#### Abstract

The reduction of nanodevices has given recent attention to nanoporous materials due to their structure and geometry. However, the thermophysical properties of these materials are relatively unknown. In this article, an expression for thermal conductivity of nanoporous structures is derived based on the assumption that the finite size of the ligaments leads to electron-ligament wall scattering. This expression is then used to analyze the thermal conductivity of nanoporous structures in the event of electron-phonon nonequilibrium.

#### 1. Introduction

The continued reduction in characteristic length scales of nanodevices has driven the need to understand the thermal characteristics of low-dimensional structures [1]. For example, the quasi-one-dimensional geometry of nanowires makes them ideal for applications such as field effect transistors (FETs), which are based on the transport of charge. However, the heat generation from the charge transport and subsequent thermal management of these FETs pose an ever-growing problem in further development and design due to thermal phenomena that arise as a result of the reduction of the characteristic lengths of the materials [2]. With the use of nanowires in FETs, a significant reduction in thermal conductivity results [3], which can drastically limit operational frequencies and powers since the removal of the generated heat is reduced compared to bulk. In addition, an increase in operational frequencies and powers in FETs can result in a large electric field experienced by the electrons which can throw the electrons out of equilibrium with the lattice resulting in another form of thermal resistance [4]. In this report, thermal processes in nanoporous gold films are considered. The nanoporous gold is essentially a random matrix of Au nanowires that exhibit a reduction in thermal conductivity due to scattering at the wire surfaces. During nonequilibrium electron-lattice processes, these scattering events must be considered to accurately predict heat flow through the structure. The rate of energy loss by an electron system out of equilibrium with its lattice is measured in nanoporous Au [5–7] with a pump-probe transient thermoreflectance (TTR) technique [8].

#### 2. Thermal Conductivity Reduction in Nanowires

The electron lattice
nonequilibrium, which drives electron-phonon coupling, has been the focus of
several studies [8–20] and recently this interest
has extended to nanowires [21–23]. In the case of bulk materials, at room
temperature, the resistance to electron transport is dominated by phonon
scattering. However, when the
characteristic length, *l*, of a
material is on the order of the mean free path of the electrons, , then the resistance becomes
affected by scattering of electrons at surfaces. In the case of the nanoporous Au, the
electron-surface scattering has been attributed to electronic coupling of
adsorbates to the conduction electrons which gives rise to the change in
resistance exploited in several applications [7].

This subsequent reduction in
electron system thermal conductivity (from here on will simply be referred to
as the thermal conductivity or conductivity since the focus of this work is
metal systems) that is associated with surface scattering is given by [3] where *k _{w}* is the reduced thermal
conductivity,

*k*is the conductivity of the corresponding bulk material, and

_{b}*u*is the ratio of the wire diameter,

*d*, to the electron mean-free path in the wire, . In the case of nanoporous Au, the wire diameter in (1) refers to the average ligament size. The corresponding bulk conductivity can be calculated from kinetic theory by , where

*C*is the electronic heat capacity, which at the temperatures of interest ( K) can be calculated by with

_{e}*being the Sommerfeld constant [20, 24], and*

*γ**v*is the Fermi velocity. The mean free path of the electrons in bulk is dominated by electron and phonon scattering, so can be estimated by , where is calculated with Matthiessen’s rule by taking into account electron-electron scattering, , and electron-phonon scattering, , where is the electron-electron scattering time, is the electron-phonon scattering time,

_{F}*T*is the electron temperature,

_{e}*T*is the lattice (or phonon) temperature, and

_{p}*A*and

*B*are scattering coefficients [25–27] that are weakly dependent on temperature in Au [28]. Matthiessen’s rule assumes that there are multiple physically distinguishable sources of scattering (e.g., electron-electron or electron-phonon), and the presence of one scattering mechanism does not alter the way in which the other mechanisms function [29].

Equation (1) is a result of an
exponential integral expansion that is valid as long as . Predicting the
reduction in wire thermal conductivity requires significantly more
consideration when [3]. However, by considering electron-boundary
scattering in *k _{b}*,

*k*can be easily estimated for all

_{w}*u*. When calculating in a nanowire, the effects of electron-boundary scattering can be incorporated via Matthiessen’s rule. This can be estimated as assuming complete diffuse scattering [30]. Note that is considered temperature independent [22]. Therefore, the conductivity of a nanowire can be expressed as which is dependent on electron and phonon temperatures and the wire diameter, and therefore can be used to examine thermal conductivity of nanowires in the event of electron-phonon nonequilibrium heating. Figure 1(a) shows the ratio as a function of Au wire diameter for various electron temperatures. In these calculations, the effects of electron phonon nonequilibrium are studied, therefore

*T*is assumed as 300 K. In Figure 1(b), the thermal conductivity is calculated as a function of wire diameter for various electron temperatures assuming . These results are compared to calculations using (1) when and simulation results from Au nanowires assuming nm when . Note that the results from [3] for and (1) for agree well with (2) over the entire range of wire diameters when

_{p}*T*equals 240 K, which corresponds to a of 45 nm assuming no boundary scattering. Figure 2 presents the electron thermal conductivity as a function of temperature for several different Au wire diameters. Similar to Figure 1, Figure 2(a) assumes an electron-phonon nonequilibrium with , and Figure 2(b) assumes . Conductivity data on bulk Au [31] in Figure 2(b) agrees well with (2) when . The nonequilibrium effects on thermal conductivity are apparent by comparing Figures 1(a) and 2(a) to Figures 1(b) and 2(b). Obviously, at high temperatures, electron-phonon nonequilibrium will result in a higher conductivity than equilibrium. In the nonequilibrium case, the lattice is colder than the equilibrium case which results in less electron-phonon scattering thereby increasing . Note in the case of very small

_{e}*d*, the equilibrium and nonequilibrium situations are the same since is restricted by electron-boundary scattering.

**(a)**

**(b)**

**(a)**

**(b)**

#### 3. Thermal Conductivity Reduction in Nanoporous Materials

Although (2) gives the reduction
in thermal conductivity due to boundary scattering, there will still be a
further reduction in conductivity due to the porous nature of the film. As the electrons are conducting through the
Au structure, the electron-boundary scattering is significantly increased since
the “nanowires” that comprise the Au mesh cannot be considered as 1D
conducting channels. In actuality, the
path of the conducting electrons has several random turns and kinks as a result
of the fabrication process. This
further conductivity reduction can be estimated by considering the percent
porosity of the nanostructure [32, 33]. By treating the pores as randomly sized
spheres, the reduction in thermal conductivity of the porous film can be calculated
with the Bruggeman assumption [34, 35] and estimated by where *f* is the porosity,
*k _{w}* is the thermal conductivity of the solid material, in
the case of the nanoporous Au, the reduced thermal conductivity due to the
ligament size being on the order of must be used, and

*k*is the reduced thermal conductivity of the nanoporous Au. Combining (2) and (3) results in an expression for electron thermal conductivity in nanoporous metal composites. Figure 3 gives the calculations for conductivity of nanoporous gold as a function of electron temperature for four different porosities,

_{p}*f*, of 0%, 15%, 30%, and 50% and three different wire diameters,

*d*, of 10 nm, 100 nm, and 1m assuming K. As expected from (3), an increase in porosity continues to reduce the overall thermal conductivity of the nonporous Au. During electron-phonon nonequilibrium, a change in the thermal conductivity would change the time it takes for the electrons and phonons to equilibrate. If the conductivity is significantly reduced, the energy density of the electron system would remain large near the heat source causing a change in the thermalization time [14, 36]. The relationship between change in electron temperature and thermal conductivity and electron-phonon coupling is given in the two-temperature model (TTM) [9]. Therefore, the effect that the change in electron temperature has on wire diameter and electron phonon coupling is given by using the thermal conductivity as defined by (2) and (3) in the TTM.

**(a)**

**(b)**

**(c)**

#### 4. Effects on Electron-Phonon Coupling Measurements

To examine this dependence, the
TTR technique was used to measure the change in electron temperature during
electron-phonon coupling on a 2m nanoporous Au film. Details of the TTR experimental setup are
given by Hopkins and Norris [14]. The Au film was grown on an oxidized Si
substrate by chemically dealloying a 40%Au–60%Ag composite with the fabrication processes
outlined in Seker et al. [37]. The film used in this study
was not actively annealed after the dealloying which resulted in a ligament
thickness of about 100 nm estimated from the SEM micrograph, seen in Figure 4. Figure 5 shows a cross-sectional SEM
micrograph of the porous Au sample. It
is apparent in Figure 5 that the limiting dimension of the porous Au sample is
the ligament diameter. This corresponds
to a porosity, *f*, of about 35% [37]. The electron-temperature
dependence after short pulsed laser heating on a 2m nanoporous film with 35% porosity from
TTM calculations is shown in Figure 6 for four different ligament
thicknesses. Bulk thermophysical
properties of Au were used in the calculations with an incident pump fluence of
10 J m^{-2} [14, 38], W m^{-3} K^{-1} was assumed for the electron-phonon coupling factor [12, 39], and an optical penetration
depth of 12.5 nm was calculated from optical constants (The optical penetration depth of Au was calculated by the typical equation , where is the incident photon wavelength (800 nm) and *k* is the complex part of the index of refraction of Au at 800 nm ()). The conductivity change from a wire diameter
of 100 nm to 1m is minimal, but still slightly
observable. A greater change from *d* of 100 nm to 10 nm to 1 nm is
evident. This is expected due to the
changes in thermal conductivity associated with *d* reduction in Au shown in Figure 3. The TTM calculations were then fit to the
experimental TTR data during the first 4.0 picoseconds after pulsed laser
heating. The same assumptions and a
similar fitting routine as Hopkins and Norris [14] were used, the difference
being that the wire diameter, *d*, in
the expression for thermal conductivity, was used as the fitting
parameter. The results of the fit compared to the experimental data are shown in Figure 7.
Seven TTR data scans were taken at different locations on the surface of
the porous Au
film, and the average best fit of the TTM was achieved with a wire diameter of
162 nm with a standard deviation of 11.3 nm, which is in good agreement with
the approximate ligament sizes observed in the SEM analysis. Using (3) (calculations of (3) are shown in
Figure 3) and assuming % which
was used in the TTM fit, the best fit wire diameter of 162 nm corresponds to a
thermal conductivity of about 323 W m^{-1} K^{-1} assuming a
maximum electron temperature of 820 K (from the TTM fit) and a cold lattice at
300 K. This value of conductivity
corresponds to overall thermal conductivity on the porous structure. For this situation ( nm), the conductivity in each ligament is 622 W m^{-1} K^{-1} (see (2)), but since the probe in the TTR experiments measures
the conductance averaged over a probe spot size of ~10m [14], the porous aspect
of the sample which reduces the overall conductivity is observed. In a bulk Au sample, the thermal conductivity
during this electron-phonon nonequilibrium predicted via (2) (when is negligible) is 733 W m^{-1} K^{-1}.

#### 5. Conclusions

In summary, the thermal
conductivity in nanowires during electron-phonon nonequilibrium was studied and
applied to Au nanoporous structures.
Based on kinetic theory, a simple expression to predict the thermal
conductivity in nanowires was derived by taking into account electron-boundary
scattering. This expression agrees very
well with results from other works. By
introducing an expression to take into account the porosity of the nanocomposite,
an expression for the thermal conductivity of the nanoporous Au was developed. This expression was then used in conjunction
with the two-temperature model to study the change in electron temperature
during short-pulsed laser heating. The
TTM with *k _{p}* was fit to
transient thermoreflectance data taken on a nanoporous Au film with a best fit
wire diameter that agrees with estimates of the wire diameter based on SEM
micrographs. Therefore, the TTR
technique can be used to characterize thermophysical properties of nanoporous
materials and is sensitive to reductions in thermal conductivity that would
arise due to the structure and geometry of nanoporous structures.

#### Acknowledgments

The authors are greatly appreciative to W.-K. Lye at UVa for insight into nanoporous Au fabrication and G. Chen at MIT for helpful discussions on electron transport in nanowires. P. Hopkins was supported by the NSF Graduate Research Fellowship Program. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed-Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.