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, λ, 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]
(1)
where kw is the reduced thermal
conductivity, kb is the
conductivity of the corresponding bulk material, and u is the ratio of the wire diameter, d, to the electron mean-free path in the wire, λ.
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 kb = CevFλ/3, where Ce is the electronic heat capacity, which at the
temperatures of interest (Te < 5000 K) can be calculated by Ce = yTe with γ being the Sommerfeld constant [20, 24], and
vF is the Fermi velocity. The mean free path of the electrons in bulk
is dominated by electron and phonon scattering, so λ can be estimated by λ = vFτ, where τ is calculated with Matthiessen’s rule by
taking into account electron-electron scattering, , and electron-phonon scattering, 1/τep = BTp, where τee is the
electron-electron scattering time, τep is the electron-phonon
scattering time, Te is the
electron temperature, Tp is the lattice (or phonon) temperature, and 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 u > 1. Predicting the
reduction in wire thermal conductivity requires significantly more
consideration when u < 1 [3]. However, by considering electron-boundary
scattering in kb, kw can be easily estimated
for all u. When calculating λ in a nanowire, the effects of
electron-boundary scattering can be incorporated via Matthiessen’s rule. This can be estimated as 1/τBD = vF/d assuming
complete diffuse scattering [30]. Note that τBD is considered
temperature independent [22]. Therefore, the conductivity of a nanowire can
be expressed as
(2)
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 kw/kb as a function of Au wire
diameter for various electron temperatures.
In these calculations, the effects of electron phonon nonequilibrium are
studied, therefore Tp 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 Te = Tp. These results are compared to calculations
using (1) when u > 1 and
simulation results from Au nanowires assuming λ = 41.7 nm when u < 1. Note that the
results from [3] for u < 1 and (1) for u > 1 agree well with (2) over the entire range of wire diameters when Te equals 240 K, which
corresponds to a λ 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 Tp = 300, and Figure 2(b) assumes Te = Tp. Conductivity data on bulk Au [31] in Figure 2(b) agrees well
with (2) when d ≫ λ.
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 λ.
Note in the case of very small d,
the equilibrium and nonequilibrium situations are the same since λ is restricted by electron-boundary scattering.
Ratio of wire thermal conductivity, (2), to bulk thermal conductivity versus wire diameter for several different electron temperatures (a) during electron-phonon nonequilibrium when Tp = 300 and (b) when Te = Tp. The calculations shown when Te = Tp = 240 K, corresponding to an electron mean free path of 45 nm ,agree well with previous results using λ = 41.7 nm.
Ratio of wire thermal conductivity, (2), to bulk thermal conductivity versus wire diameter for several different electron temperatures (a) during electron-phonon nonequilibrium when Tp = 300 and (b) when Te = Tp. The calculations shown when Te = Tp = 240 K, corresponding to an electron mean free path of 45 nm ,agree well with previous results using λ = 41.7 nm.
Wire thermal conductivity versus electron temperature for several different wire diameters (a) during electron-phonon nonequilibrium when Tp = 300 and (b) when Te = Tp. In the case of a bulk, (2) used to predict wire diameter agrees well with experimental data of thermal conductivity data taken on bulk Au at relatively low temperatures.
Wire thermal conductivity versus electron temperature for several different wire diameters (a) during electron-phonon nonequilibrium when Tp = 300 and (b) when Te = Tp. In the case of a bulk, (2) used to predict wire diameter agrees well with experimental data of thermal conductivity data taken on bulk Au at relatively low temperatures.
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
(3)
where f is the porosity,
kw 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 λ must be used, and kp 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, f, of
0%, 15%, 30%, and 50% and three different wire diameters, d, of 10 nm, 100 nm, and 1 μm assuming Tp = 300 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.
Thermal conductivity of porous gold material versus electron temperature for different porosities and different ligament thicknesses. A continued decrease in conductivity is observed with increasing porosity in addition to decreasing diameter, as expected from the expression for kp which is given by (2) and (3).
Thermal conductivity of porous gold material versus electron temperature for different porosities and different ligament thicknesses. A continued decrease in conductivity is observed with increasing porosity in addition to decreasing diameter, as expected from the expression for kp which is given by (2) and (3).
Thermal conductivity of porous gold material versus electron temperature for different porosities and different ligament thicknesses. A continued decrease in conductivity is observed with increasing porosity in addition to decreasing diameter, as expected from the expression for kp which is given by (2) and (3).
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 2 μm 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 2 μm 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], 2.2 × 1016 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 δ = λ/(4πk), where λ is the incident photon wavelength (800 nm) and k is the complex part of the index of refraction of Au at 800 nm (k = 5.125)). The conductivity change from a wire diameter
of 100 nm to 1 μm 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 f = 35% 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 (d = 162 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 ~10 μm [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 vF/d is negligible) is 733 W m-1 K-1.
Change in electron temperature after short-pulsed laser heating in the event of electron-phonon nonequilibrium. Using the expression for kp and a porosity of f = 35%, the sensitivity of electron cooling to ligament thickness and subsequent thermal conductivity is observed. It is apparent that electron-cooling is not extremely sensitive to the thermal conductivity reduction experienced during electron-phonon coupling.
TTM using kp fit to TTR experimental data taken on a 2 μm Au nanoporous film with ligament thicknesses around 100 nm. The fit was achieved by iterating d, the ligament, or “nanowire” thickness. A best fit was achieved with a thermal conductivity calculated using
d = 162 nm, in relatively good agreement with the ligament thickness of the nanoporous Au film that was tested.
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 kp 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.
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