Spatially resolved thermoreflectance techniques for thermal conductivity measurements from the nanoscale to the mesoscale

Thermoreflectance techniques, namely timeand frequency-domain thermoreflectance (TDTR and FDTR, respectively), are ubiquitously used for the thermophysical characterization of thin films and bulk materials. In this perspective, we discuss several recent advancements in thermoreflectance techniques to measure the thermal conductivity of solids, with emphasis on the governing length scales and future directions in expanding these advances to different length scales and material structures. Specifically, the lateral resolution of these techniques, typically on the order of several micrometers, allows for an understanding of the spatially varying properties for various materials. Similarly, limitations of TDTR and FDTR with respect to their volumetric probing regions are discussed. With a recently developed steady-state thermoreflectance (SSTR) technique, these limitations are overcome as probing volumes approach spot sizes. Finally, recent pushes towards the implementation of these techniques without the use of a thin metal transducer are presented, with guidelines for future avenues in the implementation under these specimen configurations.

viding insight into the vibrational mean free path spectra of solids (i.e., relating to the "thermal conductivity accumulation function") [47][48][49][50][51][52][53][54][55] and the spectral coupling of phonons across interfaces (i.e., relating to the "thermal boundary conductance accumulation function"). 10 Thermoreflectance techniques, by definition, rely on the principle of thermoreflectance, 1,56-58 measuring a material's change in reflectivity due to the change in its temperature. In a typical modulated pump-probe thermoreflectance experiment such as TDTR or FDTR, a pump beam is used to thermally excite the surface of the sample at some frequency, f . The change in intensity of a reflected probe beam is related to the temperature change on the surface as function of either f or the delay time between the pump and probe pulses in the case of short pulsed-based experiments. Where TDTR utilizes short, typically sub-picosecond, pulses to monitor the thermoreflectance decay as a function of delay time after pump pulse heating as well as the phase shift induced from the modulated temperature change at f , FDTR can utilize a variety of pulsed or continuous wave (cw) lasers to monitor the phase shift in thermoreflectance signals solely as a function of f . When f becomes low enough, the material of interest will reach steady-state conditions during on periods of the modulation event. In this regime, a third technique has recently emerged. "Steady-State Thermoreflectance" (SSTR) operates like FDTR only in the low frequency limit, 59 monitoring the thermoreflectance of the surface at increasing pump powers and inducing a Fourier-like response in the material. Ulitmately, SSTR offers an alternative method to measure the thermal conductivity of materials via optical pump-probe metrologies. The characteristic pump excitations and responses for each of these techniques is presented in Fig. 1. We review the recent advances in SSTR in Section IV.
In addition to their non-contact nature, these optical metrologies are advantageous relative to many other thermometry platforms in the relatively small volume and near-surface region in which they measure. By using proper laser wavelengths to ensure nanoscale optical penetration depths, the thermal penetration depth (i.e., the depth beneath the surface in which these techniques measure the thermal properties), δ thermal , can be limited to the focused spot size, or much less, depending on the modulation frequency. Further, given the pump and probe spot sizes can be readily focused to length scales on the order of micrometers, thermoreflectance techniques allow for spatially-resolved surface measurements of thermal properties with micrometer-resolution, and the ability to create thermal property areal "maps" or "images". We review the pertinent length scales of TDTR, FDTR, and SSTR in Section II, followed by the advances towards areal thermal property "mapping" in Section III.
The change in reflectivity of a given material is related to both the change in temperature of the material (i.e., the thermoreflectance, which is ultimately of interest for the measurements of temperature changes and thermal properties) and the change in the number density of the carriers excited by the optical perturbation. Thus, the total photorelfectance signal change that is measured in TDTR, FDTR, or SSTR can be expressed as the sum of these two components: 60 where ∂R/∂T is the temperature reflectance coefficient and ∂R/∂N is the free-carrier reflectance coefficient, and ∆T and ∆N are the changes in temperature and free carriers from the pump excitation, respectively. In the majority of thermoreflectance measurements reported in the literature, samples interest are coated with a thin metal film transducer, in which the change in reflectivity is directly related to the change in temperature. This comes with the advantage that the optical penetration depth in metals is confined to < 20 nm over a wide range of optical wavelengths, resulting in a "near surface" heating event. An additional advantage of using a metal film transducer is that the ∂R ∂N ∆N term in Eq. 1 is much smaller than ∂R ∂T ∆T , except for when pump excitations induce interband transitions and their contributions become comparable. Even so, the contribution to ∆R from ∂R ∂N ∆N lasts only for ∼1 ps for most metals. 61,62 This is unlike non-metals where not only temperature, but also conduction band carrier population can change the reflectivity significantly for lifetimes much longer than a picosecond. [63][64][65] However, if the optical and thermal penetration depths of the pump and probe beams are properly accounted for, and the thermoreflectance signal in the measured photoreflectance response can be isolated from the free-carrier response (i.e., if the first term on the right hand side in Eq. 1 can be separated from the second term on the right hand side), then TDTR, FDTR, and SSTR can be conducted without the use of a metal film transducer, giving rise to increased measurements sensitivities to different thermal properties of materials that are not necessarily accessible in the traditional metaltransducer configuration. We review recent advances in TDTR and FDTR without a metal film transducer, and provide guidelines for the implementation in SSTR in Section V.
The remainder of this perspective is structured as follows. Section II will discuss the pertinent length scales associated with thermoreflectance techniques, with emphasis on the thermal penetration depth, and therefore the probing volume, in these experiments. Section III will discuss the lateral resolution associated with these techniques, abstracting them to a spatial understanding of variations in thermoreflectance and therefore thermal properties. Section IV will review the newly-developed SSTR technique, offering future avenues of exploration. Finally, Sec. V extrapolates the use of these techniques and their implementation without the use of a thin metal transducer. Section VI will provide a final perspective and offer guidelines for each of these implementations in the future.

II. THERMOREFLECTANCE LENGTH SCALES
The length scales associated with thermoreflectance techniques make them applicable to a wide range of material systems and geometries. Because pump and probe beams are typically collinearly aligned through objective lenses with high numerical apertures, lateral resolution in these measurements can be on the order of micrometers. The through-plane resolution is often described by the thermal penetration depth: In this equation, D is the thermal diffusivity and ω is the angular frequency at which the pump is modulated. The thermal penetration depth is mathematically defined as the depth at which the temperature decays to 1/e of its value on the heated surface as a result of a modulated heating event, and is typically considered the probing depth under the surface in modulated thermoreflectance experiments. Various works have relied upon Eq. 2 in their analysis of mean-free paths in TDTR and FDTR studies. 49,66,67 However, this expression is derived for the high frequency, one-dimensional heating of a bulk material (i.e., in the limit where the spot size is much greater than the thermal penetration depth), and fails to account for interfacial resistances that pose themselves so as to influence δ thermal or when the geometric length scales of the heating event become limiting. 68 As an example, we present the associated thermal penetration depths for a-SiO 2 and Si coated with and without an Al transducer in Fig. 2, calculated from the numerical solution to the cylindrical heat equation from Braun et al. 68 In the high frequency limit, ≥ 1 MHz, good agreement is found between Eq. 2 and the numerical results for the bulk uncoated a-SiO 2 and Si, validating δ thermal in the high frequency limit. Nonetheless, there are inherent limitations in assuming Eq. 2 for all frequencies of modulation, especially when considering samples coated with metal film transducers. As f → 0, δ thermal calculated via Eq. 2 approaches infinity, suggesting that depths much larger than the pump/probe radii can be probed. Not only is this unphysical, it is also thwarted by the complex multilayer systems typically analyzed using TDTR and FDTR. In the low frequency limit, the thermal penetration depth is limited by the 1/e 2 pump/probe radii, exemplified in Fig. 2(a) for a-SiO 2 and Si without a metal transducer.
The inclusion of a thin metal transducer also has significant implications on δ thermal . This stems from both the in-plane heat spreading in the metal film as well as the associated resistance at the metal/substrate interface. On a thermally insulating substrate, for example, the temperature gradient will be preferentially formed laterally in the metal film before entering into the substrate and decaying as normal. This is shown in Fig. 2(b) for a-SiO 2 coated with 100 nm Al. Compared to the direct heating of a-SiO 2 , the 1/e thermal penetration depth is increased as a result of heat spreading in the Al and thus an increase in the effective heater radius. The interfacial resistance at the metal/substrate interfaces begins to play a significant role when the resistance of the metal/substrate interface is comparable to the resistance of the volume probed in the substrate, i.e., R thermal ∼ δ thermal /κ, where R thermal is the resistance of the metal/substrate interface and κ is the thermal conductivity of the substrate. If we assume that R thermal = .1 m 2 K GW −1 for Al/a-SiO 2 and Al/Si, for example, the associated penetration depths required for the resistances of the two systems to be equivalent are ∼15 and ∼1500 nm for a-SiO 2 and Si, respectively. This is most impactful for Si, shown in Fig. 2(b). In the low frequency limit, a reduction in the 1/e penetration depth occurs as a result of this criteria, as the associated temperature drop at the interface is on the order of or greater than that in the Si. In the high frequency limit, the deviation between Eq. 2 and the calculated data begins at ∼1 MHz, associated with a penetration depth of ∼1 µm, for similar reasons. One would expect similar deviations at modulation frequencies approaching the GHz range for a-SiO 2 . The above discussions regarding the associated probing depths at the low frequency limit have provided a fundamental foundation for SSTR, which inherently relies on the steady-state temperature rise of a material system as a result of the low frequencies of heating, where the penetration depth is on the order of the spot size.
As a result, a new regime previously untapped regarding measurement length scales has been uncovered. Summarized in Fig. 3 are conventional optical metrologies used for the extraction of thermophysical properties, and their associated characteristic length and time scales. In time-and frequency-domain thermoreflectance techniques, the high frequencies of modulation inherently limits the probing volume to on the order of 100's of nm to 1 µm for most materials. On the other hand, sample size restrictions via the laser flash technique are typically limited to ≥ 100 µm. 69 SSTR offers a novel regime of measurement length scales, as spot sizes can be tuned between several micrometers and millimeters, power permitting. SSTR is further discussed in Sec. IV.

III. SPATIALLY MAPPING THERMAL CONDUCTIVITY WITH MICROMETER-SCALE RESOLUTION
The ability to locally probe regions of interest with these thermoreflectance techniques makes high spatial resolution mapping a logical extension. With lateral resolution on the order of the spot size, the techniques naturally lend themselves to observe local variations in thermoreflectance, and therefore local variations in thermal properties. In the following two subsections, we outline the implementation of thermoreflectance mapping via TDTR and FDTR, with key differences between the two techniques elucidated. The applicability of thermoreflectance mapping has been applied to a variety of material systems. Here, we highlight a multitude of relevant scenarios in which micrometer-scale thermal property mapping shows good applicability for understanding the underlying thermal physics.

III.A. Introduction & Implementation
Spatial monitoring of the thermoreflectance of material systems was first implemented in the early 1980s. 70,71 An example of a spatially resolved thermoreflectance line scan is shown in Fig. 4 for a patterned 75 As + implanted Si wafer. The small spot sizes of the pump/probe laser beams (∼1 µm) allow for monitoring the pattern with high spatial resolution and throughput. Similar techniques monitoring the associated thermoreflectance were presented in a variety of following works. [72][73][74][75][76][77][78][79][80] Quantification of the associated thermal properties of given material systems, however, has only recently been developed due to a broader understanding of the associated analysis with thermoreflectance techniques, 5,81,82 including the inclusion of a metal transducer to enhance sensitivity to underlying material properties and incite a near-surface heating event due to the shallow optical skin depth. 83

III.A.1. Time-Domain Thermoreflectance
In TDTR, thermal properties are extracted from the the radially symmetric heat diffusion during a pulsed heating event. 5 While readily implemented, performing full TDTR scans (time delays from 0 to ∼5 ns, which corresponds to 2-3 minutes per scan) across the surface of an entire sample to create these spatially resolved thermal property maps can thus be time consuming. To overcome this limitation, two experimental variables can be manipulated and judiciously chosen to enhance sensitivity to the parameter of interest, 1) the modulation f and 2) the pump-probe delay time. In doing so, highthroughput raster scans over a given region of the sample can be performed. Huxtable et al. 84 first presented this thermal conductivity mapping procedure for TDTR in an effort to better describe a combinatorial Nb-Ti-Cr-Si system. Full TDTR scans from their work are shown in Fig. 5(a) for regions composed primarily of Cr, Nb, and Ti. At intermediate delay times, 50 ≤ t ≤ 300 ps, the ratio -V in /V out approximately scales with the thermal effusivity, √ κC. For material systems in which there is large variation in the effusivity, these intermediate pumpprobe delay times are ideal, and the thermal conductivity of a particular region can then be extracted with knowledge of the its volumetric heat capacity. Additionally, at these delay times, sensitivity to the thermal boundary conductance at the metal transducer/substrate interface is relatively low. This is best exemplified by the thermal conductivity correlation curve, shown in Fig. 5(b) for the combinatorial Nb-Ti-Cr-Si system in Ref. 84. The upper and lower bounds of G suggest that small variations in G do not significantly impact the thermoreflectance signal, and therefore the extracted thermal conductivity. The final thermal conductivity micrograph of the system can be seen in Fig. 5(c).
An alternative approach to this intermediate pumpprobe delay time is appropriate when the material system of interest does not show large variations in volumetric heat capacity; this can be specific to single-phase systems or systems of varying crystalline orientation. In this scenario, knowledge of the interfacial conductance allows for the minimization of its sensitivity such that variations in G result in minute alterations in the extracted thermal conductivity. The general process includes the extraction of G and its uncertainty from a variety of full TDTR measurements. Following, a sensitivity analysis is performed, and the delay time at which the sensitivity to G is minimized is chosen as the delay time for the mapping procedure. This analysis becomes difficult when the thermal conductivity of the underlying material is low, and the sensitivity to G does not go to zero for the given delay times. This can be overcome by reverting to the previous option, whereby the delay time is fixed to short, intermediate delay times, or by implementing an FDTR-like approach to extract G and κ simultaneously via a multi-modulation frequency approach, as discussed below. Multiple maps can also be performed with alterations in modulation frequency and delay time to extract additional parameters.

III.A.2. Frequency-Domain Thermoreflectance
In FDTR, the radially symmetric heat diffusion equation is compared to a pulsed-pulsed or cw-cw frequency sweep to extract parameters of interest. If the lock-in amplifier is permitting, a multi-modulation frequency approach can be used to extract multiple parameters simultaneously. This was first exemplified in Yang et al.'s work 85 using an FDTR configuration, whereby the pump was modulated at 6 independent frequencies to extract both the G and κ of a pattered Au/Ti interface, summarized in Fig. 6. Figures 6(a) and 6(b) show the FDTR data and best fit and sensitivity to G and κ, respectively, while Fig. 6(c) shows the phase micrographs at the 6 frequencies. Little contrast can be seen in the phase micrographs at 37.9, 431.7, and 901.3 kHz, consistent with the low sensitivity to G at these modulation frequencies.
Contrarily, at 5.6, 9.73, and 16.7 MHz, the large sensitivities to G result in large contrasts in the associated phase micrographs. Thus, at 6 independent frequencies, the sensitivity to G and κ is fully captured, and allows for a robust characterization of the material system. Using a similar mapping procedure, Hatori et al. 86 have shown that a single modulation frequency can be used to extract the thermal effusivity in a fiber composite material, suggesting that simultaneous modulation of the pump is not necessary.

III.A.3. Implementation
The implementation of a thermoreflectance mapping technique, be it TDTR-or FDTR-based, requires interfacing between components of the imaging system, most importantly the motorized stage for sample positioning and lock-in amplifier, to allow for a high throughput imaging technique. Because alterations in the alignment of either of the aforementioned systems is not ideal, samples are typically rastered with respect to the pump and probe beams rather than relying on galvomirrors that would alter the detection aspect of these methods. The method of rastering is dependent on the user and the available components. This can stem from a stepperlike action or a method that scans continuously. Both require careful choices of the lock-in amplifier acquisition parameters, with knowledge of the scan velocity in the latter case. We emphasize that the spot size must be continuously in focus to extract an appropriate thermore- flectance signal in both of these cases. Focal changes on the order of the spot radius have a quantifiable impact on the effective pump/probe size, and therefore the extracted parameter of interest. Since spot sizes approaching ∼1 µm become the most sensitive parameter in these thermoreflectance techniques, 6,87 defocusing of the pump and probe spots can and should be remediated. Here are possible mitigation strategies to avoid this source of uncertainty, if present: 1. If the size of the spot is large, or if the spot does not change focus over the duration of the map, then a focus at the beginning of the mapping procedure should suffice. The user should ensure, however, that no focal changes are taking place with a reference sample or by some other means.
2. An autofocus procedure can be performed at each pixel, usually by maximizing the magnitude of the thermoreflectance signal. While simple, the procedure is time consuming, as the autofocus procedure may take on the order of several seconds to complete. This adds unwanted waiting time in the mapping procedure, and greatly increases the overall acquisition time.
3. An autofocusing procedure can be performed at the beginning of the image in the center of the map. This ensures that the majority of the map, depending on its dimensions, is in focus. Depending on the orientation of the sample and the motorized stage, this usually results in a defocused spot size towards the edges of the map, and therefore erroneous thermoreflectance signals in these regions that do not adequately describe the specimen of interest.
4. An autofocusing procedure that is performed at the corners of the image prior to the acquisition. This allows for the user to extract the "plane" of focus, allowing for adjustments to be made during the mapping procedure. This ensures that the beam is in focus for the entirety of the map, and is far more ideal as compared to options (2) and (3).
Care should be taken to monitor any potential changes in the environment and electronics when a mapping procedure is being performed. This includes, but is not limited to, changes in local temperature gradients which can alter the focus of the sample, drift of power for the pump and probe laser, changes in the pump phase, and diffusely scattered light caused by surface inhomogeneities at the sample surface. The last of these can be overcome by monitoring the DC reflected probe signal, as diffuse scattering will result in a reduced DC signal, and highlights regions in which the data is not to be analyzed.
In the next sub-section, the applicability of thermoreflectance mapping procedures is presented on a variety of applicable systems.

III.B. Applicability to Material Systems
Thermal property microscopy has applications in a variety of material systems. It is particularly applicable in systems in which changes in composition, layering, homogeneity, etc. are present. Here we highlight a number of works that utilize a mapping procedure to extract a variety of properties of interest. First, the mapping procedure is employed in combinatorial material systems, whereby multiple phases of materials are present to extract their thermal properties of interest. Further, the application towards thermal barrier coating systems consisting of various coatings is shown. In more recent works, the procedure has been pushed towards the characterization of high thermal conductivity and 2D materials to provide a better understanding of the underlying phonon physics.

III.B.1. Combinatorial Materials
Combinatorial materials present a unique opportunity to extract thermal properties of novel materials and phases, and the variation of these thermal properties around heterogeneous material interfaces. With robust deposition and characterization techniques, the fabrication of these material systems has become increasingly rapid and reliable. Here, we highlight works that examine the combinatorial thermal nature of a Nb-Ti-Cr-Si (Ref. 84) and Cr-Pd-Pt-Rh-Ru (Refs. 88 and 89) diffusion multiples via a TDTR mapping approach.
A summary of the results for the Nb-Ti-Cr-Si diffusion multiple are shown in Fig. 5. The delay time was set to 92 ps to extract the normalized effusivity of the material system. Specifically, the extraction of the Laves phase of TiCr 2 of varying Cr stoichiometry was extracted. For a Ti content increase from 33.5 at. % to 36.1 at. %, there is a corresponding decrease in the thermal conductivity from 42 to 20 W m −1 K −1 . Additionally, the b.c.c. phase of a Ti alloy (85.5 at. %) with Cr (14.5 at. %) is extracted, yielding a thermal conductivity of 7.5 W m −1 K −1 . The thermal conductivity mapping procedure creates a visual representation of the thermal conductivity variations in this Ti/Cr system, and is much higher throughput than standard TDTR. Especially for the Laves phases of TiCr 2 , nearly 20 points comprise the region from 33.5 at. % Ti to 36.1 at. %. With an acquisition time of ∼3 s per pixel, this takes approximately 60 seconds. Contrarily, performing full TDTR scans in each of the 20 regions would have resulted in 40 minutes of acquisition time, assuming 2 minutes for a given scan. An additional avenue of interest is presented at the triple junction of Nb-Si-Ti in this same work. 84 Specifically, grains of the (Ti,Nb) 3 Si phase are present with differing thermal conductivities. The differences in observed thermal conductivity are attributed to differing orientations, suggesting the ability to extract the thermal conductivity tensor of a material system via thermal conductivity mapping.
Using combinatorial material systems has the advantage of creating multiple phases on a single sample. This allows for the robust characterization of various alloys that would otherwise be fabricated on numerous separate samples, adding the potential for additional complexities, uncertainties, and time for complete characterization. Nickel-based super alloys are typically used in high temperature applications in a wide range of compositions. In fabricating Cr-Pd-Pt-Rh-Ru diffusion multiples, Zheng et al. 89 were able to examine the electronic contribution to thermal conductivity in a wide range of Ni alloys, where ∼50 Ni(Cu,Fe) alloys had been measured on a single wafer. In fabricating the diffusion multiple, 125 data points over a 250 µm length were taken with relative ease. At 3 points per second, 89 this measurement procedure takes about 6 minutes. We reiterate the fact that using conventional TDTR performed at each of these data points would have taken an order of magnitude longer in acquisition time. Ultimately, the fabrication of this diffusion multiple allowed for the robust and highthroughput validation of the Wiedemann-Franz law in Ni alloys over a wide range of compositions.

III.B.2. Coatings
Coatings are an integral component in a variety of research applications, whereby the coating enables some level of environmental or thermal protection, or some additional functionality, enhancing the performance of the underlying media. For example, thermal barrier coatings in aerospace applications preferably have a low thermal conductivity to protect the underlying turbine blade, allowing for an increase in gas engine operating temperatures and efficiency. 92 On the other hand, in nuclear power generation contingent on UO 2 kernals, the efficiency is dictated by heat transfer into the coolant from the fuel particle, 93,94 and thus a high thermal conductivity is preferable so as to not induce large temperature gradients.
Work in the literature is sparse regarding the thermal conductivity mapping of thermal barrier coatings. However, considering the possibly multi-phase, anisotropic materials that constitute these coatings from various application processes, more research in this field is deserv- ing. To date, few works have rigorously examined the spatial profile of thermal conductivity in barrier coating components. Figure 7 shows an example of a yttriastabilized zirconia (YSZ) thermal barrier coating system on a two-phase Nb silicide composite substrate bonded by a multiphase bond coat. 95 As is evident, the thermal conductivity of the YSZ is low as compared to the Nb silicide composite, which is preferable in this system. While contrasts in the thermal conductivity of this specific YSZ barrier coating are not large, the procedure presents itself to a wide variety of applications in the barrier coating community. This can include structural influences on the thermal conductivity of the barrier coating itself, or understanding the effect of cycling on other components, such as the thermally grown oxide that is necessary for the fabrication of these systems. Similar implementations of such a mapping procedure have also been performed on pyrolitic carbon and silicon carbide coatings for UO 2 . 96

III.B.3. 2D Materials
Two-dimensional and quasi two-dimensional materials present themselves with a wide range of applications including supercapacitors and batteries, 97 THz emitters/detectors, 98 and field effect transistors, 99 to name a few. Beyond their intrinsic properties, this is in large part due to the vast capabilities of introduc-ing dopants so as to exude properties of interest. [99][100][101][102] Because of their two-dimensional nature, measurements of the thermal properties become increasingly difficult due to the time constant associated with heat transport in these materials. 103 Nonetheless, various works have examined thermal transport in and through relevant material systems with success, both with thermoreflectance techniques and other techniques. [104][105][106][107] In thermoreflectance techniques, this has been primarily limited to extracting conductances at interfaces with these materials. This is due to the general lack of sensitivity to cross-and in-plane transport in 2D materials via these techniques. The spatial implementation of these techniques is less common throughout the literature, but still provides important insights in various material systems.
Yang et al. 40 presented a work focusing on the examination of in-plane conductances in few-layer graphene, and implements a spatial mapping procedure with FDTR to extract a distribution of the basal-plane thermal conductivity of few-layer graphene. This was achieved by implementing a multi-modulation frequency approach to capture all potential sensitivity to both the cross-and inplane conductance in these layers. At their spot sizes (∼1 µm effective pump/probe radii), in-plane heat spreading due to the Au has a more pronounced effect, and thus its thermal conductivity must be rigorously determined. An example of the final result is presented in Fig. 8, where the in-plane conductance of a graphene flake of several layers is present alongside the basal-plane thermal con- ductivities of 1-and 2-layer graphene. The large uncertainties associated with the extracted values can be attributed with the minute changes in phase resulting from the contribution to in-plane heat spreading from the graphene layer as compared to the Au transducer. The cross-plane thermal boundary conductance, on the other hand, can be extracted more easily.
Several other works have examined the conductance at 2D material interfaces and contacts via a thermoreflectance mapping procedure. 108,109 However, at a single pump-probe delay time and modulation frequency, it is difficult to adequately extract parameters of interest in these systems, namely the conductance at these contacts. The majority of works report on 2D materials on a thermally grown oxide. Thus, the sensitivity to the conductance is greatly subdued due to the low thermal conductivity of the thermal SiO 2 . This has shown to impact the results of Brown et al. 109 on MoSe 2 , whereby there is a disparity between the mapped conductance and the conductance extracted via a standard TDTR measurement. While the trends were comparable, the absolute values differed quite significantly, and can be attributed to overall lack of sensitivity to the conductance for a single delay time and single modulation frequency. Ideally, this can be remediated by a multi-modulation frequency approach and fitting to a model in the frequency domain.

III.B.4. Thermal Management
The management of heat from a dynamic and static perspective is a consideration that must be taken into account for a variety of applications. From a static perspective, heat sinks typically have a large thermal conductivity to better manage large temperature rises in devices. 110 Thus, materials such as diamond, 111 SiC, 112 BAs, [113][114][115] to name a few, are all at the forefront of thermal management from a static point of view. Dynamic thermal management additionally has a variety of applications. These range from thermally driven logic, 116 thermoelectric efficiency enhancement, 117-119 and improved heat sinking for device cooling, 120,121 but the implementation is much more difficult than dynamic electrical management. Spatially resolved thermoreflectance techniques have good applicability in understanding materials that are used in thermal management from both a static and dynamic perspective.
An important consideration in the fabrication of devices from a thermal perspective is the inclusion of grain boundaries, which have the potential to significantly impact the thermal conductivity of a material system. 122,123 When the phonon mean free paths in a material are on the order of, or greater than, the spot size, these length scales can now be probed via thermoreflectance mapping techniques, which will provide a more robust understanding of the physics surrounding grain boundary scattering. This was exemplified in polycrystalline diamond, whereby a reduced thermal conductivity was observed within 10 µm of an individual grain boundary. 90 A better visualization of this is presented in Fig. 9, which shows electron-backscatter diffraction and thermal conductivity maps of a diamond polycrystal. Also included in the figure is the analysis showing a reduction in thermal conductivity as the measurement approaches the grain boundary.
Thermoreflectance imaging has also been implemented in the lithiation and delithation processes of single crystal 2D layered MoS 2 . 91 In this processes, the electrochemical intercalation of Li ions is induced and reversed by the application of a voltage. The distribution of Li ions for a given MoS 2 crystal is not uniform, 124,125 and thus naturally lends itself as an opportunity for thermoreflectance mapping. Shown in Fig. 10 are micrographs during lithiation and delithiation processes of a ∼10 nm MoS 2 flake via TDTR. Evidently, the thermal conductance is varied drastically as a result of the lithiation/delithiation process, with an order of magnitude change in conductance. Further, while traditional TDTR can successfully track the intercalation process, spatially understanding the inhomogeneity of the process allows for an understanding of hot spot generation in devices contingent on the intercalation process.

III.C. Summary and Future Directions for Spatially-Resolved Thermal Conductivity Measurements with Thermoreflectance
In this section, we have presented the basis of a thermoreflectance mapping procedure in TDTR and FDTR configurations. The ability to monitor the thermoreflectance spatially allows for high-throughput acquisition capabilities of the thermal properties of interest. Specimens in which there are large surface inhomogeneities due to compositional variations, localized grain orientations, or when extrinsic influences impact the observed thermal conductivity are ideal for these kinds of mapping procedures. We provided several examples, including diffusion-multiples, barrier coating materials, and 2D materials. Thermoreflectance mapping with micrometer scale areal resolution has large implications for the highthroughput extraction of the thermal properties of these materials, and can be readily implemented with the addition of standardized stages.
Despite these significant advances in using thermoreflectance techniques to spatially resolve variations in thermal conductivity, there are several future avenues that can be taken. We have already discussed the significance of the lateral resolution associated with TDTR and FDTR, and its importance for this abstraction of the technique. The lateral resolution, however, is inherently limited by diffraction using conventional lenses and objectives. Oil immersion objectives with numerical apertures greater than unity, as well as artificially engineered metamaterials 126 offer unique, untapped avenues in achieving resolution beyond the diffraction limit. Indeed, a push towards shorter wavelengths offers an increase in resolution, but the associated thermoreflectance coefficients at these wavelengths must also be taken into consideration. 127 Beyond the associated lateral resolution in thermoreflectance techniques, the "depth profiling" applicability leaves much to be studied. We discussed extensively the characteristic length scales in Sec. II, and how the fundamental probing volume associated with TDTR and FDTR techniques cannot extend beyond the spot size in the cross-plane direction. Even so, the high modulation frequencies associated with these techniques limit these volumes to even nearer to the surface, limiting the available information to be garnered. SSTR offers promising capabilities, as probing volumes are typically on the order of the spot size. For polycrystalline systems, the analysis becomes increasingly difficult as a result of the multitude of orientations probed, but will, however, provide a better fundamental understanding of variations of thermal conductivity in the through-plane direction. Combined with the excellent lateral resolution associated with the technique, the implementation will work on the basis of deriving a three-dimensional description of the thermal conductivity for a provided region.
In the next section, we expand on the capabilities of TDTR and FDTR, departing from the transient and modulated heating regime and entering into a steadystate regime in SSTR. 59 SSTR has a multitude of advantages as compared to the aforementioned techniques, as discussed below.

IV. STEADY-STATE THERMOREFLECTANCE
A recent experimental extension of these thermoreflectance-based metrologies is the SSTR technique. 59 In both TDTR and FDTR, an amplitudemodulated heat source induces a both steady-state and modulated temperature rise in a material. 128 The premise of TDTR and FDTR is to capture the modulated temperature rise, so that the steady-state component is ignored. 5 In contrast, the premise of SSTR is to capture this steady-state temperature rise.

IV.A. Principles of Operation
The principles of SSTR are discussed extensively in Braun et al.'s work, 59 and thus only the basics will be presented here. The configuration of SSTR is akin to that of FDTR, where an amplitude-modulated pump heats the specimen of interest, while the probe monitors the thermoreflectance. A key difference between the two, however, is the frequency of modulation of the pump. In FDTR, these modulation frequencies are typically on the order of kHz to MHz, where the phase lag between the pump and probe signals is compared to the heat diffusion equation to extract properties of interest. Example FDTR data are shown in Fig. 11, where the probe phase is presented for fused silica, sapphire, boron arsenide, and diamond. 113 Indeed, a significant phase lag is present at these modulation frequencies for materials of low thermal conductivity, i.e., in fused silica and sapphire. For BAs and diamond, on the other hand, the pump and probe remain in-phase with one another until ∼1 MHz. This occurs as a result of the highly conductive nature of these materials, as a steady-state regime is maintained until such high modulation frequencies are reached. Contrarily, for materials less conductive by nature, the phase lag as measured by the probe is indicative of a lack of steady-state regime. Thus, the premise of SSTR is to perform measurements at lower modulation frequencies (< 1 kHz) to capitalize on the steady-state temperature rise in these material systems. Several advantages arise as a result of performing these thermal conductivity measurements at lower modulation frequencies in the steady-state regime as compared to conventional TDTR and FDTR techniques. Typically, the penetration depth, and therefore the probing volume, of the latter two is on the order of several hundred nanometers due to the transient nature of the heating event. 68 This primarily enhances sensitivity to the crossplane properties of the material system. Contrarily, at modulation frequencies typically used in an SSTR configuration, the penetration depth approaches that of the spot size (c.f., Fig. 2), and is fundamentally sensitive to all components of the thermal conductivity tensor for a given material system. Additionally, the use of thin transducers generally enhances sensitivity to properties of the underlying media in SSTR, where sensitivity to the tranducer thickness, thermal conductivity, and interfacial resistance at the transducer/substrate interface are less as compared to TDTR and FDTR. Finally, transient techniques are fundamentally sensitive to the ther-mal effusivity of bulk materials at conventional modulation frequencies. The steady-state temperature profile, on the other hand, is solely dependent on the thermal conductivity, and can be determined independently from the volumetric heat capacity.

IV.B. Implementation
The implementation of SSTR is nearly identical to that of FDTR, and will therefore not be discussed in depth. A pump of sufficient power should be utilized so as to induce a steady-state temperature rise at the sample surfacefor high thermal conductivity materials, this may be as much as 1 W depending on the spot size. Modulation of the pump can be achieved through a variety of methods, either internally to the laser source or externally via an electro-optic or acousto-optic modulator, or mechanical chopper. The mechanical chopper is preferable due to the high damage thresholds, as generally larger powers are used as compared to TDTR and FDTR. Prior to being combined with the probe, a portion of the pump is directed into a photodetector to extract the magnitude of the pump modulation, ∆P . The rest of the pump power is reflected by a cold mirror, and directed to the sample surface. The probe is split into two paths for the implementation of a balanced photodetector scheme, which greatly enhances signal-to-noise at the pump modulation frequencies. After passing through the cold mirror and focused through the objective lens along with the pump, the reflected probe signal is directed to a balanced photodetector with a path-matched reference beam, minimizing common noise in the probe.
There are two approaches to performing an SSTR mea-surement: (1) by using a lock-in amplifier (LIA) to monitor changes in reflectance as the pump power is systematically increased, or (2) by using a periodic waveform analyzer (PWA) to extract both the pump and probe waveforms to extract the thermal conductivity. The extracted signals can be related by the equation where ∆V is the magnitude of the probe signal, V is the probe DC signal, ∆P is the magnitude of the pump signal, and ∆T (κ)/∆|Q| is calculated using the thermal model. γ is a proportionality constant that encompasses the thermoreflectance at the sample surface, differences in objective magnifications and losses, as well as any differences associated with the electronics. Ultimately, it relates the associated changes in magnitude of the pump and probe to the temperature excursion for a given heat flux at the sample surface. Typically, γ is extracted via the use of a calibration sample, whose thermal conductivity is known from other metrologies. Knowledge of γ is paramount in SSTR, and thus to account for any variations in transducer film quality during deposition processes, a reference specimen should be placed near to the specimen of interest during the deposition of the transducer to confirm the proportionality constant.

IV.C. Measurements
Example SSTR data in the two configurations are shown in Fig. 12. Specifically, Fig. 12(a) shows change in probe photodetector voltage (∆V /V ) as a function of the pump photodetector voltage (∆P ) using the LIA approach. Linear trends in the data are observed for a wide range of thermal conductivities, from that of a-SiO 2 to that of diamond. The slope of the data are fit to the heat diffusion equation to extract thermal conductivity, again independently of volumetric heat capacity. Using the PWA approach is much more simplified, as just the pump and probe waveforms for a given pump power need to be acquired. Example pump waveforms for Al on a-SiO 2 and Si are shown in Fig. 12(b). Note that a larger pump magnitude is observed on Si as opposed to a-SiO 2 , due to the increased pump power necessary to induce a steady-state temperature rise of similar magnitude. This is evidenced by the probe waveforms, shown in Fig. 12(c). Due to the low thermal conductivity of a-SiO 2 , a steady-state regime is not achieved until near the end of the heating event, whereas steady-state is reached nearly instantly in the case of Si due to its highly conductive nature. With knowledge of the pump and probe magnitudes, Eq. 3 can be used to extract the thermal conductivity of the two substrates. Summarized in Fig.  12(d) are the thermal conductivities of materials measured in the study. SSTR is capable of measuring materials with a wide range of thermal conductivities, and has been demonstrated in the range of 1-2000 W m −1 K −1 , independent of the volumetric heat capacity. While currently SSTR has not been applied to systems with thermal conductivities less than 1 W m −1 K −1 , in principle, there is no limitation to extending SSTR to this regime of lower thermal conductivity materials.

IV.D. Summary and Future Directions for SSTR
In this section, we provide a brief overview of the steady-state based thermoreflectance metrology, SSTR. 59 While TDTR and FDTR are contingent on the modulated temperature response, the analysis in SSTR is greatly simplified. At low modulation frequencies, the steady-state thermoreflectance response is dominated by the thermal conductivity, independently of the volumetric heat capacity. Additionally, this ensures that the depth of penetration is on the order of the spot size, larger than that of TDTR and FDTR, but smaller than the permissible sample dimensions in bulk techinques, such as laser flash. 69 Thus, power permitting, SSTR allows for the attainment of length scales not captured by current optical techniques, both transient and steady-state, allowing for a broader spectrum of specimen lengths to be measured.
SSTR has shown to be an excellent method to characterize the thermal conductivity for a variety of material systems. Yet, the newness of the technique allows for the potential for a variety of upgrades. For one, the technique relies on a single low modulation frequency to extract the thermal conductivity. This is contingent on the characterization of the calibration coefficient, γ, to be able to relate heat flux to a temperature rise at the sample surface. Potentially, this can be remediated by modulating at higher frequencies in an FDTR-like approach, where sensitivity to the thermal effusivity of the material system is gained over simply the thermal conductivity. In this manner, γ and it's requirement can be remediated. Beyond the requirement of the calibration coefficient, the technique lends itself to a variety of implementations that cannot currently be reached by traditional implementations of TDTR or FDTR. As discussed earlier, the characteristic probing volume associated with SSTR is governed by the 1/e 2 pump/probe radii. Thus, so long as the spot size can be tuned from the typical several micrometers to several hundreds of micrometers in a controlled manner, the associated probing volumes can be tuned as well. As a result, systems in which interfaces are "buried" have a larger potential for being characterized. As an example, vertical diodes are generally fabricated with through-plane dimensions on the order of several micrometers to several tens of micrometers, a parameter which must be optimized for associated breakdown voltages. [129][130][131][132][133] By tuning the spot size, sensitivities to different properties of interest, namely the thermal conductivities of layers and the associated conductances between them, can be altered to extract these relevant parameters. Additionally, barrier coating systems for gas-turbine engines consist of a top-layer environmental barrier coating (EBC), typically on the order of 100 µm. 134,135 The thermomechanical stability of the turbine blade is contingent on the silicon bond coat applied between the turbine blade and the top EBC coat. SSTR offers the capability of characterizing this delicate interface, power permitting, making it a potential diagnostic tool for aerospace applications.
In the next section, the implementation of the aformentioned transient thermoreflectance techniques is shown in sample configurations that lack a typical transducer layer.

V. THERMOREFLECTANCE TECHNIQUES WITHOUT THE USE OF A THIN FILM OPTO-THERMAL TRANSDUCER
Thermoreflectance techniques have shown their vast range of capabilities in the measurement of thermal properties of material systems, including interfacial conductances, 136-138 thermal conductivities and their associated anisotropies, 139,140 and volumetric heat capacities. 19,141 While a non-destructive technique, typical implementations of these techniques require the deposition of a thin metal transducer to capitalize on the larger thermoreflectance signals associated with these materials. 127,142 Additionally, due to the small optical skin depth associated with metals over a range of wavelengths, 83 analysis using a surface heating event is invoked and greatly simplified compared to volumetric heating events. Furthermore, the thermoreflectance of these metal transducers is linearly proportional to temperature under small perturbations, whereas semiconductors have an additional component to their thermoreflectivity, excited carrier populations, that convolutes the thermoreflectance signal. However, the deposition of a thin metal film inherently modifies the specimen of interest, and introduces an additional interfacial thermal resistance that may not be ideal for realization of the underlying thermal properties.

V.A. Implementation
Initially, transducerless measurements via thermoreflectance techniques showed strong applicability in the ion implantation community. Smith et al. 71 first exemplified the application of an FDTR configuration at a fixed modulation frequency of 100 kHz for the characterization of annealed and non-annealed ion implanted silicon via photothermal reflectance. Specifically, they monitor this signal for boron implanted Si for doses ranging from 10 11 to 10 15 B + ions cm −2 at an accelerating voltage of 175 keV. They observe an increase in the measured thermal wave signal with increasing ion dose, suggestive of an overall decrease in the thermal conductivity of the under- lying Si, consistent with the more recent work by Scott et al. 143 Several other works applied the technique for the characterization of semiconductor wafers, [72][73][74][75][76][77][78]80 although no quantitative discussion regarding the thermal properties are provided.
The experimental extrapolation for transducerless thermoreflectance measurements is identical to that of conventional ones -only considerations in the analysis must be taken into account. However, the choice of the pump and probe wavelengths, if possible, should be chosen such that the heating event induced by the pump and optical penetration depth of the probe are near surface, which greatly simplifies the overall analysis. Yang et al. 145 have extensively modeled optical absorption in measurements for TDTR and FDTR, considering the pump as a volumetric heating source. Briefly, they solve the heat conduction equation in the frequency domain, assuming an exponentially-decaying heat generation term. Both the absorption depths of the pump, δ 0 , and probe, δ 1 , must be considered in this analysis. If δ 0 and δ 1 are not known for a given material system, then the effective optical penetration depth, δ eff = δ 0 + δ 1 , can be applied as a fitting parameter given additional known values in a given material system. We emphasize that direct-probing bulk specimens is primarily sensitive to the associated in-plane transport of the system, as opposed to it's cross-plane component with the addition of a metal transducer. 87 This occurs as a result of the generally larger temperature gradients associated with the inplane direction as opposed to the cross-plane direction for measurements performed at modulation frequencies in which the thermal penetration depth is much smaller than the spot size. Specific examples of the implementation of transducerless thermoreflectance measurements are presented below.

V.B. Measurements
To date, few measurements have been performed without a transducer via thermoreflectance techniques. Khaifzov et al. 144 exemplified the possibility of performing transducerless measurements on proton irradiated silicon using TDTR. Specifically, they employ a probe offset approach to monitor the phase lag as a function of pump/probe separation, 146-148 similar to approaches for measuring the anisotropic thermal conductivity of materials via a beam-offset approach. 139,149 Figure 13 shows representative data and best fits for polished and protonirradiated Si at 125 K. 144 The difference in phase lag between the two specimens is suggestive of a change in the thermal conductivity as a result of the irradiation process. Indeed, they recover values of 210 ± 24 W m −1 K −1 and 137 ± 4 W m −1 K −1 for polished and protonirradiated specimens, respectively, at 125 K. While departing from the conventional TDTR approach (concentrically aligned pump/probe beams), the analysis offered by Khafizov et al. exemplifies the ability to employ advanced TDTR techniques that account for optical absorption so as to extract thermal parameters of interest in the absence of a metal transducer.
Further, Yang et al. 145 performed FDTR measurements applying the transducerless formulation to 490 nm amorphous silicon (a-Si) on fused silica and silicon systems, employing cw pump and probe lasers at 785 and 532 nm, respectively. This implementation ensures that δ 0 > δ 1 . Both the thermal conductivity of the a-Si top layer and δ eff are chosen as fitting parameters. Figure  14(a) shows the data and best fit for the a-Si/SiO 2 system when the volumetric heating term is not considered. Evidently, the model fails to capture the measured data when fitting for the thermal conductivity of the a-Si top layer. With the implementation of the depth-dependent heating term, however, the model is in excellent agreement with the data, as shown in Fig. 14(b). It is important to note, however, that both the thermal conductivity of the a-Si layer and δ eff are chosen as fitting parameters, where 1.3 ± 0.1 W m −1 K −1 and 296 ± 10 nm are extracted, respectively. The thermal conductivity of the a-Si layer agrees well with the literature, 150-152 but fitting for δ eff leaves for much to be desired.
Wang et al. 87 extended measurements in absence of a metal transducer to a variety of bulk semiconductors to extract their associated thermal conductivities using a combined FDTR/TDTR approach. Because TDTR is contingent on the use of pulsed lasers, pump/probe delay times, pump modulation frequencies, and photon energies can be judiciously chosen such that the thermoreflectance signal is dominated by thermal carriers, and not that of excited electronic carriers. In their implementa- tion, Wang et al. 87 perform pulsed FDTR measurements at a fixed pump/probe delay time to extract the radial thermal conductivity in Si, GaAs, and InSb. Most importantly, the optical penetration depth is not used as a fitting parameter in this implementation, but is rather taken from the literature for the given pump and probe wavelengths. 83 The importance of pump/probe delay time choice is also emphasized, as earlier delay times are indicative of the excited carrier contribution to reflectance in addition to the contribution from temperature. Example data and best fits from Wang et al. 87 on GaAs at 200 ps and 5 ns are shown in Fig. 15. In the case of 200 ps, the model fails to capture the acquired data, attributed to the additional contributions to thermoreflectivity via excited carriers. At 5 ns, however, far from the associated relaxation time of these carriers, the extracted data and model agree very well with one another, and a thermal conductivity of GaAs that agrees well with the literature is recovered. However, despite performing transducerless thermoreflectance measurements on semiconducting materials, with emphasis on the implementation of a volumetric heating source in the analysis, the similarity in optical and heat penetration depths in GaAs and InSb at the pump wavelength reduce the analysis to that of a surface heating event. Thus, for materials in which these  Measurements via thermoreflectance techniques in a transducerless configuration have shown promising results. With knowledge of the optical constants of the material system being probed, analysis using the heat diffusion model that accounts for volumetric heating and probing can be readily implemented. Nonetheless, adjusting system parameters such that near-surface heating events are induced is ideal, as with a typical metal transducer. Thus, a push towards UV-based pump/probe systems that ensure near-surface absorption is promising, as the analysis is greatly simplified and not obfuscated by the associated pump/probe optical penetration depths and their contributions to the thermoreflectance signal.
Practically, a pump/probe laser system with variable wavelengths should be implemented, so as to allow for a variety of absorption depths for a given material system. While we have emphasized that reducing the optical ab-sorption depth greatly simplifies the analysis, increasing this parameter offers the potential to enhanced sensitivity to underlying media that cannot be probed during a near-surface heating event. While we have not discussed the implementation of SSTR in a transducerless configuration, the implementation is akin to that of TDTR and FDTR, as discussed earlier. Modeling should be done in a similar fashion; however, SSTR fundamentally differs from the latter two in that it encapsulates the thermal conductivity tensor, whereas primarily the in-plane components are probed in the two transient techniques. A variety of avenues can thus be undertaken, offering the possibility for novel measurements to better understand device design and fabrication.

VI. CONCLUSION
Thermoreflectance-based techniques have shown good applicability for the characterization of the thermal properties of a variety of material systems. Due to the inherently micrometer-scale spot sizes associated with focusing through high numerical-aperture objective lenses, diffraction limited spots can be achieved, vastly improving the spatial resolution of the measurement system. For materials with inhomogeneities or phase contrasts, rastering the specimen greatly increases throughput and yields insight into novel mechanisms of thermal transport. Nonetheless, at typical modulation frequencies, the probing volumes in TDTR and FDTR are inherently limited due to the modulated nature of the pump heating event. SSTR offers exciting new possibilities, probing depths on the order of the spot size while independently measuring thermal conductivity from volumetric heat capacity. Because the probing volume is on the order of the spot size, depths from several micrometers to millimeters, power permitting, can be achieved, allowing for an understanding of the thermal conductivity of the underlying media at these depths. A push should, however, be made towards UV-based thermoreflectance systems. Currently, modification of specimens via the deposition of a metal transducer can be alleviated by probing the specimen directly. However, the inherent absorption of the pump and probe and their contributions to thermoreflectance must be accounted for. UV-based systems ensure near surface absorption, greatly simplifying the analysis and allowing for a greater slew of materials to be accessed. Overall, thermoreflectance offers exciting capabilities in the field of thermophysical metrologies, and the unique characterization of the fundamental heat transfer mechanisms using the variations described in this perspective will drive the micro-and nanomaterial industries. ACKNOWLEDGEMENT We appreciate support from the Office of Naval Research, Grant Number N00014-18-1-2429 and the Army