Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials

Thermal conductivity K is an important physical property that measures the capability of a material in conducting heat from high temperatures to low temperatures. Fourier's law of heat conduction states that the heat flux is proportional to the temperature gradient by the thermal conductivity. The thermal conductivity of a material is usually temperature-dependent and can be directionally dependent, i.e., anisotropic. A temperature difference might also exist when heat flows across an interface of two materials. Interfacial thermal conductance (denoted as G) is thus defined as the ratio of the heat flux to the temperature drop across the interface to characterize the resistance to the heat flow.

The thin film or membrane form of many solid materials with a thickness ranging from one atomic layer to hundreds of micrometers has been extensively used in engineering systems to improve their mechanical, optical, electrical, and thermal functionalities, for example, in microelectronics,¹ photonics,² optical coatings,³ solar cells, and thermoelectrics.⁴ When the thickness of a thin film is smaller than the mean free paths or wavelengths of its heat carriers (electrons or phonons), the thermal conductivity of the film is significantly different from its bulk counterpart due to the geometric constraints. As a result, the thermal conductivities of most crystalline thin films are thickness-dependent and anisotropic, even if the thermal conductivities of their bulk counterparts are isotropic. Many other factors such as sample preparation/processing also strongly affect the thermal conductivity of thin films. Over the last two decades, there have been quite a few textbooks,^5,6 monographs,^7,8 and review articles^9–15 published, indicating the vibrant research in this field driven by the technology needs.

Measuring thermal properties of materials is not only of fundamental importance in understanding the transport processes of energy carriers but also of practical interest in their wide applications relating to energy and information systems. Extensive efforts have been made since the 1950s for the characterization of thermal conductivity and thermal contact resistance in bulk materials.^16–21 However, most of the conventional thermal conductivity measurement techniques lack the spatial resolution to measure the temperature gradient/difference or the heat flux across a length scale below tens of micrometers.

Over the past two decades, ultrafast laser-based transient thermoreflectance (TTR) methods have emerged and evolved into a powerful and versatile technique^22–25 for characterizing thermal properties of a large variety of samples including both thin films^26–31 with a thickness down to ∼20 nm^26,27 and bulk materials.^32–35 In particular, time-domain thermoreflectance (TDTR) technique has been applied for measuring materials with thermal conductivity values ranging from a high end at ∼2000 W m⁻¹ K⁻¹ (like diamond³⁶ and graphite^24,37) to as low as 0.03 W m⁻¹ K⁻¹ (e.g., disordered WSe₂ films²⁶). The heat capacity of new materials^38,39 and thermal conductance of various solid/solid interfaces^40–47 have also been frequently measured using TDTR. As a pump-probe technique, TDTR has many advantages over other thermal conductivity measurement techniques. For example, it requires minimal sample preparation and no delicate design of electrical heaters or temperature sensors, and it works equally well in regular ambient conditions or through the window of a vacuum chamber.⁴⁸ Significant efforts have been devoted to advance the TDTR technique itself as well as its applications in thermal and phonon property characterization of various kinds of samples.

This tutorial reviews the fundamentals as well as the advanced configurations of the TDTR technique for measuring thermal properties of bulk and thin film materials, which should be of interest to the first-year graduate students as well as veteran researchers who are interested in developing new materials, understanding fundamental physics of thermal transport, or learning measurement techniques. This paper is organized as follows. Section II discusses the basics of the time-domain thermoreflectance technique, serving as a guideline in understanding the basic principles and implementations of this technique. Section III discusses several variations of the standard TDTR technique that have their own unique features while sharing the similarity with TDTR. Section IV discusses a variety of advanced TDTR configurations that have been developed to meet different measurement conditions of various kinds of samples. Finally, Sec. V gives a summary that discusses both the limitations and directions for future development.

TDTR measures thermal properties through the reflectance change with temperature, known as the thermoreflectance. Samples are usually coated with a thin metal film acting as a transducer [see Fig. 1(a) for an illustration], whose surface reflectance changes linearly with temperature when the temperature rise is small (typically <10% of the absolute temperature or <10 K, whichever is smaller). The thermoreflectance technique was first developed in the 1970s and 1980s, where continuous wave (CW) light sources were used for the heating and sensing.^49,50 With the advancement of pico- and femtosecond pulsed laser in the 1980s, this technique was widely used for studying non-equilibrium electron-phonon interaction,^51–56 coherent phonon transport,^27,57–61 picosecond acoustics,^62–72 optical properties,^73–75 thermal expansion coefficient,⁷⁶ and thermal transport properties of thin films^77,78 and interfaces.^{40,41,43,44,46,47,79–85} Among these, Paddock and Eesley⁷⁷ were the first to measure thermal diffusivity of metal films using picosecond transient thermoreflectance, and Humphrey Maris' group have done a series of original research in developing picosecond ultrasonics using picosecond light pulses.^62–71 This technique has been further developed over the last two decades for measuring anisotropic thermal conductivity^24,37,86,87 and probing spectral phonon transport.^88–92 The transient thermoreflectance technique can be implemented as both the TDTR method^22,24,25 and the frequency-domain thermoreflectance (FDTR) method^93,94 (see Sec. III A for more details of FDTR). The TDTR method measures the thermoreflectance response as a function of delay time between the arrival of the pump and probe pulses on the sample surface, where the pump beam deposits a periodic heat flux on the sample surface and the probe beam detects the corresponding temperature change through the reflectance change. A schematic diagram of a typical TDTR setup is shown in Fig. 1(b), while similar systems can also be found in other Ref. 23, 25, 39, 95, and 96.

In a typical TDTR setup, a mode-locked Ti:sapphire laser oscillator is used as the light source, which generates a train of 150 fs laser pulses at an 80-MHz repetition rate, with wavelengths centered around 800 nm. Other types of pulsed lasers may also be used as the light source, such as a dye laser with wavelengths centered at 632 nm⁹⁷ and a Yb:doped fiber laser with wavelengths centered at 1030 nm.⁹⁸ However, the Ti:sapphire laser is commonly used in the literature because of its ultrafast nature and excellent beam quality. A broadband Faraday optical isolator is installed at the outlet of the laser oscillator to prevent the laser beam from reflecting back into the oscillator. A half-wave plate installed before the isolator can be used to adjust the laser power for TDTR measurements. The laser beam is then split into a pump beam and a probe beam through a polarizing beam splitter (PBS), with the pump and the probe beams cross-polarized to each other. Another half-wave plate before the PBS can be used to adjust the power ratio between the pump and the probe beams. The pump beam is usually modulated at a frequency in the range 0.2–20 MHz using an electro-optic modulator (EOM) before being directed onto the sample through an objective lens. Alternatively, some TDTR setups use acousto-optic modulators (AOM) but with a generally limited modulation frequency of ≤1 MHz^78,97,99,100 due to the much longer rise time of AOM. The EOM modulation frequency serves as the reference for lock-in detection. The probe beam is delayed with respect to the pump beam via a mechanical delay stage before being directed onto the sample through the same objective lens. The probe beam is usually expanded before the delay stage to minimize divergence over the long propagation distance. In some other configurations, the optical delay between the pump and the probe is achieved by advancing the pump instead.²² This introduces a phase shift of $exp(i2πfmodtd)$ in the signal, where f_mod is the modulation frequency and t_d is the delay time. Advancing the pump rather than delaying the probe has the advantage that the reflected probe beam as received by the detector would not be affected by the movement of the delay stage.²² 

The reflected probe beam is collected by a fast-response photodiode detector, which converts the optical signals into electrical signals. A radio-frequency (RF) lock-in amplifier is then used to pick up the signal from the strong background noise. In some early versions of TDTR systems, an inductor (not shown in the diagram) was inserted in the signal line between the photodiode detector and the lock-in amplifier with a 50-Ω input resistance. The reason is that the pump beam was usually modulated by a square-wave function (for example, using the Model 350–160 EOM and 25D amplifier from Conoptics) and all the undesired odd harmonics of the square wave were detected by the lock-in amplifier using square-wave multipliers (such as the Model SR844 from Stanford Research Systems). The inductor thus serves as a resonant bandpass filter to remove the higher harmonics of the square-wave modulation function.¹⁰¹ This kind of resonant filter becomes unnecessary if the pump beam is modulated by a sine-wave function instead, or if a digital lock-in amplifier with clean sine-wave multipliers (such as the Model HF2LI from Zurich Instruments) is used for the lock-in detection, both of which are inherently free of unwanted harmonics. Alternatively, the thermal responses from both the fundamental frequency and the higher harmonics of the square wave can be both detected and simulated, which works equally well to derive thermal properties of the sample, see Ref. 102 for an example. The RF lock-in amplifier has outputs of an in-phase (V_in) signal and an out-of-phase (V_out) signal at the modulation frequency. In some TDTR systems, a double-modulation scheme is implemented,^23,39 i.e., a mechanical chopper is added in the probe path and two computer-based audio-frequency lock-in detections are implemented on the outputs of the RF lock-in amplifier. This double lock-in scheme removes coherent pick-up contributions to the signal at the radio frequency.³⁹ 

To make the lock-in detection effective, the reflected pump beam must be blocked from the photodiode detector. Since the pump and probe beams are cross-polarized, the PBS between the objective lens and the detector can suppress >99% of the reflected pump beam. However, due to the small value of thermoreflectance coefficient dR/dT (∼10⁻⁴ K⁻¹), a tiny amount of reflected pump beam even with an intensity less than 0.01% of the reflected probe beam is significant enough to distort the TDTR measurements.⁹⁵ Optical techniques are used to further suppress the reflected pump beam. One commonly used approach is the spatial separation of the pump and the probe beams, as depicted in the schematic in Fig. 1(b). In this approach, the pump and the probe beams are in parallel and vertically separated by ∼4 mm when entering the objective lens so that the reflected pump beam can be blocked by an aperture while only the specularly reflected probe beam is allowed to pass through the aperture and be received by the photodiode detector. Care should be taken to make sure that the reflected probe beam is not clipped by the aperture; otherwise, it would induce an amplitude modulation of the probe beam due to thermal expansion of the sample surface and affect the measurements. This approach has the merit of simple configuration and works well for optically smooth samples with surface roughness <15 nm.¹² For optically rough samples, the pump and the probe beams need to be spectrally separated so that the reflected pump beam can be eliminated using highly efficient optical filters. Spectral separation of the pump and the probe can also be accomplished through the “two-tint” approach²³ using sharp-edged filters that separate the spectra of the pump and probe by a small amount of ∼7 nm in the wavelength, or through the “two-color” approach^25,103 using second harmonic generation to double the frequency of either the pump or the probe beam. Both the two-tint and the two-color approaches can effectively filter the reflected pump beam but at a significant sacrifice of the laser power. Such a laser power loss is a critical issue for measurements of highly conductive materials such as diamond, where a sufficiently high laser power intensity is desired to generate a high enough signal-to-noise ratio (SNR). In addition, the two-tint method has a high demand on the laser stability in both the intensity and the wavelength (like the one reported in Ref. 23). Otherwise, the fluctuations in the laser wavelength would translate into fluctuations in the power intensities of the pump and probe beams and appear as noise in the TDTR measurements (like the one reported in Ref. 95). Alternatively, Sun and Koh⁹⁵ developed a simple empirical method to correct artifact signals induced by the reflected pump beam. Sun and Koh⁹⁵ observed that the leaked pump beam mostly affects the V_out signal linearly as $Vout=Vout,0+aVleak$⁠, where $Vout,0$ is the out-of-phase thermal signal unaffected by the leaked pump beam, a is a proportional constant, and $Vleak$ is the detected signal solely caused by the leaked pump beam. The constant a, which depends on factors including the photocurrent of the detector and the Q-factor of the resonant circuit, can be determined from a calibration experiment by measuring V_out as a function of V_leak and taking the gradient as $a=ΔVout/ΔVleak$⁠. The TDTR signals can thus be conveniently corrected from the artifacts by monitoring the V_leak signal during TDTR experiments. This approach was claimed to work well for all rough samples, as long as the Gaussian laser profiles on the sample surface are not distorted by the surface roughness.⁹⁵ 

In most TDTR experiments, the signals are taken as the phase of the detected temperature response as carried by the reflected probe beam, computed as $φ=tan−1⁡(Vout/Vin)$⁠, or equivalently the ratio $R=−Vin/Vout$⁠, to derive the thermal properties of the sample. Using the phase signal for data reduction has the advantage over the amplitude signal as it does not require normalization.²² In addition, the normalized amplitude signal loses sensitivity to the thermal properties of the sample especially when the V_out signal is small. However, cautions need to be taken to correct the additional phase shift introduced by cables, electronic instruments, and optical components into the measured signals. Usually, this phase shift due to the instrumentation can be conveniently canceled based on the fact that the V_out signal in TDTR experiments should be constant across the zero delay time.

For a Ti:sapphire oscillator centered around 800 nm, an Al thin layer with a thickness of ∼100 nm is a common choice as the transducer layer due to its strong absorption (the 99% absorption depth is <60 nm)¹⁰⁴ and an exceptionally large thermoreflectance coefficient dR/dT∼10⁻⁴ K⁻¹ at the 800-nm wavelength.^98,105,106 The thin metallic transducer layer not only makes it possible to assume a surface heat flux boundary condition and thus simplifying the analysis but also improves the overall SNR due to its large thermoreflectance coefficient.

In TDTR experiments, besides the unknown thermal properties of the sample layer to be derived, there are also many other input parameters (such as the laser spot size, thermal conductivity of the transducer layer, thickness and heat capacity of each layer in the sample stack) that affect the signals and need to be pre-determined as accurately as possible. The commonly adopted approaches to determine these input parameters are described briefly below.

For the characterization of the laser spot size, while many techniques such as the knife-edge method^25,107 and patterned sample image method⁹⁴ can be used, an accurate and easy-to-use approach for TDTR is to measure the spatial correlation between the pump and probe focal spots.³⁹ In this method, the focused pump spot is swept across the probe spot at a positive delay time of ∼100 ps and a high modulation frequency of 10 MHz. The root-mean-square (RMS) average of the pump and probe spots $w0$ is derived by fitting the V_in profile to a Gaussian function $Vin∼exp⁡(−xc2/w02)$⁠, where x_c is the offset distance. This simple approach works well for TDTR experiments because the TDTR signals are affected by the RMS average of the pump and probe spot sizes rather than their individual sizes, see more details in the thermal modeling in Sec. II B. The thermal conductivity of the metal transducer layer can be characterized using a four-point probe measurement of the electrical resistance and the Wiedemann-Franz law as $Km=LmT/RC′hm$⁠, where L_m is the Lorenz number for the metal film, T is the temperature, R is the electrical resistance measured using four-point probe, C′ is a constant related to the sample lateral size and the spacing between the probe needles. The metal transducer thickness h_m can be determined by picosecond acoustics.⁷² To measure the electrical resistance R, a large and clean glass slide is usually placed next to the sample when metal film is deposited on the sample and the four-point probe measurement is performed on the metal film deposited on the glass slide instead. In most cases, the four-point probe measurements are performed only at room temperature. To obtain thermal conductivity of the metal film at other temperatures, the additional electrical resistivity of the metal film as compared to its pure bulk counterpart is assumed to be dominantly caused by impurity scattering, which is temperature-independent. The additional electrical resistivity of the metal film can thus be determined by the four-point probe at room temperature, which, combined with the temperature-dependent thermal conductivity of its bulk counterpart from the literature, gives the information of the temperature-dependent thermal conductivity of the metal film. The heat capacities of the metal transducer film and the sub-layers are usually assumed to be the same as their bulk counterparts⁵ and can be obtained from the literature. For those samples whose heat capacities are not readily available from the literature or not measurable using standard techniques such as the differential scanning calorimetry, frequency-dependent TDTR experiments can simultaneously determine the thermal conductivity and heat capacity of the sample, see details in Sec. IV A, advanced TDTR configurations.

An estimation of the steady-state temperature rise is also important for TDTR experiments, as the temperature rise needs to be small (usually <10 K) to ensure the reflectance change is linearly proportional to the temperature change. Cahill²² has derived the expression of steady-state temperature rise for a semi-infinite solid as $ΔT=A02πw01K$⁠, where A₀ is the laser power absorbed by the sample. For thin layers. and interfaces, the temperature rise can be estimated as $ΔT=A0πw02hK$ and $ΔT=A0πw021G$⁠, respectively. Note that these formulas are only for a rather crude estimation. A more in-depth discussion on the steady-state temperature rise in thermoreflectance experiments can be found in Ref. 108.

The determination of thermal transport properties by TDTR is typically accomplished by adjusting free parameters (the unknown thermal properties) in a thermal transport model to obtain the best fit between the model prediction and the experimental data. Although significant advances have been made on observing the interesting or even counter-intuitive phenomena in nanoscale thermal transport, most studies still use the effective thermal conductivity of nanostructured materials as a common language to interpret the findings and to communicate across different communities in physics, materials, and engineering. Similarly, while TDTR can be used to study non-equilibrium phenomena such as quasi-ballistic phonon transport^{90,91,109–111} and electron-phonon coupling⁵⁵ due to its ability to detect temperature evolutions at micrometer-scale and picosecond-scale resolutions, the majority of the experimental work in the literature deduced thermal properties based on Fourier's law of heat conduction with an effective thermal conductivity. However, care should be taken when applying a diffusive thermal transport model to systems where a significant fraction of heat is carried by phonons with long mean-free paths (MFPs).^90–92,112

The early versions of TDTR data reduction schemes can only be applied to some simplified cases with many assumptions.^77,78 The key advance in the data analysis of TDTR experiments was made by Cahill in 2004.²² The three-dimensional heat diffusion equation through multi-layered structures was solved with the consideration of the so-called “pulse accumulation” effect, i.e., the response of a new pulse should account for the previous pulse that has not fallen to a negligible value. Schmidt²⁴ further extended the model for anisotropic heat conduction where thin films are very likely anisotropic due to the size effect of energy carriers. While more modeling efforts are needed for TDTR measurements for various sophisticated situations (see Sec. IV for details), the thermal model for the most commonly encountered case in TDTR experiments is outlined here. A schematic of the sample structure in TDTR experiments is shown in Fig. 2. In this basic model, both the laser profile and the thermal properties of the sample are assumed to be axisymmetric rather than in-plane anisotropic (see Sec. IV C); both the heating and sensing occur on the top surface of the sample; and the different phonon modes are assumed to be in thermal equilibrium so that effective thermal properties K_r, K_z, C, and G can be assumed for each layer. The thermal modeling of TDTR experiments involves two steps: one is to solve heat diffusion in a multilayer stack, and the other is to model the data acquired from TDTR experiments.

The complexity of the thermal model described above potentially also enables TDTR for determining many thermal properties involved in the model, including the heat capacity C, in-plane thermal conductivity K_r, cross-plane thermal conductivity K_z, and interface conductance G. However, it imposes great challenges to extract the target property with multiple unknown parameters. Therefore, the number of unknowns needs to be reduced when performing TDTR measurements. For example, the heat capacity is usually pre-determined from separate measurements using differential scanning calorimetry or known from the literature values, with the unknown parameters often reduced to K_r, K_z, and G. The number of unknown parameters can be further reduced if the samples are isotropic so that there is no need to differentiate the in-plane and through-plane thermal properties. Furthermore, most practitioners use a large laser spot size (w₀= 5–20 μm) and a high modulation frequency (f = 1–10 MHz), resulting in quasi-one-dimensional thermal transport. In this case, the TDTR signals are most sensitive to the thermal properties in the through-plane direction (K_z and G). Therefore, TDTR is often used to measure the through-plane thermal conductivity K_z of the sample and the transducer/sample interface conductance G without much elaboration.^{12,40,120–125}

Figures 4(a) and 4(b) show an example of TDTR in-phase and out-of-phase data measured on a GaN sample with a 100 nm Al transducer as a function of the delay time. The modulation frequency was f = 10 MHz and the laser spot size was w₀ = 12 μm. As mentioned before, the ratio between in-phase and out-of-phase signals, −V_in/V_out, is usually compared with the thermal model simulation over the delay time range 0.1–8 ns to extract the thermal properties, as shown in Figs. 4(c) and 4(d). The through-plane thermal conductivity K_z of the sample and the transducer/sample interface conductance G affect the ratio signal R = −V_in/V_out in different manners over the 0.1–8 ns delay time range. As indicated by the ±30% bounds of the best-fitted value in Figs. 4(c) and 4(d), K_z of the sample mainly affects the amplitude of the ratio signal, whereas the interface conductance G mainly affects the gradient of the ratio signal. Both K_z and G can thus be simultaneously determined from one single set of TDTR measurements.

The uncertainties of multiple fitting parameters can be estimated using a multivariate error-propagation formula based on Jacobian matrices derived by Yang et al.¹²⁷ This multivariate error-propagation formula has been validated by the Monte Carlo method¹²⁷ but is much more computationally efficient: it takes only a few seconds to complete the analysis on a desktop computer using MATLAB scripts rather than a few days as needed by the Monte Carlo method. Figure 4(f) shows the confidence range of K_z and G of the GaN sample coated with 100 nm Al estimated by the least-squares regression method when these two parameters are fitted simultaneously from TDTR measurements. The results show that K_z and G can be determined with an uncertainty of ∼10% and ∼5%, respectively, at both frequencies of 1 and 10 MHz. The major sources of measurement uncertainty come from the thermal capacitance (volumetric heat capacity times the film thickness, C*h) of the transducer for high-frequency measurements and from the laser spot size w₀ for low-frequency measurements.

The reliability of the TDTR technique for thermal conductivity characterization has been verified on a series of standard bulk material samples over a wide range of thermal conductivity values. Figure 5 shows the benchmark studies of TDTR conducted by Zheng et al.¹²² and Wilson and Cahill¹²⁸ over a wide range of thermal conductivity from 0.2 to 2000 W m⁻¹ K⁻¹. The through-plane thermal conductivities of these materials determined by TDTR are in good agreement with the literature values with an overall experimental uncertainty of ±8%.

Besides the bulk materials, TDTR has also been widely employed to measure the through-plane thermal conductivity of many thin film materials. Note that TDTR cannot measure any arbitrarily thin films but usually requires the films to be “thermally thick”, i.e., the film thickness should be larger than the through-plane thermal penetration depth, $hfilm>dp,z$⁠, defined as $dp,z=Kz/πfmodC$⁠. For thermally thin films with $hfilm<dp,z$⁠, the measurements of K_z of the film would be challenging due to the low sensitivity to K_z. An advanced dual-frequency TDTR configuration can be employed to improve the accuracy in determining K_z of thermally thin films, see Sec. IV B for more details.

The thermal conductivity of superlattices and nanolaminates^{27,59,78,79,123,132–138} [see Fig. 6(b) for an example] has been extensively characterized using TDTR over the last 15 years. To conduct the TDTR measurements, superlattice samples are usually configured in a scheme as shown in Fig. 6(a). The superlattice sample is treated as a homogeneous thin film with thickness h_film and an effective thermal conductivity. The through-plane thermal conductivity of superlattices has been found to depend on several factors including the interface density (=2/d_SL, where d_SL is the period thickness), the total film thickness, and the thickness ratio between the two materials. Figures 6(c)–6(e) summarize a few typical results of K_z of superlattices measured using TDTR reported in the literature. A higher interface density generally reduces the K_z of superlattices [see Fig. 6(c)] due to the interface thermal resistance introduced by the periodic boundaries.^79,137 However, the K_z of superlattices was also found to depend on the total film thickness, as shown in Fig. 6(d), indicating that a significant fraction of phonons can coherently transport across the interfaces and are only scattered by the film boundary.^27,137 Besides the two boundary scattering length scales, the thickness ratio of the two materials was also found to affect the thermal conductivity significantly, as illustrated in Fig. 6(e) for the example of InAlAs/InGaAs superlattices.¹³⁶ 

Over the past two decades, TDTR has been applied to characterize the thermal properties of a wide range of bulk and thin film materials and their interfaces. Figure 7 summarizes the thermal conductivity and interface conductance of some typical samples measured as a function of temperature using TDTR. It shows that TDTR has been applied across a broad range of thermal conductivity and interface thermal conductance. Ultra-low thermal conductivities of full-dense solid materials were reported for fullerene derivatives at room temperature (∼0.05 W m⁻¹ K⁻¹)¹²⁵ and disordered layered WSe₂ thin films at cryogenic temperatures (∼0.03 W m⁻¹ K⁻¹),²⁶ both measured using TDTR. The lowest thermal conductance was reported for the interface between Bi and hydrogen-terminated diamond, with a value of ∼8 MW m⁻² K⁻¹ at room temperature,⁴¹ while the highest thermal conductance observed to-date is for metal/metal interfaces, such as Al/Cu, with a G of ∼3700 MW m⁻² K⁻¹ at room temperature.⁴⁰ 

In addition to the “standard” TDTR configuration discussed in Sec. II, there are a large variety of related TDTR techniques. For example, instead of using femtosecond laser pulses for both the pump and probe beams, some other transient thermoreflectance techniques use picosecond or nanosecond laser pulses as the pump beam, and either the synchronized and time-delayed laser pulses or a continuous wave (CW) laser as the probe beam, which is monitored with either a lock-in amplifier or an oscilloscope.^100,145,146 In what follows, the unique features and challenges of three important variations of TDTR, i.e., frequency-domain thermoreflectance (FDTR), time-resolved magneto-optic Kerr effect (TR-MOKE), and asynchronous optical sampling (ASOPS), are briefly discussed.

Frequency-domain thermoreflectance, shorten as FDTR, is a variation of TDTR where the thermoreflectance signal as a function of the modulation frequency of the pump beam is collected instead of monitoring the thermoreflectance signal as a function of the delay between the arrival time of the pump and probe pulses. FDTR is thus much easier to implement because it avoids the complexity of a long mechanical delay for the time delay and it can use inexpensive CW laser sources. By holding the delay stage at a fixed location, an ultrafast-laser-based TDTR can also be implemented as FDTR.⁹³ Both the pulsed and CW FDTR are discussed and compared below.

The pulsed FDTR uses a similar setup as the conventional TDTR [see Fig. 1(b) for a schematic of the TDTR setup]. The only difference is that the resonant circuit used to eliminate the higher harmonic signals in TDTR cannot be used in FDTR experiments since the data are acquired as a continuous function of modulation frequency while the resonant circuits are usually at fixed cutoff frequencies. The use of the resonant filter, however, is not always necessary for TDTR, if the pump beam is modulated by a pure sine wave or if a digital lock-in amplifier using clean sine-wave multipliers is used for the lock-in detection, or if the higher harmonics are also considered in the thermal modeling.

The CW FDTR can be configured in a much easier way as shown in Fig. 8. One major challenge of CW-laser-based FDTR is the accurate determination of the phase signal in FDTR experiments. Besides the desired thermal phase signal $φtherm$⁠, an additional frequency-dependent phase shift, collectively written as $φinstrum$⁠, would be introduced by components such as the photodetector, the cables, instruments, and the different optical path lengths of the beams. In ultrafast-laser-based TDTR and FDTR, this instrumentation phase can be conveniently corrected by the fact that the $Vout$ signal should be constant across the zero delay time. For CW-laser-based FDTR, one commonly adopted approach is to split a portion of the pump beam after the EOM and send it to a reference photodetector that is identical to the primary photodetector, as illustrated in Fig. 8.⁹³ Note that the “identical” here means not only the same detector model but also the same operational parameters such as the applied reverse bias, the incident beam intensity, and the laser wavelength, all of which affect the phase shift introduced by the detector.¹⁰¹ Besides, the optical path length between the EOM and the reference detector should also be identical to the sum of the path lengths from the EOM to the sample and from the sample to the probe detector. In such a condition, the signal of the primary detector would be $φ1=φtherm+φinstrum$⁠, while the signal of the reference detector would be $φ2=φinstrum$⁠. The lock-in amplifier measures the phase difference between the two signal channels as $φtherm=φ1−φ2$⁠. An alternative approach is to use the same photodetector but to conduct two separate experiments: one is the phase signal of the probe beam with the reflected pump beam being filtered and the other is the phase signal of the pump beam with the reflected probe beam being filtered. In this case, the measured phase of the pump is $φpump=φinstrum−φref$ and the measured phase of the probe is $φprobe=φtherm+φinstrum−φref$⁠. The thermal signal can thus be deducted by subtracting the phase responses of the two experiments as $φtherm=φprobe−φpump$⁠. However, extreme care should be taken to make sure that factors such as the laser beam intensity and wavelength at the detector are identical for the two experiments to avoid unintentional systematic errors.

Figure 9 shows an example of the calculated signals for CW and pulsed FDTR measurements of a sapphire sample with a 100 nm Al transducer over the modulation frequency range of 0.05–20 MHz, using an RMS spot size of 5 μm. The delay time is fixed at 100 ps for the pulsed FDTR. The results show that both CW and pulsed FDTR configurations have almost the identical V_out signals but the V_in signals differ significantly, suggesting that the V_out signals in TDTR experiments mainly come from the continuous heating at the modulation frequency. The phase signal $φ$ of the pulsed FDTR changes relatively little over the modulation frequency range of 0.05–20 MHz, while the phase signal $φ$ of the CW FDTR changes dramatically as a function of frequency.

The sensitivity analysis for TDTR experiments can be applied similarly to FDTR. While there are cases in the literature using the phase signal $φ$ to derive thermal properties,^93,94 for consistency, the ratio signal $R=−Vin/Vout$⁠, which is equivalent to the phase signal $φ=arctan(Vout/Vin)$⁠, is used here. Figure 10 compares the sensitivities of the ratio signals from CW and pulsed FDTR to different parameters of the sapphire sample over the modulation frequency range of 0.05–20 MHz, using a spot size of w₀ = 5 μm [see Eq. (2.23) for the definition of the sensitivity coefficient]. The results show that both the CW and pulsed FDTR have similar sensitivities to the thermal conductivity and heat capacity of the substrate; however, the signals of pulsed FDTR are much more sensitive to the spot size w₀ and the heat capacitance (C_Alh_Al) of the transducer film.

Theoretically, the continuous-wave laser beam in CW FDTR can be modulated at any frequency. However, in practice, the modulation frequency is limited to be <20 MHz due to the poor SNR at higher modulation frequencies. A heterodyne technique called broadband frequency-domain thermoreflectance (BB-FDTR)¹⁴⁷ has thus been implemented to extend the modulation frequency up to 200 MHz . However, measurements at such a high modulation frequency can be challenging due to the small thermal penetration depth and severe heating effect. A schematic of the BB-FDTR setup is shown in Fig. 11. The major difference between FDTR and BB-FDTR is that an additional modulation at frequency f₂ is added on the reflected probe beam. Regardless of how high the modulation frequency of the pump beam f₁ is, the lock-in amplifier only measures the signals at a much lower frequency, f₁−f₂, which can be chosen to be in a proper range to achieve very high SNR and retain great fidelity of the thermal signal. Another benefit is that the frequency difference f₁−f₂ can also be chosen to be close to the upper limit of the frequency range of the lock-in amplifier so that the higher harmonic components are naturally excluded from the lock-in detection.¹⁴⁷ Extending modulation frequency range greatly expands the capability of FDTR. For example, it was introduced to study phonon MFP spectra in semiconductors,^92,147,148 similar to the frequency-dependent TDTR that is based on the physical picture of non-diffusive phonon transport, see more details in Sec. IV D.

TDTR can also incorporate a novel temperature sensing technique utilizing time-resolved magneto-optic Kerr effect (TR-MOKE).^34,149,150 Instead of thermoreflectance, TR-MOKE relies on the temperature-dependent transient polar Kerr rotation to detect the temperature response of a magnetic transducer under pump laser heating, from which the thermal properties of the material underneath the transducer film can be derived, following the same data reduction scheme of TDTR. Thermometry based on the thermo-magneto-optic Kerr effect allows the use of much thinner magnetic transducer films that are not necessarily optically opaque as required in the conventional TDTR method. A thinner transducer with a lower thermal conductivity can minimize lateral heat flow in the transducer layer and thus enhancing the measurement sensitivity to the in-plane thermal conductivity.^34,87 Meanwhile, the small thermal mass of the transducer layer also enables an enhanced sensitivity to the interface thermal conductance¹⁵⁰ when the thermal conductivity of the substrate is small.

Figure 12 shows the TR-MOKE signal detection scheme. To conduct TR-MOKE measurements, the sample needs to be coated with a thin perpendicularly magnetized transducer, which is magnetized with an external magnet prior to the measurements. A non-polarizing beam splitter is inserted between the steering PBS and the microscope objective lens to divert the reflected pump and probe beams toward the detection path. In the detection path, the pump beam is removed by a filter, while the probe beam passes through a half-wave plate and then is split into two orthogonally polarized components by a Wollaston prism. The half-wave plate is adjusted such that the two components have approximately the same intensity. Transient changes in the polarization of the probe beam are monitored by detecting the changes in the relative intensities on the balanced detector.

In case of non-perfect balancing with the half-wave plate, thermoreflectance signals overlap the transient Kerr rotation. Since the TR-MOKE signals would change signs for oppositely aligned magnetization states of the magnetic transducers, the TR-MOKE signals can be isolated out by subtracting the in-phase and out-of-phase signals recorded for oppositely aligned magnetization states of the transducers as $Vin=(VinM+−VinM−)/2$ and $Vout=(VoutM+−VoutM−)/2$⁠. Figure 13 shows an example of the TR-MOKE signals measured as a function of delay time for a bulk black phosphorus sample coated with a 26.9 nm TbFe transducer.¹⁵¹ Note that both the M+ and M− signals in Fig. 13(a) show observable oscillations in the short delay time range up to 500 ps [see the inset of Fig. 13(a)], which was attributed by the authors to the Brillouin scattering.¹⁵¹ The Brillouin scattering arises from the interaction between the reflected probe beam and the acoustic waves inside the black phosphorus sample caused by the birefringence originated from the in-plane anisotropy in black phosphorus.^152,153 These oscillations are also canceled out by the corrective subtraction [note that the corrected signals in Fig. 13(a) are smooth without oscillations]. This approach makes TR-MOKE thermometry less prone to errors because any spurious signals that are independent of the magnetization states of the transducer can be canceled out.

Some commonly used magnetic transducer films for TR-MOKE are Co/Pt, Co/Pd, CoFe/Pt, TbFe, GdFeCo, etc. For example, Feser et al.⁸⁷ used a Co/Pt multilayer with a total thickness of 19 nm as the TR-MOKE transducer with the nominal structure of the multilayer from top to bottom as Pt (1 nm)/[Co (0.5 nm) Pt (1 nm)] × 6/Pt (10 nm). Chen et al.¹⁴⁹ carefully evaluated the performance of four types of magnetic transducers, including TbFe, GdFeCo, Co/Pd, and CoFe/Pt, and found that TbFe with a thickness of ∼18.5 nm gives the optimal performance. Magnetic transducers as thin as 4.2 nm were also used for TR-MOKE experiments.¹⁵⁰ Given the small thermal masses of the magnetic transducers, adsorbents such as hydrocarbons and water molecules on the sample surface can possibly change the effective heat capacity of the transducer and need to be carefully considered in the thermal model. Kimling et al.¹⁵⁰ empirically made the correction by adding a 1-nm-thick transparent layer with a thermal conductivity of 2 W m⁻¹ K⁻¹ and a volumetric heat capacity of 2.8 J cm⁻³ K⁻¹ on top of the transducer layer in their thermal model for data reduction.

Besides thermal characterization, TR-MOKE has also been widely employed to study spin dynamics and ultrafast magnetization processes,^154–161 which, however, is beyond the scope of this tutorial.

In conventional TDTR, the pump and probe beams are two synchronous pulse trains and the measured data are recorded in the time domain as a function of the delay time. This approach is known as synchronous sampling, with the speed of data acquisition often limited by the traveling speed of the delay stage as well as the time required for the system to reach an equilibrium state for each new delay time (typically ∼1 s is needed to acquire one data point). Much faster data acquisition is possible with asynchronous optical sampling (ASOPS), using two different mode-locked lasers with slightly different pulse repetition rates. This automatically provides a temporally varying delay between the two pulses.

Figure 14(a) illustrates the principle of signal detection in ASOPS. The curves shown on the top of Fig. 14(a) represent the surface temperature of the sample in response to the pump input, which has a repetition rate of f_pump and a period of 1/f_pump. Each successive probe pulse, represented by the dots, is delayed with respect to the pump pulse by the time Δt = Δf/(f_pumpf_probe), where Δf = f_pump – f_probe is also known as the beat frequency. The total number of points sampled during one full period of the signal is N = f_probe/Δf. More importantly, this sampling process automatically repeats at a period of t = 1/Δf. The ASOPS technique is thus an optical analog of the electronic oscilloscope. For example, given f_pump = 80 MHz and frequency offset Δf = 1 kHz, ASOPS would achieve a delay time increment of Δt = 0.16 ps, and the measurement time to finish one full period scan is only 1 ms, which means that the ASOPS instrument can obtain 1000 acquisitions within 1 s. This fast data acquisition allows ASOPS to average many repeated measurements, significantly reducing the detection noise to a very low level. Note that the temporal resolution eventually achieved by ASOPS depends not only on the delay time increment but also on the pulse duration and the detection bandwidth. Nevertheless, a temporal resolution of 1 ps is already sufficient for TDTR experiments, while ultrafine temporal resolution as small as 50 fs has been reported for the ASOPS technique.¹⁶² 

ASOPS was first reported by Elzinga et al. in 1987,¹⁶³ who used a Nd:YAG laser with an 81.597 MHz repetition rate as the pump beam and a 10 kHz offset rate for the probe and measured the fluorescence lifetime of rhodamine B to be 2.3 ns. Fiechtner et al.^164,165 later employed ASOPS for rapid diagnosis of turbulent flames in combustion. Kafka et al.¹⁶⁶ built a two-color ASOPS system using two regeneratively mode-locked Ti:sapphire lasers and achieved a temporal resolution of 150 fs. In recent years, researchers started using the ASOPS technique for THz spectroscopy.^{162,167–170} For example, Cuffe et al.¹⁷¹ studied the lifetimes of confined acoustic phonon modes in free-standing silicon membranes, using an ASOPS system with two Ti:sapphire oscillators at 1 GHz repetition rate and a frequency offset of 10 kHz.

ASOPS clearly has many advantages over the traditional mechanical delay-based pump-probe systems: (1) ASOPS enables much faster data acquisition. The noise level can thus be significantly reduced by averaging over a large amount of data acquired within a short period of time. (2) ASOPS eliminates the mechanical delay stage and the associated systematic errors due to beam pointing instability and spot size variation. (3) ASOPS has access to the full delay range, whereas in the traditional pump-probe metrology, the delay time is limited by the length of the delay stage. Admittedly, all the advantages of ASOPS come at the cost of requiring two ultrafast lasers instead of one.

Another important feature of ASOPS is that the probing rate can be many times slower than the pumping rate, with f_pump = nf_probe + Δf, where n is an integer. This enables sampling of ultrafast phenomena at a much slower acquisition rate. For example, Pradere et al.¹⁷⁵ used this technique to sample 30 kHz thermal waves using an infrared camera at an acquisition rate of only 25 Hz.

In addition to the frequent practice of measuring the through-plane thermal conductivity K_z and the interface thermal conductance G, advanced TDTR configurations have also been developed over the past decade to measure specific thermal properties (e.g., heat capacity, anisotropic thermal conductivity, etc.) or to meet the requirements of different circumstances (e.g., measurements of liquids and rough coatings). Here, we discuss advanced TDTR configurations that demonstrate the flexibility and versatility of the TDTR technique for thermal property characterization of novel materials. These advanced TDTR configurations are essentially based on varying the modulation frequency and/or the spot size, which alters the measurement sensitivity of different target properties, making TDTR more powerful and versatile.

In TDTR measurements of thermal conductivity, the heat capacity of the sample is usually a prerequisite that affects the determined thermal conductivity values significantly and thus needs to be pre-determined accurately. Measurements of the heat capacity of nanostructured materials such as thin films, however, are very challenging.^176,177 A common practice is thus to adopt the bulk values for the thin films, where the bulk values can be easily measured using conventional techniques such as the differential scanning calorimeter. This is usually a reasonable assumption for most of the well-studied condensed materials (e.g., Al, Si, SiO₂, etc.) since the nanostructures do not significantly change the heat capacity.¹⁷⁷ However, this approach is problematic for many novel materials either because it is questionable to assume bulk heat capacity for the nanostructures^178–180 or because it is difficult to obtain the bulk form of the new materials for a conventional heat capacity measurement. TDTR has been demonstrated to determine thermal conductivity K and heat capacity C simultaneously for both bulk and thin film materials by performing multiple measurements using different modulation frequencies,^38,39 with the basic principles explained below.

Figure 15(a) shows an example of the sensitivity coefficients [defined in Eq. (2.23)] of the TDTR ratio signal to K and C of the substrate as a function of modulation frequency when the delay time is fixed at 100 ps. At a high modulation frequency of 10 MHz, the TDTR signal has the same sensitivity to K and C, suggesting that the TDTR signal –V_in/V_out is mostly determined by the thermal effusivity $e=KC$ of the substrate at high modulation frequencies. As the modulation frequency decreases to sufficiently low of 0.1 MHz, the sensitivities to K and C have the similar amplitudes but opposite signs, suggesting that the TDTR signal is mainly affected by the thermal diffusivity $α=K/C$ of the substrate at low modulation frequencies. Since the TDTR signals depend on K and C differently at high and low modulation frequencies, a K-C diagram shown in Fig. 15(b) suggests that the K-C curves at the different modulation frequencies should cross at a certain point. These two properties can thus be determined simultaneously by conducting at least two sets of the measurements at a high modulation frequency and a low modulation frequency. Figure 15(c) shows an example of the frequency-dependent TDTR approach on simultaneous measurements of K and C of bulk Si.³⁸ A good agreement of ±5% in the determined C of bulk Si was achieved with the literature value. Wei et al.³⁹ validated this approach for the heat capacity of several standard bulk samples, as shown in Fig. 15(d), with an overall accuracy of ±8%.

Besides the bulk samples, this frequency-dependent TDTR approach also applies to thin film materials. Employing this frequency-dependent TDTR approach, Liu et al.¹⁸¹ simultaneously determined the through-plane thermal conductivity and volumetric heat capacity of three types of hybrid organic-inorganic zincone thin films prepared by the atomic layer deposition (ALD) and molecular layer deposition (MLD) techniques, with the film thickness in the range 40–400 nm, see Fig. 16 for a summary of the results. The heat capacities of the hybrid films were found to be independent of the film thickness, while the thermal conductivities changes slightly with the film thickness. Very low thermal conductivities of 0.15–0.35 W m⁻¹ K⁻¹ were measured for these thin films. The ultralow thermal conductivities are due to the alternating layered structures with very different atomic configurations between the ALD atomic layers and the MLD molecular layers that strongly scatter the phonons. Wang et al.¹²⁵ also used the frequency-dependent TDTR approach to simultaneously determine the heat capacity and thermal conductivity of fullerene derivative thin films with thicknesses in the range 50–120 nm. The same method has also been employed by Xie et al.^129,182 to study both the thermal conductivity and heat capacity of 27 different polymers. Another application of this frequency-dependent TDTR approach was also conducted by Liu et al.¹⁸³ to study the film thickness effect on the effective thermal conductivity and volumetric heat capacity of polystyrene thin films over the thickness range 5–300 nm.

TDTR can measure the through-plane thermal conductivity K_z of thin films using a high modulation frequency and a large laser spot size, in which configuration the heat flow is mainly one-dimensional in the through-plane direction. However, using TDTR to measure K_z of thermally thin films on low-thermal-conductivity substrates is very challenging. The reason is that the TDTR signals at high modulation frequencies are also highly sensitive to other input parameters such as the thickness and heat capacity of the Al transducer, h_AlC_Al, the uncertainty of which would significantly affect the accuracy of the determined K_z of the thermally thin films. Here, the “thermally thin” films are referred to as the films with thickness h_film less than the thermal penetration depth in the through-plane direction, defined as $dp,z=Kz/πfC$⁠.¹² To overcome this challenge, a dual-frequency TDTR approach has been developed¹⁸⁴ that takes the advantage of the different frequency dependence in the sensitivities of the different parameters.

This dual-frequency TDTR approach was verified by measuring the through-plane thermal conductivity of a 3-μm-thick Cu film on a SiO₂ substrate.¹⁸⁴ The thermal conductivity of the Cu film is expected to be isotropic with negligible boundary scattering effect due to the much shorter mean free paths of the major heat carriers, i.e., electrons, compared to the film thickness. The K_z of the Cu film measured using dual-frequency TDTR can thus be verified against the K_r of the film determined from a four-point probe measurement using the Wiedemann-Franz law. Excellent agreement was achieved between the dual-frequency TDTR measurements and the four-point measurements of this Cu film, whereas the conventional TDTR measurements showed ∼20% discrepancy from the four-point measurements.¹⁸⁴ The dual-frequency TDTR approach has also been successfully applied to measure K_z of single crystalline Si films with thickness in the range 1–10 μm,¹⁸⁵ which has shown strong anisotropy in the in-plane and through-plane directions¹⁸⁶ due to the long MFPs of phonons in crystalline silicon.¹⁸⁷ 

Generally, in the dual-frequency approach, the high frequency should be chosen to have the highest sensitivity to K_z of the thin film, and the low frequency should be chosen to have a near-zero sensitivity to K_z of the thin film but sufficiently high sensitivity to h_AlC_Al of the transducer layer. Through detailed modeling analysis, Jiang et al.¹⁸⁴ proposed a guideline that the high modulation frequency can be chosen so that the thermal penetration depth in the film d_p,z is nearly half of the film thickness h_f, $dp,z≈0.5hf$⁠, and the low modulation frequency can be chosen so that $dp,z≈1.5hf$⁠. This dual-frequency approach has been shown to be able to improve the accuracy in determining K_z of thin films by $≈$3 times and is very useful when the thermal conductance of the transducer/film interface is high (G > K_z/h_f) and the thermal conductivity of the underlying substrate K_sub is low (K_z/K_sub > 10).¹⁸⁴ 

While the TDTR signals are most sensitive to thermal conductivity in the through-plane direction, TDTR can also be configured to measure anisotropic thermal transport properties. Several advanced approaches based on TDTR have been developed, including a variable spot size configuration,³⁷ a beam-offset configuration,^86,87 and an elliptical-beam configuration,¹¹⁹ all with their own advantages. A summary and comparison of the three different configurations are presented in Table I, with the details discussed below. Besides TDTR, some related techniques such as the 3ω method^188,189 and the transient thermal grating^190,191 can also measure anisotropic thermal transport properties.

By changing the relative size of laser spot as compared to the thermal penetration depth, as illustrated in Fig. 18(a), TDTR can measure both the in-plane and the through-plane thermal conductivities. When TDTR experiments are conducted using a laser spot size w₀ much larger than the in-plane thermal penetration depth d_p,r, the heat mainly flows in the through-plane direction. In this case, the TDTR signals are sensitive to the through-plane thermal conductivity K_z only. When TDTR experiments are conducted using w₀ similar or even smaller than d_p,r, the heat flux becomes three-dimensional. In this case, the TDTR signals are sensitive to both the in-plane and the through-plane thermal properties, K_r and K_z. Therefore, both K_r and K_z of the substrate can be determined by conducting two sets of measurements under the different heat transfer regimes. Note that since the TDTR signals are sensitive to K_z in both regimes, the validity of this approach relies on the assumption that K_z is the same for the two sets of the measurements. Considering that K_z of some materials measured by TDTR depends on the modulation frequency (see Sec. IV D for details), the same modulation frequency should be chosen for the two sets of the measurements. Caution should also be taken to make sure that the laser spot sizes are large enough not to induce spot size dependence in the measurements.

The variable spot size TDTR approach described above is limited to materials that are transversely isotropic because the thermal model assumes a cylindrical symmetry. Feser et al.^86,87 developed a beam-offset TDTR approach to measure in-plane anisotropic materials. In this approach, the pump beam is swept across the probe beam and the full-width half-maximum (FWHM) of the out-of-phase signal at a negative delay time (e.g., −100ps) is used to derive the in-plane thermal conductivity along the scanning direction.¹⁹⁴ Feser et al.⁸⁷ also recommended using a 70-nm-thick NbV film (typically with a thermal conductivity of ∼20 W m⁻¹ K⁻¹) instead of the conventional 100-nm-thick Al film (with a thermal conductivity of ∼200 W m⁻¹ K⁻¹) as the metal transducer for the beam-offset TDTR experiments. With a much-reduced thermal conductance of the metal film (K_mh_m), the FWHM signal is more sensitive to the thermal conductivity of the substrate and less sensitive to the properties of the transducer layer, thus yielding a smaller measurement uncertainty.⁸⁷ For this reason, TR-MOKE is more suitable to perform beam-offset experiments because much thinner transducer films (as thin as ∼5 nm) can be used for the measurements.

The procedure of extracting in-plane thermal conductivity from beam-offset TDTR experiments is demonstrated in Fig. 19, where a ZnO [11–20] sample coated with 100 nm Al is taken as the example. Figure 19(a) shows the V_out signal of the ZnO sample measured as a function of offset distance x_c using a laser spot size w₀ = 4.7 μm at a modulation frequency of 0.35 MHz. ZnO is a hexagonal Wurtzite crystal with a higher thermal conductivity parallel to its c-axis (55–62 W m⁻¹ K⁻¹ from the literature^37,195) than the other directions (∼44 W m⁻¹ K⁻¹ from the literature^37,195). The in-plane thermal penetration depth in ZnO, when measured at a modulation frequency of 0.35 MHz, is ∼4.2 μm, which is close to the laser spot size w₀. Such a configuration of the laser spot size and modulation frequency is expected to have a high sensitivity to K_r of the substrate. The FWHM is determined as 11.3 μm by fitting the V_out signals in Fig. 19(a) using a Gaussian function. Meanwhile, the FWHM signals are also simulated as a function of K_r, as shown by the red solid line in Fig. 19(b). K_r of the substrate along the offset direction can thus be determined by matching the measured FWHM to the simulated FWHM. The in-plane thermal conductivity of ZnO along the direction perpendicular to the c-axis is thus extracted as K_r = 38.5 W m⁻¹ K⁻¹ from Fig. 19(b).

The beam-offset TDTR approach is subject to measurement uncertainty from the FWHM signals and from the input parameters, especially the laser spot size. Alternatively, based on the physical picture that the sensitivity of the TDTR signals to the in-plane thermal conductivity depends on how the spot size $w0$ is compared to $dp,r$⁠, an elliptical-beam TDTR approach was recently developed to measure the in-plane thermal conductivity of transversely anisotropic samples.¹¹⁹ 

In the elliptical-beam TDTR approach, the experiments are conducted following the same procedure as in the conventional TDTR, i.e., the pump and the probe spots are concentrically aligned on the sample surface, the ratio signals R = −V_in/V_out acquired as a function of delay time are used to derive the thermal properties, except that the pump beam is of a highly elliptical shape. A schematic of the elliptical-beam method is shown in Fig. 20. By using a highly elliptical pump beam for TDTR experiments, a quasi-one-dimensional temperature profile that has a fast decay along the short axis of the pump beam is induced on the sample surface. The detected TDTR signal is thus exclusively sensitive to the in-plane thermal conductivity along the short axis of the elliptical beam.

In the elliptical-beam configuration, the shape and the orientation of the elliptical pump beam can be controlled using a pair of cylindrical lenses. The RMS average of the pump and probe spot sizes can be characterized using the dependence of V_in on the spatial offset of the pump and probe beams at a short positive delay time (e.g., 100 ps) at a high modulation frequency of 10 MHz.³⁹ Generally, the size of the elliptical laser spot should be chosen so that the sensitivity of the TDTR signals to the in-plane thermal conductivity is sufficiently high along the short axis direction but is effectively suppressed along the long axis direction. A general guideline is that the major radius of the elliptical spot w_x should be at least five times the in-plane thermal penetration depth in the same direction, w_x > 5d_p,x, and the minor radius of the elliptical spot w_y needs to be less than two times the in-plane thermal penetration depth in the same direction, w_y < 2d_p,y.¹¹⁹ For low-thermal-conductivity materials such as quartz, a more stringent criterion of w_y < d_p,y would be needed to have a sufficiently high sensitivity to K_y. Other than that, the data acquisition, data reduction, and uncertainty analysis in the elliptical-beam configuration are all the same as in the conventional TDTR approach. The thermal model for the beam-offset TDTR configuration can also be applied to the elliptical-beam TDTR configuration but with the offset distance set as zero, since both employ the analysis in the Cartesian coordinates.

Figure 21 shows the in-plane thermal conductivity tensor of ZnO [11–20] measured using the elliptical-beam method, compared with those measured using the beam-offset method. A modulation frequency of 0.35 MHz was chosen for both methods. The beam-offset method used a circular laser spot with radius $w0=4.7μm$⁠, while the elliptical-beam method used an elliptical laser spot with a long radius of $wx=17.3μm$ and a short radius of $wy=4.5μm$⁠. Both measurements were conducted under their optimal experimental configurations. Overall, these two methods compare relatively well with each other, with the measured in-plane thermal conductivities within the error bars. However, the data by the beam-offset method scatter more significantly, whereas the data by the elliptical-beam method show better consistency. From the elliptical-beam experiments, the thermal conductivities along the in-plane directions perpendicular to and parallel to the c-axis of ZnO are determined to be K_a = 46 W m⁻¹ K⁻¹ and K_c = 56 W m⁻¹ K⁻¹, respectively. The K_c of ZnO determined from the elliptical-beam method (56 W m⁻¹ K⁻¹) is slightly lower than a first-principles calculation in the literature (62 W m⁻¹ K⁻¹)¹⁹⁵ but is consistent with another TDTR measurement of K_z of ZnO [0001] (55 W m⁻¹ K⁻¹),³⁷ which has its c-axis along the through-plane direction.

The knowledge of phonon mean free paths (MFPs) is important in developing microscopic pictures of the heat conduction processes, especially in micro-/nano-structures and in ultrafast processes where Fourier's law of heat conduction fails. Recent advances in atomistic simulations have enabled the calculations of phonon properties including phonon dispersions and phonon lifetimes with good accuracy, even without input parameters.^196–200 However, their direct measurements remain very challenging. Based on the simple idea that the measured apparent thermal properties (thermal conductivity and interface thermal conductance) would deviate from the predictions of Fourier's law when the characteristic length is comparable to or even smaller than the phonon MFPs, some experimental work based on TDTR and its variations has been proposed to directly probe the phonon MFPs. Such characteristic thermal length scales can either be the heater size or the thermal penetration depth in TDTR experiments. This section focuses on previous attempts in the construction of phonon MFPs using TDTR and its variations.

Understanding that the spot size-dependent TDTR approach has a limit on the smallest laser spot size achievable due to the diffraction limit, Hu et al.²⁰⁶ returned to nanostructured samples similar to Siemens et al.⁸⁸ to reconstruct the MFP spectrum by performing TDTR measurements of samples using nano-patterned heaters with size D varying from 30 nm to 60 μm. The schematic of the sample structure is shown in Fig. 23(a), and an example of the nano-patterned heater is shown in Fig. 23(b). Figure 23(c) shows that the apparent thermal conductivity of SiGe determined from TDTR experiments with the hybrid nano-patterned transducers depends on the nano-heater size D. The MFP spectrum $α(Λ)$ of SiGe was reconstructed based on Eq. (4.14) from the measured apparent thermal conductivity values, shown as the symbols in Fig. 23(d), which are in good agreement with the first-principles prediction of the thermal conductivity accumulation function [the dashed line in Fig. 23(d)]. This method has also been applied to other samples including sapphire, GaN, and GaAs. The similar approach of reconstructing phonon MFPs has also been applied in other pump-probe techniques like transient thermal grating and ultrafast X-ray based pump-probe measurements. Interested readers can refer to Refs. 89, 207, and 208 for more details.

In addition to the heater size, another important length scale that affects the phonon transport regime is the thermal penetration depth $dp$⁠, which describes the characteristic length of the temperature field when the surface of a sample is periodically heated at the frequency f. Phonon transport becomes quasi-ballistic when $dp$ is comparable to the phonon MFPs. Koh and Cahill⁹⁰ were the first to observe a modulation-frequency-dependent thermal conductivity in some semiconductor alloys measured using TDTR, and they attributed it to the non-diffusive phonon transport, i.e., phonons with $Λ>dp$ (and $Λ>2dp$ in their later publication²⁰⁹) do not contribute to the apparent thermal conductivity measured by TDTR. This simple interpretation, however, works as a good approximation only for semiconductor alloys due to their extremely broad MFP distributions²⁰⁹ but not for pure crystals, as frequency dependence was not observed in TDTR measurements of crystalline Si despite its very long MFPs of phonons.^185,187 Assuming the same cutoff length of $Lc=dp$⁠, Regner et al.⁹² measured the accumulation function of silicon using a variation of TDTR, the broadband frequency-domain thermoreflectance (BB-FDTR), as described in Sec. III A. However, their results failed to compare well with first-principles predictions at cryogenic temperatures. Wilson and Cahill¹¹² later pointed out that the discrepancy could be due to an artifact created by the transducer thermalization. In the BB-FDTR experiments by Regner et al.,⁹² a 55 nm Au/5 nm Cr bilayer was used as the metal transducer, and both the heat deposition and temperature sensing were assumed to happen at the surface of the Au layer. However, Wilson and Cahill¹¹² pointed out that a significant amount of the laser energy was deposited in the Cr layer instead of the surface of the Au layer due to the weak electron-phonon coupling in Au, neglect of which would cause a considerable error. A later publication by Regner et al.²¹⁰ confirmed the non-negligible effect of the transducer type on TDTR/FDTR experiments. To better understand the frequency-dependent thermal conductivity in TDTR and FDTR experiments, Yang and Dames²¹¹ proposed a refined model based on the Boltzmann transport equation and suggested that the relaxation time $τ$ as compared to the heating period 1/f, represented by the dimensionless number $fτ$⁠, rather than the Knudsen number $Λ/Lp$⁠, is the key quantity to determine the phonon transport regime.²¹¹ Yang and Dames²¹¹ then reconstructed the accumulation function using the relaxation times of Si and achieved much better agreement with first-principles calculations.

While both the spot size dependence and modulation frequency dependence in TDTR experiments have been utilized to reconstruct phonon MFPs, it was puzzling to understand why the limiting length scale sometimes depends on the spot size but not the modulation frequency, and other times on the frequency but not the spot size. Wilson and Cahill¹¹² provided a unifying picture on the failure of Fourier's law in TDTR experiments. By comparing TDTR measurements of crystalline Si, Si_0.99Ge_0.01 alloy, and boron-doped Si, they pointed out that Fourier's law can fail to predict the heat flux correctly in either in-plane or through-plane directions or both, depending on different circumstances.¹¹² In measurements using small spot sizes, the primary effect of the small spot size is to cause a reduced in-plane heat flux than Fourier's law prediction. The thermal diffusivity of high-wavevector phonons of the substrate determines whether Fourier's law fails in in-plane or through-plane direction. Fourier's law fails more readily in through-plane direction (manifested as a frequency dependence in the measured thermal conductivity) for materials with a low thermal diffusivity for the high-wavevector phonons, and Fourier's law fails in in-plane direction (manifested as a spot size dependence) for materials with a high thermal diffusivity for the high-wavevector phonons. Besides the work by Wilson and Cahill,¹¹² there are also a bunch of other theoretical and numerical studies attempting to elucidate the nature of frequency and spot size dependences in TDTR experiments.^209–215 Interested readers can refer to these references.

In addition to the frequency-dependent thermal conductivity observed in TDTR measurements, the transducer/sample interface conductance G was also reported to significantly depend on the modulation frequency in SiGe and some layered 2D materials like MoS₂³⁵ coated with Al transducers. The simultaneous frequency-dependence of K_z and G was explained by a different physical picture based on non-equilibrium phonon transport. Interested readers can also refer to Refs. 35, 192, and 216 for more details. Meanwhile, Vermeersch et al.¹⁰⁹ also observed frequency dependence in both the apparent K_z and G for an InGaAs sample coated with a 50-nm-thick Al transducer, and they suggested that such deviations from Fourier's law of heat conduction were due to the Lévy-type superdiffusive processes. By incorporating the Lévy superdiffusion dynamics into a model, they were able to fit the TDTR ratio signals measured as a function of delay time at different modulation frequencies simultaneously, from which they extracted the intrinsic K_z and G of InGaAs along with two other fitting parameters. Interested readers can refer to Refs. 109, 110, and 111 for more details.

To close, the knowledge of phonon mean free paths (MFPs) is important in developing microscopic pictures of heat conduction process. Variations of TDTR enabled many attempts in the construction of phonon MFPs with the assistance of recent developments in atomistic simulations. However, a convincing approach has not appeared.

During the past two decades, TDTR has been developed into a powerful and versatile tool for the measurements of thermal transport properties of bulk and thin film materials. In this tutorial, we have discussed the basic principles and implementation of the TDTR technique as well as its various configurations for measuring novel materials. By utilizing different laser spot sizes and modulation frequencies for the measurements, the heat transfer regime in TDTR experiments and consequently, the sensitivity of the TDTR signals to different parameters can be controlled, enabling TDTR measurements of multiple thermal transport properties such as the through-plane thermal conductivity, in-plane thermal conductivity, heat capacity, and the thermal boundary conductance of interfaces. TDTR has been demonstrated to measure a wide range of through-plane thermal conductivity from a high value of 2000 W m⁻¹ K⁻¹ of diamond to the ultralow thermal conductivity of ∼0.03 W m⁻¹ K⁻¹ for disordered WSe₂ thin films. Featured by a sub-picosecond time resolution due to the ultrafast pulsed laser used, TDTR can also be applied for the study of heat carrier dynamics, such as phonon mean free paths and electron-phonon coupling. However, while being powerful and reliable, TDTR is still subject to several limitations, which also offers great opportunities for future developments of the TDTR technique.

1. As an optical-based pump-probe technique, TDTR requires the sample surface to be optically smooth to have the probe beam specularly reflected into the detector. A hypothesis is that the diffusely scattered probe light is most likely modulated by thermoelastic effects and would have an erroneous contribution to the signal if received by the detector.¹² However, there is still no systematic study on the effect of diffusely scattered probe light yet. A general rule-of-thumb is that the RMS surface roughness of the sample should be <15 nm.¹² An alternative way to measure rough samples using TDTR is to deposit the metal transducer on a transparent substrate first, and then bond the rough sample onto the transducer and measure from the transparent substrate side. This, however, has a stringent requirement on the bonding between the metal transducer and the sample so that the additional thermal resistance introduced by the bonding would not dominate in the measurements. Meanwhile, TR-MOKE thermometry is expected to be more tolerant to the sample surface roughness because most of the spurious signals induced by the diffusely scattered pump beam, which should be independent of the magnetization states of the transducer, can be canceled out by subtracting the two signals recorded for oppositely aligned magnetization states of the transducer. This speculation, however, has not been rigorously proven so far.

2. The range of thermal conductivity that can be measured by TDTR is limited by the workable modulation frequencies and laser spot sizes. The modulation frequency f in TDTR is usually in the range of 0.2–20 MHz. TDTR measurements at f < 0.2 MHz are usually problematic due to the poor signal-to-noise ratio caused by the 1/f noise and the large uncertainty in determining the phase caused by the strong pulse accumulation. TDTR measurements at f > 20 MHz are also challenging due to the weak out-of-phase signals and the high level of the radio-frequency noise picked up by the detector and the signal cables. Besides, the Nyquist criterion^172,217,218 also limits the modulation frequency to be less than half of the laser repetition rate, i.e., <40 MHz. Likewise, the laser spot size w₀ in TDTR is usually limited in the range 1–30 μm. The lower value in w₀ is due to the diffraction limit (in fact, the diffraction limit cannot even be reached because the back focal plane of the objective lens cannot be filled due to the thermal lensing effect²¹⁹) and the higher limit in w₀ is due to the requirement of a minimum laser power intensity. Given the limits in the lowest modulation frequency and the smallest spot size, it is difficult for TDTR to measure in-plane thermal conductivity K_r < 5 W m⁻¹ K⁻¹.³⁷ Meanwhile, the limit in the highest modulation frequency also constrains TDTR measurements of thermally thin films (with thickness h < d_p,z). For example, it is very challenging for TDTR to measure the thermal conductivity of Si films with a thickness <1 μm at room temperature due to the insufficient sensitivity at modulation frequencies <10 MHz. Expanding the workable modulation frequency and laser spot size can greatly extend the measurable thermal conductivity range. For example, measurements of small in-plane thermal conductivity K_r < 5 W m⁻¹ K⁻¹ usually requires a lower modulation frequency of <0.2 MHz to induce sufficient in-plane heat diffusion. The challenge of a poor signal-to-noise ratio at low modulation frequencies can be overcome by averaging over many repeated measurements. The problem of strong pulse accumulation can be resolved by reducing the laser repetition rate. The advancement in laser technology with an adjustable repetition rate of laser sources can make it possible to extend the lowest possible modulation frequency so that TDTR can measure even lower in-plane thermal conductivities. On the other hand, the modulation frequency of TDTR can also be extended up to a few hundred MHz using a GHz repetition rate laser. The advantage of a high-frequency TDTR is that it has a much finer depth resolution and thus it can measure thermal conductivities of extremely thin films, interface thermal conductance of even higher values, and probe phonon dynamics over a broader range. Direct measurements at such high modulation frequencies are certainly challenging due to the strong coherent noise and the low signal level, which, however, can be overcome by using the heterodyne detection, which has been implemented successfully in BB-FDTR.¹⁴⁷ It is also possible to reduce the laser spot sizes even further below the diffraction limit through the use of near-field optics,²²⁰ by applying the pump and probe light pulses to localized regions of the sample.

3. Current TDTR cannot be used to measure the thermal conductivity of monolayer or few-layer two-dimensional (2D) materials. So far, the thermal conductivities of monolayer or few-layer 2D materials have been dominantly measured by micro-Raman spectroscopy and thermal bridge method, both with their intrinsic limitations that impair the accuracy of their measurements.^221,222 While many attempts have been made on direct TDTR measurements of semiconductors without metal transducers,^223–227 significant progress is needed in thermal conductivity measurements of monolayer and few-layer 2D materials.

4. Most analyses of TDTR data to-date are based on solutions of the diffusion equation with the assumption that all the thermal excitations are in equilibrium with each other and a temperature field is well defined. These assumptions are not rigorously justified unless the couplings between different heat carriers are sufficiently strong, and the mean-free-paths of all of the significant heat carriers are small compared to the characteristic lengths (e.g., thermal penetration depth, laser spot size, etc.). The thermal conductivity of some semiconductor materials extracted from TDTR experiments was found to depend on the modulation frequency or the laser spot size, suggesting that the heat diffusion equation cannot accurately describe the experiment. A more rigorous practice is to analyze the TDTR data based on phonon Boltzmann transport equation, which, despite some recent progress,^{110,111,215,228} inevitably complicates the data reduction process.

5. TDTR measurements at low temperatures <30 K are usually challenging due to the small heat capacities of both the metal transducer and the experimental sample that do not allow a high enough laser power for the measurements without inducing too much temperature excursion. On the other hand, TDTR measurements at high temperatures are mainly limited by the chemical and physical stabilities of the metal transducer layer. TDTR measurements using Al transducer is usually limited to be at <600 K due to the low melting point of Al. Researchers are looking for alternative metal transducers for TDTR measurements at higher temperatures (with Pt and HfN²²⁹ as two examples of the potential candidates) or attempting to perform TDTR measurements without metal transducers at high temperatures.¹⁴² Meanwhile, the advent of ultrafast laser sources with tunable wavelength over a wide range (such as a system based on optical parametric oscillator^230,231) would also be a great advantage for TDTR, as the probe wavelength could then be chosen to match a peak in the thermoreflectance of the metal transducer, thus increasing the signal-to-noise ratio significantly. TDTR with a tunable wavelength laser also has some other advantages. For example, Lozan et al.²³² used a red pump beam (λ = 800 nm) in a TDTR system to generate a surface plasmon on a gold film and used a green probe beam (λ = 532 nm) to detect its propagation because gold has a large thermoreflectance coefficient at the probe wavelength.

The authors sincerely thank Professor David Cahill at UIUC for helpful comments and suggestions after the first draft of this tutorial was posted on arXiv. The authors are also very grateful to the three anonymous reviewers for their constructive comments. R.Y. acknowledges the financial support by the National Science Foundation (Grant Nos. 0635579, 0846561, and 1512776), DOD (Grant Nos. FA9550-08-1-0078, FA9550-11-1-0109, FA9550-11-C-0034, and FA8650-15-1-7524), and ARPA-E (DE-AR, No. DE-AR0000743) over the past decade on modeling and characterization of thermal transport in nanostructured materials. R.Y. also thanks the previous students and post-doctors in his research group who have made significant contributions to some of the works reviewed here, including Jie Zhu, Jun Liu, Wei Wang, Xiaokun Gu, and many collaborators worldwide who provided materials or other assistance to the experimental details.