Analysis of heat flow in layered structures for time domain
Analysis of heat flow in layered structures for timedomain thermoreflectance David G. Cahill^{a)} Department of Materials Science and Engineering and Frederick Seitz Materials Research Laboratory, University of Illinois, Urbana, Illinois 61801 (Received 11 June 2004; accepted 19 September 2004; published 10 November 2004) The iterative algorithm of Feldman for heat flow in layered structures is solved in cylindrical coordinates for surface heating and temperature measurement by Gaussianshaped laser beams. This solution for the frequencydomain temperature response is then used to model the lockin amplifier signals acquired in timedomain thermoreflectance measurements of thermal properties. Â© 2004 American Institute of Physics. [fusion_builder_container hundred_percent=”yes” overflow=”visible”][fusion_builder_row][fusion_builder_column type=”1_1″ background_position=”left top” background_color=”” border_size=”” border_color=”” border_style=”solid” spacing=”yes” background_image=”” background_repeat=”norepeat” padding=”” margin_top=”0px” margin_bottom=”0px” class=”” id=”” animation_type=”” animation_speed=”0.3″ animation_direction=”left” hide_on_mobile=”no” center_content=”no” min_height=”none”][DOI: 10.1063/1.1819431] 
I. INTRODUCTION Timedomain thermoreflectance (TDTR) is a pumpprobe optical technique that can be used for measuring the thermal properties of materials.^{1â€“5} We have previously described our implementation of this technique,^{6,7 }and our application of this method in studies of the thermal conductivity of thin films,^{8,9 }the thermal conductance of interfaces,^{8,10} spatially resolved measurements of microfabricated structures,^{10} and highresolution mapping of the thermal conductivity of diffusion multiples.^{11} In most cases, analysis of TDTR experiments requires comparisons between the data and a model of the heat transport in the system under study. Unknown thermal properties are treated as free parameters and adjusted to minimize the differences between the model and the data. We have briefly described how the frequencydomain thermal response can be used as the input to a calculation of the inphase and outofphase lockin amplifier signals in TDTR experiments ^{8,11} but we have not previously described our method for calculating the frequencydomain response. The purpose of this article is to describe the details of those calculations and provide additional discussion of our methods for analyzing TDTR data. II. FREQUENCY DOMAIN SOLUTION FOR THE SURFACE TEMPERATURE OF A SINGLE LAYER We begin with the frequencydomain solution^{12} for a semiinfinite solid that is heated at the surface by a periodic point source of unit power at angular frequency w
where L is the thermal conductivity of the solid, D the thermal diffusivity, and r the radial coordinate. This solution for the semiinfinite solid differs from the solution for the infi ^{a)}Author to whom correspondence should be addressed; electronic mail: dcahill@uiuc.edu

infinite solid by a factor of 2. Since the coaligned laser beams of a typical TDTR experiment have cylindrical symmetry, we use Hankel transforms^{13,14} to simplify the convolution of this solution with the distributions of the laser intensities. The Hankel transform of g(r) is The surface is heated by a pump laser beam with a Gaussian distribution of intensity p(r); the 1/e^{2} radius of the pump beam is w_{0} where A is the amplitude of the heat absorbed by the sample at frequency w. The Hankel transform of p(r) is The distribution of temperature oscillations at the surface (r) is the inverse transform of the product of G(k) and P(k) The surface temperatures are measured by thermoreflectance, i.e., the change in the reflectivity with temperature. This change in reflectivity is measured by changes in the reflected intensity of a probe laser beam. The probe laser beam also has a Gaussian distribution of intensity although the radius may be different than the pump beam; the 1/e^{2} radius of the probe beam is w_{1}. The probe beam measures a weighted average of the temperature distribution (r) The integral over r is the Hankel transform of a Gaussian, leaving a single integral over k that must be evaluated numerically 
As expected, Eq. (9) is unchanged by an exchange of the radii of the pump and probe beams. The upper limit of the integral can be set to without a significant loss of accuracy. Before we generalize this solution to a layered structure, we examine the low and high frequency limits of Eq. (9). In our experiments, the pump and probe have the approximately the same diameter, so we set w_{0}= w_{1}. In the low frequency limit, , Equation (11) is particularly useful for estimating the steadystate temperature rise of the probed region of the sample. For example, with A_{0}=2 mW, Si thermal conductivity of L where C is the heat capacity per unit volume; is the thermal effusivity. Equation (13) is equivalent to the solution for onedimensional heat flow with a uniform heat flux of III. FREQUENCY DOMAIN SOLUTION FOR THE SURFACE TEMPERATURE OF A LAYERED STRUCTURE Equation (9) can be generalized to a layered geometry using the algorithm described recently by Feldman;^{15} this algorithm has also been applied in the analysis of data^{16} obtained by the 3w method. The final result is obtained iteratively: we number the layers by n=1 for the layer that terminates at the surface of the solid. The iteration starts with the layer farthest from the surface; in practical applications of this method to the analysis of TDTR data, heat cannot reach the other side of this bottom layer at rates comparable to the modulation frequency; therefore, B^{+}=0 and B^{âˆ’}=1 for the final layer. 
FIG. 1. Calculated frequency response of Al/Si and Al/SiO_{2} /Si thermal Each layer n is described by three parameters; the thermal conductivity _{n}, thermal diffusivity D_{n}, and thickness L_{n}. We model an interface conductance by a layer with a small thermal conductivity and small thickness. For a layered structure, the only change in Eq. (9) is to replace G(k) for the single layer [Eq. (4)] by Example calculations using the combination of Eq. (9) and Eq. (18) are shown as Fig. 1. IV. MODELING OF DATA ACQUIRED IN TIMEDOMAIN THERMOREFLECTANCE EXPERIMENTS We can now use the frequencydomain thermal response, Eqs. (9) and (18), to calculate the changes in reflectivity that will be measured in a TDTR experiment. Because the width of the optical pulses produced by a Ti:sapphire laser, <0.5 ps, are much shorter than the time scales of interest in the thermal model, t > 50 ps, we can approximate the frequency spectrum of the laser output as a series of delta functions separated in frequency by the repetition rate of the laser 1/.The pump beam is modulated by a 50% duty cycle at frequency f. This modulation reduces the magnitude of the peaks at all multiples of 1/ and produces sharp sidebands around each of these peaks at odd multiples of f; the amplitudes of the sideband at frequency f removed from the main peaks is a factor of 2/Ï€ times smaller than the amplitudes of the main peaks. Surface temperatures are measured by the changes in reflectivity R with temperature T, i.e., the thermoreflectance dR/dT. The signal of interest is produced by the product of the temperature changes and the intensity of the probe beam; in the frequency domain this product becomes a convolution 
FIG. 2. Partial sums of the series used to calculate the reflectivity changes, see Eq. (19); dR/dT is the thermoreflectance of the surface. The solid lines are the real part of R_{M}(t) evaluated for two delay times t=100 and t= âˆ’100 ps; the dashed line is the imaginary part of R_{M}(t) and is independent of t for short delay times. The terms in the sum for the real part of R_{M}(t) have been multiplied by a Gaussian factor of the form exp (Ï€(f/f_{max})^{2}) with f_{max}=100 GHz to improve the convergence of the sum. The thermal model is for Al/SiO_{2} /Si as described in the caption to Fig. 1. 
FIG. 3. Calculated dependence of the reflectivity changes at fixed delay times of 100 and âˆ’100 ps on modulation frequency f; dR/dT is the thermoreflectance of the surface. The imaginary part of the response (dashed line) is essentially constant for these two times. The thermal model is for Al/SiO_{2} /Si as described in the caption to Fig. 1

of the frequency spectra of the thermal response and the probe beam. The time delay t shifts the relative phase of the probe frequency spectrum. The lockin amplifier picks out the frequency components of the convolution evaluated at f and âˆ’f The power absorbed by the sample from the pump beam at frequency f, Af, is related to the average power absorbed from the pump beam A_{0} by A_{f}=2A_{0}/Ï€. 
In our implementation of TDTR, the arrival time of the pump beam is advanced rather than the more typical case of increasing the delay time of the probe beam. This procedure introduces an additional phase shift with the limit of the sum set to M=10t / t. The dependence of R(t) on the modulation frequency f is illustrated by Fig. 3 for fixed delay times of t= Â±100 ps. In principle, the optimal choice of f will depend on the system under study. In practice, we prefer f1/(8)10 MHz Typically, a rf lockin amplifier incorporates a squarewave mixer and therefore the input signal from the photodetector must be filtered to remove the odd harmonics of f that are present in the intensity of the reflected probe. In our apparatus, a tunable inductor is placed in series between the output of a reversed biased pin Si photodiode and the 50 input of the rf lockin amplifier.^{17} The inductance and the modulation frequency f are adjusted to maximize the response at f; the quality factor of the resonant circuit is Q 10. The rms voltage measured by the lockin amplifier at the modulation frequency f, V_{f}(t), is related to the changes in reflectivity by where V_{0} is the average voltage output of the detector and R is the reflectivity of the sample. The real and imaginary parts of V_{f}(t) are given by the inphase and outofphase signals of the lockin amplifier, respectively. We use the fact that the imaginary part of V_{f}(t) is constant as the delay time crosses t=0 to correct for small errors in the phase of the reference 
FIG. 4. Dependence on delay time t at fixed modulation frequency of f =9.8 MHz for a TiN/MgO(001) epitaxial layer. Measured data are shown as filled circles [inphase or real part of V_{f}(t) and open circles [outofphase or imaginary part of V_{f}(t). The solid and dashed lines are the real and imaginary parts of the model calculation, respectively, see Eq. (22) We compare an example of a full calculation of V_{f}(t) /V_{0} 
have previously discussed how our approach of analyzing the ratio V_{in} /V_{out} minimizes these errors;^{6,8} the optical design of Capinski and aris^{3} provides another method for improving the accuracy of TDTR measurements at large delay times. This work was supported by NSF Grant No. CTS 0319235. Data were acquired using the equipment in the Laser Facility of the Frederick Seitz Materials Research Laboratory (MRL) at the University of Illinois at Urbana Champaign. ^{1}D. A. Young, C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, in Phonon Scattering in Condensed Matter, edited by A. C. Anderson and J. P. Wolfe (Springer, Berlin, 1986), p. 49. ^{2} C. A. Paddock and G. L. Eesley, J. Appl. Phys. 60, 285 (1986). ^{3}W. S. Capinski and H. J. Maris, Rev. Sci. Instrum. 67, 2720 (1996). ^{4} B. Bonello, B. Perrin, and C. Rossignol, J. Appl. Phys. 83, 3081 (1998). ^{5} N. Taketoshi, T. Baba, E. Schaub, and A. Ono, Rev. Sci. Instrum. 74, 5226 (2003). ^{6} D. G. Cahill, K. E. Goodson, and A. Majumdar, J. Heat Transfer 124, 223 (2002). ^{7} D. 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). ^{8} R. M. Costescu, M. A. Wall, and D. G. Cahill, Phys. Rev. B 67, 054302 (2003). ^{9} R. M. Costescu, D. G. Cahill, F. H. Fabreguette, Z. A. Sechrist, and S. M. George, Science 303, 989 (2004). ^{10} S. Huxtable, D. G. Cahill, and L. M. Phinney, J. Appl. Phys. 95, 2102 (2004). ^{11} S. Huxtable, D. G. Cahill, V. Fauconnier, J. O. White, and J.C. Zhao, Nat. Mater. 3, 298 (2004). ^{12} H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University Press, New York, 1959), p. 263. ^{13} R. N. Bracewell, The Fourier Transform and its Applications (McGrawâ€“ Hill, New York, 2000), Chap. 13, pp. 335â€“339. ^{14}Y. Ohsone, G. Wu, J. Dryden, F. Zok, and A. Majumdar, J. Heat Transfer 121, 954 (1999). ^{15} A. Feldman, High Temp. – High Press. 31, 293 (1999). ^{16} J. H. Kim, A. Feldman, and D. Novotny, J. Appl. Phys. 86, 3959 (1999). ^{17} K. E. Oâ€™Hara, X.Y. Hu, and D. G. Cahill, J. Appl. Phys. 90, 4852 (2001). 
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