# Heat Transport in Nanoscale Heterosystems:  A Numerical and Analytical Study

Authors: Boris Krenzer; Anja Hanisch; Andreas Duvenbeck; Bärbel Rethfeld; Michael Horn-von Hoegen
Journal: Journal of Nanomaterials
 (2008)

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## Abstract

The numerical integration of the heat diffusion equation applied to the Bi/Si-heterosystem is presented for times larger than the characteristic time of electron-phonon coupling. By comparing the numerical results to experimental data, it is shown that the thermal boundary resistance of the interface can be directly determined from the characteristic decay time of the observed surface cooling, and an elaborate simulation of the temporal surface temperature evolution can be omitted. Additionally, the numerical solution shows that the substrate temperature only negligibly varies with time and can be considered constant. In this case, an analytical solution can be found. A thorough examination of the analytical solution shows that the surface cooling behavior strongly depends on the initial temperature distribution which can be used to study energy transport properties at short delays after the excitation.


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## Body

## 1. Introduction

The ultrafast
dynamics in a solid subsequent to short pulse laser irradiation are of great
interest yielding insights into the microscopic processes of the energy
transfer between different degrees of freedom of a solid. After excitation, a
number of relaxation processes lead to the establishment of a thermal
equilibrium in the laser-irradiated solid. Electron-electron thermalization is
a rather fast process taking place on a subpicosecond timescale [1, 2]. Compared to this, the heat
exchange between the electron and phonon subsystems is a slow process [3–5]. The nonequilibrium between
these two subsystems can be described by the two-temperature model [3, 4].For timescales larger than the electron-phonon
coupling time, which typically lies between 1 picoseconds and 10 picoseconds,
the system can be described by the usual heat conduction
equation:(1)ρc∂T(r→,t)∂t=∇⋅[K∇T(r→,t)]+A(r→,t)withρ and c the specific
mass and specific heat capacity, respectively. In (1), Fourier's law has been used relating
the heat flux to the temperature-gradient. K is the thermal
conductivity which generally is a tensor. The source term A(r→,t) represents the
heat generation in the solid per unit time and unit volume subsuming all
microscopic steps from the absorption of a photon to the formation of the
thermal equilibrium between electrons and phonons. Equation (1) can
be solved analytically for simple forms of A(r→,t) and assuming
isotropic media with constant thermal conductivity K [6].For heterostructures, the assumption of an isotropic
medium is no longer valid and (1) has
to be solved independently for the different materials applying appropriate
boundary conditions. It is well known that the boundary between two materials
acts as a barrier to thermal heat diffusion [7–9]. The heat flow Q˙ across an
interface is given by(2)Q˙=1RK⋅ΔT,whereRK denotes the
thermal boundary resistance, which couples the heat flow across the interface
to the temperature jump ΔT at the
interface. If this heat transport is determined by phonons only, the
temperature jump arises because phonons incident on the interface are partially
reflected and only a fraction is transmitted across the interface. The simplest
models that can be used to calculate the phonon transmission probabilities are
the acoustic mismatch model (AMM) and the diffusive mismatch model (DMM)
[7, 8]. If the wavelength of the
phonons is larger than the interface roughness, the AMM can be applied for the
calculation of the transmission coefficient [10]. The AMM treats phonons as acoustic waves that are
reflected and refracted at the interface. The transmission probability is
calculated by applying the acoustic analog of the Fresnel equations in optics
[11]. In the framework
of the DMM, which applies if the phonon wavelength is comparable to or smaller
than the interface roughness, leading to a strong diffuse scattering at the
interface, the transmission probability depends on the phonon density of states
of the two adjacent media [7, 8, 10]. In the past, the above mentioned models have been
extended and refined including lattice-dynamical calculations [12–15], heat transport in
superlattices [16–18], ballistic transport [19], exact phonon dispersion
curves instead of the usually applied Debye-approximation [20], and electronic
contributions to the heat transport across the interface [21–23]. Recently, the
Kapitza-effect, that is, the formation of a temperature jump at an interface,
has also been observed in molecular dynamics simulations [24].The most simple heterosystem which can be studied is a
layered structure of two materials, that is, a thin film on a substrate. The
thermal boundary resistance of such heterosystems for a large number of different
material combinations and even the heat transport properties of nanolaminates
has been determined in the past using the time-domain thermoreflectance
technique [8, 25–29]. In a previous study, the
temporal surface temperature evolution of a thin Bismuth-film on a Silicon
substrate has been investigated by means of ultrafast electron diffraction
[30, 31]. In this technique, a short
electron pulse is diffracted at the surface with variable delays from an
initial fs-laser pulse excitation [32–36]. The surface temperature is extracted from the
diffraction spot intensity, which is affected by the Debye-Waller-effect. For
the Bi/Si-system, it was observed that the initial temperature increase is
followed by an exponential surface temperature decay. From the decay time
constant, the thermal boundary resistance has been extracted, which was found
to be in good agreement with values calculated from the AMM and DMM [31]. A direct determination of
the thermal boundary resistance from the exponential decay time constant,
however, is only possible if the substrate temperature is constant during the
experiment [8].
Generally, (1) has
to be solved separately for the two materials with a thermal boundary
resistance as an input parameter yielding the best agreement between experiment
and simulation [8, 25, 26].In this paper, we will present numerical simulations
for the temporal evolution of the surface temperature of Bi films of various
thicknesses on a Silicon substrate by applying (1). It will be shown that for the Bi/Si-system, the thermal boundary
resistance can be directly extracted from the experimentally observed decay
constant without an elaborate comparison of simulations and experimental data.
In addition, an analytical solution for the heat transport in the Bi/Si-system
will be presented and discussed. It will be shown that the analytical approach
well describes the results of the numerical simulation and observed surface
cooling behavior. Although the discussion presented here is carried out for the
Bi/Si-system, the results are applicable to other material combinations. That
the temporal substrate temperature changes need to be small compared to the
temperature changes in the film is the only restriction. If this condition is
fulfilled, even information on the heat transfer mechanisms at times shorter
than the electron-phonon coupling time can even be obtained. This constitutes a
new approach for studying such processes.


## 2. Numerical Study

The numerical integration of (1) is accomplished in a standard, explicit
FTCS-scheme (forward time centered space) in one dimension, namely, the
cross-plane, which in the following will be referred to as the
“z-direction" [37]. The spatial and temporal discretization are both
taken at finite and constant values obeying the stability criterion for the
numerical integration [37]. In the above mentioned ultrafast electron
diffraction experiment, the probed area (300 μm × 4 mm) was an
order of magnitude smaller than the excited area (spot size ≈4 mm) resulting
in a laterally homogeneous heating of the probed area. The lateral
heat-diffusion is, therefore, neglected in the following. Additionally, the
tensor properties of the heat diffusion constant K are reduced to
a single, but z -dependent
constant in this one-dimensional treatment.At the surface theNeumann-type boundary condition, ∇T=0 has been
applied. This condition is valid if no particles, that is, electrons, atoms, or
clusters are leaving the surface and radiation losses can be neglected. In an
experiment, both conditions can be fulfilled using low laser excitation energies.
In the studied Bi/Si-systems, the maximum surface temperature was below 300 K,
which means that radiative losses are negligible. Additionally, no ablation has
been observed [30, 31].At the second boundary, the back side of the simulated
volume, a constant temperature has been applied (Dirichlet-condition),
which can be considered as a heat sink (thermostat). The effect of this
boundary condition on the temporal surface temperature evolution has been
minimized by choosing a sufficiently large simulation slab.The source term in (1) for one dimension can be written as
(3)A(z,t)=I0(1−r)αexp(−αz)q(t),whereα is the inverse
absorption length and r the reflection
coefficient for light at a given wavelength λ [4, 6]. q(t) is the temporal
profile for the heat generation, that is, the temporal profile of the laser
pulses. For the simulation, q(t) has been taken
constant for a period of 45 femtoseconds according to the laser pulse duration
used in the experiment. I0 is the
integrated laser-pulse intensity which was set to 8.5×1014 W/m2 in the simulation depicting
the experimental value. Note, the above treatment of the source term is only
correct for strong electron-phonon coupling resulting in a thermal equilibrium
of electrons and phonons on a short timescale. Generally, the two-temperature
model has to be applied in order to obtain the time-dependent temperature
distributions of the electron and phonon subsystems [3, 4, 26]. In section 3, the effect of strong and weak electron-phonon coupling on
the temporal temperature evolution will be discussed in more detail. In
addition to the application of the two-temperature model, multiple reflection
of light in the thin film and internal reflection, that is, the dynamic change
of the optical properties during the excitation process, have to be considered
for the initial heating dynamics [38, 39]. As the main concern of this work is the cooling
behavior of thin Bi-films, we assume the source term (3) to be valid and use bulk values for
the optical constants.According
to (3),
the excitation of the Bi/Si-system with light of wavelength 800 nm results in a
temperature jump at the interface because of the largely different absorption
coefficients of the two materials (cf.
Table 1).Table 1
Literature values for the material constants used in
the numerical simulation.r is the
reflectivity for light of wavelength 800 nm, α is the
absorption coefficient, K is the heat
conductivity, ρ and c are the mass
density and specific heat, respectively [40].
r (%)α(106m−1)ρ (kg/m3)K (W/(K m))c (J/(K kg))Bi9058.897807.9(a)122Si300.07723281000.0(b)722(a) at 300 K.(b)at 80 K.At the interface the energy flux is given by (2) which is taken into account by
applying(4)Kf∂Tf∂z|z=d=Ks∂Ts∂z|z=d=−1RK(Tf−Ts), as Neumann-type boundary conditions [25, 26, 37]. In (4), Tf and Ts are the film- and substrate temperatures, respectively. Kf and Ks are the heat
diffusion constants in the film and the substrate. The boundary condition (4) guarantees that no heat is
accumulated at the interface: the heat flow toward the interface equals the
heat flow from the interface into the substrate. Due to the existence of a
Schottky-barrier between Bi and Si and the small density of states in the
interval up to 1.55 eV (energy of the photons with wavelength 800 nm) above the
valence band maximum in Silicon, the energy transport across the interface by
electrons is neglected [40, 41].The numerical integration uses literature values for
the material parameters that are tabulated in
Table1. It should be
noted that the material parameters can depend on the dimension of the material,
for example, the film-thickness. As their thickness dependencies are unknown,
we use bulk values for Bismuth. The thermal boundary resistance is set to RK=9.76×10−8 (K m2)/W as experimentally
determined [30, 31]. For all numerical integrations, a starting
temperature of 80 K is used.In Figure1, the results of the
numerical integration are shown. Figure 1(a) displays the spatial and
temporal temperature distributions of a 10 nm thin Bi-film. The total
simulation slab dimension in spatial direction was 110 nm, but for a better
representation only the first 30 nm are shown. The boundary between Bi and Si
is clearly visible for all delays at z=10 nm. The maximum
temperature of the Bi-film at short delays is 240 K. For larger delays, the
Bi-film cools down due to heat transport across the interface. On the contrary,
the Si-temperature is below 81 K for all times. As discussed, above the small
heat generation in Silicon for short delays is explained by the three orders of
magnitude smaller absorption coefficient for light of wavelength 800 nm
compared to Bi. In addition, at larger delays heat transmitted through the
interface is efficiently dissipated in the Si-substrate which has a thermal
conductivity that is two orders of magnitude higher than Bi (cf.Table 1). The Silicon substrate acts as a thermostat and the energy
loss in the Bi-film is given by (2). It has to be noted that the thickness of the silicon substrate
in the simulation dSi=100 nm is smaller
than the phonon mean free path in silicon (λPhSi≈250 nm) and the
validity of Fourier's law in (1) is
not fulfilled. However, as shown above, the temperature gradient in the
Si-substrate is very small. We found that larger substrate thicknesses has no
effect on the temperature evolution of the Bi-film reversely justifying the
application of Fourier's law.Result of the numerical integration
of the heat diffusion equation using literature values andRK=9.76×10−8 (K m2)/W. (a) shows a temporal and
spatial temperature of a 10 nm thin Bi-film on a Si-Substrate. (b) displays the
temporal surface temperature (distance from surface z=0 nm) evolution
for different Bi-film-thicknesses. For all numerical integrations, the
thickness of the Si-slab was 100 nm, but for a better representation only the
first 30 nm of the total integration slab are shown in (a). Note, the unusual,
nonlinear color scale in (a) is for descriptive reasons.
(a)(b)For a comparison with the experiment, the temporal
evolution of the temperature is obtained by taking slices of the integration
slab at the surface layer (z=0 nm). A
compilation of the time-dependent surface temperature for different
film-thicknesses is displayed in
Figure 1(b). The
surface temperature evolution of Bi-films with thicknesses below ∼100 nm can be
divided into two contributions: a fast temperature decay at short delays
followed by a slow decay for long delays. These different decay behaviors can
be explained by studying the spatial temperature profiles for different delays.Figure2 shows the spatial
temperature profiles at 1  picosecond and 6 picoseconds delay of a 10 nm thin
Bi-film. At 1 picosecond, delay the temperature distribution across the film is
inhomogeneous. Heat diffusion in the film drives the system into an
equilibrated state with a homogeneous temperature distribution across the film.
For a 10 nm thin Bi-film this state is reached after ∼6 picoseconds (Figure 2 dashed
line). This equilibration-time for heat diffusion in the film is smaller than
the characteristic time for diffusion obtained from (21) and from [25]. We attribute this
discrepancy to the application of Fourier's law in (1). As the phonon mean free path in Bi is λPhBi≈13 nm, the thermal
energy transport is overestimated for film-thicknesses below the phonon mean
free path resulting in a faster surface temperature decay. For such thin films
the Boltzmann transport equations have to be solved which is beyond the scope
of this study. For comparison with the experiment, we assume that the heat
diffusion can be treated using Fourier's law because the surface temperature
dynamics for these short timescales is not accessible due to the limited
experimental resolution (cf.
Section 4).Figure 2
Spatial temperature profiles at
1 picosecond (solid line) and 6 picoseconds (dashed line) delay of a 10 nm thin
Bi-film. The thermal boundary resistance in the simulation isRK=9.76×10−8 (K m2)/W. The total simulation slab
thickness is 100 nm but for a better representation only the first 30 nm are
shown. At 1 picosecond, a temperature gradient in the Bi-film is still present.
After ∼6 picoseconds
the temperature gradient has vanished and results in a homogeneous spatial
temperature distribution.With increasing film-thickness, the time required to
establish a homogeneous temperature distribution increases. Concomitantly, the
temperature level of the homogeneous state decreases as the amount of absorbed
energy is distributed over a larger volume. For larger film-thicknesses, this
results in temporal surface temperature evolutions which are similar to the
surface temperature evolution of a Bismuth single crystal. In fact, the surface
temperature evolution of the 500 nm thick Bi-film is the same as obtained for a
Bi-single crystal in the displayed range. We conclude that the fast temperature
decay at short delays is governed by heat diffusion in the film itself.The evolution at delays after a homogeneous
temperature distribution is formed, follows an exponential behavior. This is
well understood in terms of the thermal boundary resistance. Rewriting (2) results in(5)ρcd∂Tf∂t=−1RK(Tf−Ts) with d denoting the
film-thickness. ρ and c are the mass
density and heat capacity of the thin film, respectively. For a constant
substrate temperature Ts, the above
equation results in an exponential decay of the film temperature Tf with a time
constant:(6)τK=cρRKd.The validity of (6) will be further discussed insection 3 By determining the decay constant, the thermal boundary
resistance RK can be
extracted from the exponential surface temperature decay which can also be used
to crosscheck the accuracy of the numerical integration. It turns out that the
value of RK determined from
the decay constant of the numerical simulation critically depends on the size
of the spatial discretization Δz.Figure 3(a) shows a linear
relationship between the extracted thermal boundary resistance and the spatial
discretization. For finite step widths, the difference between the input value
and the extracted value for the thermal boundary resistance can be up to 25%
(for Δz=5 nm, cf. Figure 3(a)). Only for Δz→0 the extracted
thermal boundary resistance is equal to the input value. In contrast, the
temporal discretization has a relative influence on the order of only 10−4 for the extracted thermal
boundary resistance as shown inFigure 3(b).
For the simulation, we used a spatial and temporal discretization of Δz=0.25 nm and Δt=0.01 femtosecond,
respectively, which results in a thermal boundary resistance that is 2% larger
than the input value.Dependence of the thermal
boundary resistanceRK extracted from
the decay of the surface temperature obtained from the numerical integration
of (1) on
(a) the spatial discretization Δz and (b)
temporal discretization Δt. The temporal discretization in (a) is Δt=0.01 femtosecond. As
evident from the figure, the extracted RK linearly
depends on the spatial discretization Δz and reaches the
input value only for Δz→0. In (b), the numerical integration is performed with
a spatial step-width of Δz=2 nm. The
numerical integration uses the literature values fromTable 1 and RK=9.76×10−8 (K m2)/W. The Bi-film-thickness is
10 nm.
(a)(b)


## 3. Analytical Solution of the Heat Diffusion Equation after Excitation

One result of the above numerical simulation is the
negligible temperature variation of the substrate temperature in the
Bi/Si-heterosystem. For this case, that is, thin film on a substrate with an
approximately constant temperature, an analytical solution can be derived.
Consider the one-dimensional heat diffusion equation after
excitation:(7)∂∂tT(z,t)=κ∂2∂z2T(z,t) with κ=K/(ρc) the
diffusivity. Introducing the dimensionless variables:(8)θ(ξ,t˜)=T(z,t)−TsTi−Ts,(9)ξ=zd,t˜=κtd2,σ˜K=dRKK, where Ti is the initial
surface temperature (z=0), Ts the substrate
temperature, and d the
film-thickness. For the derivation of the analytical solution, we apply the
same boundary conditions used in the numerical simulation which
are(10)∂θ(ξ,t˜)∂ξ|ξ=0=0,∂θ(ξ,t˜)∂ξ|ξ=1=−σ˜Kθ(ξ,t˜).Standard procedure for solving (7) yields the series
expansion:(11)θ(ξ,t˜)=∑n=1∞Ene−λn2t˜cos(λnξ), where the
Eigenvalues λn are obtained by
evaluating the transcendental equation:(12)λntan(λn)=σ˜K. The
coefficients En in (11) are determined from the initial condition θ(ξ,t˜=0). Within the scope of this paper, the initial
condition is the state after thermal equilibration between the electron- and
phonon-subsystems. Depending on the timescale on which the electron-phonon
thermalization occurs, two limiting cases can be distinguished yielding
different initial conditions. For weak electron-phonon coupling, the energy
transport at short timescales is mediated by ballistic electrons resulting in a
fast dissipation of the excitation energy in the film [42].
Provided that these hot electrons are reflected at the interface and remain in
the film, this results in a homogeneous distribution of the absorbed energy and
the phonon temperature will be constant across the film(13)θ(ξ,t˜=0)=1,for0<ξ<1. With this
initial condition the coefficients, En are given by
[43](14)En=2sin(λn)λn+sin(λn)cos(λn). On the other
hand, if electron-phonon coupling occurs on a time scale which is faster than
energy dissipation by hot electron across the film, the heat generation can be
described by (3) and
the initial condition will be an exponential temperature distribution given by(15)θ(ξ,t˜=0)=e−α˜ξ with α˜=αd and α the absorption
coefficient used before. With this temperature distribution, the coefficients En are given
by(16)En=2λn(α˜(1−e−α˜cos(λn))+λne−α˜sin(λn))(α˜2+λn2)(λn+sin(λn)cos(λn)). This condition leads
to an analytical solution of the heat diffusion equation that is comparable to
the numerical simulation shown in Figures 1 and 2.The temporal surface temperature evolutionθ(ξ=0,t˜) of the
analytical solution (11)
including the first 200 expansion terms are shown in Figure 4(a) for σ˜K=0.01297, which corresponds to a 10 nm thin Bi-film on an
Si-substrate. For t˜>1, the surface
cooling follows an exponential behavior with the same decay constant regardless
of the temperature distribution at t˜=0. At small t,˜ the temperature
evolution strongly depends on the initial condition as can be seen in the close
up view for 0<t˜<1 shown in Figure 4(b). If the initial temperature distribution is
exponential, the result from the analytical model is equal to the numerical
simulation: a fast temperature decrease (t˜<0.3) is followed
by a slow surface cooling. Using the same arguments as above, the fast
temperature decrease is driven by heat diffusion in the film itself until a
homogeneous temperature distribution across the film is established.
Subsequently, the surface cooling is determined by the heat transport across
the interface. Consequently, an initially already homogeneous temperature
distribution directly results in the slow surface cooling (Figures 4(a) and 4(b)
dotted line).(a) Surface temperature
evolutionθ(ξ=0,t˜=κt/d2) including the
first 200 terms of the analytical solution of the heat diffusion equation for σ˜K=d/(RKK)=0.01297, corresponding to a 10 nm thin Bi-film on an
Si-substrate with an initially exponential (solid line) and constant (dashed
line) temperature distribution. (b) close up view of (a) for 0<t˜<1. In (b), the first expansion terms θλ1 corresponding
to λ1 are separately
displayed. (c) Double-logarithmic plot of Eigenvalues λ1 to λ4 versus σ˜K. The solid and dotted line represent σ˜K and π/2, respectively.
(a)(b)(c)The similarity of the exponential surface temperature
decay fort˜>1 regardless of
the initial conditions is explained by the dominant contribution of first term θλ1 of the series
expansion (11).
Independent of the initial condition θλ1 describes the
temporal evolution for t˜>0.3 as evident
from Figure 4(b). For an
initially constant temperature distribution, θλ1 even describes
the surface cooling behavior for t˜<0.3.From (12), it
is seen that the Eigenvalues λn are only
determined by σ˜K. The dependence of the Eigenvalues λ1 to λ4 are shown
in Figure 4(c), which explains
the dominant contribution of θλ1 to the series
expansion for t˜>0.3. For σ˜K<1λ1 is more than an
order of magnitude smaller, and for σ˜K>1, λ1 is at least a
factor of 3 smaller than the other Eigenvalues. On basis of this observation,
the transient surface temperature evolution can be described with a single
exponential decay characterized by a decay constant (cf. (11)):(17)τ˜=1λ12. Figure 4(c) shows that
within 5%, the Eigenvalue λ1 is equal to σ˜K for σ˜K<1 (thick solid
line). Using (9)
and (17), the decay constant of the
surface cooling τK is given
by(18)τK=τ˜⋅d2κ=1σ˜K⋅d2K/cρ=cρRKd, which is the
same result as stated previously by (6). This means that for a given set of constants ρ, c, and RK, the decay
constant linearly depends on the film-thickness as long as σ˜K<1 is fulfilled.
The upper limit for the film-thickness is(19)1>σ˜K=dRKK→d<RKK=lK with the
Kapitza-length lK defined
previously [9, 44]. The Kapitza-length is a
measure for the thermal resistance of the interface in terms of the thermal
resistance of a perfect crystal. In the case of a thin Bi-film on an
Si-substrate lK=770 nm using
literature value for the thermal conductance K and the experimentally
determined thermal boundary resistance RK. This means that the temperature difference on the
two sides of the interface is the same as the temperature difference of a stationary
temperature profile between two sites in bulk-Bi that are 770 nm apart.Forσ˜K>1, the Eigenvalue λ1 asymptotically
reaches the value π/2 (cf. Figure 4 thick dotted line). In this limit, the
decay-constant quadratically depends on the film-thickness, but is independent
of the thermal boundary resistance RK: (20)τDiff=4π2⋅d2κ=4π2cρKd2. The decay
constant obtained by (20) is associated with the heat diffusion in the film itself.The conditionσ˜K<1 implies a lower
limit of the decay constant τK for the
temperature decay which is determined by the heat transport across the
interface (cf. (18)):(21)τK>cρKd2>τDiff,which is the same result as previously stated
[25]: the thermal
boundary resistance can be extracted from the temperature decay if the decay
constant associated with the heat transport across the interface is larger than
the time constant of the heat diffusion in the film itself. By comparing (19) and (21), this means that the film-thickness has to be smaller than the
Kapitza-length lK, otherwise the surface cooling behavior is determined
by the heat diffusion in the film.The transition from a surface cooling that is
determined by the heat transport across the interface to a surface cooling
which is driven by heat diffusion in the film itself can also be seen in the
numerical results (Figure1) at a film-thickness on the order of the Kapitza-length.As an example, for a 10 nm thin Bi-filmτK must be larger
than 15 picoseconds which is well below the time constant of the numerical
simulation of τsim=1187 picoseconds and
the experimentally observed value τexp=1205 picoseconds
(cf.Section 4). The condition that the
thermal boundary resistance can only be extracted from the surface temperature
decay for delays with t˜>0.3 corresponds to
5 picoseconds for a 10 nm thin Bi-film.


## 4. Comparison of Numerical Results and Experiment

In Figure5, an example of the temporal
surface temperature evolution of a 10.4 nm thin Bi-film deposited on an Si(001)
substrate at 300 K is shown. The transient surface temperature evolution is
obtained from the (01)-diffraction spot by means of ultrafast electron
diffraction [30, 31]. An exponential fit to the data (dashed line in Figure 5) yields a time constant for the decay of τexp=(1205±70) picoseconds. As
discussed above, the thermal boundary resistance can be determined from the
decay constant which yields RK=(9.7±0.6)×10−8 (K m2)/W using bulk values for c and ρ (cf.Table 1). Within the error this value is the same as previously
determined from the surface temperature decay of a 5.5 nm thin Bi-film
[30, 31].Figure 5
Comparison of the simulation
result and experimental data for a 10.4 nm thin Bi-film deposited on an Si(001)
substrate. The experimental data are obtained from the temporal evolution of
the (01)-spot intensity that is converted to a temperature using the
Debye-Waller-effect [30, 31].
An exponential fit to the experimental data yields the decay constant τexp=(1205±70) picoseconds.
Convolving the numerical result with a rectangular function of 70 picoseconds
width has no influence on the time constant which is τsim=1187 picoseconds for
the 10.4 nm thin Bi-film.The result of the numerical integration is also shown
in Figure5 (dash-dotted curve). In
order to account for the finite temporal resolution of the experiment, the
result of the numerical integration has been convoluted with a boxed shaped
function. The temporal resolution in the experiment is limited by the velocity
mismatch between the probing electrons at grazing incidence and the pumping
laser pulses at normal incidence [30]. During the travel time of the electrons across the
surface, the resulting measured temperature is an average over the travel time
which is 70 picoseconds for the above shown experiment (electron energy: 7 keV,
sample width: 3 mm, incident angle: 5°). Convolution with a
rectangular function assumes that any spot of the sample is probed with an
equal number of electrons. Note, this procedure gives an upperlimit for the
temporal resolution. If the electron distribution is inhomogeneous across the
sample width, the temporal resolution is increased because the major part of
the electrons is diffracted from a smaller area of the sample. The transient
temperature evolution obtained from the convolution of the simulation with a
box-shaped function is also displayed in
Figure 5 (solid
line). Apart from the region of the initial temperature increase at short
delays, both sets are similar and the surface temperature decays with a time
constant of τsim=1187 picoseconds
which agrees with the experimentally determined value τexp.The differences between the convoluted and original
temperature profiles at small delays are two-fold. One is the linear increase
of the surface temperature of the convoluted profile compared to the step-like
increase of the unconvoluted profile.
The other
is, the fast decrease of the surface temperature for small
delays, driven by heat diffusion in the film itself, is leveled out by the
convolution. For thicker films, however, the fast temperature decay for small
delays is still present even after convolution with a 70 picoseconds wide
rectangular function which is demonstrated in
Figure6.Figure 6
Surface temperature evolutionθ(0,t) for a 50 nm
thick Bi-film (σ˜K=0.065). The curves
are obtained by convolving the results from the analytical solution with a
rectangular function of 70 picoseconds width in order to account for the
limited temporal resolution of the experiment (see text for detailed
discussion). Compared to the fast decay at short times resulting from the model
with an initially exponential temperature distribution, the surface of a
homogeneously heated film cools down exponentially with a time-constant τ=5823 picoseconds.
For delays larger than 200 picoseconds, both models yield the same exponential
decay.Figure6 compares the temperature
evolution of 50 nm thick Bi-film, σ˜K=0.065, obtained from the analytical solution of the heat
diffusion equation with the two different initial conditions, that is,
exponential and constant initial temperature distribution. To account for the
finite experimental temporal resolution, the two results have been convolved
with a rectangular function of 70 picoseconds width. For delays larger than 200 picoseconds, corresponding to t˜>0.53, both evolutions follow an exponential decay with
time-constant τ=5823 picoseconds.
Evidently, the cooling behavior for delays below 200 picoseconds drastically
depends on the initial condition. An initially constant temperature
distribution results in the slow exponential decay similar to delays larger
than 200 picoseconds. Compared to this slow cooling, the surface temperature
drops much faster if the initial temperature distribution is exponential since
heat diffusion in the film itself dominates the cooling behavior at these short
delays.From the discussion of the initial condition insection3, the two different surface cooling behaviors can be used to
gain information on the energy transport at short times after the excitation.
If the electron and phonon subsystems are weakly coupled, the main transport
mechanism is via hot electrons resulting in homogeneously heating of the phonon
system. In the limit of an instantaneous electron to phonon energy transfer,
the phonon subsystem is expected to be an exponential temperature distribution
across the film. To study this property, further experiments with varying
film-thicknesses are in progress.


## 5. Summary

In conclusion, we have applied the one-dimensional
heat diffusion model to the heterosystem of a thin Bi-film on an Si-substrate.
The numerical integration was carried out for timescales larger than the
typical electron-phonon coupling times when the two subsystems are in thermal
equilibrium. The Bi-surface temperature has been determined for different
film-thicknesses. For film-thicknesses below 100 nm, the surface cooling is
characterized by two different timescales. For short delays after the initial
heat pulse, the surface cooling is dominated by thermal diffusion in the film
resulting in a homogeneous temperature distribution across the film.
Subsequently, the surface temperature evolution is determined by the heat
transport across the interface resulting in a slow exponential temperature
decay. For these films, the decay constant linearly depends on the
film-thickness. For film-thicknesses larger than 100 nm, the surface cooling is
virtually determined by the heat diffusion in the film itself.Because of its thermal properties, the Silicon
substrate temperature stays constant for all delays, which allows the
derivation of an analytical solution for the heat diffusion equation. The
examination of the analytical solution shows that the linear dependence of the
decay-constant is valid up to film-thicknesses on the order of the
Kapitza-length. For thicknesses larger than the Kapitza-length, it is shown
that the decay-constant quadratically depends on the film-thickness and the
thermal boundary resistance cannot be determined from surface temperature
decay. Additionally, the analytical study yields insight into the initial
dynamics of energy dissipation. Two limiting cases were studied: strong versus
weak electron-phonon coupling. These two cases result in different initial
temperature distributions across the film which have a strong effect on the
surface cooling behavior. We have shown that for certain sets of parameters,
the experimentally obtained temporal resolution is sufficient to resolve the initial
surface cooling behavior and we motivate further experimental studies on the
ultrashort dynamics of energy dissipation. Due to its large Kapitza-length, the
Bi/Si-heterosystem is an ideal candidate for such studies.



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*Source: 590609-2008-05-29.xml*