Calculation of the heat capacity of a thin membrane at very low temperature
O. V. Fefelov,
1,
* J. Bergli,
1
and Y. M. Galperin
2,3,4
1
Department of Physics, University of Oslo, P.O. Box 1048, Blindern, 0316 Oslo, Norway
2
Department of Physics and Center for Advanced Materials and Nanotechnology, University of Oslo, P.O. Box 1048, Blindern,
0316 Oslo, Norway
3
A. F. Ioffe Physico-Technical Institute of Russian Academy of Sciences, 194021 St. Petersburg, Russia
4
Argonne National Laboratory, 9700 S. Cass Avenue, Argonne, Illinois 60439, USA
Received 18 December 2006; published 1 May 2007
We calculate the dependence of heat capacity of a freestanding thin membrane on its thickness and tem-
perature. A remarkable fact is that for a given temperature, there exists a minimum in the dependence of the
heat capacity on the thickness. The ratio of the heat capacity to its minimal value for a given temperature is a
universal function of the ratio of the thickness to its value corresponding to the minimum. The minimal value
of the heat capacitance for a given temperature is proportional to the temperature squared. Our analysis can be
used, in particular, for optimizing support membranes for microbolometers.
DOI: 10.1103/PhysRevB.75.172101 PACS numbers: 65.40.Ba, 95.55.Rg
I. INTRODUCTION
Thin freestanding membranes are extensively used for
sensing and detecting, in particular, for mounting of
microbolometers.
1
Thermal and heat transport properties of
such membranes are very important for the sensitivity of
such bolometers and their time response. At low tempera-
tures, the wavelength of thermal phonons responsible for the
heat capacity and conductance can exceed the membrane
thickness, b. In this case, the vibrational modes significantly
differ from those in bulk materials.
2
In particular, the lowest
vibrational mode the bending mode has a quadratic rather
than a linear dispersion law.
3
The finite-size effects,
4
as well
as the role of bending modes for membranes
2
and long mol-
ecules in polymer crystals,
5
were discussed. However, the
dependence on thickness of the heat capacity of membranes
was not explicitly considered.
The contributions of the low-energy modes to the low-
frequency density of vibrational states increase with decrease
of the membrane thickness. As a result, the low-temperature
heat capacity also increases. As the thickness increases, the
heat capacity is determined by higher modes having an es-
sentially linear dispersion law. Consequently, the heat capac-
ity crosses over to that of a bulk material. As a result, the
thickness dependence of the heat capacity of thin membranes
is nonmonotonous, having a sharp minimum at some opti-
mal, temperature-dependent thickness. This minimum is
similar to the minimum in thickness dependence of the bal-
listic heat transfer power radiation predicted in Ref. 6.
However, the concrete values of the optimal thickness for the
heat capacity and the ballistic heat transfer are different.
Consequently, a proper choice of the membrane thickness
can be used for optimizing the time response of microbolom-
eters mounted on thin freestanding membranes. The present
Brief Report is aimed at the theory of low-temperature heat
capacity of thin freestanding membranes.
II. VIBRATIONAL SPECTRUM
The vibrational modes of a thin membrane are superposi-
tions of bulk longitudinal and transverse modes, their relative
weights being determined by boundary conditions—both
normal and tangential stresses should vanish. The eigen-
modes are classified as symmetric SM and antisymmetric
AM. Both are superpositions of the longitudinal and trans-
verse bulk modes with wave vectors k
l
and k
t
, respectively.
The relations between k
l
and k
t
are
3
tanbk
t
/2
tanbk
l
/2
=-
4k
l
k
t
k
2
k
t
2
- k
2
2
, 1
tanbk
l
/2
tanbk
t
/2
=-
4k
l
k
t
k
2
k
t
2
- k
2
2
, 2
for SM and AM, respectively. Here k
and k
denote perpen-
dicular and parallel components of the wave vectors with
respect to the membrane. Inserting the dispersion laws of the
bulk modes, k
l
= / c
l
and k
t
= / c
t
, where c
l
and c
t
are speeds
of transversal and longitudinal sound, into Eqs. 1 and 2,
one obtains transcendental equations for the dispersion laws,
s,n
k
, of different vibrational branches. Here the subscript
stands for the branch type, while n stands for its number.
In addition to AM and SM, there exists a horizontal shear
mode HS, which is a transversal wave with both displace-
ment and wave vector parallel to the plane of the membrane.
The HS mode dispersion law is
3
HS,n
= c
t
n/b
2
+ k
2
, 3
where n is an integer number. SM, AM, and HS modes are
the only vibrations that can exist in a membrane. The modes
possess a very important property: as follows from Eqs.
1–3, the frequencies of all modes scale as
1
s,n
=2c
t
b
-1
w
s,n
bk
. 4
Figure 1 shows the six lowest branches. Only the three
lowest branches are gapless; consequently, only they contrib-
ute to the heat capacity of a membrane at k
B
T
HS,1
0.
The lowest HS and SM are linear at k b
-1
, while the AM
dispersion law can be approximated at small k
as
2
PHYSICAL REVIEW B 75, 172101 2007
1098-0121/2007/7517/1721014 ©2007 The American Physical Society 172101-1