arXiv:1511.03312v1 [cond-mat.mtrl-sci] 10 Nov 2015
The Role of Thermalizing and Non-thermalizing Walls in Phon
on
Heat Conduction along Thin Films
Navaneetha K. Ravichandran and Austin J. Minnich
∗
Division of Engineering and Applied Science,
California Institute of Technology, Pasadena, California
91125, USA
(Dated: November 16, 2015)
Abstract
Phonon boundary scattering is typically treated using the F
uchs-Sondheimer theory, which as-
sumes that phonons are thermalized to the local temperature
at the boundary. However, whether
such a thermalization process actually occurs and its effect o
n thermal transport remains unclear.
Here we examine thermal transport along thin films with both t
hermalizing and non-thermalizing
walls by solving the spectral Boltzmann transport equation
(BTE) for steady state and transient
transport. We find that in steady state, the thermal transpor
t is governed by the Fuchs-Sondheimer
theory and is insensitive to whether the boundaries are ther
malizing or not. In contrast, under
transient conditions, the thermal decay rates are significa
ntly different for thermalizing and non-
thermalizing walls. We also show that, for transient transp
ort, the thermalizing boundary condition
is unphysical due to violation of heat flux conservation at th
e boundaries. Our results provide in-
sights into the boundary scattering process of thermal phon
ons over a range of heating length scales
that are useful for interpreting thermal measurements on na
nostructures.
∗
aminnich@caltech.edu
1
I. INTRODUCTION
Engineering the thermal conductivity of nanoscale materials has be
en a topic of con-
siderable research interest over the past two decades [1]. While ap
plications such as GaN
transistors [2, 3] and light emitting diodes (LEDs) [4] require high th
ermal conductivity sub-
strates to dissipate heat, the performance of thermoelectric an
d thermal insulation devices
can be significantly enhanced by reducing their thermal conductivit
y [5, 6]. In many of these
applications, phonon boundary scattering is the dominant resistan
ce to heat flow, making
the detailed understanding of this process essential for advancin
g applications.
Phonon boundary scattering has been studied extensively both th
eoretically and exper-
imentally. The thermal conductivity reduction due to boundary sca
ttering of phonons is
conventionally treated using the Fuchs-Sondheimer theory, which
was first derived for elec-
tron boundary scattering independently by Fuchs [7] and Reuter a
nd Sondheimer [8] and was
later extended to phonon boundary scattering in several works [9
–11]. Fuchs-Sondheimer
theory is widely used to interpret experiments but makes an importa
nt assumption that the
diffusely scattered part of the phonon spectrum at a partially spec
ular wall is at a local
thermal equilibrium with the wall - the thermalizing boundary condition
. The thermalizing
boundary condition is also a key assumption in the diffuse boundary sc
attering limit of
Casimir
'
s theory [12].
Several computational works [11, 13–16] have studied the reduc
tion in thermal con-
ductivity due to phonon boundary scattering in nanostructures b
y solving the phonon
Boltzmann transport equation (BTE). These works have consider
ed either thermalizing
or non-thermalizing boundaries but have never compared the effec
t of these two different
boundary conditions on the thermal conductivity of nanostructu
res. Several experimental
works have also studied the reduction in thermal conductivity of na
nomaterials such as
nanowires [17–19], thin films [10, 20, 21] and nanopatterned struct
ures [22] due to phonon
boundary scattering. These works have used the Fuchs-Sondhe
imer theory to interpret their
measurements. However, it is not clear if the assumptions made in th
e Fuchs-Sondheimer
2
theory are necessarily applicable for these experiments. In fact,
an analysis of the effect
of the key assumption made in the Fuchs-Sondheimer theory, that
the walls are thermal-
izing, has never been investigated due to the challenges involved in so
lving the BTE for
non-thermalizing walls.
Here, we examine the role of thermalizing and non-thermalizing walls in h
eat conduction
along thin films by solving the spectral phonon Boltzmann transport
equation (BTE) for
a suspended thin film under steady state and transient transport
conditions. We find that
steady state transport is insensitive to whether phonons are the
rmalized or not at the bound-
aries and that Fuchs-Sondheimer theory accurately describes th
ermal transport along the
thin film. In the case of transient transport, we find that the deca
y rates are significantly
different for thermalizing and non-thermalizing walls and that Fuchs-
Sondheimer theory
accurately predicts the thermal conductivity only when the therm
al transport is diffusive.
Moreover, under transient transport conditions, we find that ph
onons cannot undergo ther-
malization at the boundaries in general due to the violation of heat flu
x conservation. Our
results provide insights into the boundary scattering process of t
hermal phonons that are
useful for interpreting thermal measurements on nanostructu
res.
II. MODELING
A. Boltzmann Transport Equation
We begin our analysis by considering the two dimensional spectral tr
ansient Boltzmann
transport equation (BTE) under the relaxation time approximation
for an isotropic crystal,
given by,
∂g
ω
∂t
+
μv
g
∂g
ω
∂z
+
v
g
p
1
−
μ
2
cos
φ
∂g
ω
∂x
=
−
g
ω
−
g
o
(
T
)
τ
ω
+
Q
ω
4
π
(1)
Here,
g
ω
is the phonon energy distribution function,
ω
is the phonon frequency,
v
g
is the
phonon group velocity,
τ
ω
is the phonon relaxation time,
x
and
z
are the spatial coordinates,
t
is the time variable,
g
0
(
T
) is the equilibrium phonon distribution function at a deviational
3
temperature
T
=
T
0
+∆
T
from an equilibrium temperature
T
0
,
μ
is the direction cosine,
φ
is
the azimuthal angle and
Q
ω
is the rate of volumetric heat generation for each phonon mode.
As the in-plane (x) direction is infinite in extent, we require boundary
conditions only for
the cross-plane (
z
) direction. In the traditional Fuchs-Sondheimer problem, the bou
ndary
conditions enforce that the diffusely scattered phonons are ther
malized while also allowing
some phonons to be specularly reflected. Here, we generalize thes
e boundary conditions to
allow for the possibility of both partial thermalization and partial spe
cularity as:
For
μ
∈
(0
,
1]
,
g
+
ω
(0
, μ, φ
) =
p
ω
g
−
ω
(0
,
−
μ, φ
)
+ (1
−
p
ω
)
σ
ω
C
ω
∆
T
(
z
= 0)
4
π
−
(1
−
σ
ω
)
π
Z
2
π
0
Z
0
−
1
g
−
ω
(0
, μ
′
, φ
′
)
μ
′
d
μ
′
d
φ
′
!
For
μ
∈
[
−
1
,
0)
,
g
−
ω
(
d, μ, φ
) =
p
ω
g
+
ω
(
d,
−
μ, φ
)
+ (1
−
p
ω
)
σ
ω
C
ω
∆
T
(
z
=
d
)
4
π
+
(1
−
σ
ω
)
π
Z
2
π
0
Z
1
0
g
+
ω
(
d, μ
′
, φ
′
)
μ
′
d
μ
′
d
φ
′
!
(2)
where,
d
is the thickness in the cross-plane direction,
g
+
ω
(0
, μ, φ
) is the phonon distribution
leaving the cross-plane wall at
z
= 0,
g
−
ω
(0
, μ, φ
) is the phonon distribution approaching the
cross-plane wall at
z
= 0,
g
+
ω
(
d, μ, φ
) is the phonon distribution approaching the cross-plane
wall at
z
=
d
,
g
−
ω
(
d, μ, φ
) is the phonon distribution leaving the cross-plane wall at
z
=
d
,
C
ω
is the specific heat of a phonon mode with frequency
ω
,
p
ω
and
σ
ω
are the phonon
specularity parameter and the thermalization parameter for the t
hin film walls respectively.
The specularity parameter represents the fraction of specularly
scattered phonons at the
boundaries and the thermalization parameter represents the fra
ction of the phonon distri-
bution that is absorbed and reemitted at the local equilibrium temper
ature of the thin film
walls. For simplicity, we ignore mode conversion for non-thermalizing b
oundary condition
in our analysis.
The unknown quantities in this problem are the phonon distribution fu
nction (
g
ω
(
t, x, z, μ, φ
))
4
and the deviational temperature distribution (∆
T
(
t, x, z
)). They are related to each other
through the energy conservation requirement,
Z
ω
m
ω
=0
Z
1
μ
=
−
1
Z
2
π
φ
=0
g
ω
τ
ω
−
1
4
π
C
ω
τ
ω
∆
T
d
φ
d
μ
d
ω
= 0
(3)
Due to the high dimensionality of the BTE, analytical or semi-analytica
l solutions are only
available in literature for either semi-infinite domains [23–25] or domain
s with simple bound-
ary and transport conditions [26] or with several approximations [
27]. For nanostructures
with physically realistic boundaries, several numerical solutions of t
he BTE have been re-
ported [11, 16, 28]. However, computationally efficient analytical or
semi-analytical solutions
for the in-plane heat conduction along even simple unpatterned films
[10, 20] are unavail-
able. To overcome this problem, we solve the BTE analytically for stea
dy state transport
(section II B) and semi-analytically for transient transport along t
hin films in the TG exper-
iment [10, 20] (section II C).
B. Steady State Heat Conduction in Thin Films
In this section, we extend the Fuchs-Sondheimer relation for ther
mal conductivity sup-
pression due to phonon boundary scattering to the general boun
dary conditions described in
equation 2. To simulate steady state transport,
Q
ω
is set to 0 in the BTE (equation 1). Fur-
thermore, we assume that a one-dimensional temperature gradie
nt exists along the thin film
and
∂g
ω
∂x
≈
∂g
0
ω
∂x
. These assumptions are consistent with the conditions under which
typical
steady state thermal transport measurements are conducted
on nanostructures [19, 29, 30].
Under these assumptions, the BTE is simplified as,
v
g
μ
∂g
ω
∂z
+
v
g
p
1
−
μ
2
cos
φ
∂g
0
ω
∂x
=
−
g
ω
−
g
0
ω
τ
ω
(4)
For steady state transport, it is convenient to solve the BTE in ter
ms of the deviation from
equilibrium distribution ( ̄
g
ω
=
g
ω
−
g
0
ω
(∆
T
(
x
))). In this case, the BTE transforms into,
∂
̄
g
ω
∂z
+
̄
g
ω
μ
Λ
ω
=
−
cos
φ
p
1
−
μ
2
μ
∂g
0
ω
∂x
(5)
5
The boundary conditions (equation 2) for ̄
g
ω
now become,
For
μ
∈
(0
,
1]
,
̄
g
+
ω
(0
, μ, φ
) =
p
ω
̄
g
−
ω
(0
,
−
μ, φ
)
−
(1
−
p
ω
) (1
−
σ
ω
)
π
Z
2
π
0
Z
0
−
1
̄
g
−
ω
(0
, μ
′
, φ
)
μ
′
d
μ
′
d
φ
For
μ
∈
[
−
1
,
0)
,
̄
g
−
ω
(
d, μ, φ
) =
p
ω
̄
g
+
ω
(
d,
−
μ, φ
) +
(1
−
p
ω
) (1
−
σ
ω
)
π
Z
2
π
0
Z
1
0
̄
g
+
ω
(
d, μ
′
, φ
)
μ
′
d
μ
′
d
φ
(6)
The general solution of the BTE (equation 5) along with the boundar
y conditions (equa-
tion 6) is given by,
̄
g
+
ω
(
z, μ, φ
) =
−
Λ
ω
cos
φ
p
1
−
μ
2
∂g
0
ω
∂x
1
−
exp
−
z
μ
Λ
ω
(1
−
p
ω
)
1
−
p
ω
exp
−
d
μ
Λ
ω
+
(1
−
p
ω
) (1
−
σ
ω
)
h
A
+
ω
+
p
ω
exp
−
d
μ
Λ
ω
A
−
ω
i
1
−
p
2
ω
exp
−
2
d
μ
Λ
ω
exp
−
z
μ
Λ
ω
|
{z
}
I
̄
g
−
ω
(
z,
−
μ, φ
) =
−
Λ
ω
cos
φ
p
1
−
μ
2
∂g
0
ω
∂x
1
−
exp
−
(
d
−
z
)
μ
Λ
ω
(1
−
p
ω
)
1
−
p
ω
exp
−
d
μ
Λ
ω
+
(1
−
p
ω
) (1
−
σ
ω
)
h
A
−
ω
+
p
ω
exp
−
d
μ
Λ
ω
A
+
ω
i
1
−
p
2
ω
exp
−
2
d
μ
Λ
ω
exp
−
(
d
−
z
)
μ
Λ
ω
|
{z
}
II
(7)
for
μ
∈
(0
,
1]. Here, the terms
A
+
ω
and
A
−
ω
only depend on phonon frequency. In particular,
they are independent of the angular coordinates
μ
and
φ
. The derivation of the final expres-
sions for ̄
g
+
ω
(
z, μ, φ
) and ̄
g
−
ω
(
z,
−
μ, φ
) (equation 7) is shown in section I of the supplementary
material. The expression for the in-plane (
x
direction) spectral heat flux is given by,
q
x,ω
=
1
d
Z
d
z
=0
Z
1
μ
=
−
1
Z
2
π
φ
=0
v
x
̄
g
ω
D
(
ω
)
4
π
d
φ
d
μ
d
z
=
−
"
1
3
C
ω
v
g
Λ
ω
#
∂T
∂x
1
−
3 (1
−
p
ω
) Λ
ω
2
d
Z
1
0