Solid State Communications 129 (2004) 199–204
www.elsevier.com/locate/ssc
Specific heat and thermal conductivity of low-stress amorphous
Si – N membranes
B.L. Zink*, F. Hellman
Department of Physics, University of California, La Jolla, San Diego, CA 92093, USA
Received 20 November 2002; received in revised form 21 July 2003; accepted 25 August 2003 by R.C. Dynes
Abstract
We present values of the specific heat and thermal conductivity from 3 – 300 K of low-stress amorphous silicon-nitride thinfilms determined from measurements using a membrane-based microcalorimeter. The thermal conductivity has a temperature
dependence often seen in amorphous solids, but the magnitude is large, with the expected plateau occurring at significantly
higher temperatures than seen in other amorphous systems. Specific heat measurements show that the expected ‘peak’ in the
vibrational spectrum also occurs at relatively high temperatures. The estimated phonon mean-free-path at 300 K is <5 Å,
comparable to the inter-atomic spacing, as seen in other amorphous solids. Below < 20 K the mean free path is comparable to
or exceeds the thickness of the membrane, indicating that surface scattering dominates the thermal transport. This surface
scattering is found to be either specular or diffuse, depending on details of the membrane processing, which affects both the
thermal conductivity and specific heat below 10 K.
q 2003 Elsevier Ltd. All rights reserved.
PACS: 65.60. þ a; 66.70. þ f
Keywords: A. Thin films; A. Disordered systems; D. Heat conduction
1. Introduction
The long history of research on thermal properties of
amorphous solids details remarkably similar behavior for a
wide range of disordered materials but no comprehensive
picture of the physical origins of these similarities.
Disordered solids have low thermal conductivity, k;
compared to crystals, with a plateau generally around 10 –
50 K above which k continues to increase with temperature,
similar temperature dependence and magnitude below
< 10 K, and k / T 1:8 below 1 K [1 – 4]. The T , 1 K
behavior is currently best understood and is believed to be
caused by phonons scattering off two-level tunneling states
of incompletely understood physical origin.
Two features are nearly universally observed in the
specific heat, C; of amorphous materials. The first is a linear
* Corresponding author. Tel.: þ1-303-497-4320; fax: þ 1-303497-3042.
E-mail address: [email protected] (B.L. Zink).
0038-1098/$ - see front matter q 2003 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ssc.2003.08.048
term at T , 1 K due to a constant density of the two-level
state systems which also dominate phonon scattering and
lead to the characteristic behavior of k at these temperatures
[1,4]. The second is a broad peak or bump in C=T 3 vs. T
which occurs at the same temperature as the plateau in k:
The height of this peak, Pc ; also scales with the temperature
21:6
[5]. This bump
at which it occurs, Tmax ; so that Pc / Tmax
and the corresponding plateau in k indicate a large density of
relatively low-energy vibrational states, but the physical
origin of the scaling behavior is not yet understood. There
has been some success explaining these phenomena within
the general concept of a soft potential and/or interacting
defect model, which attempts to fit the C=T 3 peak, the k
plateau and the T , 2 K tunneling states together in a single
model. These modeling efforts are still controversial and the
details of the models and the underlying microscopic
mechanisms are not yet universally accepted.
Amorphous silicon nitride membranes are the key
structure in thin-film microcalorimeters which have proven
in recent years to be a powerful tool for measuring C and k
200
B.L. Zink, F. Hellman / Solid State Communications 129 (2004) 199–204
of thin films and other small samples over a wide temperature range [6– 10]. Recent investigations of membrane
microcalorimeters include computer simulations [11] which
provide a detailed understanding of the heat flow, allowing
extraction of the k and C of the a-Si –N membrane from
measurements of the microcalorimeter.
There have been several previous measurements of the
thermal conductivity of a-Si –N which are limited either to
T . 60 K [12– 15] or to T , 10 K [16] and show values
strongly dependent on the method used to grow the films.
Specific heat data has been reported only for sintered bulk
samples of Si – N compounds which are mostly polycrystalline [17 – 20]. To our knowledge there are no measurements
of C or k of a-Si– N in the 10 – 50 K range where the
pleateau in k and peak in C=T 3 are expected. With our
technique we can present k and C measured on the same
sample over a wide temperature range, avoiding uncertainties introduced by variation between preparation methods.
In this paper we first describe the growth and preparation
of the a-Si– N membrane. We then describe the methods
used to extract the specific heat, CSi2N ; and thermal
conductivity, kSi2N ; from 3 – 300 K, and compare the results
obtained from several calorimeters to the C and k of other
amorphous materials. We further analyze our data to
determine the phonon mean free path in the membrane as
a function of temperature. This allows us to examine the role
of surface scattering in the thermal transport in the a-Si– N
membrane.
2. Experiment
The a-Si– N film is grown by low pressure chemical
vapor deposition (LPCVD) onto silicon-dioxide coated Si
wafers. The a-Si– N is grown at 835 8C using ammonia
(NH3) and dicholorosilane (SiH2Cl2) reagent gases. The
residual stress is related to the ratio of Si to N in the
deposited film, and is controlled by adjusting the NH3:SiH2Cl2 ratio in the furnace during deposition. If the stress is low,
the film forms a free-standing membrane when the Si
substrate is removed from beneath it by etching in KOH.
Low-stress a-Si– N films are approximately 50 at.% Si and
50 at.% N, which is silicon-rich compared to the
stoichiometric compound Si3N4 [21,22]. Typical growth
rates for this process are <45 Å/min. The film thickness is
measured with optical interferometry, and is typically
between 1800– 2200 Å. The uniformity in thickness across
the surface of a single wafer is normally very good, with
deviations #20 Å.
The silicon-dioxide underlayer is either grown from the
wafer using a wet oxidization process at 1000 8C (often
called a thermal oxide) or deposited by LPCVD at 450 8C
(commonly referred to as low-temperature oxide, LTO).
The LTO can be made thicker (1.5 mm), but the thermal
oxide is flatter and less porous (thicknesses are 4000–
6000 Å).
After the a-Si– N layer is deposited, a 500 Å thick Pt
layer is sputtered and patterned into leads, heaters and hightemperature thermometers. For measurements below 50 K
we use amorphous Nbx – Si12x thermometers which are
patterened on the membrane using a Cu lift-off procedure. A
0.25 £ 0.25 cm2, ,2000 Å thick thermal conduction layer
on the center of the membrane keeps the heater, thermometers and sample isothermal.
The heat capacity, c; of the microcalorimeter is the result
of contributions from the a-Si– N membrane, the thermal
conduction layer (typically Al, Cu, or Au), the Pt leads,
heater and thermometer, and the a-NbxSi12x thermometers.
The contribution of the Pt, conduction layer and a-NbxSi12x
must be determined and subtracted from c to obtain cSi – N :
An example is shown in Fig. 1. This figure shows the total
measured heat capacity for a microcalorimeter, and the
various contributions. The dominant contribution for most
temperatures is the metallic thermal conduction layer (an Al
layer for the calorimeter shown in Fig. 1). The inset shows
the same plot at low temperatures on a log scale. The heat
capacity of the a-NbxSi12x is approximately an order of
magnitude smaller than that of the metal layers and is not
shown in the inset. The area of each of the features
contributing to c is precisely known from the photolithography; the thickness is determined either by profilometry or
inferred from growth parameters and comparison to
neighboring devices. We used literature values for the
specific heat of Al, Cu, and Pt [23,24] and approximated
the specific heat of a-NbxSi12x with a similar composition
of a-YxSi12x [25]. Calculating the contribution to c of the
conduction layer, heater and thermometers is straightforward,
Fig. 1. The total heat capacity of the microcalorimeter, and
the contributions which were subtracted to give cSi2N : The error on
the measurement is <2% based on uncertainty in calibration of the
thermometers on the membrane. The inset shows the same data to
20 K on a log scale. This microcalorimeter’s a-Si –N membrane was
grown on a LTO underlayer.
B.L. Zink, F. Hellman / Solid State Communications 129 (2004) 199–204
as these features are isothermal to good accuracy and
therefore contribute 100% of their heat capacity. The
contributions of the Pt leads and the a-Si – N membrane
are more complicated, as a thermal gradient exists across
these features. 2d heat flow simulations [11] indicate that
24% of the two-dimensional a-Si –N membrane outside the
sample area and 30% of the approximately one dimensional
Pt leads contribute to c: The contributions of the Pt and
a-NbxSi12x features are small compared to the membrane
and thermal conduction layer, and any error resulting from
deviations of the real material’s specific heat from the values
used here is negligible. Subtracting these contributions
results in a heat capacity of a-Si– N, cSi2N (in J/K), which is
converted to CSi2N (in J/g K) using the geometry of the aSi – N (100% of the 0.25 £ 0.25 cm2 central sample
area þ 24% of the ((0.5 £ 0.5) 2 (0.25 £ 0.25)) cm2 membrane border area) and a density r ¼ 2:9 g=cm3 [14,16].
Measurements of devices using different metals for the
conduction layer (Al, Cu, Au) give the same values of C for
a-Si– N, giving confidence in this method of extracting the
specific heat of the silicon-nitride [25].
Fig. 2 shows the thermal link, K (in W/K), connecting
the central area of the membrane to the Si frame. K is
determined from a steady-state measurement of DT
resulting from heating power P dissipated in the heater on
the membrane; K ¼ P=DT: At low temperature K has
contributions only from the Pt leads and a-Si– N membrane.
However, above 100 K heat losses from radiation must be
subtracted (a procedure we describe at length in another
publication [8]). The contribution of the Pt, KPt ; is
determined from the Wiedemann – Franz law ðk=s ¼ L0 TÞ
where s is the electrical conductivity, determined for each
calorimeter by measuring the resistance of the Pt heater. The
Pt films used here for leads and heater are deposited at room
temperature in a relatively poor vacuum and hence are
“dirty.” Using the Wiedemann – Franz law to determine their
contribution to K introduces an uncertainty of < ^ 15% in
KPt [26,27] that dominates the uncertainty in KSi2N between
approximately 20 K (where the Pt film reaches its residual
resistivity limit) and 100 K (where the uncertainty in the
radiative contribution dominates).
Correcting for radiation and subtracting KPt gives KSi2N :
The thermal conductivity is given by kSi2N ¼ KSi2N =at
where t is the film thickness determined by optical
interferometry and a ¼ 10:33 is a geometry factor given
by the heat flow simulations [11] for this 2d square
membrane.
3. Results
Fig. 3 shows C of low-stress a-Si– N in J/g K vs. T,
extracted from several calorimeters, some with Al and some
with Cu conduction layers. C of vitreous silica is shown for
comparison [2]. We have chosen to show C plotted in units
201
Fig. 2. The measured thermal link, K of the microcalorimeter (open
circles) shown with the contributions from radiation and from the
elements of the device. The contribution of the a-Si-N membrane
(here grown on thermal oxide) is the result of subtracting these
contributions from Ka. The high T error is dominated by the
uncertainty in the radiation term. Below 100 K the error is < 5%,
dominated by uncertainty in kPt
of J/g K both because it is the most common unit used for
amorphous materials and because it is the least ambiguous
for these non-stoichiometric materials. For reference, the
per average atomic weight (g/mol-average-atom, often
called g-atom) is 21 for a-Si – N and 20 for SiO2, and the
respective densities are 2.9 and 2.2 g/cm3. Plotting in units
of J/g-atom K would somewhat reduce the difference
between SiO2 and a-Si – N seen in Fig. 3, but the largest
difference is due to the enormously different Debye
temperatures of these two materials.
The inset shows the same plot for three microcalorimeters at low T: The error bars indicated are dominated by
the uncertainty in the thickness of the conduction layer, and
are 15– 20% below 4 K, and ,10% at higher T: The results
for all samples are within these error bars at all but the
lowest temperatures. Below 6 K, membranes grown on
thermal oxide have C which is as much as 5 times larger
than the C for membranes grown on LTO [28].
Fig. 4 shows the low-stress a-Si – N data on a log – log
C=T 3 vs. T plot. Our experimental results (B, a-Si– N grown
on LTO, X, a-Si –N grown on thermal oxide) are compared
to results for several sintered bulk crystalline Si3N4 samples
(open symbols [17 – 19]), and a sintered Si3N4 nominally
crystalline sample which the authors state is dominated by
the presence of a glassy phase ( £ ’s [20]). The vitreous
silica data is also shown. The solid line is the Debye
function for crystalline Si3N4 ðQD ¼ 1130 KÞ [17]. The
short dotted line near the y-axis indicates the low
temperature value of a Debye specific heat calculated for
a-Si– N using the density of atoms, which we estimate from
the mass density ðr ¼ 2:9 g=cm3 Þ and an average molar
mass (21 g/mol) and an average sound velocity, v; giving
QD ¼ 985 K: v is determined from room temperature
202
B.L. Zink, F. Hellman / Solid State Communications 129 (2004) 199–204
likely to be a result of different film microstructures. All
measurements of LPCVD a-Si – N indicate a very large k at
high temperatures for an amorphous solid. The shape of the
curve is typical, with a plateau beginning at < 20 K but
extending to a relatively high < 100 K. The low temperature
values are within the range of previously measured
amorphous solids [3]. The inset in Fig. 5 shows the low T
k on a log scale for a-Si – N grown on thermal oxide and
LTO underlayers.
4. Discussion
Fig. 3. The specific heat of the a-Si –N membrane. The specific heat
of vitreous silica [2] is shown for comparison. INSET: C vs. log T
below 10 K.
ultrasonic measurements which indicate two transverse
modes with vt ¼ 6:2 £ 105 cm=s and one longitudinal mode
vl ¼ 10:3 £ 105 cm=s [16,29].
The thermal conductivity of amorphous solids is roughly
spanned by vitreous silica [2] and a-CdGeAs2 [30]. Fig. 5
compares our measured k to amorphous Si– N films grown
with various techniques and these representative amorphous
solids. Our results agree well with other measurements of
LPCVD a-Si – N [16,14,13]. The difference in k between the
LPCVD films and those grown by plasma-enhanced CVD
(PECVD) or atmospheric-pressure CVD (APCVD) [12] is
Fig. 4. C=T 3 vs. T on a log–log plot comparing our measurements of
a-Si –N (B, LTO underlayer; X, thermal oxide underlayer) with
vitreous silica [2], bulk Si3N4 (W, [17,18]; S and f, [19]; £ , [20]),
and a Debye function with the QD ¼ 1130 K literature value for
bulk Si3N4 [17]. The short dotted line is the low T limit of C=T 3
using uD;amorphous ; calculated from the sound velocity in a-Si –N.
The open arrow indicates the likely location of the C=T 3 peak for aSi–N.
In Fig. 4, the peak around 10 K seen in vitreous silica is
the characteristic peak seen in amorphous solids [5] which
occurs in the same temperature range as the plateau seen in
k; <20 – 100 K in a-Si– N. The T 21:6 scaling behavior
suggests a peak at 60 K would have a maximum of Pc <
3 £ 1027 J=g K4 : As shown in Fig. 4, C=T 3 for the a-Si – N
levels off at this value before increasing again as
temperature decreases. C=T 3 for a-Si– N grown on thermal
oxide continues to increase down to 3 K. This indicates nonT 3 behavior similar to that seen in other amorphous
materials at lower T (due to TLS). The material grown on
LTO shows significantly reduced values below < 6 K, a
temperature similar to that where k deviates for the two
types of a-Si –N. The various bulk crystalline Si3N4 samples
show large sample dependence at low temperatures, with
derivations from the simple Debye function which suggest a
somewhat complicated low-energy vibrational excitation
spectrum. This is likely due to disorder in the crystalline
structure giving rise to impurity modes. Note that the
Fig. 5. The thermal conductivity of a a-Si –N film grown on thermal
oxide (B, labeled kSi – N ) compared to kSi –N values reported by other
groups grown by LPCVD (single £ , LP1 [14]; W, LP2 [13]; A, LP3
[16]; single N, LP4 [15]), PECVD (PE [12]), and APCVD (AP
[12]), as well as vitreous silica [2] and CdGeAs2 [30]. INSET: k vs.
T below 20 K. Above 20 K films grown on thermal oxide and LTO
have the same kSi2N :
B.L. Zink, F. Hellman / Solid State Communications 129 (2004) 199–204
specific heat of the bulk sample which matches the a-Si– N
grown on LTO below 10 K is dominated by the presence of
a glassy phase [20].
Fig. 6 shows our estimation of the phonon mean free path
in our a-Si – N membranes vs. T on a log – log plot.
Following the literature [2,3] we determine ‘ from ‘ ¼
3k=CDebye v; where k is the measured thermal conductivity, v
is the average sound velocity and CDebye is the specific heat
resulting from acoustic plane waves in the material,
approximated by the Debye specific heat of the crystalline
analog. The measured C is not used to determine ‘ because
it includes large contributions from local excitations that do
not carry heat. However because our amorphous film is Sirich, simply using the CDebye from bulk Si3N4 is questionable. We therefore use a Debye specific heat with QD ¼
985 K determined from the average sound velocity in
similar membranes. This specific heat contribution is close
to the value one would calculate by considering a
hypothetical crystalline analog as a two-phase mixture of
Si and Si3N4 each with the crystalline QD : Fig. 6 shows
results for two membranes, one grown on a thermal oxide
underlayer and one grown on an LTO underlayer, along with
the result for vitreous silica [2]. At high T all curves
approach the inter-atomic spacing (< 2.5 Å for a-Si– N),
and increase rapidly as the temperature decreases. Below
20 K ‘ for both a-Si– N membranes exceeds the 2000 Å
membrane thickness. The two different membranes differ
below this point. ‘ continues to increase for the membrane
grown on the thermal oxide underlayer, while ‘ in the
membrane grown on LTO is reduced and appears to reach a
limiting value of 1 mm. This behavior indicates that diffuse
surface scattering dominates the phonon transport in the aSi –N grown on LTO. Values of ‘ seen in the a-Si– N grown
on thermal oxide which exceed the membrane thickness
indicate that specular scattering occurs at the surfaces.
Values of ‘ calculated for Holmes’ data for thicker a-Si– N
membranes (shown in Fig. 5 and labeled “LP3”) are similar
to values for a-SiO2 below 1 K, with ‘ continuing to
increase as T drops. This behavior is consistent with a mean
free path not limited by surface scattering in these thicker
membranes.
We suggest that the difference in k and ‘ for the two
types of a-Si– N membrane (shown in Figs. 5 and 6) is due
to the different bottom surface created by growing on
different oxides. The top surface of the membrane is likely
less affected by the choice of oxide underlayer. We expect
that when a-Si– N is grown on the flatter thermal oxide, a
smoother surface is left behind when the oxide is etched
away. Scattering off this smoother surface would be at least
partially specular, while scattering off the rougher surface
left after removal of the LTO would cause more diffuse
scattering, reducing ‘ in the temperature range below 20 K
where the intrinsic ‘ exceeds the membrane thickness. The
effect on thermal conductivity of surface scattering in
different limits has been studied for bulk materials. Thin
films deposited on very clean surfaces of pure Si wafers can
203
Fig. 6. The phonon mean-free path as a function of temperature on a
log–log plot. INSET: The same plot at low temperatures shows the
deviation in MFP between microcalorimeters fabricated with a
thermal oxide underlayer and those with a LTO underlayer.
disturb specular surface scattering, causing reduction of the
‘ and k [31]. There is some indication that similar physics
occurs in thicker a-Si –N membranes [16]. Calculations of
scattering processes [32] assuming that phonon scattering is
diffuse at both surfaces give an expected limiting mean free
path of ,7000 Å. The 1 mm mean free path observed in the
a-Si– N grown on LTO is plausible for diffuse scattering
from the bottom surface and specular scattering from the
top, while the steadily increasing ‘ for the thermal oxide
suggest specular scattering from both surfaces.
Above 6 K a-Si– N grown on thermal oxide and LTO
underlayers have the same C: Below 6 K, the membranes
grown on LTO have a reduced C. Though the oxide is
removed in both cases and the nominal deposition
conditions of the a-Si– N are the same, it is possible that
there are subtle microstructural differences which cause the
different C (and k) for the two types of a-Si– N. However,
we suggest that it is possible that the different low
temperature C is related to the different k for the two
types of a-Si– N, and specifically to the difference in surface
roughness. In the dominant wavelength approximation [1,3],
the phonons which contribute maximally to C at a given T in
the Debye model have wavelength ldom ¼ v=nmax : Here v <
7:6 £ 105 cm=sec is the average velocity and nmax ¼
4:25kB T=h , 90 GHz £ T is the dominant frequency. This
at 6 K and 290 Å at 3 K. While this is
gives ldom ¼ 140 A
still significantly less than the membrane thickness, the
number of allowed modes in the thickness direction will be
substantially reduced from the 3d limit. However, even the
3d calculated Debye specific heat is more than an order of
magnitude less than the total specific heat, indicating that in
this temperature range C is completely dominated by nonacoustic modes. The membrane which shows specular
scattering in k has a classic low T behavior in C; which is
204
B.L. Zink, F. Hellman / Solid State Communications 129 (2004) 199–204
usually thought to be dominated by tunneling modes. The
membrane which shows diffuse scattering and a reduced k
has significantly reduced C; with an atypical temperature
dependence compared to other amorphous materials. It is
possible that the rougher surface introduces a cut-off of
some type into the tunneling mode spectrum. Measurements
of membranes with different surface preparations extending
to lower temperatures where tunneling modes clearly
dominate are required to test this speculative statement.
5. Conclusion
We presented measurements of the specific heat and
thermal conductivity of low-stress amorphous Si– N membranes from 3 to 300 K. The thermal conductivity is large at
high temperatures (above , 30 K), is within the range of
previously measured amorphous solids at lower temperatures, with the characteristic plateau at rather high
temperatures (< 60 K), and agrees well with previous
measurements of LPCVD a-Si– N at high and low T. The
specific heat shows the expected weak maximum in C=T 3 at
a similar temperature as the plateau in k: The phonon mean
free path estimated from our measurements approaches the
inter-atomic spacing at room temperature and below 20 K is
dominated by surface scattering which can either be diffuse,
with ‘ limiting at a value consistent with scattering from one
rough surface and resulting in a reduced k; or specular, with
‘ continuing to increase as T drops. Membranes with diffuse
scattering also show reduced C below 6 K, with an atypical
temperature dependence. We have also developed microcalorimeters with < 1.5 mm thick a-Si– N membranes for
measuring the specific heat of small bulk samples
(<200 mg) [33]. In these thicker samples surface scattering
will begin to affect the thermal properties at lower
temperatures. This will allow us to better isolate the effects
of surface roughness from the measurements of C and k at
low temperatures.
Acknowledgements
We would like to thank B. Revaz for numerous
contributions, D. Cahill, B. Pohl, A. Migliori, S. Huxtable,
C. Surko and R. Dynes for helpful discussions, the staff and
students of the UC Berkeley Microfabrication Lab, and the
NSF and UCDRD for support.
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