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A new spatial-scan thermoreflectance method to measure a broad range of anisotropic in-plane thermal conductivity

In-plane thermal conductivities of small-scale samples are hard to measure, especially for the lowly conductive ones and those lacking in-plane symmetry (i.e., transversely anisotropic materials). State-of-the-art pump-probe techniques including both the time-domain and the frequency-domain thermoreflectance (TDTR and FDTR) are advantageous in measuring the thermal conductivity of small-scale samples, and various advanced TDTR and FDTR techniques have been developed to measure transversely anisotropic materials. However, the measurable in-plane thermal conductivity (k_in) is usually limited to be>10 W/(m K). In this work, a new spatial-scan thermoreflectance (SSTR) method has been developed to measure a broad range of k_in of millimeter-scale small samples, including those lacking in-plane symmetry, extending the current limit of the measurable k_in to as low as 1 W/(m K). This SSTR method establishes a new scheme of measurements using the optimized laser spot size and modulation frequency and a new scheme of data processing, enabling measurements of in-plane thermal conductivity tensors of a broad range of k_in values with both high accuracy and ease of operation. Some details such as the requirement on the sample geometry, the effect of the transducer layer, and the effect of heat loss are also discussed. As a verification, the k_in of some transversely isotropic reference samples with a wide range of k_in values including fused silica, sapphire, silicon, and highly ordered pyrolytic graphite (HOPG) have been measured using this new SSTR method. The measured k_in agree perfectly well with the literature values with a typical uncertainty of 5%. As a demonstration of the unique capability of this method, the in-plane thermal conductivity tensor of x-cut quartz, an in-plane anisotropic material, has also been measured.


Introduction
In-plane thermal conductivities of bulk and thin-film materials are of great interest in a wide variety of applications such as thermoelectrics 1 , power electronics 2 , and lithium-ion batteries 3 . Accurate measurements of the in-plane thermal conductivities of these materials are not only of great scientific value but also critically important for their engineering applications.
In recent years, pump-probe techniques including both the time-domain thermoreflectance (TDTR) and frequency-domain thermoreflectance (FDTR) have become popular in measuring the anisotropic thermal conductivity of materials. [4][5][6][7] TDTR is a powerful technique based on an ultrafast laser for thermal characterization. [8][9][10] The high modulation frequency of TDTR (in the MHz range) gives it many advantages such as short thermal diffusion length (and thus high spatial resolution) and high signal-to-noise ratio (SNR). However, because of the short thermal diffusion lengths (0.1~10 µm) compared to the laser spot size, TDTR is most readily sensitive to thermal conductivities in the crossplane rather than the in-plane direction. Although several TDTR-based advanced approaches such as beam-offset TDTR 11,12 , variable spot size TDTR 13 , and elliptical-beam TDTR 14,15 have been developed to measure in-plane thermal conductivities ( ), the measurable is usually limited to be >10 W/(m • K) . Measuring lower would require a very low modulation frequency, which, however, is prohibited in TDTR due to the strong pulse accumulation 16 .
FDTR is a variation of TDTR. Unlike TDTR, FDTR usually monitors the thermoreflectance signals as a function of the heating frequency. 17,18 Most FDTR systems are based on continuous-wave lasers, which place no restrictions on the modulation frequency. 19 Theoretically, a very low modulation frequency can be used in FDTR to measure low in-plane thermal conductivities. In the literature, a beam-offset FDTR approach has been developed [20][21][22][23] to enhance the sensitivity of the frequency-dependent phase signals to the in-plane heat diffusion by offsetting the pump and probe spots at a fixed distance. Recently, Tang and Dames 24 extended the capability of beam-offset FDTR to measure the full thermal conductivity tensor of transversely anisotropic materials. However, one major difficulty of the FDTR experiments is that the reference phase of the lock-in amplifier needs to be carefully characterized to prevent the miscellaneous non-thermal phase shifts from interfering with the thermal phase signal. Besides, the current beam-offset FDTR approaches typically have their modulation frequencies limited to be >5 kHz, making this method only capable of measuring > 10 W/(m • K). 22,23 Although a transducerless FDTR approach has been developed to measure a lower , 25 this approach could only apply to semiconductors with strong optical absorption and thus has a limited application.
Alternatively, a spatial-scan thermoreflectance method can also measure . This method uses the same instrument as FDTR, and the phase signal as a function of pump-probe offset distance at a fixed modulation frequency is analyzed to derive . Great progress has been achieved over the past twenty years in developing this spatial-scan thermoreflectance technique, alternatively also known as spatial-domain thermoreflectance 26 , thermal wave microscopy 27 , or modulated thermoreflectance (MTR) [28][29][30][31] . However, most of the prior efforts limit their applications to isotropic samples, and the potential of this technique to measure the full tensor of transversely anisotropic materials is not revealed. Besides, in the past efforts thus far, the absolute phase signals are analyzed to derive the thermal properties, which is still plagued by the same problem in FDTR that miscellaneous non-thermal phase signals easily interfere the desired thermal phase signal. This paper is organized as follows. First, details of the experimental setup and the basic measurement procedures are presented. Then more details regarding this method including the sensitivity analysis, the uncertainty estimation, and the effects of the transducer layer, sample geometry, and heat loss will be fully discussed. This method will then be verified by measuring in-plane thermal conductivities of some transversely isotropic reference samples including silica, sapphire, silicon, and HOPG. Finally, the capability of this method in measuring the full in-plane thermal conductivity tensor of transversely anisotropic materials will be demonstrated through both a simulated numerical experiment and a real measurement of x-cut quartz.

Experimental setup and basic measurement procedures
The improved spatial-scan thermoreflectance experiments have been conducted independently by different systems in two labs. The experimental system in the lab of the University of Pittsburgh is shown in Fig. 1(a). This system uses a 532 nm green laser as the pump beam and a 671 nm red laser as the probe beam, at which wavelength some low-thermal conductivity metals such as Ti can be used as the transducer for the thermoreflectance measurements. As will be discussed in more detail later, a metal transducer with a lower thermal conductivity is more beneficial in measuring of the substrate, especially for lowly conductive materials. The experimental system in the lab of Huazhong University of Science and Technology is shown in Fig. 1(b). This is a commercial FDTR system purchased from Fourier Scientific, designed and built by Dr. Aaron Schmidt. 32 Note that Fig. 1(b) is a simplified diagram of the system as some features designed for FDTR but irrelevant for the current method, such as a sampler for measuring the pump reference phase, are not shown here.
This system uses a 488 nm blue laser as the pump beam and a 532 nm green laser as the probe beam, at which wavelength the metal Au has a very high thermoreflectance coefficient and can be used as the transducer to achieve a high SNR. Besides, this system uses a balanced photodetector, which has a large transimpedance gain and can easily achieve low noise and significantly reduce the signal-acquisition time for the same level of SNR compared to the first system. The typical laser spot size 0 (the root-mean-square average of the 1/ 2 radii of the pump and the probe laser spots) of the first and the second system is ~12 μm and ~4 μm, respectively. In the new SSTR experiments, both the phase signal and the amplitude are recorded as a function of the offset distance between the pump and probe spots measured using a low modulation frequency . The frequency is recommended to be low enough to induce a long in-plane thermal diffusion length , at least 3 times the laser spot size along the scanning direction, , ≥ 3 . Here , is defined as , = √ / , with being the directional in-plane thermal conductivity of the substrate in the scanning direction and being the volumetric heat capacity of the substrate. This criterion usually guarantees a decent SNR for the phase signal even at a large offset distance of = 4 so that a phase gradient d /d , which is essential for measuring / of the sample, can be reliably extracted from the phase signal in the offset range 2 < < 4 . In practice, the phase difference Δ = − ( = 0) as a function of is used to derive the in-plane thermal conductivity in the scanning direction with as a known input, and the normalized amplitude signal = / ( = 0) as a function of is used to derive the laser spot size by best-fitting the measured signals with a thermal model prediction. The thermal model of anisotropic heat conduction in multilayered systems has been well developed for beam-offset FDTR in the literature 22,24 and can be directly adopted here. An example of the processed Δ and signals for a 100 nm Ti/silica sample measured using a modulation frequency of 150 Hz and a laser spot size of 11.5 μm is shown in Fig. 2. By fitting the Δ signals in Fig. 2 (a) and with the volumetric heat capacity taken as = 1.66 × 10 6 J/(m 3 • K) for silica from the literature 33 , the in-plane thermal conductivity of silica along the scanning direction was determined as 1.4 W/(m • K) , in excellent agreement with the literature 34 .
Meanwhile, the laser spot size was determined as 11.5 µm by fitting the signals in Fig.   2 (b). , from which the laser spot size long the offset direction was fitted as 11.5 μm. The dash-dot curves are the ±30% bounds of the best fitted .
One novelty of the current work is the newly proposed approach of data processing. By analyzing the phase difference signal Δ rather than the absolute phase signal , this new SSTR method exempts the necessity of characterizing the reference phase of the lock-in amplifier and thus is more convenient to practice. Another important feature of this method is that the laser spot size can be determined at the same time as the thermal conductivity measurement rather than requiring a separate experiment to be pre-determined, which not only simplifies the experiment but also increases the reliability since the effect of any chance that the laser spot size should change between different experiments due to unintentional laser beam defocusing could be eliminated. Most importantly, the breakthrough of this method is that it successfully measures as low as 1.4 W/(m • K) with great accuracy, breaking the current limit of > 10 W/(m • K) measurable by TDTR and FDTR.
Although the whole range of the Δ signal was fitted to derive in Fig. 2   Sensitivity coefficients of (a) the phase gradient signal ∇ and (b) the FWHM of to different parameters of a 100 nm Ti/silica sample measured by the improved spatialscan thermoreflectance method using a modulation frequency of 150 Hz and a laser spot size of 11.5 µm. In this analysis, the scanning direction is along . The analyzed parameters include the anisotropic thermal conductivities , , and the volumetric heat capacity of both the metal layer and the substrate, the thickness of the metal layer ℎ , the interface thermal conductance between the metal layer and the substrate, and the laser spot sizes and in the -and -direction, respectively. The parameters with a subscript "m" are for the metal transducer layer, while the unspecified parameters are for the substrate. This fitting process is sufficient for transversely isotropic materials. When measuring the full in-plane thermal conductivity tensor of transversely anisotropic materials, the abovedescribed fitting process needs to be iteratively repeated to simultaneously fit three sets of signals measured in different scanning directions, details of which will be demonstrated through a simulated numerical experiment in Section 3.2.1.

Uncertainty analysis
The uncertainties of the fitted and values could be estimated using a full error propagation formula. This formula fully accounts for the uncertainties of all the input parameters as well as the quality of the least-squares fitting of the experimental data and the experimental noise. Similar applications of this full error propagation formula were also found in the previous TDTR and FDTR experiments. 25,[35][36][37][38] In the present case, since is extracted from the Δ signal and is extracted from the signal through an iterative fitting process, this fitting process can be mathematically expressed as seeking to minimize the squared difference between the experimental data and the model predictions: where (  Since the input parameters have uncertainties, each input parameter has a normal distribution with the mean value being its nominal value and two times the standard deviation 2 being its uncertainty. Let us denote * as a random group of the possible control parameters and * as the corresponding group of fitting parameters that make the best fit. The uncertainties of the unknown properties could be revealed from the distribution of all the possible * . The covariance matrix of * could be explicitly expressed after some analytical derivation, details of which can be found in Ref. 35 where the elements on the principal diagonal are the variance of the unknown parameters. The standard deviation of each unknown parameter can be simply determined as the square root of its variance, and the uncertainty of the unknown parameter is two times its standard deviation. In practice, we only analyze the experimental data away from the pump spot center with | | > because the data around the center are bound to be near 0 for Δ and 1 for and have little sensitivity to the unknown properties. Assuming an uncertainty of 10% for , , , , , , , and , respectively, 20% for , 5% for ℎ , and 3% for , , and , respectively, using the full error propagation formula we calculate that these errors of the control parameters would propagate to an uncertainty of 3% for and 0.5% for for the silica data in Fig. 2. In addition to that, the experimental noise for the silica data in Fig. 2 would contribute another 1.9% uncertainty for and 1.4% for . The total uncertainty for of silica is thus √3 2 + 1.9 2 % = 3.5% ( = 1.4 ± 0.05 W/(m • K)) and the uncertainty of is 1.5% ( = 11.5 ± 0.2 μm).
Note that the well-developed TDTR and FDTR techniques typically have an uncertainty of ~10% in determining the through-plane thermal conductivity and an even larger uncertainty in determining the in-plane thermal conductivity. Due to the MHz-range high modulation frequencies used in TDTR and FDTR, the phase signals are also highly sensitive to the heat capacitance ℎ of the metal transducer and the metal-substrate interface thermal conductance , the uncertainties of which would propagate and cause a large uncertainty in determining the thermal conductivity of the substrate. This problem, however, is absent in this new SSTR method. Because a low enough modulation frequency ensuring , ≥ 3 is used for the measurement, the phase gradient and the normalized amplitude signals acquired in the SSTR measurements are not sensitive to the transducer layer or the metal-substrate interface (see Fig. 2 for an example), thus significantly improving the uncertainty of the measured thermal conductivity.

Some other factors to consider 2.3.1. Effect of the metal transducer layer
If the transducer layer has a much higher thermal conductivity than the substrate, heat tends to spread out in the transducer layer first before diffusing into the substrate. In this case, From both plots, we see that the sensitivity of the ∇ signal to of the substrate starts to reduce dramatically when / > 10. Therefore, ideally, a metal transducer with a thermal conductivity < 10 is recommended to measure of the substrate to achieve high accuracy.
Note that this conclusion is reached on the condition of a 100 nm thick metal transducer layer, which is usually required to prevent transmission of the laser beam through the metal layer and thus guarantee a surface heat flux boundary condition. Since mainly ℎ governs the in-plane heat spreading in the transducer layer, the requirement on the highest possible thermal conductivity of the metal transducer layer can be less stringent if the metal transducer layer could be thinner, such as in the time-resolved magneto-optic Kerr effect (TR-MOKE) setups. 39,40 Fig. 4. Sensitivity coefficients of the phase gradient signal ∇ to different parameters of (a) a 100 nm metal/silica system at 200 Hz, and (b) a 100 nm metal/Si system at 20 kHz, when measured using a spot size of 10 µm by the improved spatial-scan thermoreflectance method for different metal-to-substrate thermal conductivity ratios / .

Requirement on the sample geometry
In the new SSTR experiments, the profile of the in-phase signal in is mainly determined by the laser spot size whereas the profile of the out-of-phase signal out is mainly determined by the in-plane thermal diffusion length , . We find that the out signal would drop to <1% of its peak value at an offset distance of ≥ 3 , . Therefore, the distance from the center of the pump spot to the sample edge should be at least > 3 , to minimize the sample edge effect. Combining the basic criterion of , ≥ 3 recommended for this SSTR experiments, we require that the diameter of the sample should be 2 > 6 , ≥ 18 . Therefore, if using a laser spot size of 0 < 5 μm for the measurement, this improved spatial-scan thermoreflectance method can easily measure small-scale samples with a lateral diameter as small as 0.1 mm.
This method allows for measurements of both bulk and thin-film samples due to the sophisticated 3D anisotropic thermal modeling. Sensitivity analysis shows that if the film thickness ℎ is 3 times larger than , , the sensitivity of the phase signal Δ to the film thickness ℎ or the thermal properties of the substrate beneath the film would be <0.01, i.e., a 10% change in ℎ would cause <0.1% change in the phase signal. In such a case, the film sample could be considered as thermally bulk with its thickness being unimportant; otherwise, the thickness of the sample needs to be pre-determined carefully as an input parameter in the thermal model. A summary of the sample geometry requirement for this new SSTR experiment is illustrated in Fig. 5.   Fig. 5. Sample geometry required for the improved spatial-scan thermoreflectance experiment.  are also included in this summary. All the data are also tabulated in Table I,

A simulated numerical experiment
A numerical experiment is conducted to illustrate the iterative fitting process and demonstrate the unique capability of this new SSTR method for extracting both the full inplane thermal conductivity tensor and the laser spot size, even if the laser spot shape is not ideally circular. Here we consider a hypothetical sample with a thermal conductivity tensor of process is also illustrated in Table 2. Table 2. An illustration of the step-by-step fitting process for the new spatial-scan thermoreflectance method to simultaneously extract the anisotropic in-plane thermal conductivity tensor and the elliptical laser spot sizes. The framed parameters are freely adjustable and the others are the fixed inputs. The subscript n represents the parameters from the nth iteration of the fitting.   In the above fitting process, can be arbitrarily set as any reasonable value as its effects on the Δ and signals are negligibly small. Note that we have aligned the x-axis of the coordinate system to be parallel to the principal axis of the elliptical laser spot, which reduces the parameter of the tilting angle of the elliptical laser spot and thus significantly simplifies the thermal model. Therefore, in real experiments when the laser spot shape is not ideally circular, the major and minor radii of the elliptical laser spot need to be characterized first, and then the coordinate system needs to be redefined with the x-axis aligned either along with or perpendicular to the principal axis of the elliptical laser spot to be compatible with the thermal model.

x-cut quartz
An x-cut quartz (<110>-oriented) sample, which is transversely anisotropic, is coated with a 100 nm Ti layer and measured by the system in Fig. 1(a) to validate the capability of this method in measuring anisotropic in-plane thermal conductivity tensor. The laser spot size is approximately 10 μm, and the modulation frequency is set as 1 kHz to meet the recommended criterion of , ≥ 3 .
During the measurement, the sample is randomly positioned and the laser spot shape could be a tilted ellipse. Therefore, on the coordinate system defined by the sample stage, signals of 12 different scanning directions of = 0°, 30°, 60°… 330° are first acquired and independently fitted with the assumptions of isotropic in-plane thermal conductivity and circular laser spot shape, with the preliminary results of ( ) and ( ) for each scanning direction plotted as the solid symbols in Fig. 9(a) and 9(b), respectively. Best-fitting these spot size data using a tilted ellipse function, the major and minor radii of the elliptical laser spot are determined as 11.6 μm and 9.8 μm, respectively, with a tilting angle of 0.23°, which is negligibly small and is thus ignored. Therefore, the coordinate system defined by the sample stage coincides with the one preferable for the thermal model. The next step is to iteratively fit the signals acquired for three different scanning directions of = 0°, 30°, and 90° using the same fitting process described in Section 3.2.1.
They are shown by the shadowed regions in Fig. 9(a) and 9(b), respectively.  Fig. 9(b). Overall, these two sets of measurements are consistent. However, much better quality is evident for the current measurements as they have smaller measurement uncertainty and more consistent results between different scanning directions as compared to the literature measurements by the BO-FDTR method 24 .

IV. CONCLUSIONS
In summary, a new spatial-scan thermoreflectance method has been developed to measure the in-plane thermal conductivity tensor of millimeter-scale small samples over a broad range of thermal conductivity values from 1-2000 W/(m • K), extending the current limit of the measurable by other thermoreflectance methods to as low as 1 W/(m • K). Details of this method including the experimental configuration, sensitivity analysis, and uncertainty estimation are fully described. This method has been validated by measuring several in-plane isotropic reference samples including fused silica, sapphire, silicon, and HOPG, with a typical uncertainty of ~5%. The capability of this method in measuring the in-plane thermal conductivity tensor of transversely anisotropic materials even using a slightly elliptical laser spot shape has been demonstrated through both a simulated numerical experiment and a real measurement of x-cut quartz, with the results agreeing perfectly well with the literature values.

ACKNOWLEDGMENTS:
P.J. acknowledges the financial support by the R.K. Mellon Postdoctoral Fellowship during his research in Pitt and the HUST startup grant #3004120159.

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding author upon reasonable request.