Numerical simulation of the heat transfer in amorphous ... - Physics

Rev. Sci. Instrum., Vol. 74, No. 10, October 2003
Simulation of membrane calorimeter
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a function of temperature, and is shown in Fig. 7 along with
c 2D,s /c 2D,m vs T for c 2D,s calculated for an 1800-Å-thick Cu
conduction layer. The specific heat derived by this means of
the low stress a-Si–N is discussed in a separate
publication; 23 we note here only that the molar specific heat
is low at all temperatures, indicative of a high Debye temperature
D for crystalline Si 3 N 4 , D is 1130 K, 25 so a high
value for the amorphous material is not surprising.
FIG. 7. C (J/mol K) for 1800-Å-thick low-stress LPCVD Si–N membrane
density 2.9 g/cm 3 and 21 g/mol were used and c 2D,Cu /c 2D,m shown in
inset vs T for 1800-Å-thick Cu conduction layer based on literature values
for Cu see Ref. 24. C for Si–N calculated from measurements of c tot for
devices with Cu conduction layers from c tot A s (C Cu t Cu C m t m )
XA b C m t m c Pt c Nb–Si with X0.24 as shown in Fig. 9. A b is the area
of the border portion of the membrane. Pt leads and Pt thermometer and
Nb–Si thermometer contributions were calculated and subtracted from c tot
but are small 15% total; see Ref. 23. t m is measured by ellipsometry, t Cu
is measured by Dektak profilometry on a neighboring sample and is the
largest source of uncertainty. In Ref. 23, several different devices with different
conduction layers Cu and Al were compared and shown to give the
same C m .
sity of 2.9 g/cm 3 for low stress low-pressure chemical vapor
deposition LPCVD a-Si–N from Refs. 21 and 22. This is
significantly smaller than the specific heat of silicon oxide at
20 K. The solid line in Fig. 5 shows the relaxation calculated
using this value; the difference between this calculation and
experiment is shown in the inset of Fig. 6.
We will use the 20.3 K value found here for c 2D,m for the
simulations discussed below, but in order to extend the results
to the entire temperature range of measurement,
c 2D,m (T) and c 2D,Cu /c 2D,m (T) is needed. One approach
would be to measure and simulate the relaxation curves for
the bare membrane at all temperatures. Instead, by using a
high thermal conductivity layer such as Cu for c 2D,s , measurements
of the addenda heat capacity membrane with Cu
conduction layer were analyzed using results derived in the
next section to give the membrane heat capacity from measurements
of P/T and from a simple fit to e (t/) . With
the Cu conduction layer, to an absolute accuracy of a few %
discussed in the next section, the total heat capacity c tot
P/T can be written as a sum of the heat capacity of the
Cu plus the membrane including a contribution from the
membrane border to be numerically analyzed in the next
section plus the Pt and Nb–Si heaters and thermometers.
The total contribution of Pt and Nb–Si is less than 15% of
c tot at all temperatures. 23 The specific heat of Cu can be
directly measured by adding a second sample layer of Cu,
or from the literature values for Cu, 24 allowing extraction of
c 2D,m to 10% accuracy, which is more than sufficient to
make an estimate of c 2D,s /c 2D,m needed in the following
sections. Using this procedure, c 2D,m has been determined as
2. Calorimeter with sample
To model the standard experimental relaxation method,
time-dependent relaxation curves were simulated for the
membrane with sample layers of various specific heat and
thermal conductivity. For sample layers where a single exponential
decay provides a reasonably good description of the
results, the quality of the fit and the contributions of the
membrane border and the Pt leads are obtained and discussed
Sec. IIIB2a. In Sec. IIIB2b, we consider the differential
method in which the sample heat capacity is determined
from the difference between two measurements: the membrane
with a first sample layer typically, Cu, which provides
the addenda, and the same membrane with the first
layer and an added sample layer. The second sample layer
was considered to have either similar properties to the first
layer as in a calibration measurement where both are Cu, or
significantly lower thermal conductivity as in a standard
measurement of a sample of interest. The membrane specific
heat and thermal conductivity were held constant their values
at 20.3 K were used but we reiterate that it is the ratios
of k 2D,s /k 2D,m and c 2D,s /c 2D,m that are important.
a. Single sample layer and contributions of membrane
border and Pt leads In the first set of simulations, we took
the specific heat of the sample as fixed values for a typical
metal were used and studied the effect of changing its thermal
conductivity and thickness which changes both k 2D,s
and c 2D,s , and more importantly k 2D,s /k 2D,m and
c 2D,s /c 2D,m ). The temperature at the position of thermometer
T2 0;0.1 was calculated for 150 equally spaced times.
The initial conditions at t0 were set using the calculated
static distribution of temperatures as described in the preceding
section. In order to more clearly observe the dependence
of the parameters obtained on the ratio of k 2D,s /k 2D,m ,we
performed this set of simulations primarily without the Pt
contributions i.e., with k 2D,Pt c 2D,Pt 0); tests of the Pt
contribution will be discussed below.
The values used for the calculations are k 2D,m
0.2 W/K, c 2D,m 0.3 J/K cm 2 , and sample specific heat
c 2D,s from 0.1 to 300 J/K cm 2 giving a range of
c 2D,s /c 2D,m from 0.3 to 1000; a typical metal at 20.3 K
might have C0.055 J/K cm 3 ; this then corresponds to
thicknesses from 180 Å to 45 m. The 2D sample thermal
conductivity k 2D,s was allowed to range over five decades,
from equal to the membrane thermal conductivity to 10 5
greater. The steady-state results for P/T are shown
above in Fig. 3 and are not affected by c 2D,s .
The time-dependent temperature relaxation of T2 was fit
to a single exponential e (t/) between 100% and 0.3% of the
initial t0 value. The quality of this fit as well as the value
for were determined. The total specific heat from the simu-
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