Fig. 1. Schematic test structure used for the 3ω measurement.A metallic strip with four contact pads serves as the heater and

A novel technique for extraction of thermal conductivity in metallic thin films

Zhao-Xiang Zong, Zhi-Jun Qiu**, and Ran Liu* Key Lab of ASIC & System

School of Microelectronics, Fudan University 200433 Shanghai, China

* Corresponding author: e-mail rliu@fudan.edu.cn ** e-mail zjqiu@fudan.edu.cn

Abstract—The frequency-dependent thermal response in 3ω measurement is investigated using transient electro-thermal coupling simulations. Furthermore, the heat flow ratio m(ω), quantifying the deviation degree of Cahill’s model, is exploited for extraction of the thermal conductivity of the heater strip itself, which extends the capability of 3ω method to high-thermal-conductivity thin films.

I. INTRODUCTION Current microfabrication technologies have allowed the

semiconductor industry to produce smaller electronic devices and components with improved functionality and speed. Power density continues to increase exponentially with such new technology generation, posing a major challenge for thermal management. At the same time, a number of electronic films with high thermal conductivities are attractive for microelectronic devices because they offer enhanced thermal or thermoelectric properties [1]. For example, a multilayer structure containing several metallic materials (copper wires/barrier layer/capping layer) forms the interconnect system in the integrated circuits. As another example, the polycrystalline diamond film has been attracting interest as a promising candidate for enhanced heat removal from power circuits and for fast thermal sensors [2]. However, the thermal properties of these high-thermal-conductivity films at nanoscale dimensions can differ dramatically from those of bulk samples owing to the dependence of the material structure and purity on film processing conditions and to the scattering of heat carriers at material boundaries. Therefore, accurate characterizations of high-thermal-conductivity thin films are becoming essential for thermal management, reliability and optimal design of the integrated circuits.

Several experimental methods have been developed to enable capabilities for nanoscale thermal metrology. Among them, laser heating technique avoids contact with the sample film, but it is hard to precisely control the Joule heating intensity absorbed by the sample film, resulting in a relatively large uncertainty [3]. Besides, the drawbacks of this approach also include the time-domain noise from the detector and complicated data analysis to decouple the responses from the near-surface temperature rise and acoustic propagation. The Joule power generated by heating source can be accurately

measured and controlled through transient electrical heating technique, which is so-called “3ω method”. A harmonic electrical current at frequency ω in the metal narrow strip yields heating and temperature fluctuations at frequency 2ω which are detected through the third harmonic (3ω) of the voltage signal, as schematically illustrated in Fig. 1. Based on the analytical formulas proposed by Cahill et al. [4], the amplitude of the temperature oscillation calculated from average electrical resistance rise in the in the heater strip partially depends on thermal properties of thin film/substrate. However, when applied to high-thermal-conductivity specimens, 3ω method which is mainly designed for oxides needs additional fabrication steps to make nearly free-standing sample structure.

In this present work, transient thermal response of 3ω measurement is investigated using analytical and numerical approaches, revealing the correlation between thermal conductivity of metallic heating strip and a frequency-dependent thermal heat flow ratio. Utilizing this dependence, we present a novel technique for out-plane thermal conductivity measurements of high-thermal-conductivity thin films. The feasibility of this approach is then verified by experiments and numerical simulations.

978-1-4577-0708-7/11/$26.00 ©2011 IEEE 343

Fig. 2. Typical temperature development history in one time-cycle at low and high frequencies, respectively. T=1/(2f), with T as the heater temperature oscillation period which is half of that for the ac current

II. FREQUENCY-DEPENDENT THERMAL RESPONSE It should be noted that few efforts have been made to

illustrate the transient temperature profile of heater strip in previous investigations of 3ω method. The heater temperature is calculated from the root-mean-square (rms) voltage signal, thus 3ω measurement itself fails to provide such information. We switched to utilize Finite-Element-Method (FEM) software [5] to perform electro-thermal coupling analysis to the whole test structure. Four typical temperature oscillation profiles of the heater during one time cycle are given from low frequency (1 Hz) up to high frequency (1000 Hz), as shown in Fig. 2. At low frequency, the temperature oscillation amplitude varies significantly at z-coordinate (along length direction), thus yielding a longitudinal temperature gradient. This temperature inhomogeneity is weakened with increasing frequencies. While the frequency approaching to 1000 Hz, temperature oscillation amplitude is almost constant at any arbitrary z-position. Since the temperature gradient is the root cause for heat transfer process, according to Fourier’s diffusion law [6], the above transient temperature analysis actually demonstrates that the longitudinal heat flow inside heater strip should not be ignored in low frequency regime, although such heat flow will be prohibited at high frequencies. More important, the longitudinal heat diffusion partly depends on the thermal properties of the heater material, thus providing a possible solution to extract thermal conductivity of metallic thin films from longitudinal heat flow.

III. EXTRACTION OF HEATER THERMAL RESISTANCE According to the generalized analytical model we have

recently proposed in Ref. 7 , decoupling and quantifying of the longitudinal heat flow can be implemented by introducing a frequency-dependent parameter m(ω) as the average ratio of the heat flow along the heater strip to that in the substrate.

Thereby, 3ω voltage component ω3−zV corresponding to the longitudinal heat diffusion in metal heater strip can be expressed by [7]:

( )2

2

00

012

2

2

23

3

2)()(

)(

1

4

ωωρ

κπ

ρπα

ω

+⎥⎥⎦

⎤

⎢⎢⎣

⎡+

×

=−

mqrKdCqrqK

LD

CSLRIV

mpmm

s

mm

pmmmm

TCRz

(1)

where the effective voltage value ω3−zV denotes the root-mean-square (rms) value of )(3 tVz ω− , and I denotes that of

)cos(0 tI ω . Subscripts “s” and “m” refer to substrate and metal heater material, respectively. Dm, ρ m, Cpm are the thermal diffusivity, mass density and specific heat capacity of the heater. Lm, dm, r0, Sm are the heater’s length, thickness, half width and cross-section area, respectively. sκ denotes thermal conductivity of the substrate material and temperature coefficient of resistance

TCRα is 3.83×10-3 K-1 for aluminum heater. )( 00 qrK and )( 01 qrK refer to zeroth-order and first-order Bessel function in which 1−q refers to thermal penetration

depth defined as ω21sDq =−

.

In fact, it is hard to directly extract such signal ω3−zV since

the measured third harmonic voltage ω3V is the sum of two parts caused by longitudinal and radial thermal diffusion, respectively. As has shown in Fig. 2, longitudinal temperature gradient exhibits a strong dependence on current frequency. So, ω3−zV also shows the similar frequency-dependent response.

According to the generalized 3ω model, the modified extraction formula for thermal conductivity is given by:

.)(2 33

2

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ−−

=−

sTCR

zmm

ff

TIR

VVwL

RdI

α

κωω

(2)

where fd is the specimen film thickness. The temperature rise at the substrate surface sTΔ can be calculated using Cahill model. Given the thermal conductivity value of the underlying sample film fκ , the plot of ω3−zV versus frequency can be derived from fitting extraction formula to the experimental data ω3V at different frequencies. Afterwards, due to the high sensitivity of m(ω) to the variation of

ω3−zV , it is feasible to obtain a series of m(ω) values which are partly dependent on the thermal conductivity of the heater material.

As a result, the heat flow ratio m(ω), also in relation to heater thermal resistance, can be used to estimate the thermal properties of heater material as:

m

m

m SL

Pm

κω 1)( ∞ (3)

344

Fig. 3. Comparison of extracted heat flow ratio m(ω) from FEM numerical simulation and experimental fitting.

where P is Joule heating power andmmm SL κ is heater’s thermal

resistance.

IV. EXPERIMENTAL AND NUMERICAL VERIFICATION The sensitivity of m(ω) to the variation of thermal

resistance of the metal heater strip are experimentally testified by performing 3ω thermal measurement with different heater strip structures. In the measurements, a thin silicon dioxide film (SiO2) is deposited on silicon bulk substrate (ds=500μm). A narrow aluminum strip is sputtered on its top and then patterned by photolithography. The length (Lm) of metal strip heater between two current pads is 8 mm and that (lm) between two voltage pads is 2 mm. Specially, to illustrate the capability of nanoscale thermal metrology, thicknesses of aluminum strips in measurements are chosen at 1000, 200nm, respectively. The other detailed sample information is listed in Table I. The applied ac current frequency is varied from 0.1-1000Hz. Experimental uncertainty was reduced by controlling the temperature fluctuation inside the heat shield box within 0.4 K.

TABLE I. STRUCTURE PARAMETERS FOR THE EXPERIMENT

Based on the generalized thermal model, both numerical and experimental estimations of m(ω) were performed to the two heater structures with different cross-sectional area (in Table 1) which means different thermal resistance. As can be seen in Fig. 3, the heat flow ratio m(ω) derived from measurements is found to be highly sensitive to the difference in thermal resistance of the heater especially at 1Hz, which provides a novel means to measure the thermal conductivity of high-thermal-conductivity thin films under test by using them as the heater and comparing their m(ω) values with that of the calibrating metallic material at frequency 1Hz. The experimentally extracted ratio of two thermal resistances (Sample a to Sample d) is about 0.5, error between experimental result and the theoretical expectation (~0.2) can be partly attributed to using different input current conditions in our measurements. Therefore, applying the same input electrical current to the specimen film and calibrating film can further reduce the experiment uncertainty.

TABLE II. FEASIBILITY OF EXTRACTION OF THERMAL CONDUCTIVITY .

Moreover, good agreement is obtained in Fig. 3 between two quantification methods over wide frequency range, suggesting that numerical simulations also provide the reliable values of m(ω). Therefore, simulations can be used to testify the feasibility of such experimental solution in advance. As an example, the numerical extracted thermal conductivity of Cu ( Cuκ ) and Ta ( Taκ ) strip for heating were compared with that of calibration Al heater (

Alκ ), with results shown in Table 2. Consequently, low error rate from simulations indicates that the proposed technique based on heat flow ratio m(ω) can extend the capability of 3ω method to characterize high-thermal-conductivity thin films which are not appropriate previously to them using Cahill’s 3ω model.

I. CONCLUSIONS In summary, we have performed the electrical-thermal

coupling transient analysis to 3ω measurement structure, showing a frequency-dependent thermal response behavior. Meanwhile, the relationship between the extracted heat flow ratio m(ω) and the low-frequency thermal response is revealed based on the generalized thermal analytical model we have proposed in previous work, implying that the heat flow ratio m(ω) can be probably used for extraction of the thermal conductivity of nanoscale high-thermal-conductivity material, such as the metallic thin films. With this possibility, one can avoid the complicated fabrication process for suspended bridge structures commonly used in other electrical heating techniques [8]. The detailed experimental technique based on analytical thermal model is presented and the feasibility is then verified by experiments and numerical simulations.

ACKNOWLEDGMENT The authors gratefully acknowledge Michael Rennau of

Chemnitz University of Technology for his assistance with sample fabrication and thermal characterization. This work

Sample

Parameters Thin film

thickness df (nm)

Heater thickness dm(nm)

Heater width

wm(μm)

Electrical resistance

(Ω)

Input current I(mA)

a 1000 1000 20 16.61 110

d 1000 200 20 3.258 29

κ(sample) /κ(calibration

material)

Numerical estimation

Theoretical expectation

Error to theoretical

values

κCu/κAl 1.78 1.692 5.02%

κTa/κAl 4.35 4.157 4.64%

345

was supported by the Special Funds for Major State Basic Research Projects of China (Grant Nos. 2006CB302700 and 2011CBA00603), the International Research Training Group Program on “Materials and Concepts for Advanced Interconnects” between Chinese MOE and German DFG and the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China (Grant No. 708033)

REFERENCES [1] K. E. Goodson and Y. S. Ju, “Heat conduction in novel electronic

films,”Annu. Rev. Mater. Sci. Vol. 29, 1999, pp.261-293. [2] MN.Touzelbaev and KE Goodson, “Applications of micron-scale

passive diamond layers for the integrated circuits and microelectromechanical systems industries,” Diamond Rel. Mater.Vol.7, 1998, pp.1-14.

[3] JL Hostetler, AN Smith, and PM Norris, “Thin-film thermal conductivity and thickness measurements using pico- second ultrasonics,” Microscale Thermophys. Eng. Vol. 1, pp.237-244.

[4] DG Cahill, “Thermal conductivity measurement from 30 to 750 K: the 3omega method,” Rev. Sci. Instrum. Vol.61, 1990, 802-808..

[5] COMSOL Multiphysics, COMSOL Ab., Stokholm, Sweden (2005). Available online at http://www.comsol.com/.

[6] HS Carslaw and JC Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford,1959.

[7] ZX Zong, ZJ Qiu, SL Zhang, R Streiter,and R Liu, “A generalized 3x method for extraction of thermal conductivity in thin films,” J. Appl. Phys. Vol. 109, 2011, 063502.

[8] L.Lu, W.Yi, and DLZhang, “3 omega method for specific heat and thermal conductivity measurements,” Rev. Sci. Instrum. Vol.72, 2001, pp.2996-3003.

346