A "2-omega" technique for measuring anisotropy of thermal conductivity

Anisotropic thermoreflectance thermometry: A contactless frequency-domain thermoreflectance approach to study anisotropic thermal transport

We developed a novel contactless frequency-domain thermoreflectance approach to study thermal transport, which is particularly convenient when thermally anisotropic materials are considered. The method is based on a line-shaped heater geometry, produced with a holographic diffractive optical element, instead of using a spot heater as in conventional thermoreflectance. The heater geometry is similar to the one used in the 3-omega method, however, keeping all the technical advantages offered by non-contact methodologies. The present method is especially suitable to determine all the elements of the thermal conductivity tensor, which is experimentally achieved by simply rotating the sample with respect to the line-shaped optical heater. We provide the mathematical solution of the heat equation for the cases of anisotropic substrates, thin films, and multilayer systems. This methodology allows an accurate determination of the thermal conductivity and does not require complex modeling or intensive computational efforts to process the experimental data, i.e., the thermal conductivity is obtained through a simple linear fit ("slope method"), in a similar fashion to the 3-omega method. We demonstrate the potential of this approach by studying isotropic and anisotropic materials in a wide range of thermal conductivities. In particular, we have studied the following inorganic and organic systems: (i) glass, Si, and Ge substrates (isotropic), (ii) β-Ga₂O₃ and a Kapton substrate (anisotropic), and (iii) a 285 nm thick SiO₂ thin film deposited on a Si substrate. The accuracy in the determination of the thermal conductivity is estimated as ≈5%, whereas the temperature uncertainty is ΔT ≈ 3 mK.

A new elliptical-beam method based on time-domain thermoreflectance (TDTR) to measure the in-plane anisotropic thermal conductivity and its comparison with the beam-offset method

Materials lacking in-plane symmetry are ubiquitous in a wide range of applications such as electronics, thermoelectrics, and high-temperature superconductors, in all of which the thermal properties of the materials play a critical part. However, very few experimental techniques can be used to measure in-plane anisotropic thermal conductivity. A beam-offset method based on time-domain thermoreflectance (TDTR) was previously proposed to measure in-plane anisotropic thermal conductivity. However, a detailed analysis of the beam-offset method is still lacking. Our analysis shows that uncertainties can be large if the laser spot size or the modulation frequency is not properly chosen. Here we propose an alternative approach based on TDTR to measure in-plane anisotropic thermal conductivity using a highly elliptical pump (heating) beam. The highly elliptical pump beam induces a quasi-one-dimensional temperature profile on the sample surface that has a fast decay along the short axis of the pump beam. The detected TDTR signal is exclusively sensitive to the in-plane thermal conductivity along the short axis of the elliptical beam. By conducting TDTR measurements as a function of delay time with the rotation of the elliptical pump beam to different orientations, the in-plane thermal conductivity tensor of the sample can be determined. In this work, we first conduct detailed signal sensitivity analyses for both techniques and provide guidelines in determining the optimal experimental conditions. We then compare the two techniques under their optimal experimental conditions by measuring the in-plane thermal conductivity tensor of a ZnO [11-20] sample. The accuracy and limitations of both methods are discussed.

Time-domain thermoreflectance (TDTR) measurements of anisotropic thermal conductivity using a variable spot size approach

It is challenging to characterize thermal conductivity of materials with strong anisotropy. In this work, we extend the time-domain thermoreflectance (TDTR) method with a variable spot size approach to simultaneously measure the in-plane (Kr) and the through-plane (Kz) thermal conductivity of materials with strong anisotropy. We first determine Kz from the measurement using a larger spot size, when the heat flow is mainly one-dimensional along the through-plane direction, and the measured signals are only sensitive to Kz. We then extract the in-plane thermal conductivity Kr from a second measurement using the same modulation frequency but with a smaller spot size, when the heat flow becomes three-dimensional, and the signal is sensitive to both Kr and Kz. By choosing the same modulation frequency for the two sets of measurements, we can avoid potential artifacts introduced by the frequency-dependent Kz, which we have found to be non-negligible, especially for some two-dimensional layered materials like MoS2. After careful evaluation of the sensitivity of a series of hypothetical samples, we provided guidelines on choosing the most appropriate laser spot size and modulation frequency that yield the smallest uncertainty, and established a criterion for the range of thermal conductivity that can be measured reliably using our proposed variable spot size TDTR approach. We have demonstrated this variable spot size TDTR approach on samples with a wide range of in-plane thermal conductivity, including fused silica, rutile titania (TiO2 [001]), zinc oxide (ZnO [0001]), molybdenum disulfide (MoS2), hexagonal boron nitride (h-BN), and highly ordered pyrolytic graphite.

An electrical probe of the phonon mean-free path spectrum

Most studies of the mean-free path accumulation function (MFPAF) rely on optical techniques to probe heat transfer at length scales on the order of the phonon mean-free path. In this paper, we propose and implement a purely electrical probe of the MFPAF that relies on photo-lithographically defined heater-thermometer separation to set the length scale. An important advantage of the proposed technique is its insensitivity to the thermal interfacial impedance and its compatibility with a large array of temperature-controlled chambers that lack optical ports. Detailed analysis of the experimental data based on the enhanced Fourier law (EFL) demonstrates that heat-carrying phonons in gallium arsenide have a much wider mean-free path spectrum than originally thought.

A 3 omega method to measure an arbitrary anisotropic thermal conductivity tensor

Previous use of the 3 omega method has been limited to materials with thermal conductivity tensors that are either isotropic or have their principal axes aligned with the natural cartesian coordinate system defined by the heater line and sample surface. Here, we consider the more general case of an anisotropic thermal conductivity tensor with finite off-diagonal terms in this coordinate system. An exact closed form solution for surface temperature has been found for the case of an ideal 3 omega heater line of finite width and infinite length, and verified numerically. We find that the common slope method of data processing yields the determinant of the thermal conductivity tensor, which is invariant upon rotation about the heater line's axis. Following this analytic result, an experimental scheme is proposed to isolate the thermal conductivity tensor elements. Using two heater lines and a known volumetric heat capacity, the arbitrary 2-dimensional anisotropic thermal conductivity tensor can be measured with a low frequency sweep. Four heater lines would be required to extend this method to measure all 6 unknown tensor elements in 3 dimensions. Experiments with anisotropic layered mica are carried out to demonstrate the analytical results.

Reusable bi-directional 3ω sensor to measure thermal conductivity of 100-μm thick biological tissues

Accurate knowledge of the thermal conductivity (k) of biological tissues is important for cryopreservation, thermal ablation, and cryosurgery. Here, we adapt the 3ω method-widely used for rigid, inorganic solids-as a reusable sensor to measure k of soft biological samples two orders of magnitude thinner than conventional tissue characterization methods. Analytical and numerical studies quantify the error of the commonly used "boundary mismatch approximation" of the bi-directional 3ω geometry, confirm that the generalized slope method is exact in the low-frequency limit, and bound its error for finite frequencies. The bi-directional 3ω measurement device is validated using control experiments to within ±2% (liquid water, standard deviation) and ±5% (ice). Measurements of mouse liver cover a temperature ranging from -69 °C to +33 °C. The liver results are independent of sample thicknesses from 3 mm down to 100 μm and agree with available literature for non-mouse liver to within the measurement scatter.