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Takahiro Baba, [Tetsuya Baba](https://orcid.org/0000-0001-5820-3757), [Takao Mori](https://orcid.org/0000-0003-2682-1846)

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[Development of Fourier Transform Ultrafast Laser Flash Method for Simultaneous Measurement of Thermal Diffusivity and Interfacial Thermal Resistance](https://mdr.nims.go.jp/datasets/190d0d38-a0d6-43bf-9b00-eca22c8cbc1b)

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Development of Fourier Transform Ultrafast Laser Flash Method for Simultaneous Measurement of Thermal Diffusivity and Interfacial Thermal ResistanceVol.:(0123456789)International Journal of Thermophysics (2024) 45:27https://doi.org/10.1007/s10765-023-03324-w1 3Development of Fourier Transform Ultrafast Laser Flash Method for Simultaneous Measurement of Thermal Diffusivity and Interfacial Thermal ResistanceTakahiro Baba1,2 · Tetsuya Baba1 · Takao Mori1,2Received: 13 November 2023 / Accepted: 26 December 2023 / Published online: 1 February 2024 © The Author(s) 2024AbstractThe thermoreflectance technique is one of the few methods which can measure ther-mal diffusivity of thin films as thin as 100 nm or thinner in the cross-plane direc-tion. The thermoreflectance method under rear-heat front-detect configuration is sometimes called ultrafast laser flash method because of its similarity to laser flash method. Up to now it has typically only been possible to attempt to evaluate the interfacial thermal resistance between the thin films by preparing and measur-ing several samples with different thicknesses. In this study, a method to directly determine interfacial thermal resistance by a single measurement of a thin film on substrate is represented, by analyzing the shape of thermoreflectance signals with analytical solutions in frequency domain and time domain. Thermoreflectance sig-nals observed from metallic thin films on sapphire substrate with different thickness steps were analyzed by Fourier analysis and fitted by analytical equations with four parameters: heat diffusion time across the first layer, ratio of virtual heat sources, characteristic time of cooling determined by interfacial thermal resistance and rela-tive amplitude of the signal. Interface thermal resistance between the thin film and substrate was able to be determined reliably with smaller uncertainty.Keywords  Fourier transform · Interfacial thermal resistance · Periodic pulse heating · Thermal diffusivity · Thermoreflectance · Thin film *  Takao Mori   MORI.Takao@nims.go.jp1  Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Namiki 1‑1, Tsukuba, Ibaraki 305‑0044, Japan2  Graduate School of Pure and Applied Science, University of Tsukuba, Tennodai 1‑1‑1, Tsukuba, Ibaraki 305‑8671, Japanhttp://crossmark.crossref.org/dialog/?doi=10.1007/s10765-023-03324-w&domain=pdf  International Journal of Thermophysics (2024) 45:271 327  Page 2 of 181  IntroductionIn order to control and manage heat transport in devices and modules, it is impor-tant to consider both the transport within materials and the transport across inter-faces between materials [1]. In particular, it is primarily important to reduce thermal resistance of interfaces inside of devices such as central processing unit (CPU), light-emitting diode (LED), laser diode (LD) and optical recording media for acceleration of their operating speed, reduction of energy consumption and improvement of reliability [2, 3]. On the other hand, large thermal resistance might enhance performance of multi-layered insulation materials, and it is criti-cal to evaluate them accurately for thermoelectric thin films and devices [4, 5]. In response to these requirements, several investigations have been performed to measure interfacial thermal resistance [6–20]. Investigation of direction depend-ence of the interfacial thermal resistance was reported by performing picosec-ond thermoreflectance method under both front-heat rear-detect and rear-heat front-detect configurations [13, 14]. Dependence of the interfacial thermal resist-ance on twist angle of crystal axis between the Si thin film and the Si substrate was measured by the offset 3 omega method [15]. As a challenging approach, nanoscale heat pathways have been visualized by combination of electron beam heating and nano-thermocouple and interfacial thermal resistances on the path-ways have been measured [21, 22]. However, it has been difficult to measure the interfacial thermal resistance between thin films or that between thin film and its substrate because they are much smaller than cross-plane thermal resistance of adjacent thin films even if they are as thin as 100 nm. Reliable data of the inter-facial thermal resistance are limited, and the mechanism by which the interfacial thermal resistance occurs is still under investigation [23, 24].The strict definition of interfacial thermal resistance, also called boundary thermal resistance, varies depending on the field of science and technology [25]. The term interface thermal conductance, which is the reciprocal of the interfa-cial thermal resistance, is also used [26–29]. For example, from the case where it means thermal resistance of a joint between bulk materials or glued bulk materi-als [24] to an interface between atomic layers such as an interface between thin films formed by sputtering [30–32]. In this report, thermal resistance between substrate and thin films which are deposited on substrates by sputtering, molecu-lar beam epitaxy, or vacuum evaporation are considered. Even with such an inter-face at the atomic layer-level, it is not possible to obtain a completely uniform state on both sides of the interface. For example, in the case of epitaxial growth, deformation occurs on both sides of the interface due to lattice spacing mismatch. Even if the local thermal conductivity and thermal diffusivity can be defined, they are not supposed to be uniform in the thin film in the vicinity of the interface. In order to define the interfacial thermal resistance even in such a situation, the thermal conductivity and thermal diffusivity in the thin film or substrate sand-wiching the interface should be approximated to be constant in the mathematical model. The interface is assumed to have thermal resistance without finite thick-ness and heat capacity of interface is 0 [33]. This is because the mathematical 1 3International Journal of Thermophysics (2024) 45:27  Page 3 of 18  27model which assumes the heat diffusion time across the interface is 0 reproduces the temperature response well. The thermal resistance of such atomic layer-level interfaces is smaller than 10–8  m2·K·W−1 in most cases [30–32, 34], and this amount is comparable to that of quartz glass (1 W·m−1·K−1) of 10 nm (10–8 m) thick. Therefore, to measure thermal resistances smaller than this, measurements faster than 100 ps (10−10 s), which corresponds to the heat diffusion time across around 10 nm quartz glass, are desirable.At present, the picosecond (or femtosecond) heating thermoreflectance is one of suitable methods [2, 35]. Thus, picosecond thermoreflectance method under front-heat front-detect method (time domain thermoreflectance: TD-TR) have been applied for this purpose [26–28]. More quantitative present possible approach to determine interfacial thermal resistances using thermoreflectance method is by pre-paring and measuring thin films with different thickness. This is rather cumbersome and contains some ambiguity whether the prepared thin films of different thickness intrinsically keep the exact same characteristics. In this approach, each thin film of different thickness is sandwiched between two metallic thin films of constant thick-ness. The ultrafast laser flash method (picosecond thermoreflectance method under rear-heat front-detect configuration [35]) is applied to the series of samples and thermoreflectance signals are analyzed by areal thermal diffusion time method [33]. Areal heat diffusion time is an area surrounded by the temperature response curve and horizontal line at the maximum temperature rise (normalized to 1) when a sam-ple is pulse-heated, which can be expressed by an analytical expression based on the thermophysical property values of each layer in the sample. Since this area has a dimension of time, a systematic analysis of heat diffusion across a multilayer film is realized by defining “areal heat diffusion time”. The reason why the areal heat dif-fusion time is introduced is because the half-time method (half-value time method) [36], used to calculate the thermal diffusion time in the thickness direction of an insulated homogeneous material, does not provide an explicit analytical solution for multi-layered thin films.In this report, an approach to directly determine interfacial thermal resistance by a single measurement of a thin film on substrate is presented, by analyzing the shape of thermoreflectance signals with analytical solutions represented in frequency domain and time domain. For this study, two mode-locked fiber lasers are used for heating light and temperature measurement light, and by electrically controlling the timing of transmission of the two lasers, thermoreflectance signals were acquired over the entire pulse interval of 50 ns [35]. We have previously presented an ana-lytical solution of temperature response expressed by Fourier series for one-dimen-sional heat diffusion after periodic pulse heating [37].It is well-known that the flash method which was invented by Parker et al. in 1961 [36], and established as the reliable, standard and popular method for measuring thermal diffusivity of bulk materials [38–42]. Our thermoreflectance method based on Fourier transform is named as Fourier transform ultrafast laser flash method, since this method is natural evolution of the laser flash method for measuring ther-mal diffusivity of bulk materials [43]. This method was advanced in this work, and the signals were analyzed to be able to quantitatively determine the interfacial ther-mal resistance between thin film and substrate by curve-fitting.  International Journal of Thermophysics (2024) 45:271 327  Page 4 of 182 � ExperimentsFigure 1 shows a schematic diagram of the thermoreflectance technique in rear-heat front-detect (RF) configuration and the conventional time domain thermoreflectance technique in front-heat front-detect (FF) configuration. In the case of RF configura-tion, a laser beam is incident on the interface between the thin film and substrate to heat up the thin film. Another laser beam is incident on a thin film and reflection from its surface is detected by photodetector in order to probe its surface tempera-ture. The geometrical configuration of the thermoreflectance technique in RF con-figuration is the same as that of laser flash method, which is standard measurement method to measure the thermal diffusivity of bulk materials [36, 40, 42]. Thermal diffusivity of thin films can be calculated from thickness of a thin film and heat dif-fusion time across the thin film [35, 44, 45].The wavelength of pump beam is 1550 nm and that of probe beam is 775 nm. Detailed information about our thermoreflectance apparatus is explained in our pre-vious works [37, 46]. The method we developed to determine interfacial thermal resistance will be described in detail in the next section.In thermoreflectance measurements of this study, heat transfer across thin films can be regarded as one-dimensional because diameter of pump beam, which is typically from 50 to 100 μm, is far larger than thickness of thin films, which is thinner than 1 μm in most cases. Although, the heat eventually spreads in-plane and cross-plane in 3D, it takes much longer time than the observation timescale of 50 ns. Deviation from one-dimensional heat diffusion in 50  ns is negligibly small. Thus, temperature response from a sample can be derived from one-dimensional heat equation. In two-layer model which assumes a single-layered film on semi-infinite substrate, the relationship between the temperature at front surface of thin film Tf  , the temperature at interface between Fig. 1   Diagram of thermoreflectance technique. (a) rear-heat front-detect: RF. (b) front-heat front-detect: FF1 3International Journal of Thermophysics (2024) 45:27  Page 5 of 18  27thin film and substrate Ts , the heat flux across surface of thin film qf  and the heat flux across interface qs is expressed as follows in Laplace domain [33]where �f  is heat diffusion time across thin film, df  is thickness of thin film, �f  is ther-mal diffusivity of thin film, bf  is thermal effusivity of thin film and � is a complex variable. To consider interfacial thermal resistance between thin film and substrate, we assumed an additional layer between thin film and substrate as shown in Fig. 2.In this case, the temperature at interface between the additional layer and thin film is named T̃s1(�) , which corresponds to the temperature at the rear end of thin film. On the other hand, the temperature at interface between the additional layer and substrate is named T̃s2(�) , which corresponds to the temperature at the front end of substrate. When there is an additional layer between thin film and substrate, the relationship between temperature and heat flux can be extended by cascading quadrupole matrices as follows [33]where �A is heat diffusion time across the additional layer, dA is thickness of the additional layer, �A is thermal diffusivity of the additional layer and bA is thermal (1)�q̃s(�)T̃s(�)�=�cosh�√�f ��−bf√�sinh�√�f ��−1bf√�sinh�√�f ��cosh�√�f ����q̃f (�)T̃f (�)�,(2)�f =df2�f(3)�q̃s2(�)T̃s2(�)�=⎡⎢⎢⎣cosh�√�A��−bA√�sinh�√�A��−1bA√�sinh�√�A��cosh�√�A��⎤⎥⎥⎦�q̃s1(�)T̃s1(�)�,(4)�A =dA2�AFig. 2   Schematic diagram of two layer (thin film and substrate) model with interfacial thermal resistance  International Journal of Thermophysics (2024) 45:271 327  Page 6 of 18effusivity of the additional layer. If the additional layer has zero thickness and infi-nitely small thermal conductivity �A , elements which include �A in Eq. 3 converge as followswhere cA is specific heat capacity of the additional layer, �A is density of the addi-tional layer and R is thermal resistance which has finite value. Therefore Eq. 3 can be expressed as followsThus, the following relationship between heat flux and temperature can be obtained from Eq. 8.When thickness of substrate can be regarded as semi-infinite, the temperature of the front end of the substrate T̃s2(�) in Laplace transform is expressed as fol-lows [33]where bs is thermal effusivity of substrate.In the actual measurement, the interface between thin film and substrate is pulse-heated, and heat diffuses from the interface to both sides as shown in Fig. 3. The pulse heating is assumed to be Dirac delta function �(t) , whose Laplace transform is 1. Considering energy conservation law, the following equation is satisfied.Heat flow at the front surface pf (t) and its Laplace transform p̃f (�) can be substi-tuted by 0 since the front surface of thin film is insulated.If leftward heat flux is assumed to be positive, the relationship between T̃f (�) and T̃s1(�) can be expressed as follows(5)limdA→0,�A→0cosh�√�A��= 1(6)limdA→0,�A→0bA√�sinh�√�A��= 0(7)limdA→0,�A→0sinh�√�A��bA√�= limdA→0,�A→0√�AbA= limdA→0,�A→0dA∕√�A√�AcA�A= limdA→0,�A→0dA�A= R,(8)[q̃s2(�)T̃s2(�)]=[1 0−R 1][q̃s1(�)T̃s1(�)](9)q̃s2(�) = q̃s1(�)(10)T̃s2(�) = T̃s1(�) − Rq̃s1(�)(11)T̃s2(�) =1bs√�× q̃s2(�),(12)p̃s1(�) + q̃s2(�) = 11 3International Journal of Thermophysics (2024) 45:27  Page 7 of 18  27By considering Eqs. 9, 10, 11 and 13, T̃f (�) can be expressed as followsBy replacing hyperbolic functions with exponential functions, T̃f (�) is finally expressed as followswhere � is ratio of virtual heat sources of the mirror image method and �r is char-acteristic time of interfacial thermal resistance (cooling time constant) [33]. The temperature T̃f (�) can be regarded as proportional to the reflectance unless the tem-perature change is significantly large. Thus intensity of reflected probe beam, ther-moreflectance signal, can be expressed as follows(13)�0T̃f (�)�=�cosh�√�f ��−bf√�sinh�√�f ��−1bf√�sinh�√�f ��cosh�√�f ����1 − p̃s1T̃s1(�)�(14)T̃f (�) =1bf√�×1 + Rbs√��1 + Rbs√��sinh�√�f ��+bsbfcosh�√�f ��(15)T̃f (�) =1bf√�×�1 +�+√�r�1+√�r��exp(−√�f �)1 −�+√�r�1+√�r�exp(−2√�f �)(16)� =bf − bsbf + bs(17)�r = R2(bf bsbf + bs)2,Fig. 3   Schematic diagram of heat flux across interface between thin film and substrate  International Journal of Thermophysics (2024) 45:271 327  Page 8 of 18where k is a proportional constant. By redefining the proportional constant as k� = k∕bf  , Eq. 18 can be expressed as followsAs we explained in the previous work, regression analysis can be applied to Fou-rier coefficients of thermoreflectance signals in frequency domain [37]. Fourier coefficients of thermoreflectance signals are calculated by discrete Fourier transform (DFT).Figure 4 shows theoretical temperature response curves under the RF configu-ration after periodic pulse heating derived from Eq. 19 with heat diffusion time normalized by the period ΔT  , Φf = �f∕ΔT  is 0.01, which is close value to heat diffusion time across a 100  nm-thick metallic thin film when ΔT  is 50  ns. The timescale in Fig. 4 is normalized by ΔT  . The characteristic time �r is also nor-malized by ΔT  , and a dimensionless parameter Φr = �r∕ΔT  is introduced. Tem-perature rise is normalized by the maximum temperature rise when the thermal effusivity of the substrate is 0 ( � = 1 ). The continuous curves in Fig. 4 show tem-perature response curves with different normalized characteristic time of interfa-cial thermal resistance Φr with � = 0 , where thermal effusivity values of the film and the substrate are same. On the other hand, the dotted curves in Fig. 4 show (18)Ỹ(�) = kT̃f (�) =kbf√�×�1 +�+√�r�1+√�r��exp(−√�f �)1 −�+√�r�1+√�r�exp(−2√�f �),(19)Ỹ(𝜉) =k�√𝜉×�1 +𝛾+√𝜏r𝜉1+√𝜏r𝜉�exp�−√𝜏f 𝜉�1 −𝛾+√𝜏r𝜉1+√𝜏r𝜉exp�−2√𝜏f 𝜉� ,Fig. 4   Theoretical temperature responses depend on � and normalized characteristic time of interfacial thermal resistance Φr ( Φf = 0.01)1 3International Journal of Thermophysics (2024) 45:27  Page 9 of 18  27temperature response curves with different ratio of virtual heat sources � with Φr = 0 . It can be seen that �r affects shapes of temperature responses differently from � . Therefore, it is expected that both � and �r can be determined simultane-ously from the shape of observed thermoreflectance signal. Please note that, if the first layer is metallic layer like platinum, Φr = 0.01 with � = 0 corresponds to the interfacial thermal resistance of around 4 × 10−9 (m2·K·W−1).We measured a molybdenum thin film and a platinum thin film with a ther-moreflectance apparatus PicoTR (NETZSCH-Gerätebau GmbH) under RF config-uration. Each of the films was deposited on sapphire substrate by combinatorial sputtering system CMS-6420 (COMET Inc.). Figure 5a shows the structure of the films. Table 1 shows the conditions for synthesizing the films. The substrates are sapphire supplied by Furuuchi Chemical Cooperation. The surfaces are perpen-dicular to c-axis. Both sides are polished to flat surface with Ra (Averaged value of roughness) of 0.5 nm for the substrate of the Mo thin film. Thin film side was polished for the substrate of the Pt thin film. Both films have five steps and varies in thickness df  (about 100 nm, 125 nm, 150 nm, 175 nm, 200 nm) according to these steps. The five steps were synthesized by translation of a mask as shown in Fig. 5b. Thicknesses were measured by contact profilometer. We designed such Fig. 5   (a) Structure of thin film deposited on sapphire substrate. (b) Mechanism of combinatorial sputter-ing systemTable 1   Conditions for synthesizing the filmsFilm Deposition cham-ber pressure (Pa)Introduced gas Film synthesiz-ing pressure (Pa)Film synthesizing temperature (°C)Target power supplyMo 4.6 × 10−6 Ar 40 sccm 0.3 25.2 DC (100 W)Pt 5.0 × 10−6 Ar 40 sccm 0.3 24.1 RF (100 W)  International Journal of Thermophysics (2024) 45:271 327  Page 10 of 18a particular sample for this work, because in previous conventional cases where different thickness films were prepared separately and measured, additional varia-tions can potentially be introduced because of using different substrate pieces and having separate sputtering sessions for the fabrication.3 � ResultsThe film (Fig.  5) was heated through the transparent sapphire substrate and its temperature response was detected at the open surface of the thin film as shown in Fig. 1a. Figure 6a–c show normalized thermoreflectance signals observed from 102  nm-thick step, 154  nm-thick step and 204  nm-thick step of Mo film, respec-tively. Figure  6d–f show normalized thermoreflectance signals observed from 104 nm-thick step, 158 nm-thick step and 213 nm-thick step of Pt film, respectively. The signals were sampled at 100 GHz sampling rate, thus the sampling interval is 10 ps. The interval of the periodic pulse ΔT  is 50 ns. We calculated the Fourier coef-ficient Yn from the thermoreflectance signal using DFT in the range of 0 s to 50 ns. After obtaining Yn , we applied the regression analysis to the absolute value of the Fourier coefficient |Yn| as followswhere �n is frequency, k̂′ , �̂f  , �̂  and �̂r are estimates of fitting parameters k′, �f  , � and �r . After estimating fitting parameters by regression analysis, �f  , bf  , bs and R are determined by following equations.where cf  is specific heat capacity of thin film and �f  is density of thin film. Figure 7 shows the absolute value of the Fourier coefficient |Yn| obtained from the signal in Fig. 6d and the regression curves in the frequency domain. This time, we omitted high frequency components above 4 GHz, which are relatively noisy. Tables 2 and 3 show estimates of fitting parameters �̂f  , �̂  and �̂r by applying least squares method to the absolutes of Fourier coefficients |Yn| as shown in Fig.  7, thermal diffusiv-ity of thin film �f  and interfacial thermal resistance R determined by the estimates (20)||Yn|| =||||Ỹ(�n, k̂�, �̂f , �̂ , �̂r)|||| + �,(21)�f =df2�̂f(22)bf =√�f cf �f(23)bs = bf1 − �̂1 + �̂(24)R =√�̂rbf + bsbf bs,1 3International Journal of Thermophysics (2024) 45:27  Page 11 of 18  27of the fitting parameters. The specific heat capacity cf  and density �f  are assumed to be 251 (J·(kg·K)−1) and 10200  (kg·m−3) for molybdenum, while 133(J·(kg·K)−1) and 21500  (kg·m−3) for platinum, respectively [47]. The red lines in Figs. 6 and 7 Fig. 6   (a) Normalized thermoreflectance signal under RF configuration from the 102  nm-thick step of Mo film and regression curves in time domain. (b) Normalized thermoreflectance signal under RF configuration from the 154  nm-thick step of Mo film and regression curves in time domain. (c) Nor-malized thermoreflectance signal under RF configuration from the 204  nm-thick step of Mo film and regression curves in time domain. (d) Normalized thermoreflectance signal under RF configuration from the 104  nm-thick step of Pt film and regression curves in time domain. (e) Normalized thermoreflec-tance signal under RF configuration from the 158 nm-thick step of Pt film and regression curves in time domain. (f) Normalized thermoreflectance signal under RF configuration from the 213 nm-thick step of Pt film and regression curves in time domain (Color figure online)  International Journal of Thermophysics (2024) 45:271 327  Page 12 of 18show regression curves derived from Eq. 19 in time domain and frequency domain, respectively. On the other hand, the green lines show regression curves in the case that fitting parameters k′ , �f  and �r are estimated but � is fixed at − 1, which assumes that the substrate has infinite thermal effusivity. The blue lines show regression curves in the case that fitting parameters k′ , �f  and � are estimated but �r is fixed at 0, which assumes there is no interfacial thermal resistance between the thin film and substrate. It can be seen that contributions from both � and �r need to be considered in order to reproduce the signal with smaller deviation.   Fig. 7   Absolute of Fourier coefficients ||Yn|| of thermoreflectance signal obtained from the 104 nm-thick step of Pt film and regression curves in frequency domain (Color figure online)Table 2   Estimates of fitting parameters (values in parenthesis are fixed) from Mo film and thermal prop-erties determined by the parametersdf  (nm) �̂f  (s) �̂ �̂r (s) �f  (m2·s−1) R (m2·K·W−1)102 4.85 × 10−10 − 0.0959 2.93 × 10−92.13 × 10−52.65 × 10−91.57 × 10−10 (− 1) 4.21 × 10−86.57 × 10−59.89 × 10−95.30 × 10−10 − 0.568 (0) 1.95 × 10−5 0154 9.95 × 10−10 − 0.0621 2.36 × 10−92.37 × 10−52.32 × 10−96.00 × 10−10 (− 1) 2.62 × 10−83.93 × 10−51.01 × 10−88.73 × 10−10 − 0.279 (0) 2.70 × 10−5 0204 1.53 × 10−9 0.0137 3.68 × 10−102.72 × 10−52.92 × 10−91.12 × 10−9 (− 1) 2.57 × 10−83.71 × 10−51.03 × 10−81.30 × 10−9 − 0.0918 (0) 3.18 × 10−5 01 3International Journal of Thermophysics (2024) 45:27  Page 13 of 18  274 � Discussion4.1 � Sensitivity AnalysesFigure 8 visualizes the sensitivity analysis for the characteristic time of interfacial thermal resistance �r . The red curve in Fig. 8a corresponds to the regression curves in Fig. 6d. When �r is deviated by ± 10 %, ± 30 % and ± 50 % from the fitted value, the theoretical curve is shifted as green curves, blue curves and cyan curves in Fig. 8, respectively. Figure 8b shows the residual expression of sensitivity analyses in Fig. 8a.4.2 � Reproducibility of MeasurementReproducibility and uncertainty are discussed to confirm the validity of the meas-urement of this article for the Mo thin film and the Pt thin film sequentially. Fifteen sets of thermal diffusivity and interfacial thermal resistance values were obtained for the Mo thin film in total from each of the films as shown in Tables 4 and 5 since each of the five steps was measured three times. These tables also show average val-ues, standard deviations and relative values of standard deviations (= SD/Average) in three measurements. It can be seen that the determination of thermal diffusivity of Mo thin film is consistent in all the steps and relative values of standard deviations are less than 2 %. The determination of interfacial thermal resistance between Mo thin film and sapphire substrate has smaller standard deviation when the film layer is relatively thin, whereas it becomes more uncertain as the layer becomes thicker. This is because, if the film layer is thick and has large heat capacity, the thermal effusion from the layer eclipses the contribution from the interfacial thermal resistance.Fifteen sets of thermal diffusivity and interfacial thermal resistance values were obtained for the Pt thin film in total as shown in Tables 6 and 7 in the same way as the Mo thin film. It can be seen that the determination of thermal diffusivity of Pt thin film is also consistent in all the steps and relative values of standard deviations Table 3   Estimates of fitting parameters (values in parenthesis are fixed) from Pt film and thermal proper-ties determined by the parametersdf  (nm) �̂f  (s) �̂ �̂r (s) �f  (m2·s−1) R (m2·K·W−1)104 8.30 × 10−10 0.00559 1.91 × 10−91.30 × 10−58.55 × 10−96.23 × 10−10 (− 1) 3.35 × 10−81.73 × 10−51.54 × 10−89.55 × 10−10 − 0.129 (0) 1.13 × 10−5 0158 1.92 × 10−9 0.00919 1.86 × 10−91.30 × 10−58.43 × 10−91.62 × 10−9 (− 1) 2.72 × 10−81.54 × 10−51.47 × 10−81.77 × 10−9 0.0273 (0) 1.42 × 10−5 0213 3.33 × 10−9 − 0.135 3.32 × 10−91.36 × 10−59.63 × 10−93.09 × 10−9 (− 1) 2.11 × 10−81.46 × 10−51.33 × 10−82.82 × 10−9 0.0470 (0) 1.60 × 10−5 0  International Journal of Thermophysics (2024) 45:271 327  Page 14 of 18Fig. 8   (a) Sensitivity analysis of the characteristic time of interfacial thermal resistance �r (devia-tion ± 10 %, ± 30 %, ± 50 %). (b) Residual expression of the sensitivity analysis of the characteristic time of interfacial thermal resistance �r (deviation ± 10 %, ± 30 %, ± 50 %) (Color figure online)Table 4   All the thermal diffusivity values obtained from the whole measurement of Mo filmdf  (nm) �f  (m2·s−1) Reproduc-ibility (%)1st 2nd 3rd Average SD102 2.13 × 10−52.10 × 10−52.07 × 10−52.10 × 10−53.00 × 10−7 1.43129 2.27 × 10−52.29 × 10−52.33 × 10−52.30 × 10−53.06 × 10−7 1.33154 2.37 × 10−52.41 × 10−52.44 × 10−52.41 × 10−53.51 × 10−7 1.46181 2.60 × 10−52.64 × 10−52.66 × 10−52.63 × 10−53.06 × 10−7 1.16204 2.72 × 10−52.66 × 10−52.62 × 10−52.67 × 10−55.03 × 10−7 1.891 3International Journal of Thermophysics (2024) 45:27  Page 15 of 18  27are less than 10 %. Although the tendency is semi-quantitatively similar to the Mo thin film, reproducibility of the interfacial thermal resistance of the Mo thin film is remarkably better because of the better signal-to-noise ratio of the observed signal, for example Fig. 6a of Mo film and Fig. 6d of Pt film. The difference in signal-to-noise ratio can be attributed to the difference in polishing condition between the rear surfaces of the sapphire substrates.An experimental result of interfacial thermal conductivity between Pt and Al2O3 by the TDTR method has been reported [11]. The experimental setup was usual configuration of the TDTR measurement using a Ti sapphire laser with generative amplifier for heating and detection with optical delay system. The reported value in this study is 8.4 × 10−9 (m2·K·W−1) at 296 K, which is digitized Table 5   All the interfacial thermal resistance values obtained from the whole measurement of Mo filmdf  (nm) R (m2·K·W−1) Reproduc-ibility (%)1st 2nd 3rd Average SD102 2.65 × 10−92.55 × 10−92.53 × 10−92.58 × 10−96.43 × 10−11 2.50129 2.69 × 10−92.65 × 10−92.60 × 10−92.65 × 10−94.51 × 10−11 1.70154 2.32 × 10−93.10 × 10−93.16 × 10−92.86 × 10−94.69 × 10−10 16.39181 2.61 × 10−92.91 × 10−92.36 × 10−92.63 × 10−92.75 × 10−10 10.48204 2.92 × 10−92.49 × 10−93.00 × 10−92.80 × 10−92.74 × 10−10 9.78Table 6   All the thermal diffusivity values obtained from the whole measurement of Pt filmdf  (nm) �f  (m2·s−1) Reproduc-ibility (%)1st 2nd 3rd Average SD104 1.30 × 10−51.26 × 10−51.29 × 10−51.28 × 10−52.08 × 10−7 1.62130 1.33 × 10−51.15 × 10−51.22 × 10−51.23 × 10−59.07 × 10−7 7.36158 1.30 × 10−51.39 × 10−51.29 × 10−51.33 × 10−55.51 × 10−7 4.15183 1.34 × 10−51.31 × 10−51.25 × 10−51.30 × 10−54.58 × 10−7 3.53213 1.36 × 10−51.31 × 10−51.30 × 10−51.32 × 10−53.21 × 10−7 2.43Table 7   All the interfacial thermal resistance values obtained from the whole measurement of Pt filmdf  (nm) R (m2·K·W−1) Reproduc-ibility (%)1st 2nd 3rd Average SD104 8.55 × 10−98.60 × 10−99.33 × 10−98.83 × 10−94.37 × 10−10 4.95130 8.94 × 10−98.82 × 10−97.71 × 10−98.49 × 10−96.78 × 10−10 7.99158 8.43 × 10−91.00 × 10−86.09 × 10−98.17 × 10−91.97 × 10−9 24.1183 8.92 × 10−91.04 × 10−93.28 × 10−94.41 × 10−94.06 × 10−9 92.0213 9.63 × 10−93.70 × 10−91.01 × 10−87.81 × 10−93.57 × 10−9 45.7  International Journal of Thermophysics (2024) 45:271 327  Page 16 of 18from Fig. 3 in Ref. [11]. This value agrees with the interfacial thermal resistance from the 104 nm-thick step of Pt film in our study 8.8 × 10−9 (m2·K·W−1), which was measured at 296 K, with smaller deviation than 5 %.As described in the discussion of reproducibility the interfacial thermal resist-ance values measured by five steps of different thickness of Mo/sapphire sample, interfacial thermal resistance values agreed each other within 5 %. Since observed signal is proportional to temperature change under one-dimensional heat diffu-sion condition, and the fitted equation is exact analytical solution of one-dimen-sional heat diffusion after periodic impulse heating, it is difficult to consider that this agreement is realized by chance. As shown in Fig. 6a–f, the observed signals are successfully fitted by the analytical which has only four parameters including amplitude over the entire pulse period. These consistency and agreement can be considered as supporting the validity of measurement, analysis equation and fit-ting procedure.5 � ConclusionIn this paper, we investigated a new analytical solution for thermoreflectance method, which considers the contribution from interfacial thermal resistance between thin film and substrate. By applying Fourier expansion analysis to ther-moreflectance signals, we could determine quantitatively interfacial thermal resistance and thermal diffusivity of thin film at the same time by single measure-ments. Metallic (Mo and Pt) thin films with five steps (about 100 nm, 125 nm, 150 nm, 175 nm, 200 nm) were deposited on sapphire substrate to evaluate the reproducibility of determination. The reproducibility of thermal diffusivity meas-urement is within 10 % in standard deviation for all the steps of five thicknesses, whereas reproducibility of interfacial thermal resistance measurement is within 10 % in standard deviation for two steps (about 100 nm, 125 nm). This approach can be easily extended to analyze more complicated samples like multi-layered thin films.Author Contributions  TB (Takahiro Baba): methodology, analysis, investigation and writing. TB (Tet-suya Baba): conceptualization, review and editing, supervision. TM: conceptualization, review and edit-ing, supervision, funding acquisitionFunding  The authors acknowledge support from JST Mirai Program Grant Number JPMJMI19A1. Taka-hiro Baba was also supported by JST SPRING, Grant Number JPMJSP2124.Data Availability  The data that support the findings of this study are available from the corresponding author upon reasonable request.Declarations Conflict of Interest  One Japanese patent application (Appl. No. 2018-516599, Patent No. 6399329), one PCT patent application (PCT/JP2018/012324), and one United States patent application (16/620.294) re-lated to the work described here.1 3International Journal of Thermophysics (2024) 45:27  Page 17 of 18  27Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permis-sion directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.References  1.  C. Goupil, Continuum Theory and Modeling of Thermoelectric Elements (Wiley, Hoboken, 2016), p.125  2.  T. Baba, Proceeding of the 10th International Workshop on Therminic (2004), pp. 241–249  3.  T. Nakai, S. Ashida, K. Todori, K. Yusu, K. Ichihara, S. Tatsuta, N. Taketoshi, T. Baba, Opt. Data Storage 5380, 464–473 (2004)  4.  X. Chen, Z. Zhou, Y.H. Lin, C. Nan, J. Materiomics 6, 494–512 (2020)  5.  T. Hendricks, T. Caillat, T. Mori, Energies 15, 7307 (2022)  6.  Y. Xu, H. Wang, Y. Tanaka, M. Shimono, M. Yamazaki, Mater. Trans. 48, 148–150 (2007)  7.  R. Kato, Y. Xu, M. Goto, Jpn. J. Appl. Phys. 50, 106602 (2011)  8.  D. Zhao, X. Qian, X. Gu, S.A. Jajja, R. Yang, J. Electron. Packag. 138, 040802 (2016)  9.  N. Poopakdee, Z. Abdallah, J.W. Pomeroy, M. Kuball, ACS Appl. Electron. Mater. 4(4), 1558–1566 (2022) 10.  Y.R. Koh, J. Shi, B. Wang, R. Hu, H. Ahmad, S. Kerdsongpanya, E. Milosevic, W.A. Doolittle, D. Gall, Z. Tian, S. Graham, P.E. Hopkins, Phys. Rev. B 102, 205304 (2020) 11.  P.E. Hopkins, R.N. Salaway, R.J. Stevens, P.M. Norris, Int. J. Thermophys. 28, 947–957 (2007) 12.  Z. Cheng, Y.R. Koh, H. Ahmad, R. Hu, J. Shi, M.E. Liao, Y. Wang, T. Bai, R. Li, E. Lee, E.A. Clinton, C.M. Matthews, Z. Engel, L. Yates, T. Luo, M.S. Goorsky, W.A. Doolittle, Z. Tian, P.E. Hopkins, S. Graham, Commun. Phys. 3, 115 (2020) 13.  N. Oka, R. Arisawa, A. Miyamura, Y. Sato, T. Yagi, N. Taketoshi, T. Baba, Y. Shigesato, Thin Solid Films 518, 3119–3121 (2010) 14.  Y.J. Wu, T. Yagi, Y. Xu, Int. J. Heat Mass Transf. 180, 121766 (2021) 15.  D. Xu, R. Hanus, Y. Xiao, S. Wang, G.J. Snyder, Q. Hao, Mater. Today Phys. 6, 53–59 (2018) 16.  R.J. Stevens, A.N. Smith, P.M. Norris, J. Heat Transf. 127, 315–322 (2005) 17.  A. Giri, P.E. Hopkins, Adv. Funct. Mater. 30, 1903857 (2020) 18.  L. Guo, S.L. Hodson, T.S. Fisher, X. Xu, J. Heat Transf. 134, 042402 (2012) 19.  G.D. Mahan, Phys. Rev. B 79, 075408 (2009) 20.  T. Lu, J. Zhou, T. Nakayama, R. Yang, B. Li, Phys. Rev. B 93, 085433 (2016) 21.  N. Kawamoto, Y. Kakefuda, T. Mori, K. Hirose, M. Mitome, Y. Bando, D. Golberg, Nanotechnol-ogy 26, 465705 (2015) 22.  N. Kawamoto, Y. Kakefuda, I. Yamada, J. Yuan, K. Hasegawa, K. Kimoto, T. Hara, M. Mitome, Y. Bando, T. Mori, D. Golberg, Nano Energy 52, 323–328 (2018) 23.  R. Berman, Thermal Conduction in Solids (Clarendon Press, Oxford, 1976), p.107 24.  C. Monachon, L. Weber, C. Dames, Annu. Rev. Mater. Res. 46, 433–463 (2016) 25.  J. Chen, X. Xu, Rev. Mod. Phys. 94(2), 025002 (2022) 26.  R.M. Costescu, M.A. Wall, D.G. Cahill, Phys. Rev. B 67, 054302 (2003) 27.  D.G. Cahill, Microscale Thermophys. Eng. 1, 85–109 (1997) 28.  H.K. Lyeo, D.G. Cahill, Phys. Rev. B 73, 144301 (2006) 29.  J. Zhu, D. Tang, W. Wang, J. Liu, K.W. Holub, R. Yang, J. Appl. Phys. 108, 094315 (2010) 30.  T. Yagi, K. Tamano, Y. Sato, N. Taketoshi, T. Baba, Y. Shigesato, J. Vac. Sci. Technol. A23, 1180 (2005) 31.  N. Oka, K. Kato, T. Yagi, N. Taketoshi, T. Baba, N. Ito, Y. Shigesato, Jpn. J. Appl. Phys. 49, 121602 (2010)http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/  International Journal of Thermophysics (2024) 45:271 327  Page 18 of 18 32.  S. Kawasaki, Y. Yamashita, N. Oka, T. Yagi, J. Jia, N. Taketoshi, T. Baba, Jpn. J. Appl. Phys. 52, 065802 (2013) 33.  T. Baba, Jpn. J. Appl. Phys. 48, 05EB04 (2009) 34.  B. Hinterleitner, I. Knapp, M. Poneder, Y. Shi, H. Müller, G. Eguchi, C. Eisenmenger-Sittner, M. Stöger-Pollach, Y. Kakefuda, N. Kawamoto, Q. Guo, T. Baba, T. Mori, S. Ullah, X.-Q. Chen, E. Bauer, Nature 576, 85–90 (2019) 35.  T. Baba, N. Taketoshi, T. Yagi, Jpn. J. Appl. Phys. 50, 11RA01 (2011) 36.  W.J. Parker, R.J. Jenkins, C.P. Butler, G.L. Abbott, J. Appl. Phys. 32, 1679 (1961) 37.  T. Baba, T. Baba, K. Ishikawa, T. Mori, J. Appl. Phys. 130, 225107 (2021) 38.  F. Righini, A. Cezairliyan, High Temp. High Press. 5, 481 (1973) 39.  A. Cezairliyan, T. Baba, R. Taylor, Int. J. Thermophys. 15, 317–341 (1994) 40.  T. Baba, A. Ono, Meas. Sci. Technol. 12, 2046–2057 (2001) 41.  L. Vozár, W. Hohenauer, High Temp. High Press. 35–36(3), 253–264 (2003) 42.  M. Akoshima, T. Baba, Thermal Conductivity 28, Thermal Expansion 16 28, 497–506 (2006) 43.  T. Baba, Thermal Conductivity 35. Proceedings of 35th International Thermal Conductivity Confer-ence (DEStech Publications, Inc., Boston, 2022) pp. 129–141. 44.  N. Taketoshi, T. Baba, A. Ono, Jpn. J. Appl. Phys. 38, L1268-1271 (1999) 45.  N. Taketoshi, T. Baba, A. Ono, Meas. Sci. Technol. 12, 2064–2073 (2001) 46.  Y. Kakefuda, K. Yubuta, T. Shishido, A. Yoshikawa, S. Okada, H. Ogino, N. Kawamoto, T. Baba, T. Mori, APL Mater. 5, 126103 (2017) 47.  J.R. Rumble, CRC Handbook of Chemistry and Physics, 99th edn. (CRC Press, LLC, Boca Raton, 2018)Publisher’s Note  Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Development of Fourier Transform Ultrafast Laser Flash Method for Simultaneous Measurement of Thermal Diffusivity and Interfacial Thermal Resistance Abstract 1 Introduction 2 Experiments 3 Results 4 Discussion 4.1 Sensitivity Analyses 4.2 Reproducibility of Measurement 5 Conclusion References