A Square Pulse Thermoreflectance Technique for the Measurement of Thermal Properties

Abstract

We report on a laser-based square pulse thermoreflectance (SPTR) technique for the measurement of thermal properties for a wide range of materials. SPTR adopts the pump-probe thermoreflectance principle to monitor the evolution of local temperature after square pulse excitation. The technique features a compact setup, high spatial resolution, and fast data collection. By comparing the acquired SPTR signals with a continuum heat transfer model, material thermal properties can be obtained. Taking advantage of various spot sizes and modulation frequencies, SPTR can measure both the thermal diffusivity and thermal conductivity of poorly to moderately conductive materials and the thermal conductivity of conductive materials with satisfactory accuracy, with potential to be applied to more conductive materials. The technique was validated on three materials: fused silica, single crystal CaF₂ and single crystal nickel (with conductivities ranging from 1 W·m⁻¹·K⁻¹ to 100 W·m⁻¹·K⁻¹) with typical measurement errors of 5 % to 20 %. The leading sources of error have been identified by Monte Carlo simulations, and the primary limitations of SPTR are discussed. The compact, fiberized platform we describe here will allow instruments based on this methodology to be deployed in complex, multi-analytical environments for the type of high-throughput correlative analyses that are key to materials design and discovery.

Appendix

Ideally, a square wave can be expressed by Eq. 1. Each harmonic has an equal phase of zero and a 1/(2n − 1) amplitude. In an actual experiment, however, the signal is processed by various electrical circuits which inevitably introduce distortions. These distortions are most often frequency dependent. To conveniently capture these effects, we use a lumped transfer function to account for the changes to both amplitude and phase.

First, the imperfect square pulse is measured by removing the short-pass filter normally placed in front of the detector to reject any pump light, shutting down the probe laser, and collecting the pump signal reflected by sample surface using the photodetector.

Next, to derive the transfer function, we express the collected non-ideal square wave as

$$x{^{\prime}}\left(t\right)={\sum }_{m=-\infty }^{\infty }{P}_{m}\frac{\text{exp}\left(im\omega t+{i\phi }_{m}\right)}{m},$$

(10)

where m are odd integers. Taking advantage of the relation $\int \text{sin}\left(mt\right)\text{sin}\left(nt\right)dt={\delta }_{nm}$ ($\delta$ is Kronecker delta function), we can obtain the following relations

$$\begin{gathered} P_{m,1} = \mathop \int \limits_{0}^{\frac{1}{f}} mx^{\prime}\left( t \right)\cos \left( {m\omega t} \right)dt \hfill \\ P_{m,2} = \mathop \int \limits_{0}^{1/f} mx^{\prime}\left( t \right)\sin \left( {m\omega t} \right)dt, \hfill \\ \end{gathered}$$

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from which the factors $P_{m}$ and $\phi_{m}$ can be expressed as

$$\begin{gathered} P_{m} = \sqrt {P_{m,1}^{2} + P_{m,2}^{2} } \hfill \\ \phi_{m} = {\text{atan}}\left( {\frac{{P_{m,2} }}{{P_{m,1} }}} \right). \hfill \\ \end{gathered}$$

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