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:

m.meyyappan@nasa.gov.

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.

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