### Analysis of heat flow in layered structures for time-domain 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 Gaussian-shaped laser beams. This solution for the frequency-domain temperature response is then used to model the lock-in amplifier signals acquired in time-domain 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=”no-repeat” 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

Time-domain 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 high-resolution 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 frequency-domain thermal response can be used as the input to a calculation of the in-phase and out-of-phase lock-in amplifier signals in TDTR experiments8,11 but we have not previously described our method for calculating the frequency-domain 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 frequency-domain solution12 for a semi-infinite solid that is heated at the surface by a periodic point source of unit power at angular frequency v

where L is the thermal conductivity of the solid, D the thermal diffusivity, and r the radial coordinate. This solution for the semi-infinite solid differs from the solution for the infinite solid by a factor of 2. Since the co-aligned laser beams of a typical TDTR experiment have cylindrical symmetry, we use Hankel transforms13,14 to simplify the convolution of this solution with the distributions of the laser intensities. The Hankel transform of gsrd is

The surface is heated by a pump laser beam with a Gaussian distribution of intensity p(r); the 1/e2 radius of the pump beam is w0.

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 0(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/e2 radius of the probe beam is w1. The probe beam measures a weighted average of the temperature distribution 0(r)

The integral over r is the Hankel transform of a Gaussian, leaving a single integral over k that must be evaluated numerically

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