Heat transport across metal–semiconductor (dielectric) structure under steady state conditions

https://doi.org/10.1016/j.ijheatmasstransfer.2012.03.068Get rights and content

Abstract

With continued size reduction in microelectronic devices, the thermal boundary conductance between two materials becomes the main channel on thermal dissipation. In this work, we present a theoretical work on heat transport in two-layered systems consisting in metal and a semiconductor (dielectric) and considering the role of the interface thermal conductivity between them.

In a metal because the electrons are preferentially scattered by phonons with phonon momentum smaller than the average electron momentum (electron Fermi momentum), the electron–phonon collisions are more efficient in terms of energy relaxation than the phonon–phonon relaxation frequency and, in this case the phonon system can be described by the same temperature as the electron gas. Therefore the heat diffusion equation in a metal is solved in one temperature approximation with appropriate boundary conditions at the surface and at the interface. On the other hand, in the semiconductor the heat diffusion transport is described by the two-temperature approximation model.

Introduction

Thermal conductivity (or diffusivity) of solid film has been an important factor in the design of thin-film devices used in microelectronics and high-energy lasers systems. There the thin-film components are normally subjected to transient thermal energy due to resistive or inductive heating and optical absorption, respectively. With low thermal conductivity in the film or poorly matched film and substrate, the film may suffer excessive temperature and temperature gradients that can lead to improper operation or even permanent damage. The characteristics of heat conduction in materials are governed by their thermal conductivity (under steady-state conditions) and thermal diffusivity (under transient or periodic conditions). The two properties are related by α=κ/ρc, where α is the thermal diffusivity, κ is the thermal conductivity, c is the heat capacity and ρ is the density. Therefore, it is possible to determine the thermal conductivity from the thermal diffusivity measurements, provided c and ρ can be readily determined.

New applications in the electronic industry, with continuous trends of miniaturization and increasing power electronic devices, require materials with thermal conductivity exceeding 250–300 W/mK [1]. Recently, diamond–metal composites have drawn attention due to their enhanced thermal conductivity of up 670 W/mK which makes these composites attractive for demanding applications in microelectronics and semiconductors [2]. As a matter of facts diamond features as extreme hardness, high thermal conductivity (up to 2200 W/mK) and low thermal expansion, diamond is considered as an important material for the heat transfer in modern electronic devices. Therefore metal–diamond composites have become an opened and challenging area for researchers. However, the understanding of the heat conduction mechanism across the metal–diamond and most general metal–semiconductor interface remains an open question, systematic investigations across metal–semiconductor interface are scarce.

The rapidly growing use of femtosecond laser pulses (10−14–10−13 s) in practical applications and fundamental materials research has increased the demand for realistic kinetic model of transient nonequilibrium phenomena in metals. The ultrashort laser pulse absorbed in a metal raises the electron temperature considerably higher than the lattice temperature because of the difference in their specific heats. Due to the small electron heat capacity, the electron temperature in the irradiated metal can be transiently brought to values comparable to the Fermi temperature [3]. Subsequent electron cooling results mainly from two processes, namely electron–lattice thermal relaxation and electron heat transport. These are usually modeled with a set of coupled thermoconductivity equations for the electron and lattice components. These equations are nonlinear and can generally be solved numerically yielding the electron temperature distribution in the metal.

It is well known that in a closed electrical circuit electrons dominate heat conduction in metals whereas in semiconductors both, electrons and phonons conduct thermal energy. However in a non-uniformly heated conductor in the absence of charge current the heat flux carried out by electrons is negligible as compared with heat conducted by phonons [4].

The effect of thermal resistance on heat transfer problems in layered media has been investigated in many research works. Lor and Chiu [5] used the thermal wave model to study the effect of interfacial thermal resistance on the temperature distribution in multilayer thin film. Reflection and transmission occur when the initial wave impacts the contact system of dissimilar material. In addition, the radiation-boundary-condition model based on acoustic mismatch model or diffuse mismatch model had been adopted in their analysis. Liu [6] further analyzed the effect of interfacial thermal resistance and the ratios of the thermophysical properties of dissimilar materials, which are predicted by the radiation boundary condition model, on the dual-phase-lag thermal behavior in two layered thin films. Theoretical investigation of transient heat diffusion in layered systems has been presented in Ref. [7]. The temperature distribution was calculated in each layer using the time-dependent one-dimensional heat diffusion equation including the effect of the thermal resistance at the interface between the two layers, also see Refs. [8], [9], [10], [11], [12].

Interfaces between materials become increasingly important on small length scales. The interface thermal conductance (the reciprocal of thermal resistance), of many solid–solid interfaces have been studied experimentally but the range of observed interface properties is much smaller than the predicted by simple theory in the range of nanoscale length [13].

The interface thermal resistance is associated with the heat flow across all solid–solid as well as solid–liquid interfaces. Eventually Majumdar and Reddy [14] postulated that for heat transfer to occur across metal–nonmetal interfaces, energy transfer must occur between electrons and phonons. There are two possible pathways namely: (a) coupling between electrons of the metal and phonons of the nonmetal through anharmonic interactions at the metal–nonmetal interface, and (b) coupling between electrons and phonons within the metal, and then subsequently coupling between phonons of the metal and phonons of the nonmetal. Those authors derived expressions to estimate the conductance and its temperature dependence for electron–phonon coupling within the metal and also to estimate the total thermal conductance of metal–nonmetal interfaces. Ju et al. [15] further modified the theory and developed the two-fluid model to calculate the thermal resistance for the metal of finite thickness in terms of the phonon temperature.

Recently the effective thermal conductivity of the aluminium–diamond (Al–diamond) composites was determined using a longitudinal heat flow technique by measuring the temperature gradient across the composite sample in the steady state regime [16]. The experimental Al–diamond interface thermal resistance was calculated from the effective thermal conductivity data by implementing the Hasselman–Johnson scheme and physically confirmed using the continuum two-fluid model. More recently, Ordonez-Miranda et al. [17] considered the effect of the electron–phonon coupling on the effective thermal conductivity of metal–nonmetal multilayers. In this latter work under an external applied gradient of temperature, the two temperature model was used in order to describe heat transport in metal and one temperature approximation in the nonmetal layer. However this approximation in the metal layer is only valid in the presence of electron current density J. Otherwise, if J = 0 the two temperature model introduce spurious unphysical results on the heat transport across the metal–nonmetal layers.

In this work we present a theoretical investigation of heat transport across the metal–nonmetal interface under steady state conditions i.e. a constant heat flux is applied on surface the composite. The temperature distribution is calculated in each layer by using the time-independent one-dimensional heat diffusion equation with appropriate boundary conditions according to the experimental conditions. In particular in a metal, as it be will shown, in the absence of charge current heat is carried out by phonons while in the semiconductor heat is transported by both electrons and phonons. Under this approximation the effective thermal conductivity of the composite is discussed in terms of the one effective homogeneous layer with the same boundary conditions at the surfaces of both samples (composite and homogeneous samples).

Section snippets

Theoretical background

It is well known that the description of the conduction of heat and electricity is the concern of transport theory. In materials of ordinary size at ordinary temperatures, normal transport theory based on Boltzman equation applies. This is basically a semiclassical approach, which can fail if the physical size of the system becomes small enough. Then, from Boltzamn equation the electrical density current and the heat flux associated with electrons are given by the following expressions [4], [18]

Special cases

We now turn to the discussion of the results obtained so far and compare with previous theories on heat transport in steady state regime in layered systems. As can be observed in Eq. (13) the spatial distribution of the temperatures en each layer has different rates because of the different magnitudes of the metal, electron and phonon thermal conductivities and interfacial thermal resistance. It follows from expressions (13) that the finite energy exchanges between the electrons and phonons in

Conclusions

A theoretical analysis of heat diffusion across metal–semiconductor (dielectric) interfaces under steady state heat conditions has been performed. Solving the heat diffusion equation with appropriate boundary conditions, we obtain the electron and phonon temperature distribution in the semiconductor layer. Whereas in the metal layer because heat is carrier out by phonons (electrical current density vanishes), we used the one temperature approximation. It is shown that distribution temperature

Acknowledgments

This work has been partially supported by Instituto de Ciencia y Tecnologia de la Ciudad de Mexico (ICyTDF) and Consejo Nacional de Ciencia y Tecnología (CONACYT-Mexico).

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