of TBC of

straight chains can be almost twice that of curved chains. Hu et al.13 carried out NEMD

simulations to calculate the TBC of gold/SAM/silicon interface. Their results suggested that the

TBC decreases slightly with increasing chain length, except for the shortest one whose chain

length is only three carbon atoms. They claimed that this behavior attributes to the interference

effects between gold and silicon surfaces, reducing the transmission coefficient of phonons of

SAMs below specific chain length. Wang et al.18 implemented the NEMD simulations to study

the TBC of diamond/SAM/diamond interface. The results revealed that the chain-length

dependence of TBC exhibits a peak value for SAMs with four carbon atoms and saturates for

SAMs with more than ten carbon atoms. By analyzing the overlap between vibrational spectra

and the phonon transmission coefficients, they concluded that the vibrational coupling between

neighboring chains in SAMs scatters the propagating modes, resulting in the degradation of

TBC. These findings provide important insights into the effect of chain length of SAMs on the

TBC between two solid surfaces. However, most of the studies focus on the interface between

6

the same materials. The thermal transport through SAMs between two materials with

distinctively different vibrational properties has not be well-studied.

The present work aims to investigate the TBC between copper and carbon crystals with

different lattice structures, i.e. diamond and graphite. The improvement in TBC by introducing

SAMs at the interface is also of interest. It has been demonstrated that the phononic contribution

to TBC at metal/dielectric interface dominates over the electric contribution,5,22,23 so only the

phononic contribution to the TBC is considered. To that end, NEMD simulations24 are performed

to calculate the TBC at copper/diamond and copper/graphite interfaces. For copper/diamond

interface, the effects of SAMs with different chain lengths on TBC are also studied. Finally,

underlying mechanism of the differences in TBC is analyzed from the viewpoint of vibrational

spectrum.

7

SIMULATION DETAILS

NEMD simulations are performed to study the interfacial thermal transport across

copper/diamond and copper/graphite interfaces. For copper/diamond interface, the interfacial

bonding chemistry is modified by introducing SAMs sandwiched between the copper and

diamond surfaces. SAMs with various chain lengths, S(CH2)n (hereafter called Cn, where n=3, 7,

11, 15, an 19), are examined to understand the effect of chain length on interfacial thermal

transport.

The molecular model systems of copper/diamond and copper/graphite are shown in Figure 1(a)

and Figure 1(b), respectively. Initially, the copper crystal is generated with face-centered cubic

structure with a lattice constant of 0.362 nm and the lattice constant of diamond crystal is 0.357

nm. The graphite crystal is created in an ABAB stacking sequence with an interlayer distance of

0.335 nm. In each graphene sheet, the carbon atoms are arranged in a honeycomb lattice with

separation of 0.142 nm. After the relaxation, the cross-sectional areas are 5.0×5.0 nm2 and

5.3×5.1 nm2 for copper/diamond and copper/graphite systems, respectively, and the thickness of

the copper and diamond layer is about 30 nm each. The SAMs are placed with a packing density

of 0.51 nm2 per molecule.

The interactions between copper atoms are modeled using the embedded atom method (EAM)

potential.25 The interactions of carbon atoms within diamond and graphene sheet are modeled

using the Tersoff potential.26 The interlayer van der Waals interaction between each graphene

sheets are modeled using the Lennard-Jones (LJ) potential with parameters taken from the

previous study.27 For the interactions between copper and diamond surfaces and copper and

graphite surfaces, the van der Waals interaction in the LJ potential form is used. The LJ

8

parameters have been developed to describe the interaction between copper atoms and carbon

nanotubes.28 The previous study of copper/graphene nanocomposites29 using this LJ potential

parameters has demonstrated that the vibrational properties at copper/graphene interface can be

captured. The optimized potentials for liquid simulations (OPLS) all-atom force field30 is

implemented to describe the atoms within the SAMs. The LJ potential parameters for SAMs and

other atoms are obtained by the combination rule. All the simulations are performed using large-

scale atomic/molecular massively parallel simulator (LAMMPS) molecular dynamics package31

and visualized with the software PyMOL.32 The cutoff lengths of all the LJ interactions are set to

1.0 nm. The equations of motion are integrated with a time step of 0.5 fs.

Due to the initial structural mismatch of each surface, the system is initially relaxed in the

isothermal−isobaric (NPT) ensemble with periodic boundary condition applied to all three

directions to remove the internal stress. After the equilibrium structures and system sizes are

achieved, two 0.3 nm thick layers at the two ends in the z direction are fixed to stabilize the

system in the z direction. Then, an equilibrium run in the canonical (NVT) ensemble at 300 K is

performed to ensure the system temperature. The system size in the z direction is adjusted

gradually according to the normal stress to remove the normal stress imposed at the interface.

After the systems reached equilibrium, NEMD simulations with constant heat exchange

algorithm33 are performed in the microcanonical (NVE) ensemble to calculate the TBC at the

interfaces. The heat source and heat sink regions are defined as a 1.0 nm thick layer next to the

fixed layers, as shown in Figure 1. The temperature gradient is established by adding heat to the

heat-source regime and subtracting heat from the heat-sink regime. The magnitude of the heat

flux, J, imposed is 0.6 GW/m2. The temperature difference, ΔT, at the interface can be observed

after the steady temperature profiles are obtained. A linear regression of the temperature profile

9

of each surface was applied (red dashed lines in Figure 1). The data points near the thermostat

regimes were excluded from the fitting procedure. The temperature difference was obtained from

the extrapolation of the linear fit of temperature profile to the interface. The value of TBC, G, is

then evaluated by G =J/ΔT. Additional 5 ns run is performed after the steady state was achieved

for the detailed analysis.

10

Figure 1. Schematics and temperature profiles of the simulation system of (a) copper/diamond

and (b) copper/graphite. The copper atoms are colored in yellow and the carbon atoms are colored

in gray. The heat source region is colored in red, and the heat sink region is colored in blue.

11

RESULTS AND DISCUSSION

The temperature profiles of copper/diamond and copper/graphite are shown in Figure 1(a) and

1(b), respectively. The significant discrepancy of temperature profiles for two interfaces can be

observed. The temperature difference at copper/diamond interface is much greater than that at

copper/graphite interface. The resultant values of TBC for copper/diamond and copper/graphite

interfaces are 9.06±0.16 MW/m2K and 81.76±14.85 MW/m2K, respectively. The value for

copper/graphite interface is consistent with the previous theoretical predictions of 84 MW/m2K

for copper and multilayer graphene nanocomposite.29 The result indicates the TBC at

copper/graphite interfaces is almost an order of magnitude greater than that at copper/diamond

interface. The mechanism of this will be discussed later.

It has been demonstrated that the TBC can be modified by varying the interfacial bonding

chemistry using SAMs.7 Thus, the SAMs are introduced by covalently bonding them to both

surfaces to improve the thermal transport for the copper/diamond interface. Three different chain

lengths of SAMs, C3, C7, C11, C15, and C19, are investigated to understand the chain-length

dependence of TBC. The resultant temperature profiles of copper/diamond interface with C3,

C7, C11, C15 and C19 are shown in Figure 2. The temperature difference at the interface

decreases with increasing the chain length, indicating the long-chain SAMs promote the

interfacial thermal transport. The resultant values of TBC with SAMs of various chain lengths

are plotted in Figure 3. The data at n=0 refers to the TBC at copper/diamond and copper/graphite

interfaces without SAMs. Compared with the bare copper/diamond interface, the four times

enhancement in TBC can be achieved by introducing the C3 SAMs at the copper/diamond

interface. For the chain length studied in this work, the value of TBC increases monotonically

with the chain length for chain length shorter than C15 and reaches a plateau afterward. The

12

highest value of TBC approaches to that of bare copper/graphite interface. The observed chain-

length dependence of TBC is inconsistent with the previous studies13,16,18 where the chain length

dependence of TBC is slight and maximum TBC exists for a certain chain length. In the studies

by Luo and Lloyd16 and Wang et al.18, the interface is formed by two solid surfaces of the same

material, i.e. gold/gold and diamond/diamond. In the work by Hu et al.13 only non-bonded

interactions are considered between SAMs and silicon surface. In the present study, the SAMs

are covalently bonded to two different surfaces with distinct vibrational properties, copper and

diamond, which may have led to the different chain-length dependence of TBC.

13

Figure 2. Temperature profiles and schematics of interface of (a) copper/C3/diamond, (b)

copper/C7/diamond, (c) copper/C11/diamond, (d) copper/C15/diamond, and (e)

copper/C19/diamond. For SAMs, the sulfur atoms are colored in yellow, the carbon atoms are

colored in green, and the hydrogen atoms are colored in white.

14

Figure 3. Thermal boundary conductance as a function of chain length of SAMs. n=0 refer to the

interfaces without SAMs.

The vibrational density of states (VDOS) is an important quantity to characterize the TBC

through the various phonon vibrational modes. To probe the mechanisms of interfacial heat

transfer, the VDOS of various components adjacent to the interface, including copper, diamond,

graphite, and SAM, are calculated. The VDOS, P(f), is computed by taking the Fourier transform

of normalized velocity autocorrelation functions of atoms, given by34

1 2

1

( ) (0)

( ) lim

(0) (0)

n

i i i

i i ft

n

i i i

i

m t

P f e dt

m

= −

−→

=

=

v v

v v

(1)

15

where f is the frequency, n is the number of atoms, mi is the mass of the ith atom, vi is the velocity

vector of the ith atom, <···> indicates an ensemble average, and t is the time. Because not all

components are isotropic, the total VDOS is further decomposed into the in-plane (x and y

directions) and out-of-plane (z direction) contributions.

As shown in Figure 4(a), there is almost no difference between the in-plane and out-of-plane

VDOS for copper and diamond surfaces because both surfaces are isotropic. The VDOS of

cooper surface is distributed in the regime of 0-10 THz; while the VDOS of diamond surface is

distributed in the regime of 20-45 THz. The mismatch of phonon vibrational modes between

copper and diamond surfaces leads to the poor heat transfer at interface, i.e. low TBC. In the case

of graphite, because of the anisotropic two-dimensional shape, there is a significant difference

between the in-plane and out-of-plane VDOS, as shown in Figure 4(b). The in-plane VDOS of

graphite surface is distributed in the regime of 25-75 THz. On the other hand, the out-of-plane

VDOS of graphite surface is much softer and distributed in the regime of 0-30 THz. The

discrepancy is mainly due to the different atomic interactions in the in-plane and out-of-plane

directions of graphite surface. The carbon-carbon covalent bonds are in the in-plane direction;

whereas the van der Waals interactions between each graphene sheets are in the out-of-plane

direction. The out-of-plane VDOS of graphite surface provides the overlap in phonon vibrational

modes with that of copper surface. In view of elastic energy transport, which is usually the main

contribution to interfacial heat conduction, the overlap results in the higher value in TBC

compared with the copper/diamond interface.

The VDOS of copper/SAM/diamond interfaces with different chain lengths of SAMs are

shown in Figure 5. The two peaks of VDOS of SAMs located around 90 THz is due to the

carbon–hydrogen covalent bonds. Again, in view of elastic energy transport, the high-frequency

16

vibrational modes of carbon–hydrogen bonds make no contribution to the heat conduction across

the copper/diamond interface because the vibrational modes of copper and diamond surfaces are

below 50 THz. The distribution of VDOS of backbone chain of SAMs locates in the regime of 0-

50 THz, covering the VDOS of both copper and diamond surfaces. From a spectral point of

view, the SAMs can serve as a bridge of phonon vibrational modes between copper and diamond

surfaces, contributing to the enhancement in TBC at copper/SAM/diamond interface.

The vibrational analysis is implemented to understand the chain-length effect on TBC. With

the increase in chain length of SAMs, the VDOS, including both in-plane and out-of-plane

components, shift slightly from low-frequency regime (below 25 THz) to high-frequency regime

(above 25 THz). This shift implies that the spectral overlap between copper surface and SAMs

will decrease; while the spectral overlap between diamond surface and SAMs will increase with

the increasing chain length of SAMs. In order to quantify the spectral overlap of two

components, the following equation is introduced35

surf SAM

surf/SAM

surf SAM

( ) ( )

( ) ( )

P f P f dfO

P f df P f df=

(2)

where Osurf/SAM is the spectral overlap between copper or diamond surfaces and SAMs, Psurf(f) is

the VDOS of copper or diamond surfaces, and PSAM(f) is the VDOS of SAMs. The calculated

spectral overlaps for various chain lengths of SAMs are plotted in Figure 6. The chain-length

dependence of Odiamond/SAM is consistent with that of TBC; while that of Odiacoppermond/SAM shows

an opposite trend. It is important here that, according to the temperature profiles in Figure 2, the

temperature difference at copper/SAM interface is much greater than that at diamond/SAM,

reflecting the TBC is dominantly governed by the copper/SAM interface instead of the

17

diamond/SAM interface. Therefore, the above analysis of overall spectral overlap does not

explain the chain dependence. This is understandable since the energy transmissivity should

depend on the modes even in the frequency overlap regime.

Figure 4. Vibrational density of states of various components of (a) copper/diamond and (b)

copper/graphite interfaces. The total spectra are decomposed into the in-plane and the out-of-plane

components.

18

19

Figure 5. Vibrational density of states of various components of (a) copper/C3/diamond, (b)

copper/C7/diamond, (c) copper/C11/diamond, (d) copper/C15/diamond, and (e)

copper/C19/diamond interfaces. The total spectra are decomposed into the in-plane and the out-

of-plane components.

Figure 6. Overlap of vibrational density of states of copper/SAM and diamond/SAM as a function

of chain length of SAM.

To take the frequency-dependent transmissivity into account, the spectral interface

conductance36–38 is implemented to further understand the spectral characteristics of interfacial

thermal transport.

2

R L

2( ) Re lim ( ) (0) i ft

ij i

j is

g f t e dtA T

−

−→

=

F v (3)

20

where A is the interfacial area, ΔTs is the temperature difference at the interface, Fij is the force

acting on atom i from atom j, and R and L refer to the right and left regimes of interface,

respectively. According to the temperature profiles in Figure 2, the TBC is dominated by the

copper/SAM interface. Thus, the imaginary interface of spectral interface conductance

calculation is set at the copper/SAM interface.

Moreover, considering the VDOS matching between the copper and SAMs are limited within

0-10 THz, we only focus on the chain-length dependence of interfacial conductance at the low-

frequency regime (highlighted in yellow in Figure 7). The interfacial conductance at the low-

frequency regime (1-5 THz) is higher for long-chain SAMs. The VDOS of SAMs with different

chain length at the low-frequency regime is shown in the inset of Figure 7 for reference. The

results of VDOS indicate the vibrations of long-chain SAMs are preferable at the low-frequency

regime than that of short-chain one, leading to the higher interfacial conductance. This

vibrational behavior can be explained by the structural distinction of SAMs with different chain

lengths. Figure 8(a) shows the density distributions of SAMs with different chain length

perpendicular to the copper/SAM interface. The peaks of density distributions are sharper for the

short-chain SAMs, implying the structures of short-chain SAMs are more ordered than the long-

chain ones. The vibrational frequency is higher for the more order short-chain SAMs, leading to

the reduction of interfacial conductance at the low-frequency regime. The distributions of

backbone dihedral angle of SAMs with different chain length are shown in Figure 8(b). The

trans conformation is characterized by a 180˚ dihedral angle. The higher fraction of trans

conformation indicates the molecular chain is more straight. The fraction of trans conformation

increases with increasing chain length when chain length is shorter than C11 and it decreases

slightly afterward. The flexible long-chain SAMs form curved chains more easily than the rigid

21

short-chain ones. For chain length shorter than C11, the TBC is better for the more straight

chains, which agrees well with the previous observation that the TBC is better for straight chain

owing to the ballistic transport.11 But not only the ballistic transport, the frequency-dependent

transmissivity also plays an important role in the interfacial thermal transport. Thus, although the

C11 SAMs are the most straight chain, the highest value of TBC is achieved with C15 SAMs.

Figure 7. Spectral interface conductance at the copper/SAM interface for SAMs with different

chain length. Inset: Vibrational density of states of SAMs with different chain length at low-

frequency regime.

22

Figure 8. (a) Density profiles of SAMs with different chain length in the direction perpendicular

to the copper/SAM interface, where zero in distance refers to the position of outmost copper

surface. (b) Distributions of backbone dihedral angle of SAMs with different chain length.

23

CONCLUSION

The mechanism of thermal transport across the copper and carbon materials is investigated by

performing NEMD simulations. The distinct difference in TBC is observed for copper/diamond

and copper/graphite interfaces. The value of TBC at copper/graphite interface is almost one order

of magnitude larger than that at copper/diamond interface. The poor TBC at copper/diamond

arises from the severe mismatch in vibrational modes. Due to the overlap of vibrations modes in

the out-of-plane direction, the thermal transport at copper/graphite interface is superior to that at

copper/diamond interface.

In order to enhance the thermal transport at copper/diamond interface, the SAMs are

introduced by covalently bonding them to copper and diamond surfaces. The TBC at

copper/SAM/diamond interface is improved significantly compared with that at the bare

copper/diamond interface. The TBC shows a strong dependence on chain length that the TBC

increases with the increasing chain length for chain length shorter than C15 and reaches a plateau

afterward. By applying the vibrational analysis, it is found that the chain-length dependence of

spectral overlap between SAMs and diamond surface is consistent with that of TBC while that of

spectral overlap between SAMs and copper surface shows an opposite trend. The further account

for frequency-dependent interfacial transmission, spectrally decomposed interfacial conductance,

is calculated, and it reveals that the chain-length dependence mainly comes from the

contributions in the low-frequency regime. The vibrational and structural analysis suggests that

these low frequency vibrational modes becomes softer for longer SAMs because of the gradual

disordering in the collective structure of SAMs, improving the spectral overlap with copper. The

results provide practical guidelines to further reinforce metal matrix composite material for

higher thermal conductivity.

24

AUTHOR INFORMATION

Corresponding Author

*shiomi@photon.t.u-tokyo.ac.jp

Notes

The authors declare no competing financial interests.

ACKNOWLEDGMENT

This work is partially supported by Japan Society for the Promotion of Science KAKENHI

19H00744. S. H. appreciates the financial support from Postdoctoral Fellowship for Overseas

Researchers Program of the Japan Society for the Promotion of Science.

25

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