Thermal Property Estimation of Thin-Layered Structures by Means of Thermoreflectance Measurement and Network Identification by Deconvolution Algorithm

Laser-induced forward transfer (LIFT) is a promising technology in microscale digital printing of metals, and its feasibility is proved in a wide variety of applications such as wiring and repairing of electronic circuits [1–6]. In the metal LIFT process, a pulsed laser is used as the driving force propelling the thin metal film of the donor toward the receiving substrate (i.e., receiver), and the projected metal is deposited on the receiver. The shape and morphology of the metal deposited on the receiver can be modified by the laser heating process [5,6]. Kim et al. showed that the interfacial layer with a large electron–phonon coupling coefficient such as titanium contributes to the selective liftoff of the gold (Au) layer in the femtosecond laser ablation, thereby suppressing thermal damages on the donor samples [7]. Thus, understanding the thermodynamic responses and the structure of the donor is of paramount importance for applications of the metal LIFT on electronics repair and packaging.

The time-domain thermoreflectance (TDTR) method is an effective noncontact and nondestructive tool for studying heat transfer in thin films. This method enables the measurement of the interfacial thermal resistances in addition to the thermal diffusivities of thin-layered materials. In the TDTR measurement, a sample is coated with a thin metal film, which acts as a transducer. The thin metal film is heated by a pulsed laser (i.e., pump laser), and another laser (i.e., probe laser) monitors the reflectivity of the metal transducer as a function of time. Because reflectivity depends on the temperature of the metal surface, the temperature decay curve can be obtained by monitoring the temporal intensity change of the reflected probe beam. The thermal properties are extracted from the measured temperature change by means of an appropriate theoretical model [8–10]. However, one should know the structure of the sample before the measurement, and it is difficult to extract accurate thermophysical properties when the sample has unintentional damage (e.g., cracks in a chip) and highly temperature-dependent thermal properties, which are the typical cases in the laser processing. To overcome this constraint, the network identification by deconvolution (i.e., NID) algorithm was proposed by Székely and Van Bien [11] and became well-known for the thermal characterization of electronic components and associated materials in the field of thermal management [11–14]. This technique can be utilized not only to extract thermal properties but also to identify the structure of the sample. In this method, the temperature response is transformed to the time constant spectrum by deconvolution, and then the spectrum is transformed to the cumulative structure function and differential structure function. The cumulative structure function is the variation of cumulative thermal capacitance as a function of the cumulative thermal resistance along the heat flow in the sample. The differential structure function can be obtained by taking the derivative of the cumulative structure function with respect to the cumulative thermal resistance. The thermal resistances and thermal capacitances of each part can be identified from those functions [11,12]. Ezzahri and Shakouri demonstrated the feasibility of assessing the thermal properties of nm-scale samples by combining the numerical simulation of the TDTR experiments and NID analysis [15]. More recently, Mitterhuber et al. made structure functions from the relative temperature change obtained in the TDTR experiments on a 166 nm thick Nb₂O₅ sample. In addition, they utilized the structure functions to obtain quantitative thermal properties by introducing the scaling parameter determined from the comparison with the structure function made from the numerical simulation results [16].

In this study, we investigated the thermal properties of the sample without the use of numerical simulations or analytical solutions. We obtained the quantitative temperature response by dividing the temporal reflectivity change in the TDTR experiment by the thermoreflectance coefficient. We determined the thermal properties of the samples by analyzing the structure functions made from the temperature response in the nanosecond (ns) TDTR experiment, assuming the LIFT process with a semi-infinite substrate coated with tens of nm of a metal film heated by an ns-pulsed laser. To prepare the sample, we deposited a 100 nm-thick gold transducer layer on Al₂O₃ (sapphire) and Si substrates. The 5-nm-thick Cr and Ti adhesion layers were used to bond the gold layer to Al₂O₃ and Si substrates, respectively. We determined the interfacial thermal resistance between the Au film and the substrate layer and the thermal conductivity of the substrates after ns-pulsed laser heating from the structure functions given by the NID. These properties are important to investigate the LIFT phenomena especially the thermal damage of the substrates. Our study sheds light on the feasibility of the structure functions in assessing the thermodynamic properties of the multilayered samples not only for laser processing but also for thermal management of the electronics.

In the ns-TDTR experiments, the temperature response of the thin metal film surface is obtained by measuring the reflectivity change after excitations by the nanosecond pump laser. Figure 1 shows the schematic of the experimental setup. In this study, we used Nd:YAG pulsed laser with a wavelength of 1064 nm, pulse duration of 5.5 ns, and repetition rate of 10 Hz as the pump laser to heat the samples (Fig. 1). The probe laser was a 532-nm continuous wave laser. The beam diameters of the pump and the probe were 2.4 mm and 0.12 mm, respectively. The power of the incident probe beam is approximately 3.0 mW, and its power density is negligible compared to the peak power density of the pump laser. We acquired 10,000 times the reflectivity response after pump laser heating, and the averaged data were used to make the structure functions. The repetition rate is low enough for the sample to reach room temperature before adding another pump excitation. In addition, we directly monitored the temperature decay by the CW probe laser, without using either modulation or lock-in technique.

Generally, normalized reflectivity change in the thermoreflectance method is fitted by normalized simulation results or analytical model to extract the thermal properties. However, the structure function needs the actual temperature change as a function of the heating power. Therefore, we estimated the temperature change during the experiment by dividing the reflectivity change by the thermoreflectance coefficient of Au, $2.36×10−4 K−1$ [17]. The incident pump energy per pulse was approximately 3 mJ, and the maximum temperature rise of the samples was about 30 K.

In this study, we made the cumulative structure function and differential structure function of the sample by the NID algorithm using the ns-TDTR experimental result (Fig. 3).

where $Q$ is the energy per pulse given by the pump laser, and $R$ is the reflectivity of Au at the wavelength of the pump laser (1064 nm).

Figures 4 and 5 show the cumulative structure function $CΣ$ and the differential structure function $dCΣ/dRΣ$⁠, respectively. The two-step structure of the cumulative structure function in Fig. 4 and the two peaks of the differential structure function in Fig. 5 correspond to the two layers of the sample.

where $ρ, c$ are the density and the specific heat, respectively.

The material properties employed in Eqs. (15)–(17) are summarized in Table 1 [20].

In this study, we adjusted the parameter $P$ of the NID algorithm, a measure of the absorbed energy, so that the thermal conductivity of the Al₂O₃ substrate of the Au–Cr–Al₂O₃ sample obtained from the structure functions matched the literature values, ensuring the algorithm's calibration. Due to the variation in the literature values of the thermal conductivity of Al₂O₃, calibration was performed using three literature values [20–22]. Using the NID calibrated with each literature value of the thermal conductivity of Al₂O₃, the thermal conductivity of Si was extracted from the experimental results of the Au–Ti–Si as shown in Fig. 6. Those thermal conductivities of Si were in good agreement with a range of literature values which vary in the range of $110–148 W/(m K)$ at room temperature [20,23–26]. The error bars were calculated based on the standard deviation of the pump energy measurements.

Additionally, Fig. 7 shows the fitted interfacial thermal resistance between Au and the substrates by using the calibrated NID. The interfacial thermal resistance between gold and silicon is around $8.4–14×10−9 m2 K/W$ [27,28] in good agreement with our results without significant dependence on $λAl2O3$ used in the calibration.

We conducted experiments by changing the pump fluence (Fig. 8) to investigate the perturbative regime. The measured Si thermal conductivity is not significantly affected by heating energy when the temperature rise is within 30 K. Here, the NID algorithm was calibrated using the thermal conductivity of Al₂O₃ set at 46 W/(m K).

In this study, we analyzed the transient temperature variation obtained in the ns-TDTR experiment using the cumulative structure function and the differential structure function of a semi-infinite Al₂O₃ and Si substrate covered with 100 nm of Au thin film obtained by NID algorithm. We calibrated the NID algorithm by comparing the experimental result of the Au–Cr–Al₂O₃ sample and the literature values of the thermal conductivity of Al₂O₃. After that, we were able to extract the thermal conductivity of the Si substrate in good agreement with the literature values. We expect that the NID algorithm can be used to characterize the thermal properties of multilayered systems for design and inspection of the electronics packaging and investigation of laser ablation phenomena with an ability to visualize the structure functions.

Part of this research has been conducted under the support of the Grant-in-Aid for Scientific Research from MEXT/JSPS 21H01260.

• Grant-in-Aid for Scientific Research from MEXT/JSPS (Grant Nos. 21H01260 and 23K13260; Funder ID: 10.13039/501100001691).

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$a(t)$ =

step temperature response (K W⁻¹)

$A$ =

cross-sectional area of the heat flow path (m²)

$ai$ =

magnitude corresponding to $τi$ (K W⁻¹)

$c$ =

differential thermal capacitance (J K⁻¹ m⁻¹)

$c$ =

specific heat (J kg⁻¹ K⁻¹)

$CCi$ =

C component in the Cauer representation (J K⁻¹)

$CFi$ =

C component in the Foster representation (J K⁻¹)

$CΣ$ =

cumulative thermal capacitance (J K⁻¹)

$K$ =

value of $dCΣ/dRΣ$ in the differential structure function (W² s K⁻²)

$N$ =

the number of time constants of a system

$P$ =

power of the step excitation (W)

$Q$ =

energy per pulse of the pump laser (J)

$r$ =

differential thermal resistance (K W⁻¹ m⁻¹)

$R$ =

reflectivity

$R(z)$ =

time constant spectrum

$RCi$ =

R component in the Cauer representation (K W⁻¹)

$RFi$ =

R component in the Foster representation (K W⁻¹)

$Rint$ =

interfacial thermal resistance (m² K W⁻¹)

$RΣ$ =

cumulative thermal resistance (K W⁻¹)

$T$ =

temperature rise (K)

$W(z)$ =

weight function

$z$ =

natural log of time

$λ$ =

thermal conductivity (W m⁻¹ K⁻¹)

$ρ$ =

density (kg m⁻³)

$τi$ =

ith time constant (s)

$Φ$ =

energy absorbed by metal transducer

$χ$ =

volumetric specific heat (J m⁻³ K)