A self-heating 2ω method for Seebeck coefficient measurement of thermoelectricmaterialsTingting Miao, Weigang Ma, Xing Zhang, and Zhen Li Citation: Review of Scientific Instruments 82, 024901 (2011); doi: 10.1063/1.3544019 View online: http://dx.doi.org/10.1063/1.3544019 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/82/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Fast Seebeck coefficient measurement based on dynamic method Rev. Sci. Instrum. 85, 054904 (2014); 10.1063/1.4876595 High-accuracy direct ZT and intrinsic properties measurement of thermoelectric couple devices Rev. Sci. Instrum. 85, 045107 (2014); 10.1063/1.4870278 Data analysis for Seebeck coefficient measurements Rev. Sci. Instrum. 84, 065102 (2013); 10.1063/1.4807697 ac heating–dc detecting method for Seebeck coefficient measurement of the thermoelectric micro/nano devices J. Vac. Sci. Technol. B 30, 051804 (2012); 10.1116/1.4750497 A hot probe setup for the measurement of Seebeck coefficient of thin wires and thin films using integral method Rev. Sci. Instrum. 79, 024302 (2008); 10.1063/1.2869039

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REVIEW OF SCIENTIFIC INSTRUMENTS 82, 024901 (2011)

A self-heating 2ω method for Seebeck coefficient measurement ofthermoelectric materials

Tingting Miao, Weigang Ma, Xing Zhang,a) and Zhen LiKey Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department ofEngineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China

(Received 4 November 2010; accepted 30 December 2010; published online 16 February 2011)

A novel and reliable self-heating 2ω method has been developed to measure the Seebeck coeffi-cient of the microscale/nanoscale thermoelectric materials. Based on the analytical solution of thetransient heat-conduction equation of the specimen heated by a harmonic current, two measurementmodes have been developed: (1) the Seebeck coefficient can be directly extracted from the ratioof experimentally measured 2ω Seebeck voltage to theoretically predicted 2ω temperature drop os-cillation; and (2) the Seebeck coefficient can be steadily extracted from the measured 2ω and 3ω

voltages. This approach has been applied to a 25.4 μm thick K-type thermocouple and the measuredSeebeck coefficient corresponds well with the nominal value. © 2011 American Institute of Physics.[doi:10.1063/1.3544019]

I. INTRODUCTION

With the energy crisis and the rise of environmentalconsciousness, the thermoelectric conversion system hasattracted considerable attention due to its excellent merits,1

i.e., lightweight, small, no moving parts involved,2 no noise,no pollution, easy maintenance and long service life,3 andpromising applications in aerospace, microelectronic devices,and energy conservation devices.4 However, the thermoelec-tric devices are still limited by low thermoelectric conversionefficiency, which is determined by the materials’ dimension-less figure of merit Z or “ZT” (T is the absolute temperature).Approximately 30% of Carnot efficiency (comparable tohome refrigeration) could be reached by a device with a ZTof only 4. However, the best bulk thermoelectric materialsreported by Poudel et al.5 available today for devices thatoperate near room temperature have a ZT of about 1.2.Increasing ZT to a factor of 4 has remained a formidablechallenge. Nanotechnology provides a new promisingapproach for developing high performance thermoelectricmaterials. However, because of nanoscale and anisotropiccharacteristics, traditional measurements and devices cannotbe well applied to measure the thermoelectric characteristicsanymore. Therefore, the measurement and representation ofthermoelectric properties of nanoscale materials are a greatchallenge in present research.

A materials figure of merit is defined as: Z = σ eS2/λ,where S, σ e, and λ are the Seebeck coefficient, electrical, andthermal conductivities, respectively.6–8 The figure of merit isproportional to the square of the Seebeck coefficient, whichtherefore, is one of the most important performance param-eters. Hence, an accurate measurement of the coefficientis of great significance to study and evaluate the propertiesof new thermoelectric materials. However, experimentaldetermination of the Seebeck coefficient of thin films,defined as: S = −�V/�T, is difficult because it requires the

a)Author to whom correspondence should be addressed. Electronic mail:x-zhang@tsinghua.edu.cn.

simultaneous determination of the Seebeck voltage �V andtemperature drop �T across ultrathin films (∼nm). Yanget al.9 measured the Seebeck coefficient of thin film byapplying a heat source to the specimen and measuring thetemperature drop and 2ω voltage signal with microprobe.Völklein et al.10 measured the Seebeck coefficient of a singlenanowire by applying a reference film with known Seebeckcoefficient as a comparison. However, these measurementseither focus on the cross-plane properties or are extremelydifficult to perform due to the ultrasmall size of nanofilms.For nanoscale materials, the imposed heat flux is generallyvery small and the temperature distribution is considerablydifficult to be detected, in addition, the thermal contactresistance would engender fairly great error.

Aiming at the problems presented above, this paperdevelops a self-heating 2ω method to directly measure theSeebeck coefficient of microscale/nanoscale thermoelectricmaterials. Two modes are applied to the measurement: one isextracting the Seebeck coefficient through measuring the 2ω

voltage and calculating the temperature drop, and the other isobtaining the Seebeck coefficient by measuring the variationrelationship between 2ω and 3ω voltages. It should be high-lighted that the latter mode can obtain the Seebeck coefficientwithout requirements of theoretically calculated temperature.In this study, we first attempt to implement a general theo-retical analysis based on analytical solution of transient heat-conduction equation, and accordingly propose an approach todetermine the Seebeck coefficient of specimens in differentconditions. To verify the method, it is applied to a 25.4 μmthick K-type thermocouple with known Seebeck coefficient.

II. EXPERIMENTAL PRINCIPLE

In the measurement, the specimen itself serves simulta-neously as a thermoelectric device, a heater, and a resistancetemperature sensor. Feeding an ac electric current at the fre-quency of 1ω into the specimen creates a temperature fluc-tuation at the frequency of 2ω, and according to the Seebeck

0034-6748/2011/82(2)/024901/6/$30.00 © 2011 American Institute of Physics82, 024901-1

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024901-2 Miao et al. Rev. Sci. Instrum. 82, 024901 (2011)

FIG. 1. (Color online) Physical model of the self-heating 2ω method.

effect, a Seebeck voltage fluctuation is generated at 2ω. Com-bining the measured 2ω voltage and the theoretical analysisof the transient heat-conduction equation, the Seebeck coeffi-cient of the thermoelectric material can be obtained.

Two different materials, i.e., chromel and alumel, makeup of a thermoelectric device. Figure 1 shows the physicalmodel of the present experiment. The specimen freely sus-pends across two heat sinks (also electrodes) kept with theinitial temperature. The left and right hand branches of thesample are labeled 1 and 2, respectively. The length to radiusor thickness ratio is expected to be sufficiently large to ensurethat the heat flows one-dimensionally along the axial direc-tion. When an ac electric current of the form I0 sin(ωt) heatsthe specimen, it will create three voltage components: (1) 1ω

voltage component generated by the inherent resistance; (2)2ω Seebeck voltage generated by the Seebeck effect due tothe 1ω ac electric current induced 2ω temperature oscillationon the specimen; and (3) 3ω voltage component generated bythe product of the 1ω current and resistance–temperature de-termined 2ω oscillating resistance according to Ohm’s law.The above analysis indicates that the Seebeck coefficient ofthe thermoelectric materials can be extracted from the mea-sured 2ω voltage combined with the calculated temperaturedrop or the measured 3ω voltage.

According to the physical model described above, weselect one branch of the specimen as the research object.Since all the measurements are carried out at high vacuum(∼10−4Pa) at the room temperature, the natural convectionand radiation heat losses can be ignored. Feeding an ac elec-tric current of the form I0 sin(ωt), the governing equation ofthe one-dimensional heat conduction for a Joule self-heatingprocess can be written as,11

ρcp∂T (x, t)

∂t− λ

∂2T (x, t)

∂x2

= I 20 sin2 ωt

l A

[R + R′(T (x, t) − T0)

], (1)

where ρ, cp, and R are the mass density, specific heat, andelectrical resistance of the selected branch of specimen at thesubstrate temperature T0, respectively. l and A are the lengthand cross section. The temperature dependence of resistanceis defined as: R′ = (d R/dT )T0 = βTR273, where βT is the tem-perature coefficient of resistance and R273 is the resistance at273 K.

The heating power inhomogeneity caused by resis-tance oscillation along the specimen can be neglected.11 Let

�T(x, t) denote the temperature variation from T0, i.e.,�T(x, t) = T(x, t)−T0, Eq. (1) can be simplified as

∂�T (x, t)

∂t− α

∂2�T (x, t)

∂x2= β sin2 ωt, (2)

where α = λ/(ρcp) and β = I02R/(ρcplA).

The temperature distribution of the specimen containsstatic and harmonic components, mainly at the driving currentfrequency (ω) due to Peltier effect and at doubled frequency(2ω) due to Joule effect. The solution of Eq. (2) can be de-scribed as a sum of the different terms,

�T (x, t) = �Ts(x) + �T1ω(x) sin (ωt + ψ)

+�T2ω(x) sin (2ωt + ϕ) , (3)

where �Ts(x) is the static component that corresponds to thestatic temperature difference between the specimen Ts and theambient T0: �Ts(x) = Ts(x)−T0, �T1ω(x), �T2ω(x), ψ and ϕ

represent the magnitudes and phases of the 1ω and 2ω com-ponents, respectively. The 1ω solution considers a Peltier heatsource located at the junction since Joule effect occurs at thedoubled frequency only. On the other hand, part of the heatgeneration at a frequency 2ω is also induced due to the Peltiereffect because of the temperature oscillation at the frequencyω together with the driving ac current at the same frequencyω, while other 2ω heat generation is induced by the 1ω accurrent Joule heating. Therefore, the 2ω solution is causedby both Joule and Peltier effects. However, Roh et al.12 havedemonstrated that Joule effect is dominant over the Peltier ef-fect by about 2 orders of magnitude. And hence, the tempera-ture oscillation with the frequency 2ω at the junction inducedby the ac current is mostly due to the Joule effect. We haveassumed that the electric current is turned on at t = −∞. Theinitial and boundary conditions can be described as⎧⎨

⎩�T (0, t) = 0�T (l, t) = �Ts + �T2ω sin (2ωt + ϕ)�T (x,−∞) = 0

. (4)

Solving the inhomogeneous transport equation with non-homogeneous boundary by using the impulse theorem, we canobtain the temperature distribution along the specimens as fol-lows:

�T (x, t) = �Ts + �T2ω sin(2ωt + ϕ)

lx +

∞∑n=1

× 2γ

n3π

⎡⎢⎢⎢⎣

(1− (−1)n

)β

2

(1− sin (2ωt+φn)√

1+ cot2 φn

)

+(−1)n · 2ω�T2ω · sin (2ωt+ϕ+φn)√1+ cot2 φn

⎤⎥⎥⎥⎦

× sinnπ

lx, (5)

where γ = l2/(π2α) and cot φn = 2ωγ /n2.Neglecting the influence of interface thermal resistance,

the heat flux continuity between the two branches across theinterface should be satisfied, i.e.,

λ1 A1∂T1

∂x

∣∣∣∣l1

+ λ2 A2∂T2

∂x

∣∣∣∣l2

= 0. (6)

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024901-3 Miao et al. Rev. Sci. Instrum. 82, 024901 (2011)

Taking n = 1, substituting Eq. (5) into Eq. (6), we havethe magnitude of the 2ω temperature,

�T2ω =√

G2 + J 2

(E + H )2 + F2, (7)

where

E =∑i=1,2

λi Ai

li, F =

∑i=1,2

4ωλi Aiγi

li(1 + 4ω2γ 2

i

) ,

G =∑i=1,2

2βiλi Aiγi

li(1 + 4ω2γ 2

i

) , H =∑i=1,2

8ω2λi Aiγ2i

li(1 + 4ω2γ 2

i

) ,

J =∑i=1,2

4ωβiλi Aiγ2i

li(1 + 4ω2γ 2

i

) ,

and i represents the ith branch of the sample.From Eq. (7), when the specific heat, temperature coef-

ficient of resistance, thermal and electrical conductivities areknown, we can calculate the temperature drop between thehot junction and the heat sink for a given heating current.And hence, the Seebeck coefficient of the specimen can beextracted from the ratio of the measured 2ω Seebeck voltageto the calculated 2ω temperature drop,

S = |V2ω|�T2ω

. (8)

Furthermore, it is known that the common essentialreason responsible for the 2ω and 3ω voltage componentsis Joule heating induced 2ω temperature oscillation. Thedifference is that V2ω is generated from the coupling ofthe temperature oscillation and the Seebeck effect whileV3ω is the product of the temperature variation induced 2ω

oscillating resistance and 1ω current according to Ohm’s law.And hence, the Seebeck coefficient is expected to be obtainedby analyzing the relationship of V2ω and V3ω.

Based on the calculated temperature distribution, theresistance oscillation is obtained from knowledge of theresistance–temperature characteristics of the specimen, andfurthermore the 3ω voltage can be calculated according toOhm’s law,

V3ω = V3ω,1 + V3ω,2 = I0 sin ωt · δR1 + I0 sin ωt · δR2,

(9)

where

δRi = R′i

li

∫ li

0�Ti (xi , t)dxi . (10)

Combining Eqs. (7)–(10), the relationship of |V2ω| and|V3ω| can be expressed as

|V3ω| = χ

S|V2ω| + O(ω), (11)

where

χ2 = (O2 + P2

) + 2

G2 + J 2

(−M O [(E + H ) J + FG] − M P [F J − G (E + H )] − N O [F J − G (E + H )]

+N P [(E + H ) J + FG] + (M2 + N 2)[(E + H )2 + F2]2

),

(12)

and

M =∑i=1,2

2γi I0 R′iβi

π2(1 + 4ω2γ 2

i

) , N =∑i=1,2

4ωγ 2i I0 R′

iβi

π2(1 + 4ω2γ 2

i

) ,

O =∑i=1,2

4ωγi I0 R′i

π2(1 + 4ω2γ 2

i

) , P =∑i=1,2

I0 R′i

4− 8ω2γ 2

i I0 R′i(

1 + 4ω2γ 2i

)π2

.

Equation (11) indicates that |V3ω| and |V2ω| satisfy alinear relationship, which will be confirmed later from ex-perimental measurements. Therefore, the Seebeck coefficientcan be derived from the slope of the linear fitting betweenthe 2ω and 3ω voltage components. Compared with Eq. (8),this method only needs simultaneously measure the 2ω and3ω voltage components, other than theoretically calculate thetemperature, which improves the measurements significantly.

III. EXPERIMENTAL SETUP

Figure 2 shows the schematic of the experimental setup.The function generator (Agilent 33220A), with an internal re-sistance of 50 �, provides the harmonic currents to periodi-cally heat the sample and also supplies a reference signal tothe lock-in-amplifier (LIA), which is set to detect at two andthree times the reference frequency. The rheostat is adjustedto nearly equal to the intrinsic resistance of the sample so asto cancel the intrinsic 1ω signal, which is also beneficial forsaving the dynamic reserves of LIA. The LIA (model 7265,Signal Recovery) detects the tiny harmonic voltages fromthe differential input signals of the sample and the rheostat.Hence, the 2ω and 3ω noise voltages generated by the func-tion generator itself will be effectively eliminated.

The specimen is placed in a vacuum chamber (OptistatDN-V cryostat system, Oxford Instruments), which is con-tinuously evacuated by a two-stage vacuum-pumping sys-tem composed of a roughing pump and a high-vacuum pump(turbo molecular pump) so as to obtain and maintain a high

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024901-4 Miao et al. Rev. Sci. Instrum. 82, 024901 (2011)

TABLE I. Properties of the K-type thermocouple at 293 K.

l1, l2 D Resistivity βT cp λ ρ

(cm) (μm) (μ� cm) (� K−1) (J kg−1 K−1) (W m−1 K−1) (g cm−3)

Chromel 0.558 25.4 70.6 4.1 × 10−4 449.4 19.32 8.73Alumel 0.378 25.4 29.4 23.9 × 10−4 525.0 29.82 8.6

vacuum level of 10−4 Pa, therefore, the air convection can beignored. The temperature of heat sinks is monitored and con-trolled by the intelligent temperature controller (ITC601PT,Oxford Instruments) with an accuracy of ±0.1 K.

IV. METHOD VERIFICATION

The verification of the present approach is performedon a commercial K-type thermocouple (Chromega-Alomega,Omega Engineering) with 25.4 μm in diameter at 293 K. Thedetailed physical properties of the specimen are summarizedin Table I. The current applied on the specimen is equal to thaton the series rheostat, which is determined by the ratio of 1ω

voltage to the corresponding resistance according to Ohm’slaw. Changing the frequency of the applied current while fix-ing the corresponding amplitude, we can obtain a series of ω-dependent 2ω and 3ω voltages on the thermocouple by usingthe LIA.

A. V2ω – ω mode

The frequency-dependent 2ω voltage has been detectedby fixing the current amplitude to 8.53 and 14.30 mA, re-spectively, and the measured data are presented in Fig. 3. Onthe other hand, the corresponding 2ω oscillating temperaturedrop is obtained by theoretical analysis of the transient heat-conduction equation, Eq. (7). The average Seebeck coefficientof the tested thermocouple is extracted by least square fit-ting of the measured 2ω voltage |V2ω| to the 2ω temperaturedrop �T2ω.

FIG. 2. (Color online) Block diagrams of the experimental system for theself-heating 2ω method.

From Fig. 3, it can be found that the theoretical predic-tions match well with the experimental points, which illus-trate that the theoretical analysis and experimental approachare valid. When the input currents are 8.53 and 14.30 mA, themeasured Seebeck coefficients are 39.5 and 42.4 μV K−1, andthe corresponding relative deviations are −2.5% and 4.7%,respectively, compared to the nominal value 40.5 μV K−1.

B. V2ω – V3ω mode

According to Eq. (11), we can find a linear relationshipbetween |V3ω| and |V2ω|. And hence, the Seebeck coefficientcan be derived from the slope of the linear fitting betweenthe 2ω and 3ω voltage components. Figure 4 shows the fit-ting curves of the measured |V3ω| and |V2ω|. As seen in the

FIG. 3. (Color online) Comparison of the theoretically predicted and exper-imentally measured frequency-dependent 2ω voltages at different input cur-rents: (a) I0 = 8.53 mA and (b) I0 = 14.30 mA.

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024901-5 Miao et al. Rev. Sci. Instrum. 82, 024901 (2011)

TABLE II. Summarized experimental results of Seebeck coefficient (μV K−1) of the K-type thermocouple.

Input current (mA) Nominal value V2ω– ω method Relative deviation (%) V2ω– V3ω method Relative deviation (%)

8.53 40.5 39.5 –2.5 37.6 –7.214.30 42.4 4.7 38.5 –4.9

figures, |V3ω| and |V2ω| have a good linear relationship un-der the conditions of two different input currents, which arein good agreement with the theoretical prediction, and theslopes of the fitting lines are 0.72 and 1.18, respectively.Accordingly, the extracted Seebeck coefficient is 37.6 and38.5 μV K−1, and the corresponding relative deviations are−7.2% and −4.9%, respectively.

Comparing the two modes of the present developed self-heating 2ω method and summarizing the measurement resultsin Table II, we can find that the measured Seebeck coefficientsusing either the V2ω – ω or V2ω – V3ω mode correspond wellwith the nominal value and have relative deviations within4.7% and 7.2%, respectively.

FIG. 4. (Color online) The experimentally measured relationship between|V3ω| and |V2ω|, along with the linear fitting curves at different input currents:(a) I0 = 8.53 mA and (b) I0 = 14.30 mA.

C. Influence of the thermal radiation and the thermalboundary resistance (TBR)

The thermal radiation rate is approximately estimated by

Q̇rad = π Dlεσ[(T0 + �T )4 − (T0)4

], (13)

in which D is the diameter, σ is the Stefan–Boltzmann con-stant 5.67×10−8 W m−2 K−4, and ε is the emissivity. Takethe 25.4 μm thick K-type thermocouple fed by 8.53 mA accurrent, for instance, and the emissivity is set to be 0.2 for thecommon metallic surface.13 The radiation rate is of the orderof 10−6 W and the heat generation due to self-heating is of theorder of 10−4 W. Therefore, the influence of thermal radiationcan be negligible.

The tin solder paste is used to enhance the connectionbetween the specimen and the electrode. Even with the perfectconnection between them, TBR would still exist,14 especiallyat micro/nano scale. To verify the assumption of boundarycondition in the physical model, the TBR is defined as

RTB = Ainter�Tinter

Q̇cond, (14)

where Ainter is the interfacial area, �Tinter is the tem-perature difference across the interface, and Q̇cond is theheat flow. A typical thermal boundary conductance be-tween metal and metal is around 1×109 W m−2 K−1.15

The estimated parameters used in the calculation are given:the heating power Q̇cond = 3.3 × 10−4 W, the area be-tween paste and electrode Ainter,1 = 4.0 × 10−6 m2, and thearea between paste and specimen Ainter,2 = 1.6 × 10−7 m2.The temperature difference between the paste and electrodeis �T1 = T1 − T0 = Q̇cond RTB/Ainter,1 = 8.3 × 10−8 K, andthe temperature difference between the paste and specimenis �T2 = T2 − T0 = Q̇cond RTB/Ainter,2 = 2.1 × 10−6 K. Thesufficiently small temperature differences between specimenand paste, and paste and electrode demonstrate the assump-tion of isothermal boundary condition in Eq. (4) is reasonable.

V. SUMMARY

A novel and reliable self-heating 2ω method has beendeveloped to directly measure the Seebeck coefficient of themicro/nano scale thermoelectric materials. The strategy todetermine the Seebeck coefficient is presented. Two types ofmeasurements, V2ω – ω mode and V2ω – V3ω mode, are sug-gested, especially, the V2ω – V3ω mode can directly extract theSeebeck coefficient from the slope of the linear relationshipbetween 2ω and 3ω voltages. The present method is verifiedby a 25.4 μm thick K-type thermocouple, and the relative de-viation of Seebeck coefficient measurements is within 7.2%.In addition, the influences of thermal radiation and thermalboundary resistance between the specimen and the heat sinks

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024901-6 Miao et al. Rev. Sci. Instrum. 82, 024901 (2011)

on the experimental results are discussed, and the resultsindicate that both of them are negligible. The self-heating 2ω

method is expected to provide a new approach for evaluatingthe newly developed microscale/nanoscale thermoelectricmaterials.

ACKNOWLEDGMENTS

This work was financially supported by the National Nat-ural Science Foundation of China, Grant Nos. 50730006,50976053, and 50906042.

1F. J. DiSalvo, Science 285, 703 (1999).2R. Funahashi, M. Mikami, T. Mihara, S. Urata, and N. Ando, J. Appl. Phys.99, 066117 (2006).

3S. B. Riffat and X. L. Ma, Appl. Therm. Eng. 23, 913 (2003).4L. Shi, D. Y. Li, C. H. Yu, W. Y. Jang, D. Kim, Z. Yao, P. Kim, andA. Majumdar, ASME Trans. J. Heat Transfer 125, 881 (2003).

5B. Poudel, Q. Hao, Y. Ma, Y. C. Lan, A. Minnich, B. Yu, X. Yan, D. Z.Wang, A. Muto, D. Vashaee, X. Y. Chen, J. M. Liu, M. S. Dresselhaus, G.Chen, and Z. Ren, Science 320, 634 (2008).

6D. M. Rowe, CRC Handbook of Thermoelectrics (CRC, Boca Raton,1995).

7H. J. Goldsmid, Thermoelectric Refrigeration (Plenum, New York, 1964).8T. M. Tritt, Semiconductors and Semimetals (Academic, San Diego,2001).

9B. Yang, J. L. Liu, K. L. Wang, and G. Chen, Appl. Phys. Lett. 80, 1758(2002).

10F. Völklein, M. Schmitt, T. W. Cornelius, O. Picht, S. Muller, andR. Neumann, J. Electron. Mater. 38, 1109 (2009).

11L. Lu, W. Yi, and D. L. Zhang, Rev. Sci. Instrum. 72, 2996 (2001).12H. H. Roh, J. S. Lee, D. L. Kim, J. Park, K. Kim, O. Kwon, S. H.

Park, Y. K. Choi, and A. Majumdar, J. Vac. Sci. Technol. B 24, 2398(2006).

13F. P. Incropera and D. P. DeWitt, Introduction to Heat Transfer (Wiley, NewYork, 1985).

14E. T. Swartz and R. O. Pohl, Rev. Mod. Phys. 61, 605 (1989).15B. C. Gundrum, D. G. Cahill, and R. S. Averback, Phys. Rev. B 72, 245426

(2005).

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