Thermoreflectance techniques and Raman thermometry for thermal property characterization of nanostructures Cite as: J. Appl. Phys. 128, 131101 (2020); https://doi.org/10.1063/5.0020239 Submitted: 30 June 2020 • Accepted: 16 September 2020 • Published Online: 05 October 2020 Susanne Sandell, Emigdio Chávez-Ángel, Alexandros El Sachat, et al. COLLECTIONS Paper published as part of the special topic on Photothermics ARTICLES YOU MAY BE INTERESTED IN Tutorial: Time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials Journal of Applied Physics 124, 161103 (2018); https://doi.org/10.1063/1.5046944 Analysis of heat flow in layered structures for time-domain thermoreflectance Review of Scientific Instruments 75, 5119 (2004); https://doi.org/10.1063/1.1819431 Nanoscale thermal transport Journal of Applied Physics 93, 793 (2003); https://doi.org/10.1063/1.1524305 J. Appl. Phys. 128, 131101 (2020); https://doi.org/10.1063/5.0020239 128, 131101 © 2020 Author(s). Journal of TUTORIAL scitation.org/journal/jap Applied Physics Thermoreflectance techniques and Raman thermometry for thermal property characterization of nanostructures Cite as: J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 Submitted: 30 June 2020 · Accepted: 16 September 2020 · View Online Export Citation CrossMark Published Online: 5 October 2020 Susanne Sandell,1,a) Emigdio Chávez-Ángel,2 Alexandros El Sachat,2 Jianying He,1 2,3 Clivia M. Sotomayor Torres, and Jeremie Maire2,a),b) AFFILIATIONS 1 NTNU Nanomechanical Lab, Department of Structural Engineering, Norwegian University of Science and Technology (NTNU), 7491 Trondheim, Norway 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), (ICN-CSIC) Barcelona, Campus UAB, E08193 Bellaterra, Spain 3 ICREA—Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain Note: This paper is part of the Special Topic on Photothermics. a) Authors to whom correspondence should be addressed: susanne.sandell@ntnu.no and jeremie.maire@neel.cnrs.fr b) Current address: Institut NEEL, CNRS/UGA UPR2940, 38000 Grenoble, France. ABSTRACT The widespread use of nanostructures and nanomaterials has opened up a whole new realm of challenges in thermal management, but also leads to possibilities for energy conversion, storage, and generation, in addition to numerous other technological applications. At the micro- scale and below, standard thermal measurement techniques reach their limits, and several novel methods have been developed to overcome these limitations. Among the most recent, contactless photothermal methods have been widely used and have proved their advantages in terms of versatility, temporal and spatial resolution, and even sensitivity in some situations. Among them, thermoreflectance and Raman thermometry have been used to measure the thermal properties from bulk materials to thin films, multilayers, suspended structures, and nanomaterials. This Tutorial presents the principles of these two techniques and some of their most common implementations. It expands to more advanced systems for spatial mapping and for probing of non-Fourier thermal transport. Finally, this paper concludes with discuss- ing the limitations and perspectives of these techniques and future directions in nanoscale thermometry. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020239 I. INTRODUCTION more challenging to characterize.1,2 When the characteristic dimen- sions of the structure, such as the thickness of a thin film, become The capability of a material to conduct heat is of paramount interest for many technological developments. One of the most rep- similar to or less than the mean free path or wavelength of the heat resentative issues is that of Joule heating in microelectronic compo- carriers in the structure, i.e., electrons or phonons, size effects start nents, which limits device performance due to insufficient heat to appear, and the thermal behavior is strongly altered compared to dissipation. The ability of a material to dissipate heat is primarily the bulk material. In thin films, this can be observed by reduction dependant on its thermal conductivity; Fourier’s law of heat con- in thermal conductivity, even in-plane, below a certain thickness duction states that the heat flux density along a temperature gradi- that is dependent on the material under study. At micro- and ent is the product of thermal conductivity and thermal gradient. nanoscale, many factors can impact thermal conductivity, from Although methods for measuring the thermal conductivity of bulk preparation to functionalization and design, and are of great inter- materials have long been well-established, the measurement of thin est not only to the scientific/academic community but also for films and interfaces as well as micro- and nanostructures, all of technological applications, as can be attested by numerous publica- which are extremely widespread in modern applications, is much tions, including many review articles.3–7 J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-1 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics To measure, understand, and engineer the thermal properties conduction, here given in its differential form, of these new materials and structures, many techniques have been developed and refined. Diverging from standard electrical techni- q ¼ κ∇T, (1) ques, new contactless photothermal measurement schemes have emerged. Their main advantage is the simplicity of preparation of where q is heat flux density, κ is thermal conductivity, and ∇T is the samples, without the need for contacts and electrical insulator the temperature gradient. In steady state measurements, the local layers for conducting samples. With the steady state electrical heat flux and temperature gradient are measured, and the thermal method, good thermal contacts are mandatory,8 and in some cases, conductivity can be calculated. The temperature gradient for a the extrinsic thermal contact resistances must be taken into spatial difference is found using temperature sensors, and the heat account in the data analysis. The well-established 3ω technique for flux is known and constant with time, i.e., δq/ δt ¼ 0. The family bulk and thin film samples9,10 requires an initial lithography of steady state techniques suffers from significant drawbacks, such process followed by metallic strip deposition and, finally, the as parasitic heat loss, contact resistance of temperature sensors, bonding of electrical contacts. Although measurement accuracy is long waiting times to reach steady state, and sample size require- in the order of 5%, the fabrication process can be challenging ments, as the samples usually must be larger than a few millime- depending on sample surface quality. Scanning thermal microscopy ters. Due to these drawbacks, transient methods are often preferred offers higher thermal spatial resolution,11–13 but data acquisition is for nanoscale measurements. slow and requires nontrivial analysis and modeling for the extrac- Transient measurements involve the use of a time-dependent tion of thermal conductivity. heat source with a localized temperature probe laser. Some transi- Consequently, a variety of advanced contactless techniques ent techniques include the hot-wire method, the laser flash diffusiv- have been developed, e.g., time-domain thermoreflectance ity method, the 3ω technique, the TTG technique, and transient (TDTR),14,15 frequency-domain thermoreflectance (FDTR),16 thermoreflectance techniques, including TDTR and FDTR. Due to thermal transient grating (TTG),17 photoacoustic method,18 and its non-contact, non-destructive nature, combined with easy sample Raman thermometry.19 With enhanced spatial and temporal reso- preparation and high accuracy, the thermoreflectance-based techni- lutions, photothermal techniques, and in particular thermoreflec- ques have become one of the most popular thermal characteriza- tance and Raman thermometry, have been widely used over the tion methods. past two decades to measure samples ranging from ultrathin films The time-domain thermoreflectance technique was pioneered with thicknesses down to 1 nm, to phononic crystals and novel by Paddock and Eesley in 1986,20 who used picosecond thermore- layered materials, both supported and suspended. Their implemen- flectance to measure the thermal diffusivity of thin metal films. tations have evolved over time and are now highly competitive for Since then, the technique has been used to measure thermal and a broad range of applications. acoustic properties of bulk samples,21 thin films,22,23 interfaces,24–27 This Tutorial focuses on thermoreflectance and Raman and liquids,28–30 among others. It has also been used to probe the thermometry techniques as means of measuring thermal proper- fundamentals of heat transport, such as phonon scattering mecha- ties of nanostructures and novel materials, with the added value nisms31 and non-equilibrium electron–phonon interactions.32,33 of local measurement allowing spatial temperature mapping and In addition, it has been widely applied in the field of picosecond investigation of non-Fourier thermal transport. This information ultrasonics.34 In the thermoreflectance method, the sample’s is aimed at everyone interested in the thermal properties of temperature is modulated using a pulsed or frequency-modulated novel nanostructures and materials and how to characterize laser (the pump beam) and to detect this temperature change them. This Tutorial paper is organized as follows: Section II by monitoring the reflectance of a second laser (the probe beam). introduces the basic principle and implementation of the ther- The pump beam pulse energy is optically absorbed at the sample moreflectance technique for thin films, suspended nanostruc- surface that creates a local temperature increase. At the surface, tures, electron–phonon relaxation, and picosecond ultrasonics. the temperature-induced strain alters the optical constants of the Section III introduces Raman thermometry as an alternative and surface, which leads to a change in reflectivity ΔR. This is what is widely applicable photothermal characterization method. called the thermoreflectance effect.35 Section IV expands on more advanced approaches to thermore- To ensure that the pump laser is absorbed at the surface of the flectance and Raman techniques, finally discussing non-Fourier sample, the sample surface is typically covered with a thin metal thermal transport at low spatiotemporal scales. Finally, Sec. V film called the transducer. In the transducer, the reflectivity change summarizes and compares the techniques and discusses limita- ΔR/R due to a temperature change ΔT is linear over a temperature tions and future outlooks. range of a few degrees,   ΔR 1 δR ¼ ΔT ¼ CTR ΔT, (2) R R δT II. THERMOREFLECTANCE: CLASSICAL TDTR AND FDTR where CTR is the proportionality constant or thermoreflectance A. Basic principles coefficient. CTR is usually in the range of 10−5 to 10−3 K−1. Thermal characterization techniques can be divided into The thermoreflectance of a sample depends on a number of two main categories: steady state and transient measurements. factors, including optical absorption, light polarization, interplay of Steady state measurements are based on the Fourier law of heat electronic bands and electronic transitions, and electron–phonon J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-2 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics interactions.36 Therefore, the effect is strongly dependent on the probe beam wavelength and the transducer material. If CTR is known, the measured reflectivity change of the probe beam is a proxy for the local temperature change. As thermoreflectance is a transient method, knowledge of the absolute temperature rise is not required. However, in order to achieve the maximal temperature resolution, the wavelength-transducer material couple must be chosen carefully in order to maximize CTR . The thermoreflectance technique can be implemented as either time-domain thermoreflectance (TDTR) or frequency- domain thermoreflectance (FDTR), each with basic and various advanced implementations, as described in Secs. II B–V. Although in this work we focus on configurations implemented in laborato- ries, some of the implementations, in particular those based on pico- and nanosecond pulses, have recently been made commer- cially available.37–39 B. Femtosecond TDTR implementation A typical TDTR setup is shown in Fig. 1(a). The current most common TDTR implementation relies on a Ti:Sapphire mode locked laser at a wavelength around 800 nm, with a repetition rate of ∼80 MHz and laser pulses of ∼100 fs, due to its excellent beam quality and ultrafast pulses. To protect the oscillator from backre- flections, it is possible to install an optical isolator at the output of the laser, whose intensity can be adjusted with either a λ/2 wave- plate in front of the isolator or neutral-density filters along the optical path for more flexibility. To separate the beam into two, i.e., the pump beam and the probe beam, the preferred solution is to use a polarizing beam splitter (PBS) preceded by a λ/2 waveplate to FIG. 1. (a) Schematic of a typical time-domain thermoreflectance (TDTR) adjust the amounts going to the pump and probe beam paths, as setup. (b) Schematic of a typical frequency-domain thermoreflectance (FDTR) the probe beam power focused on the sample should be weak setup. The detectors are connected to a lock-in amplifier (not shown). The enough in comparison to the pump beam to avoid self-heating of camera, commonly used to show the sample surface, is not included in the the sample. This solution simultaneously cross-polarizes the pump schematics. (a) Adapted from Abad et al., Renew. Sustain. Energy Rev. 76, and probe beams, simplifying the detection as explained below. 1348–1371 (2017). Copyright 2017 Elsevier. (b) Adapted from Schmidt et al., Rev. Sci. Instrum. 80, 094901 (2009). Copyright 2009 AIP Publishing LLC. The most common implementation consists of modulating the pump beam with an electro-optic modulator (EOM) connected to a function generator, modulating the beam at frequencies ranging from 0.2 MHz to 20 MHz. The use of different modulation frequen- beam.45 This solution can be preferred, so that the detection of the cies enhances or decreases the sensitivity to specific thermal param- probe beam is less sensitive to misalignment. eters, primarily thermal conductivity and thermal boundary The probe beam is then focused on the sample in the same conductance (TBC).40 This frequency is used as the reference for manner as the pump beam. To ensure correct interpretation of the the lock-in amplifier used for signal acquisition. It is also possible data, the pump signal should be completely filtered out and only to use acousto-optic modulators (AOM) for cost-efficiency.41–44 the probe beam should reach the photodetector. This is because the Note that an AOM operates at a fixed frequency and, therefore, reflected pump beam is ∼104 times more intense than the thermor- lacks the versatility of EOMs to study a larger variety of materials eflectance signal, due to the small value of the thermoreflectance and systems. The pump beam is then focused on the sample with a coefficient. With cross-polarized beams, a first stage of filtering lens or a microscope objective for better spatial resolution. The with >99% efficiency can be achieved by placing a PBS between the probe beam is expanded to minimize the beam divergence and objective and the detector. A second method is usually combined then delayed with respect to the pump beam with a mechanical with the first to achieve further filtering. Three common configura- delay line. Due to the long optical path of the probe beam, this sol- tions can be used to that effect: (i) Spatial filtering. Before the ution requires careful alignment of the delay stage to avoid focusing objective, the pump beam is kept parallel to that of the misalignment-induced changes in the signal that would cause mis- probe but slightly shifted so that its reflection can be filtered out interpretations of the measured data. The optical elements used in with an aperture behind which the probe beam is detected by a fast the pump and probe arms of the setup can also be interchanged, photodetector; (ii) the two-tint configuration,46 in which two i.e., advancing the pump beam in place of delaying the probe beams slightly shifted in wavelength, typically at 785 ± 5 nm, are J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-3 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics then separated with a sharp-edged wavelength filter before detec- thicker than 50 nm in order to ensure that it is optically opaque, to tion; and (iii) the two-color configuration.28,47 Second harmonic avoid spurious thermoreflectance signals generated by the tempera- generation, with non-linear optical crystals such as barium borate ture field and changes in the optical constants of the sample below (BBO), can be used to convert either beam—usually the pump to the transducer.41 Also, the transducer makes it possible to assume a keep an optimal thermoreflectance coefficient—to 400 nm, enabling surface heat flux boundary condition, which simplifies the post- easy color separation and filtering. The first configuration is easier measurement analysis. to implement but only works for optically flat samples, whereas the The pump laser can be modulated by passing it through an other two configurations reduce the available laser power, which electro-optic modulator (EOM) driven by a function generator. can make it impossible to measure some highly thermally conduc- Alternatively, the pump laser can also be modulated through the tive materials such as diamond depending on the initial power and function generator of the lock-in amplifier, which is the configura- efficiency of the second harmonic generation. tion shown in Fig. 1(b). For both modulation methods, the modu- The RF electrical signal output by the photodetector is then lation frequency range is typically 10 kHz–20 MHz. As for TDTR, fed to a lock-in amplifier synced to the EOM modulation fre- the thermoreflectance signal from the sample is small, usually in quency. Care should be given to the choice of lock-in amplifier and the range of 10−4 V as detected by the lock-in amplifier, which its use. Although higher harmonics can also be used to derive the means that the setup must be built to maintain a high signal-to- thermal properties,40 the choice of the fundamental frequency noise ratio, including the use of optical elements designed to avoids complications in the data analysis. Indeed, if the EOM mod- minimize optical losses along the beam path. When the pump is ulation output is a square wave, a lock-in amplifier with sine wave modulated by an EOM, the pump laser passes through a beam multipliers (e.g., Zurich Instruments HF2LI) is preferred to avoid sampler that splits of a small portion of the beam, typically 1%, contamination of the signal with higher harmonics of the square into a reference photodetector. This reference detector signal can wave. If the modulation of the pump beam is a sine wave, a more be sent to the lock-in amplifier as the reference signal. If the pump economical option (e.g., Stanford Research Systems SR844) can be beam is modulated directly from the lock-in amplifier signal, it can selected. Adding modulation to the probe signal can enable the use this as an internal reference signal. The pump and probe beams implementation of a double lock-in scheme to avoid coherent are sent to the sample through the same objective. The pump and pick-up in the signal at the detection frequency.48 probe beams are then reflected back through the objective, travel The system is usually completed by a removable flip-mirror or through a beam splitter, and arrive at the detector. For the same a beam splitter in front of the aperture or the filter preceding the considerations as for the TDTR method, the main detector should detector to enable the sample surface and pump and probe beams only receive the probe beam signal. This is facilitated by choosing to be observed with a complementary metal–oxide–semiconductor different wavelengths for the pump and probe beams, so that the (CMOS)/charge-coupled device (CCD) camera. This helps to pump beam can be filtered using an optical filter. Both the refer- ensure pump and probe beam overlap and focus. The extraction of ence and the main detector are amplified silicon detectors that thermal properties usually relies on a multilayer model detailed in convert the optical signals into electrical signals that are sent to the a 2004 work by Cahill.45 As it is similar to that of the FDTR tech- lock-in amplifier. nique, details are given in Sec. II D. The lock-in amplifier outputs an in-phase Vin and an out-of-phase Vout signal at the modulation frequency. These signals represent the amplitude and phase of thermoreflectance response C. FDTR implementation of the probe beam as R ¼ Vin /Vout and w ¼ tan1 (Vout /Vin ), The frequency-domain thermoreflectance (FDTR) technique respectively, which are then used in the thermal model to derive measures the thermoreflectance effect as a function of modulation the sample’s thermal properties. The advantage of using the phase frequency of the pump laser. The pump beam heats the sample response over the amplitude response is that it has increased sensi- surface periodically at a frequency f, and the probe beam measures tivity, especially when the out-of-phase component, Vout , is small. the temperature response through a proportional change in surface An important aspect of FDTR is to accurately determine the reflectivity. The setup described here is based on a typical FDTR thermal phase signal wthermal . The measured phase signal, wmeasured , setup, as shown in Fig. 1(b). A FDTR setup typically uses two dif- is really a sum of contributions, ferent continuous wave (CW) diode laser sources, one as the pump beam and the other as the probe beam. These have the advantage wmeasured ¼ wthermal þ w path þ winstrum : (3) of being cheaper than the ultrafast pulsed laser, although it is also possible to use a TDTR setup as described in Sec. II B for FDTR, The second component, w path , is introduced by different optical by holding the mechanical delay stage at a fixed position and scan- path lengths of the beam. The last component, winstrum , is intro- ning through frequencies using the EOM.16 Hereafter, we describe duced by the electronic components of the setup and the electrical the system comprising the two diode lasers. cables.16 An important exercise of FDTR is, thus, to zero out The probe laser should be chosen to have a wavelength that w path and winstrum . This is achieved in a two-step approach. gives the maximum thermoreflectance signal from the metal trans- The first step consists of eliminating w path by making sure ducer film. For a typical wavelength of λ ¼ 532 nm, the optimal the pump and probe beams have identical optical path lengths: transducer material is gold. This probe–transducer couple gives a the optical path length between the modulated pump laser and the CTR ¼ 2:36 104 K 1 at ambient conditions, which is exception- reference detector must be equal to the path that the probe beam ally high.36 The thickness of the transducer film must usually be takes from the sample to the probe detector. The error introduced J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-4 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics by optical path length differences scales linearly with frequency, so at 10 MHz modulation, a path length difference of 1 cm translates to a phase error of 0.1°. In practice, this is done by replacing the sample with a mirror, that is, a reflective surface with no thermore- flectance effect (CTR ¼ 0), and placing the reference detector on a translational stage. The reference detector can be translated along this stage until the measured phase difference is 0°. The second step is to correct for winstrum , which is introduced by other components in the setup, such as the photodetectors, cables, and instruments. This quantity is frequency-dependent. Hence, a full frequency sweep is taken using the zero- thermoreflectance mirror as the sample while recording winstrum for all frequencies. Then, this correction factor must be subtracted from all subsequent measurements to ensure that wmeasured ¼ wthermal . In order to determine the sample’s thermal properties, all other properties of the sample system should be known as accu- rately as possible. This includes the dimensions of the sample, the thermal properties of the thin transducer film, and the laser radii. The dimensions of the sample, such as the transducer film and thin sample thickness, are usually determined using Atomic Force Microscopy (AFM) or ellipsometry. The thermal properties of the FIG. 2. Example of a knife’s edge measurement of the pump laser spot in focus. The 1/e pump radius from this fit is 13.46 μm. metal transducer can be determined using the same FDTR setup, by measuring the properties of the metal film on a known sub- strate, such as quartz glass. The thermal properties of the trans- ducer are sensitive to the film’s deposition conditions, and may be by using a program such as ImageJ, from which the laser radius reduced to only 80% of the bulk value.49 Therefore, when deposit- can be extracted. This approach has typical errors of 5%–6%. ing a transducer on the sample, it is advisable to deposit in parallel When performing the FDTR measurements, it is important to on a reference quartz sample in close proximity to the sample in make sure that the local temperature increase in the sample does the deposition chamber. This reference can then be used to deter- not exceed a few Kelvin. This is important for two reasons. First, mine the transducer’s thermal properties before measuring the the thermoreflectance coefficient is temperature-dependent and sample. The thermal properties of the transducer can also be deter- valid at room temperature. Above 10 K, the linear relation between mined using a van der Pauw four-point probe measurement of reflectivity and temperature change can no longer be assumed. electrical resistance and relating it to the thermal properties using Second, the FDTR is a non-destructive technique; however, this is the Wiedemann–Franz law.50 For the underlying substrate, bulk not valid if the sample is inadvertently exposed to high-intensity thermal properties can be assumed. laser beams that change the sample by a large local temperature The laser spot radii are one of the main error sources in ther- increase. Thus, the appropriate laser intensities should be deter- moreflectance measurements.51 In order to measure the pump and mined. This can conveniently be done using the script provided by probe laser radii, various approaches are used. A common method Braun et al., which calculates the steady state temperature increase is the knife’s edge technique. In an approach similar to Yang due to pulsed or CW laser irradiation for multilayered substrates.53 et al.,52 the radius can be obtained by scanning the laser beam In order to increase the signal-to-noise ratio in FDTR mea- across a quartz glass slide with an, e.g., 80 nm Au layer patterned surements, various approaches have been made. Yang et al.52 with a transparent window. The intensity of the reflected light is implemented FDTR using balanced photodetection, which consisted measured while the slide is translated along the axis perpendicular of a balanced photodetector with two well-matched photodiodes. to the incident laser beam by a servo motor actuator with a step The post-sample probe beam was directed at one photodetector, size of 20 nm. The result from a typical measurement is shown in while the pre-sample beam was sent to the other. The output cur- Fig. 2. The beam intensity as a function of translation distance is rents of the detectors were subtracted in the detector. The signal was typically fitted using a Gaussian cumulative intensity distribution, sent through a low-noise amplifier, thus removing the common from which the Gaussian beam radius can be extracted. The mode noise in the probe beam. This significantly reduced the phase Gaussian beam intensity is (r) / exp[(r/rlaser )2 ] and the 1/e noise, which especially affects the sensitivity for low frequencies. radius of the curve is taken as the laser’s radius. This approach typi- Malen et al.51 proposed a fiber-aligned FDTR (FAFDTR) to reduce cally has an error of less than 5%. The advantage of this method is noise. In this approach, the lasers were fiber coupled diode lasers that the “knife” can also be used to measure the thermal properties that were perfectly aligned using simple fiber coupling. A common of the transducer and to measure laser radii. mode rejection scheme similar to Yang’s approach was used, where Another method is to measure the intensity profile of the the lock-in amplifier subtracts the pre-and post-sample beam signals. focused laser spot on the sample using the camera (CMOS/CCD This scheme resulted in a signal-to-noise ratio increase of one order detector) integrated into the FDTR setup,51 as described for the of magnitude. Another approach is to use heterodyne detection as TDTR method. The intensity profile of the image can be obtained employed by Regner et al.54 in their implementation of broadband J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-5 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics FDTR (BB-FDTR) with modulation frequencies up to 200 MHz. on the sample surface is Qω ¼ 12 Q pump (1 þ cos(ωt)), where ω is This approach is described in more detail in Sec. IV A. the modulation frequency and Q pump is the DC power of the pump laser at the sample. The probe beam reflected from the surface picks up a small, modulated signal originating from the modulated pump D. Thermal transport modeling and signal processing beam heating: Iω ¼ I probe Rω , where Rω is modulated reflectivity. In doing thermoreflectance measurements, the goal is typically The amplitude Iω is sensitive to laser instabilities, so the phase of the to find an unknown thermal parameter, such as effective thermal probe beam is often  used for thermal modeling. The phase is defined conductivity, of a specified layer in the sample. This is done as an as w probe ¼ tan1 Im(Iω ) Re(Iω ) , where Im(Iω ) and Re(Iω ) are the imagi- inverse problem, by fitting the measured data with a thermal con- duction model using the unknown parameter as the fitting parame- nary and real components of Iω , respectively. This is output as ter. The most significant theoretical advance in the field was w ¼ tan1 (Vout /Vin ) from the lock-in amplifier. derived by Cahill in 2004,45 who calculated the frequency-domain The thermal modeling consists in finding a numerical solution thermal response to Gaussian laser heating of a multilayered struc- to the analytical expression for the frequency domain thermal ture. Rigorous explanations of thermal modeling of both isotropic response of a multilayer sample heated by a modulated Gaussian and anisotropic thermal conductivities are described elsewhere in beam, probed by another Gaussian beam. The equation, proposed the literature.45,55 The technique has also been used to probe pho- by Cahill,45 is nonic and electronic interactions.31,32,56 ð1 ! Here, the thermal analysis used to determine the effective π 2 k2 (r 2pump þ r 2probe ) thermal conductivity of a thin film is presented. Each layer is char- ΔT ¼ 2πQω H(χ) exp χdχ, (4) acterized by three properties: thickness t, thermal conductivity κ, 0 2 and volumetric heat capacity C, as seen in Fig. 3. The effective thermal conductance of the thin film also contains TBC, G, where rpump and rprobe are the pump and probe beam radii, respec- between the thin film and its two adjacent layers. tively, and H(χ) is the Hankel transform of the frequency-domain The model assumes a Gaussian, axisymmetric profile, and the heating response for a multilayered sample heated at the surface by thermal properties are assumed to be isotropic. The metal trans- a periodic point source. Detailed derivation of H(χ) is given in ducer layer simplifies the thermal analysis, since the rapid thermal- Eqs. (14)–(18) in Ref. 45. Figure 4 shows the measured and best fit ization of the layer means that one-dimensional heating along the z calculated phase response to modulated heating by a FDTR setup axis can be assumed. A semi-infinite boundary condition is similar to the one described in Sec. II C. The samples are 3.8 nm assumed for the substrate, which is usuallypthermally ffiffiffiffiffiffiffiffiffiffiffiffi thick com- and 37 nm polymethyl methactylate (PMMA) film on a Si substrate pared to the thermal penetration depth δ ¼ 2α/ω0 , where α is the with an 80 nm Au transducer film. r pump ¼ 2:4 + 0:12 μm. and thermal diffusivity of the substrate and ω0 is the lowest frequency of the measurement. The power from the pump beam impinging FIG. 4. FDTR phase shift vs pump beam modulation frequency for a 37 nm FIG. 3. A typical sample configuration for thermoreflectance measurements. (red) and 3.8 nm (blue) PMMA film on a Si substrate. The thermal model best fit Each layer is characterized by its thickness t n , thermal conductivity κn, and is also shown together with dotted lines showing the best fit for +10% of volumetric heat capacity C n. Also shown are the TBCs Gn between adjacent thermal conductivity. Adapted from Sandell et al., Nanomaterials 10, 670 (2020). layers. Copyright 2002 Authors. J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-6 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics r probe ¼ 13 + 0:65 μm. The fit parameter for the calculation Samples with a lower thermal diffusivity or a higher increase in is the effective thermal conductivity of the film, which temperature will result in a better signal to noise ratio. is κ eff ¼ 0:1890 W m1 K1 for the 37 nm film and Similarly to FDTR measurements, temperature increase is kept κ eff ¼ 0:116 W m1 K1 for the 3.8 nm film. The overall experi- to a minimum, typically below 3 K, to avoid modifications of the mental error was 10%.57 thermal properties due to large temperature changes, as both heat capacity and thermal conductivity are temperature dependent. The temperature increase due to the probe and pump lasers can be esti- E. Nano- and microsecond TDTR for suspended mated analytically or numerically. During the TDTR measurement, structures the temperature increase of the Au layer can also be estimated by The methods as described so far have mainly been used to the relative change in reflectivity of the layer given that the ther- characterize the thermal properties of bulk, thin film samples and moreflectance coefficient of Al or Au is tabulated.72–74 multilayer materials, but they are restricted to supported samples The measured time (t) dependence of this heat dissipation due to either short timescales, limiting to high thermal diffusivity [Fig. 5(a)] can be fitted by an exponential decay, exp(t/τ), where samples, or the analytical modeling approach selected which pre- τ is the thermal decay time—the only parameter characterizing vents quantitative analysis of complex geometries. To circumvent heat conduction in each sample. To extract thermal conductivity the timescale issue, several pico- and nanosecond TDTR imple- from the experimental signal, the experiment is reproduced in mentations have emerged3,58 that have been used, for example, to COMSOL Multiphysics using the finite elements method (FEM). characterize a thermal transistor.59 The TDTR technique has also The 3D model reproduces the geometry of the measured sample, been adapted to measure suspended thin films and nanostructures. which can be imported from electron microscopy images. Due to The first such implementation used a patterned aluminum trans- the low increase in temperature confirmed from the small reflectiv- ducer on which the lasers are focused. It has been used to measure ity change, radiation losses are negligible, and convection becomes the thermal conductivities of nanowires,60,61 phononic crystals,62–68 critical for samples with low diffusivity, e.g., suspended structures. and pillar-based structures69 fabricated from a ∼100 nm thick The thermal model accounts for conduction in the material and silicon layer. This technique was also used to demonstrate the conduction/convection in air for experiments not performed in impact of the wave properties of phonons at low temperatures70 vacuum. The simulation is performed with different values of and the possibility to focus a phonon flux in a narrow region thermal conductivity of the material of interest and decay times are thanks to surface scattering of phonons in engineered structures.71 extracted by fitting an exponential decay curve to cooling. The In this method, a pulsed laser periodically heats the surface of thermal conductivity can be extracted by fitting the experimentally the sample, whereas the reflected power of a CW probe laser, mea- extracted decay time to the function linking simulated decay times sured by a standard photodetector, is sent to an oscilloscope for real- to thermal conductivity. This process is shown in Figs. 5(a)–5(b). time signal processing and analysis instead of the more classic lock-in For phononic crystals, the total error of this technique has been amplifier. This is made possible by the slow temporal dynamics of estimated to be in the range of 5%–10%.68 This estimation includes suspended structures for which the heating time is usually chosen to both the error in measurement and the error in estimating the be several microseconds. The repetition rate is similarly adjusted to structure size. Indeed, an error of a few nanometers in the measure- allow the sample to cool down to its initial temperature. ment of hole diameters, which is considered average, modifies the Since the full temporal trace is acquired in one acquisition, the final measured thermal conductivity by 1%–2%, depending on signal to noise ratio can be increased by averaging with the oscillo- the geometry. The measurement error itself is estimated to be on scope combined with post-processing in LabView. The oscilloscope the same order and includes the fitting error which is kept below in used to average up to 104 waveforms, significantly improving the 2%. Overall, measuring nominally identical structures yields an signal, from which heating and cooling can now be seen [Fig. 5(a)]. error between 5% and 8%. FIG. 5. (a) Schematics of the micro-TDTR setup. (a) Among the several decay curves with different values of sample thermal conductivity (κ nw ), one of them fits the experimental data. (b) κ(τ) is interpolated from the pairs [thermal conductivities sampled by finite element method (FEM); decay time]. A corresponding thermal conduc- tivity can be found for any experimentally measured decay time (τ exp ). Adapted from Anufriev et al., ACS Nano 12, 11928 (2018). Copyright 2018 American Chemical Society. J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-7 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics F. Measuring thermal transport at interfaces, novel dip.26 Thickness of the films can be measured using AFM, or ideally, layered materials, and ultrathin films ellipsometry, as a non-contact optical technique. With constant miniaturization of electronic devices, the density of interfaces between materials is increasing rapidly. At these inter- G. TDTR beyond thermal conductivity faces, a considerable thermal resistance arises. As the dimensions of devices reach the nanometer scale, the thermal resistance of inter- Although this Tutorial mainly focuses on thermal conductivity faces becomes a dominant obstacle for heat transport, over the characterization, TDTR also has a wider range of applications intrinsic thermal conductance of different layers. The inverse of the related to electron–phonon thermalization at short timescales and interface thermal resistance is G, the TBC. The TBC depends on a in the field of picosecond ultrasonics at intermediate timescales, combination of factors such as the vibrational overlap, the interface both of which are briefly described here. structure, and chemical bonding.75 In the last 10 years, a number of Phonons and electrons are two main energy carriers in metals experimental studies have investigated the effect of chemical bonding and semiconductors. Knowledge of their coupling is required for on thermal transport.76–79 A popular strategy has been to use func- many purposes, such as advanced engineering of solid-state tionalized self-assembled monolayers or hydrogen bonds to investi- devices86 and accurate modeling of metal manufacturing processes, gate the effect of density and strength of chemical bonds on the which involves pulsed laser processes.87 In order to find the elec- TBC, which can increase G up to an order of magnitude.79–82 Others tron–phonon coupling constant, the interactions between these have increased the TBC through increased interlayer adhesion with carrier populations are measured in real time. Due to the sub- the addition of nanometric metal films, such as chromium,75 tita- picosecond timescale at which these processes take place, the nium,83 and nickel.57 Lately, two-dimensional (2D) materials such as TDTR technique is uniquely adequate to study this phenomenon. graphene and hexagonal boron nitride (h-BN) have also been consid- When a laser beam pulse is incident on a metallic surface, the ered as strategies for improving G.81,84 Typically, thermoreflectance metal’s electrons will absorb the energy and subsequently relax into techniques are employed to probe the TBC, as described below. a Fermi distribution through electron–electron (e–e) collisions. In order to have a high sensitivity to interface properties, an This initial relaxation time is typically on the order of 10–100 fs, important aspect of sample preparation is for the adjacent thin film and the e–e interactions usually dominate over electron–phonon to be thermally thin. This involves preparing thin films on the (e–p) interactions in this timespan. When the electron population order of the Kapitza length, reaches equilibrium, the high-temperature electrons transmit their energy to the lattice through e–p scattering processes as the elec- κ trons travel diffusely away from the thermally excited region. These lK ¼ , (5) interactions eventually lead to two carrier populations reaching an G equilibrium temperature. After equilibrium is reached, thermal which is the thickness of a film of thermal conductivity κ with the transport can usually be described by the Fourier law. The time same conductance value as the TBC G of interest. A typical sample frame between the initial laser pulse hitting the surface and the e–p configuration is shown in Fig. 3 in Sec. II D. equilibrium is on the order of 1 ps for most metals. These interac- The measured quantity in a thermoreflectance measurement is tions are often described by the two-temperature model,88 which the effective thermal conductivity, κ eff , which is really a sum of con- describes the time evolution of the electron and lattice tempera- tributions, which can be approximated by a series resistance model, tures. It takes into account that the electron and phonon popula- tions have distinct heat capacities and temperatures, and the e–p  1 thermalization process is governed by the e–p coupling factor, g. t 1 1 For 30 years, this phenomenon has been studied using the TR κ eff (t) ¼ t  þ þ , (6) κ film G1 G2 technique. During this time, several corrections have been made to the two-temperature model. Also, g has been found to depend on where κ film is bulk thermal conductivity of the film, t is the film the properties of nearby interfaces32 and depends on both the thickness, G1 is the TBC between the transducer and the thin film, lattice temperature and laser fluence.89 Still, several questions about and G2 is the TBC between the thin film and the substrate, as shown the scattering processes driving thermal transport remain unan- in Fig. 3. In order to measure G from this configuration, κ eff must swered, such as follows: How do the thermal and non-thermal be measured for a range of t. Then, G can be determined from states of the electron and phonon populations contribute to the e– curve fitting. p equilibration process? Which processes contribute most signifi- An extreme case for thin film sample preparation is when the cantly to the overall rate of energy transfer?90 This is still an active film has an exceptionally low κ film , as is the case for most polymers. area of research, and femtosecond TDTR is one technique that can Then, lK becomes on the order of a few nm thick. In this case, be used to investigate the phenomenon experimentally, monitoring extreme care must be taken in order to prepare films free of defects the temperature evolution in real time. and holes and to measure thickness accurately. Film preparation can It is interesting to note that intermediate timescales, between be done from spin coating dilute polymer solutions onto a meticu- electron relaxation and phonon thermal transport, correspond to the lously clean substrate,57,85 which deposits an amorphous polymer acoustic signal, which can give geometrical and structural informa- layer. If more control over the polymer structure is needed, it is pos- tion and are, therefore, also highly relevant for interface characteriza- sible to dip-coat the sample using the Langmuir–Blodgett technique, tion.34 The ultrashort laser pulse generates a local thermal stress in which deposits a monolayer of polymer onto the sample with each the region heated by the hot carriers, usually up to 50–100 nm in the J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-8 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics FIG. 6. Schematic examples of Raman modes as thermometer: (a) Stokes, Rayleigh, and anti-Stokes signals (b) redshift, intensity reduction, and broad- ening of linewidth due to temperature increasing. metallic transducer used in the TDTR technique for sub-ps laser emitting a photon with the same energy as the incident. For pulses. This stress, in turn, launches an elastic strain pulse that will Raman scattering, on the other hand, the excited electron does not propagate and interact with buried interfaces and other features, return to its initial state, and consequently, the emitted photon will leading to reflected and transmitted acoustic waves. When the laser have an energy lower (Stokes process) or higher (anti-Stokes spot radius is much larger than the optical absorption depth, which process) than the incident. In the Stokes process, the electron is the case here with spot sizes of a few micrometers, the problem absorbs part of the energy decaying in a higher energy level. For can be mostly considered to be one-dimensional. Most often, acous- the anti-Stokes process, on the contrary, the electron starts at a tic pulse propagation is studied normal to the surface. When the higher energy level and decays to a lower level, transferring the reflected wave, either from layer interfaces or buried features, reaches excess of energy to the scattered photon. Compared to Stokes scat- the surface, the reflectivity of the transducer is modified by the pho- tering, anti-Stokes scattering is normally very weak. toelastic effect and echoes appear in the detected optical signal.91 Figure 7(a) shows a simplified scheme of the Raman spec- This technique has been widely used34 as a contactless characteri- trometer equipped with a notch or edge filter, slit, grating, and two- zation technique for both solids92 and liquids93 since its first dimensional multichannel CCD detector. The filter is used to elim- implementation by Thomsen et al.94 to detect the acoustic signal inate the laser line, either by allowing transmission of photons with in the GHz to THz range. Among applications, nonlinear ultra- wavelengths greater than the incident beam (edge filter) or by sonics involving acoustic solitons95–97 or shock waves98 is rapidly blocking photons with similar wavelength (± few nm, notch filter). expanding, as is the study of mechanical contacts99,100 and the use of optical cavities.101,102 These structural studies can be directly combined with thermal measurements and require a model adapted to the incidence of both pump and probe beams, either from the front or from the back for transparent samples.91,103 Several review papers have been dedicated to this technique, its development, and applications.34,104 III. FROM RAMAN SPECTROSCOPY TO RAMAN THERMOMETRY Raman spectroscopy is a light-scattering technique dedicated to the study of the vibrational properties of materials based on light–matter interaction. In general, a monochromatic laser is focused on a sample, and the backscattered light is collected and analyzed by the Raman spectrometer. Most of the light will be elas- tically scattered conserving its energy (Rayleigh scattering); however, a small fraction (1 part in 10 million) will be inelastically scattered with a slightly different energy (Raman scattering). Figure 6(a) shows a schematic representation of the energy levels for the scattering processes. For Rayleigh scattering, the incident FIG. 7. Scheme of (a) typical Raman setup, and its application for (b) single photon excites the electrons of the system from its ground state to and (c) two-laser Raman thermometry. a virtual state that, when it decays, returns to the initial state by J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-9 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics Once the laser light is rejected, the recollected signal is focused on energy. Otherwise, the grating of the spectrometer must move from a slit diverging directly to a concave mirror. The focus of this one peak to another, which can cause the amount of light reaching mirror is located onto the slit; hence, the reflected light is colli- the detector to vary due to this mechanical movement. mated before reaching the grating. The grating is used to disperse Additionally, in some materials, the laser excitation creates a high the signal onto the CCD detector by reflecting each wavelength at a density of photoexcited carriers (electrons and holes) which different angle. The resolution of the spectrometer is determined cascade down emitting phonons. Therefore, if the phonon lifetime by the groove density (g/mm) and the distance of the second is long enough, the process creates a large population of non- mirror and the CCD camera (focus length). In general, this dis- thermal phonons, which, in turn, affects the Stokes/anti-Stokes tance is optimized by the manufacturer company, and the user ratio.109,110 In general, the absolute intensity is a difficult parameter does not have access to this parameter. Then, the resolution of the to measure accurately and consistently. spectrometer is mainly affected by groove density. A high density Another way to estimate the temperature from the Raman results in high dispersion and thus a high resolution, but it will spectra is using the peak position and FWHM, both of which are considerably reduce the light intensity and increase the acquisition sensitive to the temperature.111,112 An increase (decrease) in the time. Finally, the Raman signal is recorded by a CCD camera and temperature causes thermal expansion (contraction) and a change presented in terms of relative energy (ERaman ¼ Elaser + Evibration ) in interatomic forces due to the anharmonicity of the lattice poten- and expressed in wavenumbers (cm−1). tial. The change in the interatomic forces is reflected in the Raman A typical Raman spectrum contains a set of peaks (Raman peak’s position, resulting in a redshift to lower wave numbers as modes) located at specific frequencies that depend on the vibra- the temperature increases. The temperature dependence of the tional properties of the material under study. Since each material phonon lifetime, which originated from the anharmonicity of the has its own set vibrations, Raman spectroscopy is commonly used lattice, results in a broadening of the linewidth of the Raman spec- as a standard technique for elementary and structural characteriza- trum. Therefore, the temperature of the sample under the focused tion of materials. In addition, Raman modes are very sensitive to laser spot can easily be obtained by fitting the spectral position and crystallographic orientation and small changes on the crystal struc- linewidth of the observed Raman mode, given a previous calibra- ture of the material. Variations induced by defects, strain, tempera- tion of how these parameters change with temperature. ture, inclusions, disorder, alloying, etc., can be also detected by this There are, however, some drawbacks to use this method. The technique.105–108 primary complication is that there can sometimes be other contri- butions to the change in peak position and linewidth, such as strain, compositional and structural disorder, impurities and con- A. Basic principles and measurements of in-plane tamination of the sample, as well as the presence of pseudo-phases thermal conductivity and mapping and deformation of the material.113–116 Raman thermometry, or the optothermal Raman method, is a For either method, the main requirement for the use of contactless, steady state technique for measuring thermal conduc- Raman thermometry is that the material should have a non- tivity based on probing of the local temperature using the Raman negligible Raman signal from one of its optical modes. For example, signal as a thermometer. Considering that photon dynamics are amorphous materials and metals exhibit poor Raman signals and, not affected by the temperature of the sample, the temperature therefore, are not good candidates to be measured with Raman ther- deduction is purely related to vibrational properties of the system. mometry. On the other hand, most inorganic and organic semicon- In this sense, any temperature-dependent phonon property can be ductors, electrical insulators, and polymers exhibit many optical used to probe the local temperature such as the ratio of Stokes (IS ) Raman modes depending on their symmetry. In any case, the tem- and anti-Stokes (IA ) intensity; peak position; intensity; and line- perature dependence of just one optical mode can serve as a local width (full width at half maximum, FWHM) of a Raman mode. temperature probe. This is particularly important in that different A schematic representation of a typical “Raman-thermometer” is materials can be measured simultaneously if their Raman peaks are shown in Fig. 6. distinguishable from each other. This feature is especially useful for The simplest approach to determine the temperature is to use supported materials, such as 2D materials on a substrate, for which the ratio of the Stokes/anti-Stokes intensity IA /IS , which is related the simultaneous knowledge of both their temperatures enables the to the temperature as follows: determination of interface thermal properties.117 Note that in a given   thin film or material layer, the temperature is considered uniform in IA n hω thickness. Similarly to the TR techniques, the determination of inter- ¼A ¼ A exp , (7) IS nþ1 kB T face thermal properties is, therefore, usually an effective value. where n is the phonon population, A is a calibration constant deter- mined experimentally, h is the reduced Planck constant (h/2π), ω is B. Single laser Raman thermometry the phonon frequency, kB is the Boltzmann constant, and T is Single laser Raman thermometry uses the excitation laser as the temperature. Figure 6(a) shows an illustration of change in the both the heat source and the temperature sensor simultaneously intensity of Stokes and anti-Stokes processes for a given T. The [see Fig. 7(b)]. The excitation laser is focused onto the surface of main advantage of this method is its simplicity. However, simulta- the sample using a microscope objective. Then, the local heating is neous measurements of the Stokes and anti-Stokes peaks require a controlled by varying the incident laser power. The temperature material whose Ran active phonon modes have low vibrational rise in the spot region depends on optical absorption and thermal J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-10 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics properties of the material. Then, the thermal conductivity of the Another advantage of single laser Raman thermometry is its sample can be extracted with a suitable heat diffusion model under sub-micrometer spatial resolution, offering the possibility to map certain assumptions. The first assumption is that the power density the thermal properties at microscale. Moreover, the spatial resolu- of heat production is equal to the absorbed laser light given by tion can be also improved using, e.g., Tip Enhanced Raman Spectroscopy (TERS) that could reach resolutions better than   2Pa 2(x2 þ y2 ) 10 nm.132,133 Kuball et al.134 showed this potential by mapping the P(x, y, z) ¼ α0 exp  exp[a0 z], (8) temperature distribution of AlGaN/GaN field-effect transistors. πσ 2 σ2 Soini et al.135 mapped the temperature rise and extracted the where Pa is the absorbed power, α 0 is the optical absorption coeffi- thermal conductivity of free-standing GaAs nanowires. A further cient, σ is the spot radius of the laser, and and x-y and z are the step was made by Stoib et al.126 measuring the spatial dependence in-plane and out-of-plane coordinates, respectively [see Fig. 7(b)]. of thermal conductivity on silicon and silicon–germanium mesopo- A second assumption can be made for materials with strong light rous films. absorption at the surface. In this case, the power source can be expressed using cylindrical coordinates as 1. Measurement of absorbed power  One of the main sources of error in this technique comes 2Pa 2r2 P(r) ¼ 2 exp  2 , (9) from the measurement of absolute absorbed power. Since the tem- πσ σ perature rise is directly proportional to the absorbed power, any deviation in this quantity will directly affect the calculated thermal and the thermal conductivity of an isotropic and semi-infinite conductivity. In order to reduce the error in the measurement of material is given by118 absorbed power, it is advisable to use the same Raman setup to do so. While this option is not available for all commercial Raman Pa κ ¼ pffiffiffi , (10) setups, especially fiber-based ones, one can build a homemade 4 πΔTσ setup using the same microscope objective. A schematic representa- tion of such a setup is shown in Fig. 8. As seen here, to measure where ΔT ¼ Tmeasured  Tsink is the temperature rise. It is impor- power absorption, three steps are needed: (a) calibration of the inci- tant to remark that the laser beam is simultaneously heating and dent power (sensor S1) with respect to a secondary sensor (S2), (b) probing the local temperature of the sample, i.e., the temperature calibration of the reflected power (sensor S3) using a highly reflective and the detection sensitivity are spatially distributed. Then, to be mirror with respect to S2, and (c) the measurement of the incident consistent with calibration, the spatially dependent temperature has (S2), transmitted (S4), and reflected (S3) power simultaneously using to be weighted by the local power density over the excited area as the sample. This calibration process accounts for the optical losses in follows:119 the system, such as absorption/reflection from the microscope objec- Ð1 tive and the beam splitter. It is important to test this setup as a T(r)r exp[r 2 /σ 2 ]dr Tmeasured ¼ 0Ð1 2 2 , (11) 0 r exp[r /σ ]dr where T(r) is the temperature field distribution given by118,120   2Pa 2r2 T(r) ¼ pffiffiffi exp  2 I0 (2r 2 /σ 2 ), (12) kσ π σ where I0 is a modified Bessel function of order zero. For materials with weaker optical absorption, light will not be fully absorbed at the surface and the power source is given by Eq. (8). A detailed semi-analytical description spatial dependence of temperature can be found in the works of Lax.118,120,121 Another popular method to extract the thermal conductivity from Raman thermometry methods is to use FEM. In these simula- tions, the heat power and thermal properties of the sample are given as input, and the resulting temperature profile is calculated. Then, by adjusting the simulated temperature profile to the mea- sured one, thermal conductivity can be estimated.122 In addition, FEM simulations facilitate the introduction of more complexity to the model, such as temperature dependence of thermal conduc- FIG. 8. Schematic configuration for the incident, reflected, and transmitted tivity,123 TBC,124,125 grain boundaries,126,127 finite size of the power measurements: (a) Calibration of incident power, (b) calibration of reflected power using a high reflective mirror, and (c) simultaneous measure- sample,128 air conduction and convection,129 or the use of other ment of the optical properties of the sample under study. heating sources as a metallic strip in place of the laser beam.130,131 J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-11 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics FIG. 9. Theoretical and experimental absorbance (A), reflectance (R), and transmittance (T), as a function of Si membrane thickness for an incident green laser light (514 nm). The solid lines are calculations obtained from Fabry–Pérot simulations. The solid dots are experimental data points. Inset: diagrammatic Fabry–Pérot effect in membranes. Adapted from Chávez-Ángel et al., APL Mater. 2, 012113 (2014). Copyright 2014 Author (s), licensed under a Creative Commons Attribution (CC BY) License. function of incident power, in order to avoid any nonlinear response analytical model can be applied according to Fourier’s law, from the optics. Finally, the optical reflectance (R) and transmittance (T) can be obtained by measuring the incident (PI ), reflected (PR ), and Pa dT ¼ κ(T) , (14) transmitted (PT ) powers after focusing the laser spot on the surface A dr of the sample. Then, the absorbed power can be calculated as where Pa is the absorbed power, A is the cross-sectional area A¼1RT of the heat flux, κ is the thermal conductivity, and T is the PR PT temperature. The absorbed power Pa is measured considering ¼1  : (13) PI PI the difference between incident and transmitted plus reflected light intensities. For a given sample in a membrane format with For the case of a film on a substrate, it is recommended to first thickness t, the cross-sectional area is A ¼ 2πrt. Taking measure the optical properties of the substrate and then the film on rdT/dr ¼ dT/d(lnr) ¼ θ(r), the following expression can be the substrate. For suspended systems with thicknesses smaller than, obtained for the thermal conductivity: or of the order of the wavelength of the incident light, the thickness dependence of the optical coefficient has an oscillatory behavior, behaving as a Fabry–Pérot optical cavity. Figure 9 shows this oscil- P0 κ(T) ¼  : (15) latory behavior of absorbance, reflectance, and transmittance of 2πtθ(r) suspended silicon membrane as a function of their thicknesses. C. Two-laser Raman thermometry Two-laser Raman thermometry is a contactless optical techni- que based on the concept of Raman thermometry; however, with the decisive difference that heating of the sample and probing of the temperature are decoupled by using two spatially independent lasers [see Fig. 7(c)]. The heating laser is focused on to the lower surface of the sample, while the low power probe laser is used to obtain the local temperature on the upper surface of the sample. The main advantage of this approach in comparison to 1-laser Raman thermometry is that no assumption of the absorbed power has to be made to obtain the thermal conductivity of suspended two-dimensional structures since it directly maps the thermal field. A representative thermal map of a 250 nm thick suspended silicon membrane is shown in Fig. 10. In the quasi-two-dimensional case, no temperature gradients FIG. 10. Example of a thermal map of a 250 nm thick Si membrane. A projec- tion of the thermal field is also shown in the lower plane. The red areas indicate are present in the third dimension. Also, assuming temperature the heating spot in both panels. ranges that exhibit a purely diffusive heat transport regime, an J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-12 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics Therefore, knowing P0 and t, the value of κ(T) can be extracted technique was proposed as a method to probe which phonon MFPs from the logarithmic temperature profile T ln(r). are most important for thermal transport.54 Within the Fourier In the case of temperature-independent thermal conductivity transport regime, a periodic heating induced by FDTR introduces in the temperature range under study, the thermal field decays as an exponentially decaying thermal wave with a thermal penetration pffiffiffiffiffiffiffiffiffiffi T(r)  ln(r) in the diffusive limit and κ is directly obtained from depth Lp ¼ 2α/ω, where α is the thermal diffusivity of the mate- Eq. (15) and the slope of the linear fit of the logarithmic tempera- rial and ω is the angular frequency of the heat source. Koh and ture profile. The analytical solution for the temperature field in the Cahill142 found that the thermal conductivity of semiconductor case of a suspended isotropic membrane is given by integrating alloys varied with heating frequency and deduced that phonons Fourier’s equation as follows: with MFPs higher than Lp did not contribute to the measured thermal conductivity at the given frequency. In BB-FDTR, by P0 sweeping ω across a broader range of frequencies than the classic T(r) ¼ T0  ln(r/r0 ) (κ ¼ κ 0 ): (16) 2πtκ implementation, a broad range of MFPs can be probed. Regner et al.54 modulated the pump beam frequency f1 from At high temperatures, the dependence of thermal conductivity with 200 kHz to 200 MHz. The large frequency range was achieved by a temperature must be taken into account in the integration of heterodyne approach, which filtered the ambient and coherent Eq. (14) for the estimation of the temperature distribution on the noise that usually prohibits measurements at frequencies above sample surface. For instance, in semiconductors, the dependence of 20 MHz. This was achieved by inserting a second electro-optic thermal conductivity with temperature usually decays as modulator that induced a secondary modulation of the probe beam κ(T)  1/T.136 at frequency f2 after reflection from the sample surface. This Therefore, the two-laser Raman technique together with the method heterodynes the signal into two separate frequencies, i.e., analytical model described above can be applied when the absorbed f1 − f2 and f1 + f2. The photodetector output was passed through a power can be considered uniform along the thickness. This is valid low pass filter that rejected f1 + f2. Then, f1 and f2 were swept to for any material exhibiting a detectable temperature-dependent maintain f1 − f2 at 86 kHz, which was the chosen frequency to min- phonon Raman scattering signal. A pre-calibration of the Raman imize noise while staying within the limit of the low-pass filter. shift vs temperature is required when the temperature coefficient at For an intrinsic c-Si sample, the frequency range was divided a given temperature range can be determined. Taking silicon as an into overlapping windows, and the thermal conductivity in each example, the spectral position of the longitudinal optical phonons section was fitted such that a frequency-dependent thermal con- (LO) at room temperature, ΔωR ¼ 520:7 cm1 , can be used as a ductivity was obtained. To interpret the results, the Boltzmann temperature reference, and a temperature coefficient can be transport equation (BTE) was solved for an FDTR-like system. At extracted from the temperature calibration of the LO phonon low heating frequency (Lp . MFP), the BTE prediction matched frequency. The temperature resolution depends on the material Fourier’s Law. However, when the heating frequency was high, under study and can be estimated through the spectral resolution. smaller length scales are probed, such that Lp , MFP. In this case, The spatial resolution is diffraction limited and can be as low as phonons can travel ballistically without scattering, and the BTE 300 nm. This technique has been implemented to measure the predicts an apparent reduction in thermal conductivity as com- thermal conductivity of suspended quasi-2D structures such as Si pared to the Fourier law. membranes of different thickness,137–139 porous membranes,140 By plotting the normalized accumulated thermal conductivity phononic crystals (PnCs),129,141 and 2D materials127 and is particu- κ acc for the corresponding MFP, as seen in Fig. 11, it is possible to larly useful for the investigation of temperature dependence of discern the phonon MFPs contributing to thermal transport. Also thermal conductivity in the most unexplored temperature range shown are similar results for Si from other techniques, such as dual (400–1000 K). frequency TDTR,143 spot size dependent TDTR,144 and TTG.145 The results are in good agreement with a first-principles calculation IV. ADVANCED CONFIGURATIONS of thermal conductivity for Si by Esfarjani and Chen.146 The results indicate that when the dimensions of Si layers decrease, such that A. Broadband frequency-domain thermoreflectance the sample’s dimensions limit the MFP, the effective κ may be sub- (FDTR) stantially lower than the bulk value. This limits the heat dissipation In insulators and semiconductors, the main heat carriers capabilities of nanometric Si structures in electronics, which (phonons) suffer from a variety of scattering mechanisms that limit may be problematic for thermal management of these devices.147 their maximum travelled distance inside the solid. Mechanism such Similar results have been obtained for other crystalline semiconduc- as impurities, boundaries, and collisions with other particles and tors, such as GaAs, GaN, AlN, and 4H-SiC.148 quasi-particles due to lattice anharmonicity set an upper limit to this distance, i.e., its mean free path (MFP). It is natural to think B. Asynchronous optical sampling that, depending on the energy of each phonon, they will propagate at different distances in a material, carrying a different amount of As explained in Sec. II D, the interpretation of the detected energy. Therefore, phonons having different MFPs will contribute signal is not completely straightforward in the classic implementa- differently to thermal conductivity. tion of the TDTR or FDTR system. Many artifacts appear in the Then, κ is the result of the cumulative contribution of lock-in signal, which can arise from residual pump signal, mis- phonons with a range of MFPs. The broadband FDTR (BB-FDTR) alignment of the pump and the probe beams, or the cumulative J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-13 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics between the pump and probe beams; another is phase correction against residual heating and other background noises. Also, fibers can be implemented in the delay line to reach longer time delays between pump and probe beams. These issues can also be eliminated by replacing the mechani- cal delay line and instead using a pump and probe beam that has slightly different repetition rates. This has the additional advantage of reducing the required acquisition time tremendously. The first implementation of this technique named asynchronous optical sampling (ASOPS) was used for spectroscopy. Later, the technique gained popularity for Terahertz spectroscopy and picosecond ultra- sonics and was more recently adapted to thermal characterization, with the possibility of modulating the pump beam. A schematic of this heterodyne picosecond thermoreflectance (HPTR) system is shown in Fig. 12, where one observes the two main differences with classic implementation. First, the delay line is replaced by a shift in the repetition rate F ¼ 1/T of the probe beam by ΔF ¼ 1/ΔT ¼ 600 Hz with respect to the pump beam. Second, an acousto-optic modulator enables the modulation of the pump beam. With this configuration, the temporal resolution is better FIG. 11. Accumulated thermal conductivity κacc measurements of Si at 300 K than 1 ps, whereas the time delay can be as long as 13 ns, which is by BB-FDTR55 (squares), TTG145 (circles), dual-frequency TDTR145 (upward particularly well suited to thermal metrology at the nanoscale. triangles), and spot size dependent TDTR144 (downward triangles). The solid As shown in the time diagrams of the signals acquired with line is a first-principles calculation of thermal conductivity of Si.146 the HPTR system in Fig. 12, the acquired signal corresponds to the envelope of the reflected probe beam. This means that the acquired effect of the pump pulses due to modulation. To counter the sys- signal has an equivalent timescale, dilated with respect to that of tematic errors that these artifacts induce, different strategies have the material response by a factor (T þ ΔT)/ΔT. The choice of the emerged. One such strategy is the use of in-phase and out-of-phase difference between pump and probe beam repetition rates is thus signals to reduce the contribution of misalignment and drifts crucial when selecting the proper components, such as FIG. 12. Time diagram of heterodyne picosecond thermoreflectance (HPTR) signals, (a) pump beam, (b) probe beam, (c) temperature, (d) reflected probe beam, and (e) envelope of the reflected probe beam delivered by the detector. Both pump and probe pulses (a) and (b) have been represented by Dirac combs for clarity reasons but their duration is 100 fs. As for S(t), each pulse duration is the detector response time sd. Adapted from Dilhaire et al., J. Appl. Phys. 110, 114314 (2011). Copyright 2011 AIP Publishing LLC. J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-14 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics photodetectors. Indeed, in the ideal case, the temporal resolution directly depends on the sampling period, i.e., on the repetition rate difference. In that case, in which the optical pulses and photodetec- tor impulses are considered Dirac functions, the chosen repetition rate difference leads to an expected maximal resolution for the material response equal to the pulse duration: ∼100 fs in the con- figuration proposed by Dilhaire et al.149 For the best compromise in terms of detection speed and resolution, the bandwidth B of the detector should be chosen as ΔF  B , F, where F is the repeti- tion rate of the pump laser. Note that the difference between the repetition rates of the pump and probe lasers (ΔF  600 Hz ) is much smaller than their absolute values (F  80 Mhz). Dilhaire et al.149 proposed a detector with B ¼ 8 MHz , leading to a tempo- ral resolution of approximately 125 ns in equivalent time and better than 1 ps in material response. The system can be further improved by modulating the pump beam. The interest is twofold: (i) the signal-to-noise ratio is increased by using a lock-in amplifier and (ii) additional information is added to the signal. This additional information is particularly useful in the case where the response of the material does not reach its equilibrium before the arrival of the next pulse, leading to a cumulative effect and loss of information. In this way, modulation enables the retrieving of this lost informa- tion. In this work, the reported signal-to-noise (S/N) ratio is better FIG. 13 Schematic representation of CCD-based thermoreflectance techniques than 10−6 for an acquisition time of a few minutes. Alternatively, using (a) homodyne and (b) heterodyne detection. by compromising the S/N ratio, it is possible to perform extremely fast acquisition of a few seconds per probe beam position, thus enabling relatively fast full mapping of thermal properties with spatial resolution diffraction limited to the waist of the probe beam. trigger, respectively, and R0 is the DC reflectivity. By triggering the camera at a frequency fc ¼ 4f and using a multichannel lock-in scheme, a two-dimensional image of the reflectivity change can be C. CCD based thermoreflectance obtained. Then, if the dT/dR calibration is known, a thermal image is generated. The choice of fc ¼ 4f allows the camera to take four The charge-couple device (CCD)-based thermoreflectance images per each period τ of temperature modulation. Each of these technique has emerged as novel contactless method for thermal images (Ij ) corresponds to the temporal integration of R(x, y, t) characterization of nano- and microdevices. Being a thermoreflec- given by153 tance method, the technique is also based on measuring changes in reflectivity induced by a change in the local temperature of the ana- ð j τ/4 lyzed sample. Unlike conventional thermoreflectance mapping, Ij ¼ R(x, y, t)dt, with j ¼ 1, 2, 3, 4: (18) where a thermal image is obtained by scanning the sample with the ( j1)τ/4 laser beam, i.e., point-by-point, the CCD-based thermoreflectance obtains a thermal image in a single shot by illuminating the sample By combining these four images, each element of Eq. (17) can be with a light emitting diode (LED) focused by a microscope objec- obtained through the following relations:151 tive150 (see Fig. 13). In general, a frequency generator is used to elec- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trically modulate the temperature of the sample, and the thermal 4π ΔR(x, y) ¼ pffiffiffi (I1  I3 )2 þ (I2  I4 )2 , image is taken in a homodyne (single frequency) or heterodyne 2τ (dual frequency) fashion. It is important to note that if the sample is 2 modulated by the Joule effect at a frequency f0 , the modulated tem- R0 (x, y) ¼ (I1 þ I2 þ I3 þ I4 ), (19) τ perature will occur at the double of this frequency f ¼ 2f0 .   In the homodyne approach, seen in Fig. 13(a), the modulated 2 I1 þ I2  I3  I4 w(x, y) þ ψ ¼ arctan : temperature is phase-locked to the CCD frame trigger, while the π I1  I2  I3 þ I4 illumination light is in continuous mode. The modulated tempera- ture will induce a modulation of reflectivity at the same frequency f This method is commonly referred to as “four-bucket” or given by “four-step” technique. The approach ensures the best use of the slow readout speed of the CCD array by accumulating images with R(x, y, t) ¼ R0 (x, y, t) þ ΔR(x, y, t)cos(2πft þ w(x, y) þ ψ), (17) identical phases in each bin (I1, I2, I3, or I4) and recording a cumu- lative average.152 where w(x, y) and ψ are the phase shifts induced by thermal modu- In the heterodyne approach, the temperature of the sample lation and delay between the modulation signal and the camera and the light source are modulated at slightly different frequencies J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-15 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics FIG. 14. Timing diagram of (a) a “pulsed boxcar” averaging scheme and (b) a modified averaging scheme with longer CCD exposure. f1 and f2, respectively. A schematic representation of this approach proposed Wang et al.154 They synchronized the period of LED is shown in Fig. 13(b). The incident light flux is given by short pulse with thermal excitation, leaving the CCD open. Then, by shifting the LED pulse by a known amount, it is possible to f1 record the full thermal decay as is shown in Fig. 14(b). In this con- f¼ (1 þ cos(2πf2 t)) (20) figuration, during the exposure time of CCD, a series of excitation 2 pulses are applied to the sample. That is, for each thermal excita- and the flux reflected back to the camera is given by tion cycle, there is one LED pulse, which makes the CCD capture a series of reflected light in one exposure. The complete measure- f1 ment of the thermal event can be obtained by thermal pictures at s(x, y, t) ¼ R0 (x, y, t) different phase delays of LED illumination pulses.159 2 f þ 1 ΔR(x, y, t)cos(2πFt  w(x, y)  ψ), (21) 2 D. Transient thermal grating technique where F is the blinking term, i.e., F ¼ f1  f2 . Using the same multichannel lock-in scheme described previously, the detection of The thermal transient grating (TTG) method is an optical the F term can be extracted by triggering the CCD camera at technique for measuring thermal155–157 and acoustic158,159 proper- fc ¼ 4F. Similar to the homodyne detection, each of the terms of ties of materials. A schematic representation of four beam TTG Eq. (21) can be extracted using the four images. apparatus is shown in Fig. 15. In this method, two pulsed lasers are In both of the described configurations, the detected signal is crossed at the sample position forming an interference pattern. in the steady state. To detect faster thermal events, i.e., transient The angle between the pump beams is controlled by splitting the behavior, a time-domain approach is needed. However, such events beams with a diffraction grating (phase mask). For a transmission are typically faster than CCD frame rates. To overcome this configuration, the pump beams are blocked post-sample, while the problem, Maize et al.153 proposed the combination of pulsed light diffracted signal from the thermal grating is recorded. This signal is emitting diode (LED) and a boxcar averaging scheme to obtain mixed with an attenuated reference beam for heterodyne detection. transient images with a temporal resolution of 10−3–10−6 s. The method consists of adjusting the time delay between the LED pulse and thermal excitation applied to the sample. It synchronizes a single light pulse with the exposure of CCD at given thermal excita- tion of the device. Later, a phase delay between the LED pulse and thermal excitation is added. Then, by changing the delay of the light pulse in regular increments, the CCD, thus, records the full thermal transient of the device with time resolution limited by the pulse width.157 A schematic representation of the timing of this technique is shown in Fig. 14(a). This approach requires that the CCD is exposed to each LED pulse. Therefore, the brightness of the light source and sensitivity, exposure time, and frame rate of the CCD FIG. 15. Schematic of four-beam Transient Thermal Grating (TTG) apparatus in camera are required to be very high. In addition, the frequent expo- reflection and transmission geometry. Adapted with permission from Johnson et al., Phys. Rev. Lett. 110, 025901 (2013). Copyright 2013 The American sure of CCD can cause serious degradation of the camera, consider- Physical Society. ably reducing its lifetime. A way to overcome these problems was J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-16 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics purely diffusive thermal-transport145 or detect the crossover from non-diffusive to diffusive phonon transport157; and (v) finally, from this crossover, it is possible to reconstruct the phonon mean free path distribution of the studied sample.157,160,161 E. Transient Raman-based methods The main limitation of Raman thermometry (apart from the need of a material with Raman active modes) is that the technique is mainly limited to steady state measurements, which requires a precise estimation of the absorbed power. The laser absorptivity for supported films or any nanostructure is very difficult to determine. However, recent developments of temporal Raman-based character- ization such as (i) laser flash Raman (LF-Raman),162–164 (ii) time- domain differential Raman (TD-Raman),165 (iii) frequency-resolved Raman (FR-Raman),166 and (iv) energy transport-state resolved Raman (ET-Raman)167,168 and frequency-domain ET-Raman FIG. 16. Typical time trace from a 390 nm thick Si membrane. The electronic (FET-Raman)169 have put this method in a similar level as com- response of the sample is seen, which decays quickly to leave the thermal pared to the conventional TR techniques. response. This decay can then be fitted to extract the decay time, which is pro- portional to thermal diffusivity. The inset shows the complete trace for the 7.5 μm grating period. Reprinted with permission from Johnson et al., Phys. 1. Laser flash Raman Rev. Lett. 110, 2013. Copyright 2013 The American Physical Society. The LF-Raman method was introduced by Liu et al. to measure the thermal properties of multi-walled carbon nanotubes (MWCNTs).163 The method consists in performing two Raman The relative phase between these beams is controlled by a phase measurements using continuous and pulsed lasers. The nanotube adjusting slide in the probe beam path. was anchored from one of its edges to a heat sink, and the rest The absorption of the light causes a spatially periodic thermal completely suspended. For the first step, a short square-wave mod- grating, which, in turn, induces an optical phase and amplitude ulation is applied to the laser, which is used both as a heater and as grating through the temperature dependence of the refractive index an excitation source for the Raman signal. On the free edge, the of the sample. Then, a probe beam is diffracted from this grating sample is heated during the laser pulse duration (heating time, τ H ) and the thermal diffusivity can be determined from the rate of and fully cooled down by keeping a relative long time between signal decay. As heat diffuses from the peak to the null of the pulses (cooling time, τ C ). The time-average temperature rise during grating, the diffraction efficiency of the optical grating decreases the heating period is measured by the band shift, which is recorded and the signal intensity decays exponentially with time, i.e., after a large number of cooling–heating periods. During the laser T(t)  exp[q2 /t] as shown in Fig. 16. flash experiment, the temperature rise of the sample is described by Finally, the thermal decay can then be characterized by a a transient heat conduction model. The second step consists in decay time τ, which is related to thermal diffusivity as follows: measuring the steady state temperature using the continuous laser. Then, by combining the heat equations resulting from steady state 1 (continuous excitation) and transient (pulsed excitation), the α¼ , (22) absorbed power is eliminated, and the thermal diffusivity of the q2 τ MWCNT is given by where q ¼ 2π/L is the grating wavevector corresponding to a  rffiffiffiffiffiffi 2 grating period L. The grating period is controlled by the angle of 3l π P2 θ(x ¼ 0, τ)jτ0H incidence θ and is given by α¼ , (24) 4 τ H P1 θ(x ¼ 0) λ where l is the distance between the heating laser and the anchored L ¼ sin (θ/2): (23) edge, P1 (P2 ) is the incident power of the pulsed (continuous) laser, 2 and θ is the measured temperature rise in the transient (θ(x, τ) and Some advantages of the TTG method are, e.g., as follows: (i) it steady state (θ(x). Using the same approach, Li et al. extended this does not require a transducer layer (i.e., the absence of thermal method by developing the theoretical model to measure the thermal contact resistances in the measurements and analysis); (ii) the abso- diffusivity of supported and suspended 2D materials.162,170,171 lute absorbed power does not need to be measured, which can be At least three variations of this method have been reported. challenging for nanoscale objects; (iii) the thermal grating is The first variation was introduced by Li et al. developing a variable defined in the plane of the sample, such that in-plane thermal spot size LF-Raman method.172 They used the spot size as an extra transport is assured; (iv) the thermal length scale can easily be degree of freedom in the LF-Raman and showed the potential of varied by changing the grating period, which is useful to ensure a this technique to measure thermal conductivity, thermal diffusivity, J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-17 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics TBC, as well as the effective laser absorption coefficients of that is used to heat the sample and excite the Raman signal simul- arbitrary layers of different 2D materials including van der Waals taneously. The method uses the same cooling and heating times heterostructures. A second variation, dual-wavelength LF-Raman (τ H ¼ τ C ), and the Raman signal is recorded for different excita- spectroscopy, was proposed by Fan et al.164 They showed that by tion frequencies. Since τ H ¼ τ C , the pulse interval is not long introducing two pulsed lasers of different wavelengths, it is possible enough for the sample to cool down at high frequencies, such that to achieve a temporal resolution of ∼100 ps. The method consists a heat accumulation effect takes place. This generates a quasi-steady in heating a Raman active sample with a given wavelength and state temperature rise. Finally, the thermal diffusivity is obtained by pulse width. Then, the temperature is probed with a second pulsed fitting Δω, IN , and AN with a 1D transient heat transfer model. laser with a different wavelength and shorter pulse width. The method is similar to TDTR; however, the change in local tempera- 4. Energy transport-state resolved Raman and ture is determined using the band shift approach. The thermal dif- frequency-domain ET-Raman fusivity is estimated by fitting the measured temperature rise for Energy transport-state resolved Raman (ET-Raman) was different time delays between the pump and probe beams. A third introduced by Wang’s group173,181 and was used to determine the variation was introduced by Liu et al.173 to measure a quasi- thermal properties of supported 2D-MoS2 flakes. The method transient temperature rise: differential LF-Raman spectroscopy. The consists of combining Raman measurements using a CW and a method is very similar to LF-Raman but with the addition of an picosecond pulsed (PP) laser and different microscope objectives. extra measurement with a pulsed laser. Two measurements with a The combination of these measurements allows the extraction of pulsed laser are carried out with different heating times τ H1 and not only the in- and through-plane thermal conductivity, but also τ H2 , where τ H2 is slightly longer than τ H1 . Then, by subtracting the specific heat capacity, hot carrier mobility, and TBC. An exten- temperature rise measured at τ H2 and τ H1 , it is possible to estimate sion of this method was introduced by Zobeiri et al.169 by devel- a quasi-transient temperature at a time τ ¼ τ H2  τ H1 . Later, the oping a frequency-domain energy transport state-resolved Raman subtracted signal is normalized by the temperature rise measured at (FET-Raman) by combining FR-Raman and ET-Raman. This continuous wave excitation, and the thermal diffusivity is obtained. method was used to measure the thermal properties of suspended The main advantage of this configuration compared to conven- 2D-MoSe2 flakes. Similar to FR-Raman, it uses a single CW laser tional LF-Raman is that the time-averaged temperature rise can be and a modulator instead of two lasers. estimated within a narrow temperature range. An extended description of both methods can be found in a recent review article by Xu et al.175 and in the works of Wang’s 2. Time-domain differential Raman group.174 Similar to LF-Raman, TD-Raman employs a pulsed laser for both heating and Raman excitation. The experiment was introduced F. Probing non-Fourier thermal transport by Xu et al.165 and consists in heating up a Si cantilever using a Fourier’s law has been used to describe thermal transport at pulsed laser with different heating times (20 μs , τ H , 30 ms) and macroscale over several decades, and most of the theories have vali- fixing the cooling time (τ C ¼ 10 ms). The τ H range was chosen to dated its accuracy in the diffusive heat transport regime. However, cover the whole transient range of the temperature rise from room it is well known that in nanomaterials and nanostructures, heat temperature to the steady state temperature. τ C was fixed at a value transfer exhibits non-diffusive behavior. The use of Fourier law large enough for the system to relax back to the room temperature, becomes inaccurate when describing thermal transport at extremely eliminating the cumulative thermal effect on the cantilever. Then, short time scales (t  τ or L  Λ), where τ and Λ are the lifetime the Raman signal was recorded for different heating times. The peak and MFP of heat carrying phonons, respectively. Recent experi- shift, linewidth, and Raman intensity were extracted and plotted ments on laser heating of atomically thin layers and simulations of against τ H . Finally, the thermal diffusivity was obtained by fitting the thermal transport in solids at nanoscale show large discrepancies band shift (Δω), the normalized intensity (IN ¼ I/τ H ) and normal- with respect to classical laws. For instance, in problems involv- ized Raman peak area (AN ¼ ARaman /τ H ) with a 1D transient heat ing both short temporal and spatial domains, the thermal trans- transfer model. The authors claimed a deviation in the thermal diffu- port in regions close to the heated surface is ballistic, and local sivity of the order ±10% approximately. The inaccuracy of this non-equilibrium effects become dominant, as was described in method was attributed to two main factors; the very large acquisition Sec. II G. In such cases, the Boltzmann transport equations are times for short heating times (∼15 min for τ H ¼ 20 ms) and the better at estimating the temperature evolution in comparison temperature-induced thermal stress in the cantilever. In both scenar- with the Fourier law.176 ios, an out-of-focus effect during the spectra acquisition could take Advancements in nanometrology have enabled interesting place, affecting the Raman measurements. observations and better understanding of non-diffusive heat trans- port phenomena at the nanoscale. In principle, to experimentally 3. Frequency-resolved Raman probe non-diffusive thermal transport, metrology tools with either This method was developed by Wang et al.166 to probe very high temporal or spatial resolution are required. For example, fast thermal responses. They pushed the temporal resolution to 5 μs thermo-electrical techniques, such as scanning thermal microscopy and claimed a further improvement to nanosecond, only limited by that offers high spatial resolution, or the thermal bridge method that laser modulation capacity. The method consists in modulating the gives high temperature accuracy, have been employed to probe ballis- temperature of a Si cantilever through a square wave (SW) laser tic heat conduction and explain deviations of nanoscale energy J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-18 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics In Table I, the discussed methods are compared in terms of measured properties, sample geometry, advantages, disadvantages, and uncertainty. This table can be used as a first basis to choose which technique to select in order to perform thermal measure- ment on micro- and nanostructures. For each technique, we have indicated a great advantage of each technique in bold lettering. Due to many variations within the TR- and Raman families of techniques, it can be challenging to provide a clear cut answer to the question of which technique is best suited for a given sample. However, in summarizing Table I and the literature presented so far, it can be concluded that TDTR undoubtedly has the highest spatiotemporal resolution, giving it a clear advantage for some FIG. 17. Illustration of thermal transport, affected by heater size and suppres- applications, such as to observe electron–phonon thermal relaxa- sion of effective thermal conductivity for individual phonon modes in the diffusive tion mechanisms on the femtosecond timescale. Also, this resolu- region (left, D  Λω ) and in the ballistic region (right, D  Λω ). Adapted with tion will typically give lowest uncertainty when measuring thermal permission from Hu et al., Nat. Nanotechnol. 10, 701 (2015). Copyright 2015 conductivity of ultrathin films. Another advantage of TR is that Springer Nature. the signals can be recorded for all types of materials. The main drawback of Raman thermometry is that it is not suitable for non-Raman active materials or with negligible Raman signals, such transport from the Fourier law.12,177,178 On the other hand, contact- as amorphous materials and metals. In addition, the technique can less thermoreflectance techniques with high temporal resolution have be very time consuming for materials with weak Raman signal, revealed quasi-ballistic effects in in-plane and cross-plane heat trans- requiring long integration time to reduce measurement uncertainty. port experiments60,179 and enabled the quantification of full spectral TR techniques are typically faster, where one measurement point contribution to thermal conductivity from all phonon modes,180 can be measured, with sufficient averaging, in a matter of minutes. such as the BB-FDTR approach described in Sec. IV A. In these The TR techniques, being optical in nature, put certain limitations experiments, the characteristic length of heat conduction was compa- on the sample. The main limitation is the requirement of a smooth rable to or smaller than the MFP of heat carriers. In particular, the surface, such that sample roughness does not contribute to the phonon spectral distribution was measured by probing quasi-ballistic probe signal through diffuse scattering effects. The typical criterion transport near heaters down to 30 nm using the TDTR technique, is that the rms roughness should be below 15 nm when using while the transfer regime was controlled by a characteristic thermal optical light. Measurement errors due to roughness can be clearly length, which was proportional to the heater size d compared to observed in experiments, but the understanding of these mecha- phonon MFPs (Λω ) (see Fig. 17). nisms and how to account for them have not been studied system- Furthermore, experimental evidence of the violation of atically.2 This is not a requirement for Raman thermometry and Fourier’s law has been reported even when the phonon MFP is thus enables Raman methods to be used for a wider range of geom- much shorter than the characteristic sample length.181 A different etries, such as for single NW measurements.183 way to determine limits to diffusive approximation is by taking TR methods require the deposition of a metal transducer into account the ratio of the phonon mean free path (Λω ) to a char- layer, which can compromise the quality of certain materials, such acteristic physical length (d), which is called the Knudsen number, as self-assembled monolayers. It also complicates the thermal anal- Kn ¼ Λω /d When Kn , 1, the heat transport is diffusive, but when ysis in some cases as it introduces an additional layer with an asso- Kn . 1, ballistic contributions need to be included. The Knudsen ciated TBC to the system. Wang et al.184 and more recently Qian number in nanostructures can be affected either due to the size et al.185 demonstrated that it is possible to do TR measurements of effect, which results in a modified phonon spectrum and Λω or the some semiconductor materials without the transducer, but this size of the heat source. For example, Hoogeboom et al. revealed a technique has only been demonstrated for a few material systems. new collectively diffusive regime using multiple heat sources.182 The Raman thermometry techniques have the advantage that they They showed that when the separation between heat sources is do not require any sample preparation, such as metal transducer small enough, long-MFP phonon contributions to heat dissipation deposition, which makes for the simplest sample system. This is could play a significant role and restore heat transfer efficiency to highly advantageous for many samples, such as for supported and near the diffusive limit. suspended 2D materials, which have been widely characterized by Raman thermometry.186 Steady state Raman thermometry does have a fairly high experimental uncertainty, for instance when mea- V. SUMMARY AND OUTLOOK suring ultrathin films. However, recent developments of transient The implementation and uses of various thermoreflectance Raman-based methods discussed in Sec. IV E have put this method and Raman thermometry methods have been discussed so far. in a similar uncertainty level as conventional TR techniques. The Here, we first compare these methods. We then discuss some of the measurement uncertainty of Raman thermometry is highly material limitations, development needs, as well as the challenging research dependent but is estimated to be 10%–20% in most cases. questions that are yet to be resolved using these methods. We finish For TR methods, the laser spot sizes and modulation frequen- by suggesting future directions for the field. cies determine the sensitivity to thermal properties, which limits J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-19 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics TABLE I. Comparison of TR- and Raman thermal measurement techniques. A significant advantage of each technique has been highlighted in bold face lettering. Technique Measured properties Sample geometry Advantages Disadvantages Uncertainty Time-domain Thermal conductivity in Bulk, supported thin - Nanometer scale - Metal transducer needed <10% thermoreflectance cross-plane and in-plane films and ultrathin thermal depth - Complex and expensive direction, TBC, e–p materials, multi-layered resolution experimental setup coupling, structural samples, suspended - High sensitivity to thin - Smooth surfaces required properties, and layer structures films and TBC - Complex post-processing thicknesses - Time resolution below - ρ and C needed 1 ps. Frequency-domain Thermal conductivity in Bulk, supported thin - Nanometer scale - Metal transducer needed 10% thermoreflectance cross-plane and in-plane films and ultrathin thermal depth resolution - Smooth surfaces required direction, TBC materials, multi-layered - High sensitivity to thin - Complex post-processing samples, suspended films and TBC - ρ and C needed structures - Cheaper and easier implementation than TDTR Single-laser Thermal conductivity, Bulk, supported thin - In- and cross-plane - Relatively high 10%–20% Raman TBC films and 2D materials, thermal conductivity experimental uncertainty thermometry multi-layered samples, measurements using a - Need to measure suspended structures, suitable model adsorbed power single NWs - No need for a smooth - Raman peak sensitive to sample surface strain/impurities in sample - No sample preparation - Steady state - ρ and C not needed - Not suitable for - Submicron resolution materials without Raman - Simultaneous active modes. knowledge of structural properties of sample and substrate temperatures - Simultaneous determination of temperatures of substrate and supported layers Two-laser Raman Thermal conductivity, Suspended structures - In-plane - Need to know intensity 15%–20% thermometry in-plane measurements profile of pump laser - No sample preparation - Need to measure - ρ and C not needed adsorbed power - Submicron resolution - Raman peak sensitive to strain/impurities in sample - Steady state - Not suitable for materials without Raman active modes Asynchronous Thermal conductivity in Bulk, thin films, - Faster acquisition - Metal transducer needed 10% optical sampling cross-plane and in-plane suspended structures time than TDTR due to - Complex and expensive direction no mechanical delay experimental setup stage - Smooth surfaces -Eliminates systematic - Complex post-processing errors due to beam - ρ and C needed instabilities for long time delays - Time resolution as low as 50 fs. J. Appl. Phys. 128, 131101 (2020); doi: 10.1063/5.0020239 128, 131101-20 Published under license by AIP Publishing. Journal of TUTORIAL scitation.org/journal/jap Applied Physics TABLE I. (Continued.) Technique Measured properties Sample geometry Advantages Disadvantages Uncertainty CCD-based Thermal conductivity in Bulk, thin films, -2D imaging of thermal -Time consuming 10% thermoreflectance cross-plane and in-plane suspended structures properties acquisition, typically 1 h direction -Non-uniformity defects for 500 × 500 pixel image of sample are instantly revealed Transient thermal Thermal diffusivity, Bulk, thin films, - In-plane - Sometimes difficult to 20% grating in-plane suspended structures measurements measure semiconductors - No sample preparation - Low efficiency of diffraction pattern Transient Raman Thermal diffusivity and Supported and - In-plane and cross - Time consuming 10%–20% thermometry conductivity suspended plane - Not suitable for nanostructures - No sample preparation materials without Raman structures - No need to measure active modes absorbed power the range of thermal conductivities that can be resolved. For fre- assuming the Fourier law while simultaneously concluding that quencies outside the typical range of 10 kHz–20 MHz, the thermal this assumption does not accurately describe the experiment. signal is lost due to the mode noise. The laser radius, typically 1–30 While BB-FDTR54 and TDTR142,194 experiments shed light on μm, is limited by diffraction in the lower limit. Meanwhile, the diffusive-to-ballistic transition for selected bulk semiconductors upper limit is imposed by the minimum required beam intensity. and superlattices, it is currently not straightforward to study the Expanding the range of both frequency and laser radii would phenomenon for all materials. greatly enhance the measuring capability. One demonstration of The interfaces between adjacent materials remain a significant this, as discussed in Sec. IV A, is the heterodyne approach applied challenge for the field of thermal characterization. TBC can be in the BB-FDTR technique, which allowed modulation frequencies measured using both TR and Raman methods, for example, up to 200 MHz. It is expected that developments in laser technol- between single layer graphene and an h-BN flake. The advantage of ogy will help to expand this frequency range. Near-field optics187 the TR method, recently demonstrated by Brown et al.,195 is that could help lower the laser radius below the diffraction limit. This the measurement can be done with a low temperature gradient at applies equally to the mapping of thermal properties. 2D mapping the interface, less than 10 K. The reported TBC value for graphene/ has been achieved using both TR and Raman methods using a h-BN was 34.5 MW m−2 K−1. However, the requirement of trans- motorized translational stage with microscale resolution, which has ducer deposition makes the sample system more complex. Chen been demonstrated using TDTR,22 FDTR,52 and thermal field et al.117 measured the TBC of the same interface using Raman ther- mapping using Raman thermometry.138 mometry, reporting a value of 7.4 MW m−2 K−1. For this measure- TR methods are sensitive to through-plane thermal conductiv- ment, the Raman temperature of the graphene layer reached 480 K, ity, which has been shown by various authors.51,188 However, many with a temperature gradient of 80 K between graphene and h-BN. bulk and thin film materials have differing thermal conductivities The large temperature increase in the sample system is a drawback depending on the plane, and various strategies have been demon- of Raman thermometry, affecting the reported values. Nevertheless, strated to measure the in-plane thermal conductivity. The both these experimental methods report a TBC value far below the approaches include beam-offset measurements,189–191 co-aligned theoretically predicted value of 187 MW m−2 K−1.196 The assump- beams varying the heating frequency,52,192 the variable spot size tion of an abrupt interface with a well-defined temperature on approach,193 and TTG.155–157 Measurement schemes for accurate either side of the interface typically breaks down for most real sur- determination of in-plane thermal transport is still an active area of faces, perhaps with the exception of semiconductor superlattices research. grown by molecular beam epitaxy. The assumption ignores nano- When interpreting TR signals, the thermal transport is typi- scale imperfections such as roughness, interdiffusion, disorder, dis- cally assumed to be purely diffusive, i.e., adhering to Fourier’s law. locations, and bonding mechanisms which all affect the measured This assumption is only valid when the MFPs of all heat carriers TBC.197 These are some of the reasons for the sometimes-large are small compared to the thermal penetration depth. The assump- variation in TBC measured for the same material system. While tion breaks down when the heating frequency is high enough to the machine learning method is a practical manner to predict the exclude the contribution of phonons with MFPs larger than the thermal conductance of heterointerfaces,198,199 advances on the thermal penetration depth of the experiment, or when the sample theoretical understanding of the issue still remain. The current size is comparable to the MFPs of the heat carriers. 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Schmidt, J. Appl. Phys. 116, 023515 ICN2 is supported by the Severo Ochoa program from the Spanish (2014). 26 E. Ziade, M. Goni, T. Sato, P. Czubarow, and A. J. Schmidt, Appl. Phys. Lett. Research Agency (AEI, Grant No. SEV-2017-0706) and by the 107, 221603 (2015). CERCA Programme/Generalitat de Catalunya. ICN2 authors 27 H. Aller, X. Yu, A. J. Gellman, J. A. Malen, and A. J. H. McGaughey, in 2018 acknowledge support from the Spanish MICINN Project SIP (No. 17th IEEE Intersociety Conference on Thermal and Thermomechanical PGC2018-101743-B-I00) and the H2020 European FET Open Phenomena in Electronic Systems (IEEE, 2018), pp. 567–571. project PHENOMEN (No. GA 713450). E.C.-A. acknowledges finan- 28 A. Schmidt, M. Chiesa, X. Chen, and G. Chen, Rev. Sci. Instrum. 79, 064902 cial support from EU Project NANOPOLY (No. GA 289061). The (2008). 29 NTNU authors are supported by the Research Council of Norway Z. Tian, A. Marconnet, and G. Chen, Appl. Phys. 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