thermal conductivity of znte nanowires...thermal conductivity of znte nanowires keivan davami,1...
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Thermal conductivity of ZnTe nanowiresKeivan Davami, Annie Weathers, Nazli Kheirabi, Bohayra Mortazavi, Michael T. Pettes et al. Citation: J. Appl. Phys. 114, 134314 (2013); doi: 10.1063/1.4824687 View online: http://dx.doi.org/10.1063/1.4824687 View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i13 Published by the AIP Publishing LLC. Additional information on J. Appl. Phys.Journal Homepage: http://jap.aip.org/ Journal Information: http://jap.aip.org/about/about_the_journal Top downloads: http://jap.aip.org/features/most_downloaded Information for Authors: http://jap.aip.org/authors
Thermal conductivity of ZnTe nanowires
Keivan Davami,1 Annie Weathers,2,a) Nazli Kheirabi,1,a) Bohayra Mortazavi,3,4,a)
Michael T. Pettes,2,a) Li Shi,2,5 Jeong-Soo Lee,1 and M. Meyyappan1,6,b)
1Department of IT Convergence Engineering, Pohang University of Science and Technology (POSTECH),Pohang, South Korea2Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas, USA3Centre de Recherche Public Henri Tudor, Department of Advanced Materials and Structures, 66,rue de Luxembourg BP 144, L-4002 Esch/Alzette, Luxembourg4Institut de M�ecanique des Fluideset des Solides, University of Strasbourg/CNRS, 2 Rue Boussingault,67000 Strasbourg, France5Center for Nano and Molecular Science and Technology, Texas Materials Institute,The University of Texas at Austin, Austin, Texas, USA6NASA Ames Research Center, Moffett Field, California 94035, USA
(Received 21 May 2013; accepted 24 September 2013; published online 4 October 2013)
The thermal conductivity of individual ZnTe nanowires (NWs) was measured using a suspended
micro-bridge device with built-in resistance thermometers. A collection of NWs with different
diameters were measured, and strong size-dependent thermal conductivity was observed in these
NWs. Compared to bulk ZnTe, NWs with diameters of 280 and 107 nm showed approximately
three and ten times reduction in thermal conductivity, respectively. Such a reduction can be
attributed to phonon-surface scattering. The contact thermal resistance and the intrinsic thermal
conductivities of the nanowires were obtained through a combination of experiments and
molecular dynamic simulations. The obtained thermal conductivities agree well with theoretical
predictions. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4824687]
I. INTRODUCTION
Thermoelectric devices without moving parts and with
relatively low pollution offer attractive new approaches for
energy harvesting and active-cooling thermal management
applications. Thermoelectricity as a way of energy produc-
tion via the conversion of thermal energy to electricity is
now achievable, however, the small efficiency of bulk ther-
moelectric (TE) materials limits their applications.1 The fig-
ure of merit for thermoelectric materials is given as
ZT ¼ S2rT=ðje þ jlÞ; (1)
where S is the Seebeck coefficient, r is the electrical conduc-
tivity, T is the absolute temperature, and je and jl are the
electron and lattice components of the thermal conductivity,
respectively. The product S2r is also known as the power
factor. Maximizing the thermoelectric figure of merit is chal-
lenging because the required parameters are interconnected;
for most existing TE materials, ZT values hardly exceed
unity. There has been a tremendous amount of research in
the last decade to improve ZT beyond �1; new approaches
are needed to achieve higher ZT values �3 necessary to
compete with mechanical energy systems.2,3 A number of
experimental and theoretical studies have suggested the pos-
sibility of obtaining improved ZT values for one dimensional
nanostructures by optimizing the phonon density of states4
and by decreasing the lattice thermal conductivity through
boundary scattering.5,6
The ZT for various II–VI and III–V semiconductors in
both bulk and NW forms has been calculated using an itera-
tive solution of the Boltzmann transport equation to model the
electronic transport and a full transmission function approach
for the prediction of the lattice thermal conductivity.7,8
However, there are only few measured thermoelectric prop-
erty data of these nanowires for comparison with the theoreti-
cal calculations.9,10 In particular, thermal conductivity
measurement results of II–VI nanowires are still lacking.
Here, we report the diameter-dependent thermal conductivity
of individual ZnTe NWs. We observe that the thermal con-
ductivity for a NW with a diameter of 107 nm is reduced by
nearly one order of magnitude compared to that of the bulk
ZnTe, similar to the theoretical predictions of Mingo.7
II. EXPERIMENTAL WORK
ZnTe NWs were synthesized using a vapor-liquid-solid
(VLS) method reported previously.11–13 NWs were then trans-
ferred to a micro-bridge device by drop-casting a
NW/isopropyl alcohol solution. Scanning electron microscopy
(SEM) and high resolution transmission electron microscopy
(HRTEM) were used to characterize the crystal structure of the
NWs. The procedure for the thermal conductance measure-
ment using a suspended microbridge device has been discussed
previously.14–16 Six 420lm long SiNx beams were used to sus-
pend two adjacent silicon nitride (SiNx) membranes. A
serpentine-shaped Pt resistor �50 nm thick, �250 nm wide,
and �350 lm long was patterned on each membrane. These
serpentine structures acted as both heater and temperature sen-
sor. The NWs were drop casted on the device several times
a)A. Weathers, N. Kheirabi, B. Mortazavi, and M. T. Pettes contributed
equally to this work.b)Author to whom correspondence should be addressed. Electronic mail:
0021-8979/2013/114(13)/134314/7/$30.00 VC 2013 AIP Publishing LLC114, 134314-1
JOURNAL OF APPLIED PHYSICS 114, 134314 (2013)
until only one NW bridged the two suspended structures. The
sample was placed into a cryostat under high vacuum in order
to eliminate the heat transfer through the air. When a bias volt-
age was applied to one of the resistance thermometers, the
temperature of the heating membrane increased due to Joule
heating.
By measuring the change in the resistance of each resis-
tor, the temperature of each side can be calculated and the
total thermal conductance (GT ¼ 1=RT)—which includes the
contributions from the nanowire itself and also two
NW/device contacts—can be obtained analytically. Detailed
measurement techniques and the calculations as well as
uncertainty analysis were presented previously.15 From the
measured total thermal conductance and considering the
dimension of the NW, it is possible to calculate the effective
thermal conductivity as jef f ¼ GtLnw=Anw where Anw, and
Lnw are the cross sectional area and length of the suspended
NW bridging between the two membranes. To quantify the
contact thermal resistance to the NW, thermal conductance
measurements for some samples were performed before and
after Pt-C deposition at the contacts using a 10 keV electron
beam similar to the description in Ref. 17. Additionally, a
classical molecular dynamics (MD) simulation was per-
formed to estimate the thermal boundary resistance between
ZnTe and platinum, a parameter which was used to calculate
the contact thermal resistance contribution, and the intrinsic
thermal conductivity of the NW denoted as Rc and jint,
respectively.
III. RESULTS AND DISCUSSION
HRTEM and selected area electron diffraction pattern
(SAED) images for a ZnTe NW with a diameter of 80 nm
are shown in Fig. 1(a). The SAED pattern exhibits one set of
Bragg reflections indicative of single crystallinity in these
NWs over the irradiated volume. The SAED pattern identi-
fies the phase as cubic FCC with a lattice constant a of
�6.1 A. The SAED values are in accordance with bulk lat-
tice parameters. The growth direction of the NW is along the
h111i direction, with a lattice spacing of d111� 3.5 A. SEM
images of the micro-bridge device with a ZnTe NW bridging
the two heater pads are shown in Fig. 1(b).
The effect of contacts on the measured total thermal
conductance of the sample needs to be determined first. The
surface of the electrodes in contact with the NW is not
smooth, which results in decreased heat transfer from the
electrodes to the NW. The measured total thermal resistance
of the sample (RT) includes the contact thermal resistance
(Rc) between the NW and the membranes at the two ends
and the thermal resistance of the NW (Rnw)
Gt ¼1
Rt¼ ðRnw þ RcÞ
�1: (2)
The thermal resistance between the NW and the electro-
des can be decreased by Pt-C deposition at the contacts.18
We deposited Pt-C at the contacts for several NWs
(Fig. 2(a)) and measured the thermal conductance before and
after Pt-C deposition. Thermal conductivity measurements
FIG. 1. (a) High-resolution TEM image of a ZnTe nanowire with 80 nm di-
ameter. The inset shows a selected area electron diffraction pattern of the
nanowire. The scale bar for the HRTEM is 5 nm. (b) An SEM picture of the
device, the suspended heaters, and a nanowire bridging the suspended
structures.
134314-2 Davami et al. J. Appl. Phys. 114, 134314 (2013)
were impossible prior to Pt-C deposition in the case of some
large diameter nanowires with high thermal contact resistance.
The results for a NW with a diameter of 145.6 nm are shown
in Fig. 2(b) where a 2 to 10% increase was observed in the
thermal conductance after Pt-C deposition, resulting mainly
from the increased contact area. Shi et al.19 estimated the con-
tribution of contacts with carbon deposition to be less than
15% for silicon nanowires. Since the nanowire is placed on
the electrode, it is not in contact with the substrate, instead in
contact with just the Pt electrodes before FIB deposition, and
with the Pt-C coating (which is supposed to have similar
properties with the Pt electrodes) and Pt electrodes after FIB
deposition. The resistance between the nanowire/Pt electrodes
(before and after FIB coating) were considered in the calcula-
tion. The resistance between Pt-C pads/substrate has been pre-
viously shown to have no effects on the results.18
The effective thermal conductivity versus temperature
for several NWs with different diameters is shown in Fig. 3.
In this figure, the error for the experimental results is less
than 5% for each point. The thermal conductivity increases
with increasing diameter and decreases slightly with increas-
ing temperature because of increased Umklapp phonon scat-
tering. The size dependent thermal conductivity and similar
decreasing trend by diameter reduction in nanowires have
been shown before both theoretically7,8 and experimen-
tally.14 The enhanced carrier-boundary/interface scattering
lowers the thermal conductivity when the characteristic
length becomes comparable to the energy carrier mean free
path. When the characteristic dimensions have similar sizes
as the dominant phonon wavelength, the dispersion relation-
ship between the phonon wavelength and wave vectors
changes. This size effect can change three main parameters,
namely: the density of states, the phonon group velocity and
the mean free path, which can further reduce the lattice com-
ponent jl of the thermal conductivity as a result of phonon-
phonon scattering.
Slack20 reported the thermal conductivity of bulk ZnTe
below room temperature and also provided a predictive rela-
tion with a very good approximation above room temperature,
which is also plotted in Fig. 3. The reported thermal conduc-
tivity for bulk ZnTe is 18 W/m K at 300 K, whereas this value
decreases one order of magnitude to 1.8 W/m K for 107 nm di-
ameter NW here. The thermal conductivity of bulk ZnTe
peaks at �25 K. A relatively small peak is observed for the
NW with d¼ 134.6 nm after Pt-C deposition at the contacts,
at a higher temperature �100 K compared to the bulk mate-
rial. The observed small peak can be attributed to static scat-
tering processes such as boundary and impurity scattering,
dominating over phonon-phonon scattering at temperatures
below 100 K. We note that although we could not detect
defects in the NWs using TEM, even single crystalline NWs
are known to have some defects and impurities;21,22 it is more
likely that the peak shift is a result of the small NW diame-
ter.14 Since nanowires have small diameters, the boundary
scattering always plays an important role in the measurement
temperature range. A lack of clear ascending/descending order
in the thermal conductivity of the nanowires with temperature
increase shows that the boundary effect is dominant over the
whole temperature range (Fig. 3). A slight decreasing trend
can be observed in the nanowire thermal conductivities when
the temperature is increased and this can be attributed to the
effect of phonon-phonon Umklapp scattering which becomes
significant at higher temperatures.
FIG. 3. The effective thermal conductivity calculated from the total thermal
conductance versus temperature for nanowires with different diameters.
Unfilled symbols and filled circles are results for NWs with and without Pt-
C deposition, respectively. Solid lines are a guide to the eye. For all the ex-
perimental results, the error is less than 5%. The error bars for d¼ 134.6 nm
with filled symbols are equal to the diameter of the markers.
FIG. 2. (a) A nanowire bridging the suspended heaters before and after Pt-C
deposition. The scale bar for the left figure is 2 lm and for the right one is
5 lm. (b) Effective thermal conductance measurement results for a nanowire
with a diameter of 145.6 nm. Unfilled symbols and filled symbols are results
measured before and after Pt-C deposition at the contacts, respectively.
134314-3 Davami et al. J. Appl. Phys. 114, 134314 (2013)
The uncertainty in the NW thermal conductivity calcula-
tion, U�j , in Figs. 2(a) and 3 was determined from the random
(P) and bias (B) uncertainties in the suspended length, L, and
the cross-sectional area, A, both determined from SEM
images, and the sample thermal conductance, Gs, via the fol-
lowing equation:
U�j ¼ j
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU �GS
�Gs
� �2
þ P �L
�L
� �2
þ B �L
�L
� �2
þ P �A
�A
� �2
þ B �A
�A
� �2s
: (3)
The random uncertainties in L and A were calculated from
the measurement of the SEM micrographs, arising mainly
due to finite pixel size. Length uncertainty is usually �1–3%
because the edge is not perpendicular and the electrodes are
raised so there is some ambiguity in the length. The bias
uncertainty, B �L , is based on the resolution of the SEM,
approximately 1.3 nm. We expect a bias uncertainty, B �A ; of
� 2% in the NW diameter determination from the SEM. The
uncertainty analysis for this measurement method and a sim-
ilar suspended device is discussed extensively by Yu et al.15
The molecular dynamics simulation was performed for
the prediction of thermal boundary resistance (TBR)
between ZnTe NWs and deposited contacts using LAMMPS
(large-scale atomic/molecular massively parallel simula-
tor).23 Though the deposited material was amorphous carbon
with grains of Pt interspersed, the deposited material is
assumed to be pure Pt. The reasons for this assumption are,
firstly, the transport properties of the Pt-C contact material
were unavailable so we could not consider the deposited ma-
terial as Pt-C in our calculations and modeling. Second, it
would be too intensive to model the deposited material as
Pt-C. The deposition was considered to be completely uni-
form and conformal and it was assumed that the properties
of the Pt electrode and the Pt-C deposited on the nanowire
are the same. These assumptions cause some deviation in our
calculations and in order to consider the resulting impact, we
included a range of values (1.55–3.55)� 10�8 K m2/W for
the interfacial thermal resistance, R00c , to calculate the uncer-
tainty arising from the above decision for the intrinsic ther-
mal conductivity in Fig. 5. Also, a 5% error was considered
in thermal contact resistance, R, in the calculation of the
intrinsic thermal conductivity as well.
We used a non-equilibrium molecular dynamics (NEMD)
scheme for the evaluation of thermal boundary resistance. The
interactions between ZnTe atoms were introduced using
Tersoff potential with a set of parameters proposed by
Kanoun et al.24 This potential was successfully used by Wang
and Chu25 for the evaluation of temperature-dependent ther-
mal conductivity of bulk ZnTe using the equilibrium molecu-
lar dynamics method. They reported a decreasing trend for the
thermal conductivity of bulk ZnTe as the temperature
increases above 300 K. Similar trend was observed here for
thermal conductance and thermal conductivity of ZnTe nano-
wires at temperatures higher than 300 K as seen in Figs. 2 and
3, respectively.
The highly cited EAM developed by Foiles et al.26 was
used to account for the interactions between the platinum
atoms. The nonbonding interactions between individual Zn-
Pt and Te-Pt atoms were introduced using Lennard-Jones
(LJ) potential which is expressed as follows:
UðrÞ ¼ 4e½ðr=rÞ12 � ðr=rÞ6�; (4)
where r is the interatomic distance between atoms, e is the
depth of the potential well and r is the equilibrium distance.
The LJ parameters for individual atoms in our simulations
are based on those for LJ interactions in the universal force
field UFF model by Rappe et al.27 The pair parameters were
then calculated using the Lorentz-Berthelot mixing rules.
The UFF-based LJ parameters are ePt�Zn ¼ 4:3 meV, ePt�Te
¼ 7:7 meV, rPt�Zn ¼ 2:457A, and rPt�Te ¼ 3:218A. We
note that Tersoff potential is commonly used for the model-
ing of material combinations with chemical interactions and
not the non-bonding forces. Since the surfaces of Pt and
ZnTe are not chemically functionalized, it is acceptable to
neglect the formation of covalent bonds between ZnTe nano-
wires and Pt substrate. Accordingly, as a common assump-
tion, the ZnTe and Pt atoms interact through van der Waals
forces, which are commonly modeled by Lennard-Jones and
Morse potentials. The Lennard-Jones parameters for individ-
ual atoms in our simulations were calculated using universal
force field UFF model by Rappe et al.27 The Debye tempera-
tures of bulk Pt and bulk ZnTe are 234 K and 222 K, respec-
tively. Using the Tersoff potential developed by Kanoun
et al.,24 the Debye temperature of bulk ZnTe was calculated
to be 184 K.25 Unfortunately, we could not find the calcu-
lated Debye temperature for bulk Pt by the embedded atom
potential (EAM).26 Since the Debye temperature values for
bulk ZnTe and bulk Pt are lower than the room temperature
(300 K), commonly, the quantum effects are not taken into
account in the molecular dynamics simulations.
The molecular dynamics simulation gage section for the
evaluation of thermal boundary resistance between ZnTe and Pt
is presented in Fig. 4. This model consists of 15606 and 10406
atoms for Pt and ZnTe, respectively. The time increments of
simulations were set at 1 fs (10�15 s). Periodic boundary condi-
tions were applied in order to remove the surface effects on the
reported results. Moreover, to have the system at accurate equi-
librium conditions without residual stresses, the number of
ZnTe lattices along the sections was chosen to be 11, which
accurately match 17 lattices of Pt atoms. Prior to applying the
loading conditions, the specimen was left to relax to zero stress
using a constant pressure-temperature (i.e., NPT ensemble) by
means of the Nos�e-Hoover barostat and thermostat method.
134314-4 Davami et al. J. Appl. Phys. 114, 134314 (2013)
After the relaxation step in our NEMD simulation, the atoms
at the two ends of the model were fixed to prevent them from
sublimation. The simulation box (excluding the fixed atoms
at the two ends) was divided into 18 slabs. Then, a tempera-
ture difference was applied between the first and 18th slabs.
In this study, the temperatures at the first (cold reservoir) and
18th (hot reservoir) slabs were set at 275 K and 325 K,
respectively, using the Nos�e-Hoover thermostat method
(NVT), while the remaining slabs were under constant
energy (NVE) simulations. In order to maintain the tempera-
ture differences at two ends, a constant positive heat flux
was applied to the hot reservoir and at the same time a nega-
tive heat flux was applied on the cold reservoir by the NVT
method. The non-equilibrium steady state heat transfer can
be achieved after 1 ns of the exchanging process, when a
temperature profile is established along the sample. The ther-
mal boundary resistance could be obtained using the follow-
ing relation:
R ¼ ADT
q; (5)
where DT is the steady-state temperature jump between the
two surfaces of ZnTe and Pt, A is the interface and q is the
heat flow across the interface. In this work, the temperature
at each slab is computed as follows:
TiðslabÞ ¼ 2
3NkB
Xj
p2j
2mj; (6)
where TiðslabÞ is the temperature of the ith slab, N is the num-
ber of atoms in this slab, kB is the Boltzmann’s constant, mj
and pj are the atomic mass and momentum of atom j, respec-
tively. The simulations were performed for 6 ns and the aver-
aged temperatures at each slab were computed. The averaged
temperature profile in Fig. 4(b) shows the establishing of a
temperature difference DT across the Pt and ZnTe interface.
As expected, due to higher thermal conductivity of Pt in com-
parison with ZnTe, the slope of the temperature profile along
the ZnTe atoms is steeper than that along the Pt atoms. By cal-
culating the heat flow along the specimen, the interface ther-
mal boundary resistance, R00c ; between ZnTe and Pt at room
temperature was obtained as ð2:5560:15Þ � 10�8 Km2=W.
The mean free path of ZnTe was calculated to be 17.1 nm by
the use of Tersoff potential.25
The MD simulation result was applied in the calcula-
tions to estimate the contribution of contacts to the measured
total thermal conductance and the intrinsic thermal conduc-
tivity value of the nanowire (jint). Based on the model
applied in similar works,15,17,28 the total thermal resistance
of the contacts at both ends can be calculated as:
Rc ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jintAnw
R0c
stanh
Lcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jintAnwR0cp
! ; (7)
where Lc the contact length is considered to be about 1 lm
for our nanowires, and R0c is the value of the thermal resist-
ance between the nanowire and the contact per unit length.
The R00c value acquired from the molecular dynamic simula-
tion for the thermal boundary resistance between ZnTe and
Pt per unit area divided by the nanowire perimeter can be a
good estimate of the R0c parameter after Pt-C deposition. As
can be seen, the contact resistance Rc itself is related to the
desired nanowire intrinsic thermal conductivity hence cannot
be directly calculated. However, in the case of a specific ref-
erence nanowire with Pt deposited on its contacts, consider-
ing Eq. (7) and applying the estimated R0c parameter, jint and
consequently Rc after Pt-C deposition for that NW can be
obtained from the following equation for the measured total
thermal resistance RT :
RT ¼ Rc þLnw
jintAnw: (8)
Assuming that jint value remains unchanged, the difference
between the RT values before and after Pt deposition for the
reference NW corresponds only to the decrease in its contact
thermal resistance. Thus, its Rc and the R0c values before the
deposition can be estimated. The jint values for any other
NW can be calculated as well using Eqs. (7) and (8).
However, in each case, the R0c parameter applied in Eq. (7)
FIG. 4. (a) Molecular dynamics model for the evaluation of thermal bound-
ary resistance between Pt and ZnTe, (b) Established temperature profile
along the molecular dynamics model showing a steady-state temperature dif-
ference between the Pt and ZnTe interface.
134314-5 Davami et al. J. Appl. Phys. 114, 134314 (2013)
should be calculated separately by scaling the value obtained
for the reference NW (before Pt-C deposition) according to
the relative contact width values. This can be done by con-
sidering a parameter denoted as b which is the contact width
of the cylindrical nanowire sitting on a flat surface due to the
van der Waals interactions with the substrate, calculated as
follows for each NW:29,30
b ¼ 4
�d2w
4pE
�1=3
kffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 1p
; (9)
where d is the NW diameter, E�1 ¼ ð1� �12Þ=E1þ
ð1� �22Þ=E2, � is Poisson’s ratio and the E is the Young’s
modulus of the materials. In the calculations, we considered
�ZnTe and EZnTe values to be 0:36 and 6:32� 1010N=m2, and
assumed �Pt and EPt to be 0:39 and 1:68� 1011N=m2,
respectively. In Eq. (9), w is the adhesion energy per unit
area calculated as w ¼ A�
16pz02 where A*¼ 10�18 J is the
Hamaker constant29 and z0 is the equilibrium separation
between the NW and the substrate estimated to be 3.5 A
from the atomic spacing of ZnTe.11 The dimensionless pa-
rameter k is obtained from:
k ¼ 4r0
ð2p2E2w=dÞ1=3; (10)
where r0 is the theoretical joint strength, r0 ¼ w0:97z0
. The pa-
rameter m in Eq. (9) can be determined from k by
1=2k3nðm2 � 1Þ
hmffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 1p
� lnðmþffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 1p
Þi
þffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 1p h ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m2 � 1p
lnðmþffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 1p
� mlnðmÞio¼ 1:
(11)
As a summary of our method, a reference nanowire with
Pt-C deposited on the contacts was considered. For this
nanowire, the parameter R0c in Eq. (7) with the units of W
m/K (the thermal resistance between the nanowire and the
contact per unit length) is estimated from the interface ther-
mal resistance per unit area, R00c (in W m2/K), obtained by the
MD simulation. The relation was assumed to be
R0c ¼R00c=ðNW DiameterÞ, since the nanowire was totally sur-
rounded by Pt-C in this case. Then Eqs. (7) and (8) were
used to calculate the R00c value for the nanowires without Pt-C
deposited on their contacts. The contact width b was needed
to obtain R0c for each individual nanowire without Pt-C depo-
sition, by scaling the calculated R00c value as R0c¼R00c=b.
Calculation results for most of the NWs revealed that
the contact thermal resistance was �20% of the total meas-
ured thermal resistance. However, this contribution
decreased to only 5% after deposition of Pt-C on the con-
tacts. The intrinsic thermal conductivities of NWs with dif-
ferent diameters at room temperature are compared with the
theoretical predictions of Mingo7 in Fig. 5. In this figure, the
effect of the contacts was calculated and extracted from
the results, and just the intrinsic thermal conductivity of the
ZnTe nanowires is shown.
The error in Fig. 5 was calculated using the following
method. The interfacial thermal resistance, R00c , using MD for
ZnTe/Pt interface was calculated. Even though the exact num-
ber for R00c for this interface was not available in the literature,
we note that the reported R00c for ZnTe/Pt interface as well as
other II-IV/Pt was in the order of 10�9 K m2/W in Ref. 31. R00cfor other metal/dielectric interfaces at room temperature was
reported to be in the range of 10�9-10�8 K m2/W.32 Also,
it is in the range of (1.5–1.8)� 10�8 K m2/W for the
Ge2SbTe5/ZnS:SiO2 interface.32 Assuming that R00c is in the
range of (1.55–3.55)� 10�8 K m2/W, and there is a 5% uncer-
tainty in thermal contact resistance, R, 3% in cross section
area, A, 3% in length, L, 5% in contact length, CL, and 5% in
contact width, CW, the following equation can be used to cal-
culate upper and lower error bars via a root sum square:
U�j ¼ �kint
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiUR00 c
R00 c
!2
þ U �R
�R
� �2
þ U �A
�A
� �2
þ U�L
�L
� �2
þUCL
CL
� �2
þUCW
CW
� �2
vuut ; (12)
where UR00c,U �R , U �A ;U�L ;UCL ;UCW are the uncertainties in the
interfacial thermal resistance, contact total thermal resistance
of the contacts (at both ends),cross section area, length, con-
tact length, and contact width, respectively.
The intrinsic thermal conductivity values of our samples
decrease for smaller NW diameters and the experimental
results agree with the theoretical model quite well. The
minor differences are due to the assumptions in the model
FIG. 5. Calculated intrinsic thermal conductivity (filled symbols), and theo-
retical values predicted by Mingo in Ref. 7 (unfilled symbols) versus diame-
ter at T¼ 300 K. Except for the NW with d¼ 280 nm, the reported intrinsic
thermal conductivity results for the rest of the NWs are calculated from the
measurements prior to Pt-C deposition.
134314-6 Davami et al. J. Appl. Phys. 114, 134314 (2013)
such as the ideal case of NW being pure ZnTe with a one to
one ratio between Zn and Te, while the VLS-grown NWs are
not exactly stoichiometric. Different compositions of the
NWs cause some discrepancy in their thermal conductivities.
As seen in this plot, the calculated intrinsic thermal conduc-
tivity for the nanowire with 145.6 nm diameter is more than
that for the diameter of 280 nm, which is attributed to the
aforementioned points.
IV. CONCLUSION
In summary, ZnTe nanowires were synthesized using a
VLS method and NWs with diameters in the range of
107 nm to 280 nm were assembled on micro-bridge devices
with integrated resistance thermometers, for measuring ther-
mal conductance at different temperatures before and after
the deposition of platinum on the contacts. The thermal
boundary resistance between ZnTe and Pt was estimated
using MD simulations, which was then used to calculate the
thermal contact resistance between the NWs and the micro-
bridge device to be around 20% and 5% of the total meas-
ured thermal resistance before and after Pt-C deposition,
respectively. The resulting intrinsic thermal conductivities at
room temperature showed strong size dependence, with an
order of magnitude decrease compared to bulk ZnTe for the
smallest NW with a diameter of 107 nm. The experimental
results are also in good agreement with theoretical predic-
tions in the literature for ZnTe NWs. The suppressed thermal
conductivity for small diameter NWs can be attributed to
phonon-surface scattering phenomena.
ACKNOWLEDGMENTS
This work was supported by the World Class University
program through the National Research Foundation of Korea
funded by the Ministry of Education, Science and Technology
under Project No. R31-2008-000-10100-0. Moreover, the
research was also partly supported by a grant (Code No. 2011-
0031638) from the Center for Advanced Soft Electronics
under the Global Frontier Research Program of the Ministry
of Education, Science and Technology, Korea. Most of this
work was done at UT Austin during Keivan Davami’s visit
and Professor Shi’s group is acknowledged for hosting the
visit. Bohayra Mortazavi greatly appreciates Dr. Toniazzo at
CRP Henri-Tudor for providing computational facilities.
1C. B. Vining, Nature 8, 83 (2009).2C. J. Vineis, A. Shakouri, A. Majumdar, and M. G. Kanatzidis, Adv.
Mater. 22, 3970 (2010).3T. M. Tritt, Annu. Rev. Mater. Res. 41, 433 (2011).4L. D. Hicks and M. S. Dresselhaus, Phys. Rev. B 47, 16631 (1993).5D. G. Cahill, W. K. Ford, K. E. Goodson, G. D. Mahan, A. Majumdar, H.
J. Maris, R. Merlin, and S. R. Phillpot, J. Appl. Phys. 93, 793 (2003).6L. Shi, NMTE 16, 79 (2012).7N. Mingo, Appl. Phys. Lett. 85, 5986 (2004).8N. Mingo, Appl. Phys. Lett. 84, 2652 (2004).9F. Zhou, J. H. Seol, A. L. Moore, L. Shi, Q. L. Ye, and R. Scheffler,
J. Phys.: Condens. Matter 18, 9651 (2006).10F. Zhou, A. L. Moore, J. Bolinsson, A. Persson, L. Froberg, M. T. Pettes,
H. Kong, L. Rabenberg, P. Caroff, D. A. Stewart, N. Mingo, K. A. Dick,
L. Sauelson, H. Linke, and L. Shi, Phys. Rev. B 83, 205416 (2011).11K. Davami, D. Kang, J. S. Lee, and M. Meyyappan, Chem. Phys. Lett.
504, 62 (2011).12K. Davami, H. M. Ghassemi, R. S. Yassar, J. S. Lee, and M. Meyyappan,
ChemPhysChem 13, 347 (2012).13K. Davami, B. Mortazavi, H. M. Ghassemi, R. S. Yassar, J. S. Lee, Y.
Remond, and M. Meyyappan, Nanoscale 4, 897 (2012).14D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar, Appl. Phys.
Lett. 83, 2934 (2003).15C. Yu, S. Saha, J. Zhou, L. Shi, A. M. Cassell, B. A. Cruden, Q. Ngo, and
J. Li, J. Heat Transfer 128, 234 (2006).16J. H. Seol, A. L. Moore, S. K. Saha, F. Zhou, L. Shi, Q. L. Ye, R.
Scheffler, N. Mingo, and T. Yamada, J. Appl. Phys. 101, 023706 (2007).17M. T. Pettes and L. Shi, Adv. Funct. Mater. 19, 3918 (2009).18A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M.
Najarian, A. Majumdar, and P. Yang, Nature 451, 163 (2008).19L. Shi, D. Li, C. Yu, W. Jang, D. Kim, Z. Yao, P. Kim, and A. Majumdar,
J. Heat Transfer 125, 881 (2003).20G. A. Slack, Phys. Rev. B 6, 3791 (1972).21U. Philipose, A. Saxena, H. E. Ruda, P. J. Simpson, Y. Q. Wang, and K. L.
Kavanagh, Nanotechnology 19, 215715 (2008).22M. I. D. Hertog, C. Cayron, P. Gentile, F. Dhalluin, F. Oehler, T. Baron,
and J. L. Rouviere, Nanotechnology 23, 025701 (2012).23S. Plimpton, J. Comput. Phys. 117, 1 (1995).24M. B. Kanouna, A. E. Merada, H. Aouragb, J. Cibertc, and G. Merad,
Solid Sci. 5, 1211(2003).25H. Wang and W. Chu, J. Alloys Compd. 485, 488 (2009).26S. M. Foiles, M. I. Baskes, and M. S. Daw, Phys. Rev. B 33, 7983 (1986).27A. K. Rappe, C. J. Casewit, K. S. Colwell, W. A. Goddard, and W. M.
Skid, J. Am. Chem. Soc. 114, 10024 (1992).28M. T. Pettes, I. Jo, Z. Yao, and L. Shi, Nano Lett. 11, 1195 (2011).29R. Prasher, Phys. Rev. B 77, 075424 (2008).30F. Zhou, A. Persson, L. Samuelson, H. Linke, and L. Shi, Appl. Phys. Lett.
99, 063110 (2011).31H. Wang, Y. Xu, M. Shimono, Y. Tanaka, and M. Yamazaki, Mater.
Trans. 48, 2349 (2007).32E. K. Kim, S. I. Kwun, S. M. Lee, H. Seo, and J. G. Yoon, Appl. Phys.
Lett. 76, 3864 (2000).
134314-7 Davami et al. J. Appl. Phys. 114, 134314 (2013)