Non-contact photothermal pump-probe methodologies such as Frequency-Domain Thermo-Reflectance (FDTR) systems facilitate the examination of thermal characteristics spanning semiconductor materials and their associated interfaces. We underscore the significance of meticulous measurements and precise error estimation attained through the analysis of both amplitude and phase data in Thermo-Reflectance (TR). The precision of the analytical estimation hinges greatly on the assumptions made before implementing the method and notably showcases a decrease in errors when both the amplitude and phase are incorporated as input parameters. We demonstrate that frequency-domain calculations can attain high precision in measurements, with error estimations in thermal conductivity (*k*), thermal boundary resistance (*R*_{th}), and thermal diffusivity (*α*) as low as approximately 2.4%, 2.5%, and 3.0%, respectively. At the outset, we evaluate the uncertainty arising from the existence of local minima when analyzing data acquired via FDTR, wherein both the phase and amplitude are concurrently utilized for the assessment of cross-plane thermal transport properties. Expanding upon data analysis techniques, particularly through advanced deep learning approaches, can significantly enhance the accuracy and precision of predictions when analyzing TR data across a spectrum of modulation frequencies. Deep learning models enhance the quality of fitting and improve the accuracy and precision of uncertainty estimation compared to traditional Monte Carlo simulations. This is achieved by providing suitable initial guesses for data fitting, thereby enhancing the overall performance of the analysis process.

## I. INTRODUCTION

Technological advancements have yielded the creation of energy-efficient devices. However, the continual growth of the global population and the depletion of traditional energy sources contribute to an ongoing surge in energy demand. Consequently, there is a substantial scientific emphasis on exploring renewable energy sources. Despite the remarkable progress in the microelectronic industry, facilitating the development of inventive and efficient devices, it remains crucial to address the challenge of energy wastage associated with heat generation during operation.^{1–3} The growing need for effective energy utilization has spurred investigations into thermoelectricity, a phenomenon with the capacity to convert heat into electric current. To enhance thermoelectric performance, the identification of materials yielding significant potential differences and efficient charge carrier flow under thermal gradients is a paramount task. Semiconductors characterized by narrow and direct bandgaps have surfaced as promising contenders in this quest, holding the promise of achieving elevated thermoelectric efficiencies. Gallium arsenide (GaAs) is a unique combination of a direct bandgap, cubic zinc blende structure, and high electron mobility, with improved thermoelectric properties, making it a versatile and valuable material for a range of thermoelectric and optoelectronic applications.^{4,5} However, GaAs is a thermally isotropic material; hence, we will consider only cross-plane thermal properties without the loss of generality.

Non-contact thermal pump-probe techniques leverage lock-in amplification (LIA) detection technology to effectively extract noisy signals from the measured thermoreflectance signal, enabling the simultaneous measurement of both amplitude and phase (AP) differences. A comprehensive discussion has been provided on the fundamental understanding of the correlation between the measured reflectivity change and the temperature response. This understanding facilitates the characterization of heat transfer properties across a diverse array of materials and structures studied in previous works.^{6–8} The uncertainty in the determination of the thermal properties obtained in the time-domain and then the transformed response in the frequency domain using the Hankel transform are validated with the experimental results.^{9–13} This is in return supported by advanced deep learning (DL) modeling. Measurement uncertainty indicates the level of confidence in the results and is closely related to the sensitivity of the parameters. This idea of using machine learning has already been introduced, but the analysis of error propagation in Time-Domain Thermo-Reflectance (TDTR) is still underdeveloped as compared to the Frequency-Domain Thermo-Reflectance (FDTR) measurements.^{14,15} Previous works under model studies suggest the accuracy of uncertainty estimation measured in traditional Monte Carlo simulations.^{16,17} The model discussed in this approach can be trained specifically for a given system configuration, considering parameters such as pre-assumed laser spot size, transducer thickness, and other properties. After training, these models can utilize both amplitude and phase lag data from an unknown material, obtained under the same configuration, to predict cross-plane thermal properties such as thermal boundary resistance (*R*_{th}), diffusivity (*α*), or anisotropic thermal conductivity (*k*) for the substrate. For the complete prediction of thermal properties, three models are necessary: one for each unknown material property. This comprises a more precise measurement method for an accurate figure of merit (ZT).

The paper presented here discusses the precision achieved in determining the thermal properties of the GaAs substrate, in the frequency domain, utilizing a comparative study between analytical methods and advanced machine learning approaches. Deep learning techniques have demonstrated their effectiveness in learning patterns from data and have shown remarkable success in determining thermal properties in our experiments.^{18–20} However, their application to thermoreflectance (TR) signal analysis has not been extensively explored in previous studies,^{21,22} which is one of the main motivations for this current work. The objective of this study is to investigate the impact of local minima on the analysis of FDTR amplitude data, particularly in relation to the uncertainty associated with thermal parameters. Although previous works have always been limited to the phase data of these signals, the main goal of this work is the use of both the amplitude and phase for the precious error propagation analysis of the isotropic semiconductor. Moreover, this research aims to demonstrate the potential of a simple deep learning model in predicting thermal conductivity, thermal diffusivity, and thermal boundary resistance accurately while analyzing the FDTR amplitude and phase. By integrating a deep learning model into the analysis, the study enhances the quality of the non-linear least square fitting process. The deep learning model's effectiveness lies in providing an appropriate initial guess for data fitting, which leads to more accurate results.

## II. EXPERIMENTAL SETUPS (PUMP-PROBE)

### A. Frequency-Domain Thermo-Reflectance (FDTR)

In Frequency-Domain Thermo-Reflectance (FDTR), thermoreflectance signals are obtained by modulating the frequency of the pump beam, unlike Time-Domain Thermo-Reflectance (TDTR), where adjustments are made to the delay time between the pump beam and the probe beam. This technique proves to be invaluable for measuring cross-plane thermal transport properties across a wide range of material systems. The schematic of FDTR, depicted in Fig. 1, illustrates the use of a Cobolt 06-MLD (modulated laser diode; *λ* = 488 nm, variable CW power) for excitation (pump beam). This laser is chosen for its suitability in OEM (Opto-electrical modulation) applications with the beam spots (1/*e*^{2}) having been measured by the beam profiler (Slit 2 um S-BR2-Si) kept constant throughout the experiment. The pump beam is modulated within the frequency range of 10 kHz–2 MHz, facilitated by a 30 MHz waveform generator (Agilent 33522A) serving as the reference signal for the lock-in amplifier. The probe beam Cobolt 06-01 (unmodulated laser diode; *λ* = 638 nm, variable CW power) utilized in this method is employed to detect changes in the temperature field, particularly the thermal response across the surface of the sample (Table I).

Domain . | Laser . | Laser power A_{1} (mW)
. | Reference scanning frequency f_{0} (kHz)
. | d_{1} (μm)
. |
---|---|---|---|---|

Frequency | Continuous | 10 (probe) 30 (pump) | 10–1250 (confidence interval) | 2.42 |

Domain . | Laser . | Laser power A_{1} (mW)
. | Reference scanning frequency f_{0} (kHz)
. | d_{1} (μm)
. |
---|---|---|---|---|

Frequency | Continuous | 10 (probe) 30 (pump) | 10–1250 (confidence interval) | 2.42 |

The experiment described in Fig. 1 uses the fixed parameters for the pump-probe setup in the frequency domain.

## III. PARAMETERS’ ESTIMATION APPROACH

^{21}for multi-layer thermal transport parameters for individual layers and for interfaces used in this paper. The three-dimensional signal can be written in terms of the Henkel variable (

*λ*) based on the models already developed as

*Q*is the laser power and

*d*

_{1}is the laser spot (1/

*e*

^{2}). Coefficients

*C*and

*D*can be calculated using

*A*

_{3},

*A*

_{2}, and

*A*

_{1}specified by

*d*is the layer thickness and

_{n}*k*is the cross-plane thermal conductivity of the

_{zn}*n*th layer, and

*ρ*is the density of the

_{n}*n*th layer,

*c*is the specific heat capacity of the

_{n}*n*th layer, and

*k*and

_{rn}*k*are the in-plane and cross-plane thermal conductivities of the nth layer, respectively, with the usual assumptions. The resistances of the thermal interfaces are described by matrices,

_{zn}*R*

_{n}_{,n−1}is the thermal interface resistance between the

*n*th and the (

*n*− 1)th layers. In this paper, we are studying thermally isotropic samples so

*k*

_{rn}_{ }=

*k*.

_{zn}### A. Monte Carlo simulation approach

The uncertainty in cross-plane measurements can be estimated using analytical methods and further validated using the widely recognized Monte Carlo simulation tool (Fig. 2). Analytical methods account for uncertainties stemming from various parameters and measurements, although they necessitate making initial assumptions or guesses. The accuracy and precision of analytical estimation are contingent upon the assumptions made throughout the process. The fitting approach aids in determining the optimal values for unknown parameters by comparing simulated results with experimental data. Without a reasonable initial guess or a robust fitting algorithm, the Monte Carlo results might not accurately capture the behavior of the underlying photothermal system. Moreover, it is crucial to recognize that samples with varying thermal properties may display different phase lags between pump and probe signals, particularly when the phase is sensitive to an associated unknown parameter. This sensitivity can lead to fluctuations in the observed phase lag across different frequency scanning points. Therefore, it is crucial to recognize and accommodate these variations when interpreting the results of Monte Carlo simulations for samples with diverse thermal properties. In the context of FDTR, it employs a trust-region reflective fitting algorithm alongside advanced fitting options offered by the Levenberg–Marquardt (LM) algorithm. This combination is instrumental in obtaining accurate and dependable results in data analysis, aiding in predicting whether the minimization problem converges to local or global minima.

### B. Model architecture for deep learning approach

The validity of the experiment hinges on the thermal model under consideration, utilizing a known GaAs sample with a 50 nm layer of gold (Au) as the metal transducer for FDTR analysis. The model is trained to discern patterns in error propagation while determining cross-plane thermal parameters. The training process automatically halts if there is no discernible improvement in performance with each iteration. In such cases, the training process will restart following parameter tuning. This cycle continues until the desired performance is ultimately attained. The measurement uncertainty analysis primarily concentrates on determining isotropic thermal conductivity (*k _{r}*), diffusivity (

*α*), and cross-plane thermal boundary resistance (

*R*

_{th}). Although cross-plane thermal conductivity can be measured in the case of time-domain analysis, this aspect is not investigated in this study. The anisotropic index for the GaAs bulk samples all through the training procedure has been considered $\eta = k r k z=1$.

### C. Model description

The standard approach for training backpropagation neural networks is commonly interpreted as an application of maximum likelihood estimation, a statistical procedure. When incorporating “weight decay,” this aligns with the concept of maximum penalized likelihood estimation. These procedures aim to identify a single set of network parameter values (weights) considered the “best” based on a provided set of training cases. Subsequently, these optimized parameter values are applied for predicting outcomes on test cases. In contrast, the Bayesian approach to the statistical inference of fitting models takes a distinctive approach. It advocates for grounding predictions for a test case in all conceivable values for the network parameters. These values are weighted by the probability assigned to each set of parameter values, considering the training data. Unlike conventional training, which is fundamentally an optimization problem, this training approach is characterized by speed, transforming prediction into an integration problem. The merits of such an approach include mitigating “overfitting,” quantifying the uncertainty in predictions, and the automatic determination of a suitable scale for network weights. This framework (trained in Keras TensorFlow) offers a more nuanced perspective, accounting for the entire distribution of potential parameter values rather than focusing on a single optimized set. The Bayesian training problem for backpropagation networks, a frequently employed approach, is to approximate the posterior parameter probabilities by a Gaussian distribution centered around the mode. Although this method has yielded impressive results, there exists a keen interest in exploring alternative approaches that do not hinge on the assumption of a Gaussian distribution.

Artificial neural network (ANN) is usually trained based on a backpropagation training algorithm. Backpropagation stands as a supervised learning algorithm designed to minimize the error or cost function linked to a neural network's predictions. Backpropagation algorithm used for the parameter estimation is strictly labeled a deep learning algorithm. This is different from the basic feedforward algorithm used in Multilayer Perceptrons (MLPs), thereby decreasing the training complexity with higher non-linearities. In the usual forward pass algorithm, the input data are fed into the network, and activations are computed layer by layer until reaching the output layer. The network's output is then compared to the ground truth labels or targets to calculate the error. It involves computing the gradient of the cost function concerning the network's parameters, encompassing weights and biases. In the parameter update phase, the algorithm utilizes the gradient values to indicate the direction and magnitude of the steepest ascent of the cost function. Consequently, it adjusts the parameters opposite to the gradient, effectively progressing toward the cost function's minimum. The primary goal is the iterative adjustment of parameters, aiming to minimize the error or cost between the predicted output and the target. Backpropagation operates within an iterative optimization process, continuously updating parameters until convergence or meeting a predefined stopping criterion. By applying the chain rule of calculus, the algorithm backpropagates the error through the network, assigning it to each thermal parameter. Overall, backpropagation plays a pivotal role in neural network training, iteratively calculating gradients to update parameters and enhance the network’s capability to discern intricate patterns in data. This architecture allows the neural network to learn complex patterns and relationships present in both the amplitude and phase channels separately (Simultaneous Processing). In other cases, it might be beneficial to process the amplitude and phase channels simultaneously, meaning that both channels are processed together by the same layer or set of layers in the neural network. This approach could involve concatenating the amplitude and phase channels and feeding them into a shared set of layers. This effectively leverages the information contained in the complex-valued data. Experimentation with different architectures and hyperparameters, as well as proper validation techniques, will help determine the most effective network configuration for your specific task.

The individual neurons take the product of the average weight and the input parameter. This value is further forwarded to the output nodes through a tanh activation function. The hyperbolic tangent (tanh) activation function is our rudimentary choice in this architecture for several features that include range (−1, 1), zero-centered, differentiability of non-linearities. Another reason for the choice of tanh as the activation function is due to its S-shaped curve similar to the cumulative distribution function (CDF) of a Gaussian distribution. This makes it well-suited for modeling non-linear relationships in data that have Gaussian-like distributions or profiles. The error generated due to the mismatch of the target values of the bias gets transmitted back to the model. This further updates the weights between the neurons. The description outlines the architecture and approach of a feedforward deep learning model used for predicting unknown material properties from experimental data. The model is designed to predict thermal properties (thermal conductance, boundary resistance, or diffusivity) based on a series of phase lags and amplitudes measured at different modulation frequencies in the range of 10 kHz–1.25 MHz. The model architecture of the thermoreflectance model is shown in Fig. 3(a), whereas Fig. 3(b) is the flow chart for the error prediction maximization stage. Noteworthy, the maximization stage of error prediction is actually the stage that determines the global minimum from the rest of the local minima.

### D. Deep learning analysis for uncertainty in the parameter estimation

^{7}

*J*is the Jacobian matrix,

*u*is the unknown parameters,

*C*is the assumed thermal parameters, and

*φ*is the measured input signal amplitude or phase. The diagonal elements of the matrix

*Var*[

*β*] is the uncertainty of the unknown parameters caused by the measuring uncertainty

_{u}*Var*[

*φ*] and also the uncertainty of the controlled parameters

*Var*[

*β*]. While carrying out this operation, the gradient is kept constant for different parameters over the desired window of interest. A known global minima for $ \beta u$ provides a very good prediction. This prediction is comparatively low if the assumed solution is not close to the global minima. One of the most usual ways is to select a set of points and the solution with the lowest residual. This way is quite helpful for unknown samples as well. The uncertainty in the controlled parameters and the sensitivity of the parameters determine the error caused by the initial guess. The controlled parameters are assumed to be of the normal distribution about the mean value with certain uncertainty. Monte Carlo histograms almost fitted by the prescribed neural network model clearly show the uncertainty during the measurement of the thermal properties of GaAs simultaneously as these three parameters are not correlated to one another.

_{C}Here, the bulk material is investigated by both the approaches—with phase (P) only and while using both phase and amplitude (AP). Table II presents trust regions made in the case of FDTR.

Domain . | Bulk material (thickness) . | Transducer (thickness) . | Frequency range (Hz) . |
---|---|---|---|

Frequency | GaAs (0.5 mm) | Au (50 nm) | 10 k–1.25 M |

Domain . | Bulk material (thickness) . | Transducer (thickness) . | Frequency range (Hz) . |
---|---|---|---|

Frequency | GaAs (0.5 mm) | Au (50 nm) | 10 k–1.25 M |

The approach involves training a model with artificial neural networks using an unevenly split dataset. Specifically, out of 1500 phase training points in FDTR, 90% correspond to high pump modulation frequency range values (ranging between 100 kHz and 2 MHz), while the remaining 10% correspond to low modulation frequency values (ranging between 10 and 100 kHz). A similar splitting has been performed in the amplitude approach with a sensitivity, which ranges in a similar ratio as for the phase. The purpose of this uneven split is to investigate whether the model produces varying levels of uncertainty for different frequency ranges in the test set, based on the training set sizes for each range in both approaches.

## IV. RESULTS

The fractional change in reflectance, which corresponds to the ratio of the in-phase and out-of-phase components of the TDTR AC signal, is observed against the time delay between the pump and probe pulses in Fig. 6. The experimental data demonstrate a good fit with the analytical model, as depicted in the figure, along with the corresponding standard errors shown in Table III. FDTR data (normalized amplitude and phase) have been plotted against sweeping frequencies between 10 kHz and 1.25 MHz (high sensitivity range for the model) in Fig. 4. Levenberg–Marquardt (LM) and trust-region (TR) algorithms are the usual options as the nonlinear complex least square fitting algorithm yields different results. The smallest residual of local minima is associated with points with the smaller distance from the trust regions. When deep learning models are implemented on the model output data, the distribution is mostly normal for both of them.

Domain . | Modulation frequency (Hz) . | Thermal boundary resistance (%) . | Thermal conductivity (%) . | Thermal diffusivity (%) . | χ^{2}
. |
---|---|---|---|---|---|

Frequency (P) | 10.4 k–1.25 M | 5.3572 (7.98 × 10^{−6} m^{2} W/K) | 3.925 (53.61 W/mK) | 6.537 (29.169 × 10^{−6} m^{2}/s) | 91.245 |

Frequency (AP) | 10.4 k–1.25 M | 4.8211 (8.043 × 10^{−6} m^{2} W/K) | 3.237 (52.7 W/mK) | 3.528 (27.012 × 10^{−6} m^{2}/s) | 93.234 |

Domain . | Modulation frequency (Hz) . | Thermal boundary resistance (%) . | Thermal conductivity (%) . | Thermal diffusivity (%) . | χ^{2}
. |
---|---|---|---|---|---|

Frequency (P) | 10.4 k–1.25 M | 5.3572 (7.98 × 10^{−6} m^{2} W/K) | 3.925 (53.61 W/mK) | 6.537 (29.169 × 10^{−6} m^{2}/s) | 91.245 |

Frequency (AP) | 10.4 k–1.25 M | 4.8211 (8.043 × 10^{−6} m^{2} W/K) | 3.237 (52.7 W/mK) | 3.528 (27.012 × 10^{−6} m^{2}/s) | 93.234 |

After fitting the experimental data with both analytical and deep learning algorithms, Table III indicates that FDTR yields much lower standard error than its time-domain analysis. The possible explanation is that FDTR can measure all thermal parameters with just two inputs—amplitude and phase with the possible weight vectors being the thermal parameters that are tuned in order to get the global minima. Now we would like to discuss the effects of the deep learning model on the error propagation analysis. Deep learning Levenberg–Marquardt (LM) model has an overall advantage when overestimating the uncertainty due to the convergence into the local minima. Hence, the deep learning model predictions are better than trust-region (TR) offsets. Although these two are the standard non-linear least square fitting algorithms, the backpropagation (AP) model is better than the previous results in both the time and frequency domain. This further confirms that for better error estimations, a better initial guess is always the best choice in linear square fits. The introduction of the chi-square test is to determine whether the association of two qualitative variables (amplitude and phase) as input parameters is statistically significant. In the case of the FDTR (both approaches), the chi-square statistics (Table III) further supports the idea of more input parameters for precise measurement during the comparison of observed values to the expected values.

The sensitivity between 10 and 100 kHz is quite low for the FDTR, and this can be considered the maximum likelihood estimation to start with. Hence, narrowing the range suggests more chances of convergence close or almost equal to the global minimum even with a poor initial guess when the model is working within the best sensitivity range. From Table III, one can deduce that introducing another channel of information, i.e., the amplitude of the thermoreflectance signal, will decrease the error estimation.

This section is based on the amplitude and phase data used as input parameters over just the phase analysis. The phase noise is extracted from the standard deviation of measurements taken over the scanned frequency, and the RMS of the phase noise is 0.1°, taken constant over the entire experiment. The tabular representation of FDTR shows lower error estimation with less complexity. Hereon, the discussion is about the effects of deep learning on the error analysis. Phase noise is extracted from the standard deviation of all the measurements at each modulation frequency, and the RMS of phase noise is measured. The best fit limits for FDTR calculations using amplitude and phase approach are between 100 kHz and 1 MHz, which is verified in Fig. 4 as well. The uncertainty in the determination of the parameters has subsequently decreased within this range. This, in turn, also answers the reduction in the correlation errors that arise due to the multiple parameters seen in the TDTR model, which is detrimental in understanding that as the smaller number of parameters are estimated in the TDTR due to the complexity of the model; hence, less uncertainties in the measurement are also taken into consideration. The error distribution caused by local minima is plotted in Figs. 5 and 6. Note that this error is in addition to the error caused by the phase measurement and uncertainty in the controlled parameters (Table IV).

Domain . | Simulation . | Modulation frequency (Hz) . | Thermal boundary resistance (%) . | Thermal conductivity (%) . | Thermal diffusivity (%) . | χ^{2}
. |
---|---|---|---|---|---|---|

Frequency (P) | Monte Carlo | 10.4 k–1.25 M | 3.457 (8.042 × 10^{−6} m^{2} W/K) | 5.614 (52.369 W/m K) | 7.895 (28.552 × 10^{−6} m^{2}/s) | 91.245 |

Deep learning | 10.4 k–1.25 M | 2.789 (8.09 × 10^{−6} m^{2} W/K) | 4.245 (53.128 W/m K) | 5.623 (29.256 × 10^{−6} m^{2}/s) | 96.155 | |

Frequency (AP) | Monte Carlo | 10.4 k–1.25 M | 3.398 (8.046 × 10^{−6} m^{2} W/K) | 3.217 (54.144 W/m K) | 5.846 (29.187 × 10^{−6} m^{2}/s) | 95.234 |

Deep learning | 10.4 k–1.25 M | 2.587 (8.114 × 10^{−6} m^{2} W/K) | 2.415 (54.144 W/m K) | 3.025 (28.752 × 10^{−6} m^{2}/s) | 97.859 |

Domain . | Simulation . | Modulation frequency (Hz) . | Thermal boundary resistance (%) . | Thermal conductivity (%) . | Thermal diffusivity (%) . | χ^{2}
. |
---|---|---|---|---|---|---|

Frequency (P) | Monte Carlo | 10.4 k–1.25 M | 3.457 (8.042 × 10^{−6} m^{2} W/K) | 5.614 (52.369 W/m K) | 7.895 (28.552 × 10^{−6} m^{2}/s) | 91.245 |

Deep learning | 10.4 k–1.25 M | 2.789 (8.09 × 10^{−6} m^{2} W/K) | 4.245 (53.128 W/m K) | 5.623 (29.256 × 10^{−6} m^{2}/s) | 96.155 | |

Frequency (AP) | Monte Carlo | 10.4 k–1.25 M | 3.398 (8.046 × 10^{−6} m^{2} W/K) | 3.217 (54.144 W/m K) | 5.846 (29.187 × 10^{−6} m^{2}/s) | 95.234 |

Deep learning | 10.4 k–1.25 M | 2.587 (8.114 × 10^{−6} m^{2} W/K) | 2.415 (54.144 W/m K) | 3.025 (28.752 × 10^{−6} m^{2}/s) | 97.859 |

Figure 6 illustrates the backpropagation simulation results for the AP model, incorporating a phase noise of 0.4°. The distributions obtained from AP and P (Fig. 5) exhibit remarkable similarity though for the thermal conductivity and diffusivity and closely resemble a normal distribution in all three parameter estimations. In contrast, the distribution from AP over P displays evident local minima for C and G, leading to an overestimation of their uncertainty shown in Fig. 5. By utilizing deep learning predictions as the initial guess, AP approach mitigates the challenge of overestimating uncertainty due to convergence to local minima, as observed in the standalone P approach. It is worth noting that the standalone AP employs slightly offset FDTR fitting results as the initial guess, which is a reasonable assumption for initial points. AP approach may predict more accurately than FDTR offset, which uses a normal distribution with MAPE (Mean Absolute Probability Error) less for the P and FDTR estimates, offering an improved initial guess for nonlinear least square fitting.

A comparative study between both the approaches (AP vs P) after initial estimations from the backpropagation algorithm is discussed in Fig. 7. Very clearly, it can be seen that the estimation performance has improved when the amplitude of the thermal response is introduced as an input parameter along with the phase. This is in turn supported by the uncertainty in one of the thermal parameters in comparison with the other, keeping all the initial guesses unchanged as shown in Fig. 8. Moreover, it is evident that as the number of iterations increases, the impact of using the amplitude as an input parameter in the deep learning model is predominant and the prediction of any of the parameters depends on the other in a non-linear manner. Usually, three different models predict three parameters, but this approach can simultaneously work on three parameters.

The model's performance and uncertainty estimation is influenced by the distribution of training data across the phase and amplitude datasets for varied pump modulation frequencies (here 100 kHz–2 MHz). By focusing the majority of the training data on the high modulation frequency range in FDTR, the essential creation of an imbalanced dataset can impact model learning and generalization. Similar techniques have been implemented for the increased iterations (Fig. 8) that indicate precise predictions with reduced scattering of estimated parameters when the amplitude is used as an input parameter along with the phase compared to the use of only the phase for the estimations.

## V. CONCLUSION

In this article, we used the amplitude of the frequency-domain thermoreflectance signal for the first time and obtained precise values of cross-plane thermal conductivity, diffusivity, and thermal boundary resistance for a GaAs substrate. By leveraging the capabilities of deep learning, this research demonstrates the potential for enhancing FDTR data analysis and the reduced complexity in the measurement of cross-plane thermal properties in the frequency domain in comparison with previous TDTR experiments. The utilization of deep learning methods (training comparison with trust-region algorithm and Levenberg–Marquardt algorithm) offers an alternative approach to nullify the challenges due to uncertainty caused by local minima and provides improved accuracy in predicting important thermal properties. Each model is trained individually under the different parameters assumed for better results. The deep learning-FDTR models are built and trained using the Keras deep learning framework. After training on data from a specific system configuration (with assumed pre-defined parameters such as laser spot size, modulation frequencies, transducer thermal properties, transducer thickness), the trained models can be used to predict the properties of unknown materials measured under the same configuration with improved precision. The uncertainty of the Monte Carlo simulations can be improved with careful tuning of this new fitting algorithm incorporating the Bayesian approach and closer initial guess of the global minimum. It is worth mentioning that while this study focuses on predicting one property at a time, it is possible to modify the architecture to output a vector and predict multiple properties simultaneously (e.g.) for multiple layer or superlattice structures. However, this specific approach does not employ that strategy.

## ACKNOWLEDGMENT

The authors acknowledge funding from the Natural Science Foundation of Heilongjiang Province of China (LH2022337)” and the Fundamental Research Funds for the Central Universities (HIT.NSRIF202340).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Ankur Chatterjee:** Conceptualization (lead); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). **Dariusz Dziczek:** Methodology (equal). **Peng Song:** Writing – review & editing (equal). **J. Liu:** Writing – review & editing (equal). **Andreas. D. Wieck:** Supervision (supporting); Writing – review & editing (equal). **Michal Pawlak:** Conceptualization (equal); Investigation (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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_{0.33}Ga

_{0.67}As thin film epitaxially grown on a heavily Zn doped GaAs using spectrally-resolved modulated photothermal infrared radiometry

*2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*