Supporting Information
Phonon Wave Effects in the Thermal Transport of Epitaxial
TiN/(Al,Sc)N Metal/Semiconductor Superlattices
Bivas Saha1, Yee Rui Koh2,4, Joseph P. Feser5, Sridhar Sadasivam3,4, Timothy Fisher3,4, Ali
Shakouri2,4, and Timothy D. Sands6
1
Department of Materials Science and Engineering, University of California, Berkeley, CA94720, US.
2
School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN-47907,
USA
3
School of Mechanical Engineering, Purdue University, West Lafayette, IN-47907, USA
4
Birck Nanotechnology Center, Purdue University, West Lafayette, IN-47907, USA
5
Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA
6
Bradley Department of Electrical and Computer Engineering and Department of Materials
Science and Engineering, Virginia Tech, Blacksburg, VA 24061, USA.
1. First-principles Density Functional Perturbation Theory Calculation
We use plane wave self-consistent field (PWSCF) implementation of the density functional
theory1 (DFT) with a generalized gradient approximation (GGA)2 to the exchange correlation
energy and ultrasoft pseudopotential3 to represent the interaction between the ionic cores and
valence electrons. Lattice dynamics calculations are performed within the framework of the selfconsistent density functional perturbation theory4 (DFPT). Plane waves with a cut-off of 40 Ry
and 750 Ry are used to represent the wave functions and change densities. These high cut-offs
are used to keep the errors in vibrational frequencies minimal. Integration over the Brillouin zone
is performed with a Monkhorst-Pack scheme5 with a 16x16x16 k-points, and the occupation
numbers are treated with a Methfessel–Paxton6 scheme with a broadening of 0.003 Ry. Force
constant matrices are calculated on a 4x4x4 q-point mesh to understand the detailed feature of the
phonon spectrum. Dynamical matrices at any arbitrary wave vector are obtained with a Fourier
transform based interpolation method.
2. Vibrational Spectra of TiN and (Al, Sc)N
Before we describe the vibrational spectra of the TiN and (Al,Sc)N layers, we mention here that,
to optimize the computational time and resources, a slightly different concentration of aluminum
nitride (AlN) is used in (Al,Sc)N layers for these calculations. In experimental analysis, 72%
aluminum nitride (AlN) is used in the (Al,Sc)N matrix to lattice match it with TiN. However, for
modeling analysis, we use 75% of aluminum nitride in (Al,Sc)N matrix, which allows us to
1
simulate an unit cell with 8 atoms. We believe such slight change in the AlN mole fractions
should not affect any of our qualitative and quantitative comparisons with experiments.
The phonon dispersion spectra of TiN (see Fig. S1(a)) suggest steep LA and TA phonon modes
with high group velocities. While the TA phonon mode disperse normally as expected, the LA
mode show anomalous dip at X=0.8-0.9 regions along the -X direction of the Brillouin zone.
Such anomaly in the phonon dispersion of TiN is characteristic of transition metal nitrides, like
ZrN and HfN, which also have similar dips in the dispersion spectra7. The optical phonon modes
in TiN represent nitrogen atomic vibrations with higher vibrational frequencies. There is a large
gap between the acoustic and optical phonon modes in TiN that spans from 300 to 500 cm-1
ranges. The vibrational spectrum of (Al,Sc)N does not show any anomaly (see Fig. 1S(b)),
however there is significant intermixing between the acoustic and optical phonon modes.
Calculations suggest that Al0.75Sc0.25N has a dielectric constant value of 6.25 that is consistent
with the experimental observations.
The vibrational densities of states (DOS) of TiN and (Al,Sc)N (presented in Fig. S1(c)) suggest
a significant mismatch. In the 300 to 500 cm-1 spectral range where (Al,Sc)N has a large number
of high velocity vibrational modes, TiN has a gap, meaning no vibrational states in that
frequency range. This mismatch in the densities of phonon states will surely reduce the crossplane thermal conductance across the interface. Fig. S1(c) also suggests that TiN optical phonon
modes have a very narrow distribution from 500 to 580 cm-1, however the optical phonon modes
of (Al,Sc)N disperse across a broad range of frequencies, creating windows of vibrational state
mismatch that are expected to impact the interface thermal conductance. The transmission
function of the phonon modes from (Al,Sc)N to TiN are calculated using a full Brillouin zone
dispersion relation and are presented in Fig. S1(d) as a function of the phonon frequencies.
Details about the calculation methodology are presented later in the SI section. Fig S1(d)
however, suggests no transmission of phonon modes from 300 to 500 cm-1 where mismatch in
densities of phonon states exist between the component materials.
2
S1. Figure Caption: (a) Vibration spectrum of TiN along the high symmetry directions of the
Brillouin zone calculated using an FCC primitive unit cell. (b) Phonon dispersion spectrum of
Al0.75Sc0.25N along the high symmetry points calculated using a cubic unit cell. (c) Densities of
phonon states are plotted as a function of phonon frequencies for TiN and Al0.75Sc0.25N plotted
together. PDOS mismatch is clearly visible in the image. (d) The transmission function of
phonons modes from (Al,Sc)N to TiN side calculated using diffused mismatch model.
3. Interface Thermal Conductance (ITC) Modeling: AMM, DMM and Full
Brillouin zone DMM
We provide a brief overview of the acoustic and diffuse mismatch models used to predict
the thermal interface conductance at (Ti,W)N/(Al,Sc)N interface. Though both the AMM and
DMM are simplified models that do not consider the bonding details at the interface, they are
however expected to provide an order of magnitude estimate of the interface thermal
conductance that can be compared with the interface conductance extracted from experimental
data. The interface thermal conductance G between materials 1 and 2 is calculated using the
Landauer formula:
3
o
f BE
 1 
G

(
q
,
p
)
v
T12 ( (q, p))d 3q
(1)
1
g 1, z
  
2


T

 p q , qz  0
where the integration over phonon wavevectors q is performed over the Brillouin zone of
material 1 and summation runs over all phonon polarizations p. ω1 and vg1,z denote the phonon
o
frequency and group velocity along the transport direction respectively. f BE denotes the Bose
3
Einstein distribution function and T12 denotes the transmission function from material 1 to 2. The
AMM and DMM are different approaches to estimate the transmission function.
In AMM, reflection and transmission of phonons are assumed to obey Snell's law and the
probability of transmission for a phonon incident on the interface at an angle θ1, refracted at an
angle θ2 is given by8
4Z1Z 2 cos 1 cos  2
t12 
(2)
( Z1 cos  2  Z 2 cos 1 ) 2
where Z1, Z2 denote the acoustic impedances of materials 1 and 2 respectively. The
acoustic impedance of a material is defined as the product of sound velocity and the mass
density. The transmission function T12 in Eq. (1) is obtained by performing an angular average of
the angular transmission function defined in Eq. (2). In the present implementation of AMM, the
group velocity (in the long wavelength limit) of the acoustic and transverse branches are
obtained from the first-principles phonon dispersion of (Al,Sc)N and (Ti,W)N. The contribution
of optical phonons to interface conductance is neglected. We assume that no mode conversion
occurs at the interface, i.e., LA, TA modes of material 1 transmit as LA, TA modes in material 2.
In DMM, phonons are assumed to lose their identity on reaching the interface and
invoking the principle of detailed balance the transmission function is derived. The transmission
function is given by9
q2  | vg 2, z |   ,q , p
q, p
T12, DMM ( ) 
(3)
q1  | vg1, z |   ,q , p  q2  | vg 2, z |   ,q , p
q, p
q, p
where Δq1,2 denote the volumes of discretized cells in the Brillouin zone and δ denotes
the Kronecker delta function. The above equation reduces to the following more commonly used
expression for transmission under the Debye approximation for acoustic phonon modes.
1
p v 2
g 2, p
T12, DMM ( ) 
(4)
1
1
p v 2  p v 2
g 1, p
g 2, p
In the present work, we obtain DMM estimates of thermal conductance using the exact
phonon dispersion of all the branches over the full Brilloun zone and also using Debye
dispersion for the acoustic modes (optical phonon modes are neglected).
4. Interface Thermal Conductance (ITC) Estimates
4
Interface thermal conductance (ITC) represents the ability of a hetero-interface to conduct heat
from one side of the interface to the other. Traditionally ITCs have been used for a single
interface between different materials that are, in principle, infinitely thick in both directions such
that all of the phonon modes encounter the interfaces as they propagate. In this study, we have
applied the concept of ITC to understand the heat transport in superlattices, which has seldom
been done for describing heat transport in heterostructures. It must be remembered that ITCs are
only valid in the incoherent transport regime where interfaces act as phonon scattering walls that
destroy phonon coherence. For the coherent transport regime, as the superlattices behave as an
effective medium with modified phonon dispersions, ITC doesn’t have physical significance.
Borrowing equation 1 from the main text, the interface thermal resistance (reciprocal of
conductance) of an single interface can be written as
𝑁𝑅𝐵
𝐿
=
1
1
𝜅𝑠𝑢𝑝𝑒𝑟𝑙𝑎𝑡𝑡𝑖𝑐𝑒
- (
1
2 𝜅𝑇𝑖𝑁
+
1
𝜅(𝐴𝑙,𝑆𝑐)𝑁
)………………. (1)
However, to quantify the ITCs from equation 1 as a function of the period thickness, we need to
know the thermal conductivity of the individual layers ((TiN and (Al,Sc)N) when their
thicknesses are half of the period thickness of each superlattices. Since most of the superlattices
studied here have very small period thicknesses, an individual layer with such small thickness
wouldn’t have enough resistance to be measured with experimental methods. Therefore, to
calculate the ITC as a function of the period thicknesses, we consider two scenarios for the
values of the thermal conductivity of the individual layers. In scenario (a) we assume that the
thermal conductivities of individual layers (irrespective of their thicknesses) are same as that of
the thermal conductivity of layers that are 240 nm thick (that we have measured in our
experiments). This assumption would imply that the phonon transport inside the superlattice
layers is fully diffusive even though the layer thicknesses are very small. Naturally, this scenario
would hold true for superlattices having large period thicknesses. On the other hand, in scenario
(b) we assume that the individual layers don’t provide any resistance to the overall thermal
transport and the entire resistance is at the interfaces. In other words, this means that thermal
resistances of the individual layers are negligible. Naturally, this assumption is expected to work
best for short-period superlattices.
The ITCs calculated from the experimental results using the above two scenarios as a function of
the superlattice period thicknesses (above the 4 nm period thickness, when most of the phonon
undergoes incoherent scattering at the interfaces) are presented in Fig. S2. In scenario (a) the
ITCs change only slightly over the entire period thickness range. However, the room temperature
ITC values are extremely high (about 4 GW/m2-K). We note that when we apply a single linear
fit in the
1
𝜅𝑠𝑢𝑝𝑒𝑟𝑙𝑎𝑡𝑡𝑖𝑐𝑒
as a function of N plot in Fig. 3(a) in the main manuscript, the ITC value
that we get is also around 4 GW/m2-K. The theoretical modeling analysis that employs the
radiation approximation representing the maximum ITC limit in this material at room
temperature is 0.9 GW/m2-K. The 4x larger conductance that we measure in the superlattices are
therefore due to the long wavelength coherent phonon modes that are not accounted for in the
modeling analysis.
5
Fig. S2 also suggest that the ITCs decrease with the increase in the period thickness when we use
scenario (b) assumptions. Under scenario (b) we are neglecting a large amount of resistance
contributed by the individual layers. Since the extent of such resistance increases with the
increase in the period thickness, the ITC values also decreases systematically.
S2. Figure Caption: Interface thermal conductance of the superlattices calculated with two
different scenarios and presented as a function of the period thicknesses.
5. Comments on Thickness Dependent ITC vs. Thickness Dependent Thermal
Conductivity in Fig. 4(a).
Though a possible explanation for the deviation of experimental data from Equation 1 in Fig.
4(a) could be attributed to the thickness dependence of the thermal conductivity of individual
layers; however, we think there exists enough indicators in the experimental data that point to a
more nuanced and insightful physical picture of phonon transport.
Figure 4(a) points to the existence of three distinct regimes in the plot of inverse thermal
conductivity vs. number of periods and a linear fit exists in these regimes. While the thermal
conductivity of the TiN and AlScN layers is expected to vary across these three regions, the
linear fits indicate that the thermal conductivity is approximately constant within these regions.
For example, we see from Fig. 4(a) that when the individual layer thicknesses (TiN and
(Al,Sc)N) are 120nm, 60nm, and 30nm respectively, a linear fit could be made with these three
6
data points that would result in an interface conductance of 2.4 GW/m2-K, which is an
acceptable value of ITC for a lattice-matched epitaxial interface. Similarly, within the other two
regimes as shown in Fig. 4(a), linear fits allows extraction of interface thermal conductance. We
believe that the observation of the three different regimes is a coarse indicator of phonon mean
free path in these nitride materials.
To the best of our knowledge, the phonon mean free path of TiN and (Al,Sc)N have neither been
measured experimentally nor any computational results for the cumulative thermal conductivity
vs. phonon mean free path are available in the literature. However, if other known materials and
their physical properties are to be considered as indicators, TiN is likely to have a mean free path
of tens of nm, while (Al,Sc)N being a substitutional solid alloy should have a much shorter mean
free path of the order of a few nanometers.
When the period thicknesses of the superlattices are large, i.e. in regime I (individual layers have
larger thicknesses, e.g., 120nm, 60nm or 30nm), the thermal conductivity of the individual layers
(TiN’s and (Al,Sc)N’s) having different thicknesses are expected to be very similar, as the
phonon mean free paths are smaller than layer thicknesses. Similarly when the period thicknesses
are very small (i.e., regime III), thermal conductivity of the individual layers with respect to their
changing thicknesses are again expected to be very similar since the mean free paths are greater
than the layer thicknesses. In Regime II, (Al,Sc)N is expected to have phonon mean free paths
shorter than the period thickness while TiN is likely to have mean free paths longer than the
superlattice period.
Given such subtle indicators of the phonon behavior in experimental data (Fig. 4(a)), we chose to
interpret the results in Fig. 4(a) as thickness dependent ITC regimes (which also implicitly
assume three-thickness dependent thermal conductivity of the individual layers).
We would also like to point out that in a separate study of heat conduction in (Ti,W)N/(Al,Sc)N
metal/semiconductor superlattices system, we found that a single linear fit suffices for the entire
set of experimental measurements of 1/ k vs. N. Since the mean free path of phonons in (Ti,W)N
is expected to be significantly smaller than TiN, (and is expected to be very similar to (Al,Sc)N
layers), we could explain the single linear regime quite well in that system.
6. HRTEM and HAADF-STEM images.
7
S3 Figure Caption: (a) High resolution transmission electron microscopy (HRTEM) image of
TiN/(Al,Sc)N superlattice. The thickness of the TiN layer is held as constant at 20 nm and the
(Al,Sc)N layer thickness is increased from 2 nm to 24 nm. 2 nm of (Al,Sc)N is clearly visible in
the image. (b) High magnification image of an interface that show cube-on-cube epitaxial crystal
growth. (c) HAADF-STEM image (adapted from our previous publication Ref. 10) of the
superlattice is presented.
7. Assumptions about the heat capacities of the materials vs. temperature.
The heat capacity values of nitride materials are reported in literature7,11-13. For the low
temperature heat capacities, which cannot be found in the literature, we used a linear relationship
to estimate the specific heat capacity. In the superlattices, we assumed the whole superlattice as a
single material in the theoretical calculation. The specific heat capacity of the superlattices is
determined by using volume fraction of AlN, ScN and TiN in the superlattices.
8. Plot of the TiN/Superlattice interface conductance versus period thickness
and temperature
8
We used ~100 nm TiN as the transducer for all of our TDTR measurements. The TiN transducer
layer was grown in situ in the sputter machine immediately after the deposition of the
superlattice that reduces the likelihood of any contamination of the superlattice top surface and
results in very high interface thermal conductance between the TiN transducer and the
superlattice. We measured the room temperature interface conductance of TiN/MgO to be 0.8
GW/m2K, consistent with R. M. Costescu et. al’s14 estimate of 0.7 GW/m2K . The interface
conductances as a function of period thickness and temperatures are presented in Fig. S4.
The figure S4(a) suggest that with increase in superlattice period thickness the interface
conductance between the transducer (TiN) and superlattice decreases by an appreciable amount.
The interface conductance also decreases as a function of the temperature as can be seen in Fig.
S4(b). Though the changes in the interface conductances in both the cases are appreciable, it is
important to note that the differences have a marginal effect in the ratio curves of the
experimental/Fourier model. Fig. S4(c) representing the sensitivity analysis of the interface
conductance suggest that for 150K and 500K measurements, every 100 MW/m2K changes in the
interface conductance affects the thermal conductivity by less than 0.5% over the entire time
range. In other words, the thermal conductivity measurements are insensitive to the interface
conductance.
2
G (x GW/m K)
1
150K
0.5
1
10
100
Period Thickness (nm)
Fig. S4(a) Interface conductance TiN/superlattices versus period thickness at 150K.
9
2
G (xG W/m K)
1
Period Thickness
120nm-120nm
1.25nm-1.25nm
0.1
100
200
300 400 500600
Temperature (K)
Fig. S4(b) Interface conductance TiN/superlattices versus temperature at different period
thicknesses.
2
S (Ratio/100MW/m K)
0.5
150K
0.0
500K
-0.5 2
10
3
t (ps)
10
Fig. S4(c) Sensitivity analysis of the interface conductance TiN/superlattices at 150K and 500K.
The fitting quality of the ratio curves (presented in Fig S4(d) and S4(e)) clearly shows that
the 500K measurements fit better than the 150K measurements. This results also show that
0.7nm/0.7nm superlattice fittings at 150K are relatively poor resulting in large error.
10
6
5
Ratio (-Vin/Vout)
4
120nm-120nm
0.7nm-0.7nm
3
10nm-10nm
2
1.7nm-1.7nm
150K
1 2
10
3
t (ps)
10
4
0.7nm-0.7nm
120nm-120nm
2
D
Ratio (-Vin/Vout)
3
1.7nm-1.7nm
10nm-10nm
500K
1 2
10
3
10
t (ps)
Fig. S4(d) and S4(e) Fittings quality of the selected ratio curve at 150K and 500K. Open circles are
experiment data, dashed lines are the best fits of the thermal model.
11
9. Discussion on the sensitivities of the ratio data to the thermal conductivity
of the sample
Sensitivity (Ratio/(W/mK))
The sensitivity analysis for the ratio data is presented in the Fig. S5. The details of the
description of the sensitivity analysis are presented in Ref. 15. We used the sensitivity
R  R2 1
S 1
, where R is the ratio and κ is the thermal conductivity.
R1 k1  k 2
10
9
150K
8
300K
7
500K
6
5 0
10
1
10
2
10
t (ps)
3
10
Fig. S5 Sensitivity analysis of the superlattices thermal conductivity at 150K, 300K and 500K.
The ratio curves of the measurements are sensitive to the change of the superlattices thermal
conductivity. A raise of 6-8% can be observed at 100-1000ps region with every increasing of
1W/mK in superlattices thermal conductivity at 150-500K. These sensitivity analysis of the
ratio curves enhance the confident level in the fittings quality of the measurements.
10. Discussion on assumptions about the sample/MgO interface conductance
Since the superlattice and thin-film sample thicknesses are around 240 nm, we selected
9.7MHz as pump modulation frequency to match the sample thickness. The heat penetration
depth might differ depending on the thermal conductivity of various samples; however the
penetration depth is always higher than the sample thicknesses. Therefore, the
superlattices/MgO interface conductance was always included in the measurements.
However the impacts of the interface conductance of superlattices/MgO are small and can be
ignored. This conclusion can be proved by using sensitivity analysis. The Fig. S6 shows the
sensitivity analysis of superlattices/MgO interface conductance. Since the impact of the
superlattices/MgO interface conductance to the ratio curve is different in superlattices with
different thermal conductivity, we used two superlattices with different thermal conductivity
in the superlattices/MgO interface conductance sensitivity analysis. From the Fig. S6, we can
12
observe that the ratio curves change only 0.05% and 0.3% for every 100MW/m2K change in
the superlattices/MgO interface conductance for superlattices having 5 W/mK and 10 W/mK
thermal conductivities respectively.
2
S (Ratio/100MW/m K)
0.5
superlattices=10W/mK
superlattices=5W/mK
0.0 2
10
3
10
t (ps)
Fig. S6.
Sensitivity analysis of the interface conductance superlattices/MgO at room
temperature.
11. Temperature dependence of the alloy thermal conductivities
The cross plane thermal conductivity of Ti0.5Al0.36Sc0.14N alloy as a function of temperature is
presented in Fig. S6. Thermal conductivity of the alloy decreases from 21 W/m-K at 300K, to 12
W/m-K at 500K, which accounts for about 50% decrease over the 200K temperature range.
Such decrease in the thermal conductivity as a function of temperature is due to high temperature
Umklapp scattering that govern the thermal conductivity mechanism at high temperatures.
Fitting the thermal conductivity vs. temperature with 𝜅 = 𝐴𝑇 𝛼 (characteristic of high
temperature Umklapp scattering) suggest that 𝛼 = −1.28 which is consistent with typically for
an alloy.
13
Ti0.5Al0.32Sc0.18Nnm(W/mK)
25
20
15
~ 50%
10
5
300
350
400
450
Temperature (K)
500
S6. Figure Caption: Thermal conductivity of the Ti0.5Al0.36Sc0.14N alloy which has equivalent
concentrations as that of the superlattice. Thermal conductivity decreases as a function of temperature
as seen in the figure.
12. Speeds of sound and acoustic impedances of the layers
Using the phonon dispersion spectrum (see SI Fig. 1(a) and 1(b)) that we have calculated using
density functional perturbation theory (DFPT), we find that the speed of sound in TiN and
Al0.72Sc0.28N are 6274 m/s and 7950 m/s respectively. The acoustic impedance (Z) of the
materials are calculated by using the formula Z=v* where v and  are the sound velocity and
density of the material respectively. The results suggest that the acoustic impedance Z has a
value of 3.38 x 107 Kg/m2-s for TiN and 3.19 x 107 Kg/m2-s for Al0.72Sc0.28N. Therefore, the
acoustic impedance of the materials are very close to each other.
13. Compositions and Thickness measurement.
Composition of the (Al,Sc)N layers have been measured by Rutherford Backscattering
Spectroscopy (RBS) at the Materials Research Laboratory of the University of Illinois at Urbana
Champaign. The measurements were performed with a 2000 eV He+2 ion-beam. Since
Al0.72Sc0.28N is metastable on TiN layers, we used a lower mole fraction of AlN in (Al,Sc)N to
estimate the deposition rate and resultant composition. A typical RBS spectrum along with the
fitting data for a Al0.255Sc0.2445Ta0.0005N0.5 sample is presented. The simulated data fit well with
the experimental data in the higher energy regions of the spectra representing Al, Sc, Mg and Ta
backscattering, while the fitting is relatively poor in the low energy channels corresponding to
He+2 backscattering from oxygen in MgO and nitrogen in AlxSc1-xN. All of the AlxSc1-xN films
have (0.01-0.02) atomic % of Ta as a contaminant, which originates from the manufacture of Sc
14
targets. Superlattice thicknesses are measured with X-ray reflectivity based measurements (Ref.
X).
Figure Caption: RBS Spectra along with the fitted data for a typical (Al,Sc)N film used to
measure the composition.
References
1.
2.
3.
4.
5.
6.
7.
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
X. Hua, X. Chen, and W. A. Goddard III, Phys. Rev. B 55, 16103 (1997).
D. Vanderbilt, Phys. Rev. B 41, 7892 (1990).
S. Baroni, S. de Gironcoli, A. D. Corso, P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).
M. Methfessel and A. Paxton, Phys. Rev. B 40, 3616 (1989).
B. Saha, J. Acharya, T. D. Sands and U. V. Waghmare, J. Appl. Phys., 107, 033715,
(2010).
8. E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989).
9. P Reddy, K Castelino, A Majumdar, Appl. Phys. Lett. 87, 211908, (2005).
10.G.
V. Naik, B. Saha, J. Liu, S. M. Saber, E. Stach, J. M. K. Irudayaraj, T. D. Sands, V. M.
Shalaev and A. Boltasseva, Proc. Natl. Acad. Sci. 111, 7546 (2014).
11. V. I. Koshchenko, Ya.Kh. Grinberg, A. F. Demidenko, Inora. Mater. 20, 11 15501553. (1984).
12. S. Tahri , A. Qteish, I. I. Al-Qasir and N. Meskini1 , J. Phys.: Condens. Matter 24
035401 (2012).
15
13. http://webbook.nist.gov/cgi/cbook.cgi?ID=C25583204&Type=JANAFS&Table=on.
14. R. M. Costescu, M.l A. Wall, and D. G. Cahill, Phys. Rev. B 67, 054302 (2003).
16