Interfacial thermal resistance: Past, present, and future

Interfacial thermal resistance (ITR) is the main obstacle for heat flows from one material to another. Understanding ITR becomes essential for the removal of redundant heat from fast and powerful electronic and photonic devices, batteries, etc. In this review, a comprehensive examination of ITR is conducted. Particular focus is placed on the theoretical, computational, and experimental developments in the 30 years after the last review given by Swartz and Pohl in 1989. To be self-consistent, the fundamental theories, such as the acoustic mismatch model, the diffuse mismatch model, and the two-temperature model, are reviewed. The most popular computational methods, including lattice dynamics, molecular dynamics, the Green’s function method, and the Boltzmann transport equation method, are discussed in detail. Various experimental tools in probing ITR, such as the time-domain thermoreflectance, the thermal bridge method, the $3\omega$ method, and the electron-beam self-heating method, are illustrated. This review covers ITR (also known as the thermal boundary resistance or Kapitza resistance) of solid-solid, solid-liquid, and solid-gas interfaces. Such fundamental challenges as how to define the interface, temperature, etc. when the materials scale down to the nanoscale or atomic scale and the opportunities for future studies are also pointed out.

Figure 1

Schematic diagram and temperature profile for an interface composed of two dissimilar segments. The red and blue boxes at the two ends denote the high-temperature heat source $T_H$ and the low-temperature heat sink $T_L$, respectively. The dashed line in the temperature profile pinpoints the position of the interface, where an abrupt temperature discontinuity $\mathrm{\Delta }T$ is observed.

Figure 2

Phonon reflection and refraction at an ideal interface. Vicinal slab 1 is located at $-{\delta }_1<z<0$ in material 1 and vicinal slab 2 is located at $0<z<{\delta }_2$. The normal direction is along the $z$ axis. ${\theta }_1$ and ${\theta }_2$ denote the incident angle and refracted angle on each side, respectively.

Figure 3

Location of atoms along the $x\text{−}z$ plane perpendicular to the interface between (a) two similar lattices and (b) two dissimilar lattices. (c) Location of atoms along the $x\text{−}y$ plane for two dissimilar fcc lattices when the interface is parallel to the $(100)$ plane. The atoms in material 1 are shown to the left and the atoms in material 2 are shown to the right. Surface unit cells are shown in boxes when the lattice constant of material 2 is twice the lattice constant of material 1. The surface unit cell is larger than the primitive unit cell in material 1 marked by the dashed line. (d) Location of atoms along the $x\text{−}y$ plane for two identical fcc lattices with different crystalline orientations. The $(100)$ plane is parallel to the interface on the left, and the $(110)$ plane is parallel to the interface on the right. The surface unit cell is marked with boxes. From [376].

Figure 4

Schematic diagram for the Green’s function method. There are two semi-infinite leads attached to the interface region. Consequently, the Hamiltonian is divided into five parts. From [269].

Figure 5

Two kinds of fitting approaches used in the transient thermal relaxation simulation. (a) Fitting of the temperature difference between two segments according to the exponential decay function in Eq. (28) for a solid-liquid interface. From [430]. (b) Fitting of the total energy in one segment according to the integral function in Eq. (31) for a solid-solid interface. From [481].

Figure 6

Wave packet simulation of a short-wavelength LA phonon transmission process across a grain boundary located at $z=0$. From [416].

Figure 7

Schematic diagrams of the heat transfer network at the interfaces. There are two types of heat carriers (Carriers 1a and 1b) in material 1. Material 2 has (a) one type of heat carrier (Carrier 2) and (b) two types of heat carriers (Carriers 2a and 2b).

Figure 8

Representative temperature profile for a metal-nonmetal interface in the TTM. $T_e$ and $T_{\mathrm{ph}}$ denote the temperatures of electrons and phonons on the metal side ($z<0$), respectively, while $T_n$ is the phonon temperature in the nonmetal side ($z>0$). $T_{\mathrm{fit}}$ denotes the linear fit of the temperature profile in the electron-phonon equilibrium region. The total temperature jump at the interface ($\mathrm{\Delta }T=\mathrm{\Delta }{T}_{e\text{-ph}}+\mathrm{\Delta }{T}_{\text{ph-ph}}$) is composed of both an electron-phonon coupling contribution $\mathrm{\Delta }{T}_{e\text{-ph}}$ and a phonon-phonon coupling contribution $\mathrm{\Delta }{T}_{\text{ph-ph}}$. From [492].

Figure 9

Signal detection mechanism in TDTR and FDTR. (a) Modulated pump pulsed by a sine wave from electro-optic modulator in TDTR. (b) Modulated pump in FDTR. (c) Temperature signal at the sample surface (solid line) and the probe signal at a fixed delay time ($t_d$) in TDTR. (d) Temperature signal of the in-phase component and out-of-phase component in FDTR. (e) Signal detected by a rf lock-in amplifier. (f) $V_{\mathrm{in}}$ and $V_{\mathrm{out}}$ signal vs delay time in TDTR. From [214].

Figure 10

(a) Scanning electron microscope (SEM) image of suspended MEMS devices. The red rectangle indicates the 2D materials that are ready to measure. The scale bar is $20\mu \mathrm{m}$. (b) Measurement stage (electron-beam self-heating method) and chip carrier adapted inside the SEM sample holder in Tongji University.

Figure 11

Electron-beam self-heating method. (a) Schematic and (b) equivalent thermal resistance circuit of electron-beam self-heating method. (c) Measured $R(x)$ when the electron beam is scanned across a metal-semiconductor interface and a sudden jump, representing ITR, is observed. From [286].

Figure 12

Spectral heat flux from experiment and the DMM across the interface. From [198].

Figure 13

Experimental results of ITC with increasing adhesion layer thickness in a Au/sapphire interface. The solid lines indicate the results predicted by the DMM. From [210].

Figure 14

Measured ITC vs vibrational mismatch between metals in a metal-SAM-metal junction. From [310].

Figure 15

Measured ITC vs pressure. From [190], [191], and [502].

Figure 16

Room-temperature ITC predicted by DMM (solid line) and measured by experiment for the Al/Si interface or ${\mathrm{Ge}}_x{\mathrm{Si}}_{1-x}/\mathrm{Si}$ interface with the quantum-dot nanostructure as a function of roughness rms. From [189], [183], and [91].

Figure 17

Ratio of bulk modulus and ratio of atomic mass for (a) metal-sapphire interfaces and (b) metal-diamond interfaces. The contour lines are the guides for the eye. From [565].

Figure 18

(a) Schematic of the band structure and surface states at the metal-dielectric interface. (b) Phonon emission at the interface due to the surface states. From [296].

Figure 19

The experimentally measured ITC of $\mathrm{Pt}/\mathrm{Cu}$, $\mathrm{Pt}/\mathrm{Au}$, $\mathrm{Al}/\mathrm{Cu}$, $\mathrm{Cu}/\mathrm{Nb}$, and $\mathrm{Pd}/\mathrm{Cr}$ interfaces. The data are adapted from [158], [489], [504], and [51].

Figure 20

Four different categories of interfaces with low-dimensional structures. (a) Multilayers and superlattices with separation distance $d_s$. (b) Sandwich structures with 2D (or quasi-2D) thin layers of thickness $d_s$. (c) Nanoscale contact between two 1D nanowires with contact area $A_n$. (d) One-dimensional heterostructure inside a single nanowire with cross-section area $A_n$.

Figure 21

Thickness-dependent ITC in $\mathrm{metal}/2\mathrm{D}\mathrm{m}\mathrm{aterials}/{\mathrm{SiO}}_2$ interfaces. Data are adapted from [268]. From [238], and [529].

Figure 22

(a) Nanoscale contact between two nanowires in a thermal bridge measurement. From [488]. (b) Diameter-dependent normalized contact thermal conductance per unit area between MWCNTs. The labels denote the diameters of the two segments forming a crossed contact. From [531].

Figure 23

ITC of the $\mathrm{Si}/{\mathrm{NiSi}}_2$ interfaces in a 1D heterostructure. From [286].

Figure 24

Kapitza resistance of the $\mathrm{Si}/{}^4\mathrm{He}$ interface as a function of temperature at saturated vapor pressure. The experimental data are from [217] (dashed line A) and [14] (solid line B).

Figure 25

Relationship between the interfacial thermal transport and characteristic quantities at the solid-liquid interface. (a) ITC of the SAM-water interface as a function of $1+\cos \theta$. From [431]. (b) Normalized ITR of the graphene-water interface vs the reduced water density peak ${\rho }_r$. From [8]. (c) ITC for various Si/water interfaces vs the density depeletion length $\delta$. From [393]. (d) Normalized ITC of the $n$-perfluorohexane/Au interface $G_r$ vs the reduced water density peak ${\rho }_r$ at different pressure values. From [164].

Figure 26

Spectral analysis for the vibrational modes of a solid-liquid interface. (a) Vibrational DOS of the solid monolayer at a LJ solid-liquid interface with weak (black line, $\epsilon =0.0103\mathrm{eV}$) and strong (red line, $\epsilon =0.103\mathrm{eV}$) solid-liquid interaction strength. The gray shaded region shows the bulk DOS of an inner solid monolayer for comparison. From [42]. (b) Spectral ITC $g(\omega )$ at a LJ solid-liquid interface (left axis) with the interfacial interaction strength $e_{ls}=e_{ll}$ and the spectral thermal conductivity $\kappa (\omega )$ of the bulk solid (right axis). From [405]. (c) Spectral ITC $g(\omega )$ at the left solid-liquid interface with various interfacial interaction strengths $e_{ls}$. The shaded regions denote the contribution to the total $g(\omega )$ in the direction normal to the interface. From [405].

Figure 27

(a) Measured equilibrium thermal accommodation coefficient (TAC) of the inert gases helium, neon, argon, krypton, and xenon on tungsten. The data are adapted from [460] and [246]. (b) ITR of the inert gas–tungsten interface at 1 atm calculated from Eq. (58).

Figure 28

Numerical calculations of (a) the thermal accommodation coefficient and (b) the ITC of a Pt/Ar interface vs the solid-gas interaction strength ${ϵ}_{\mathrm{sf}}$ from molecular dynamics simulations. From [273].

Figure 29

Density distribution of Ar on Pt surface at 300 K for different solid-gas interaction strength conditions. Insets: corresponding atomic configurations. From [273].

Figure 30

Schematic of the Knudsen cell. From [148].