Feasibility evaluation method for simultaneous measurement of multiple thermophysical parameters based on frequency domain thermoreflectance method

Abstract

The present invention discloses a method for evaluating the feasibility of simultaneously measuring multiple thermophysical parameters based on a frequency-domain thermoreflection method. The method obtains the nominal values of multiple thermophysical parameters to be measured and their sensitivity parameters relative to the measurement signal in the experimental frequency domain after fitting the measurement data of the frequency-domain thermoreflection method experiment through a multi-layer heat conduction model. The method calculates the logarithmic relative sensitivity of two of the thermophysical parameters, determines whether the difference between the maximum and minimum values of the logarithmic relative sensitivity in the experimental frequency domain exceeds a preset threshold, and then directly evaluates whether the two thermophysical parameters to be measured can be accurately measured simultaneously, without the need for additional parameter measurements and avoiding complex mathematical calculations. The method is simple to operate, greatly reduces the time cost required for experimental data analysis of multivariate thermophysical parameter measurements in the frequency-domain thermoreflection method, and effectively improves the efficiency and accuracy of experimental data analysis and measurement.

Description

Technical Field

[0001] The present invention relates to the field of data analysis and evaluation, and in particular to a method for evaluating the feasibility of simultaneously measuring multiple thermophysical property parameters based on a frequency domain thermoreflectance method. Background Art

[0002] Frequency Domain Thermal Reflectance (FDR), based on the thermal reflectivity of a material's surface, is one of the most advanced methods for measuring thermal properties of materials today. It is capable of measuring both bulk and micro- and nanoscale materials. This method heats the surface of the material being measured with lasers of varying frequencies, measures the temperature-dependent reflectivity of the sample surface at different frequencies, collects the temperature response signal, and uses a known thermal conductivity model to fit the experimental data to determine the unknown thermal properties of the material. The material being measured often has multiple unknown thermal properties. As the measured properties of the sample and experimental conditions change, the measurable parameters in a single FDR experiment also change. Ideally, only one unknown thermal property can be measured by fitting the data to a theoretical model. Modern electronic devices often consist of heterogeneous nanolayers with multiple unknown thermal transport properties. If these unknown parameters are interdependent, fitting multiple thermophysical parameters simultaneously in a single experiment can produce unreliable results. While some experimental analysis methods attempt to determine whether multiple thermophysical parameters can be accurately measured simultaneously in a single experiment by directly comparing differences in their sensitivity curves, such evaluation methods are not rigorous and can even fail in simple cases. Currently, there is a lack of simple, general analytical methods for fitting and measuring multiple thermophysical parameters, making efficient and accurate simultaneous measurement of multiple thermophysical parameters difficult. Therefore, accurately measuring multiple thermophysical parameters simultaneously has become a pressing challenge in the field of thermophysical measurement. Summary of the Invention

[0003] To address the shortcomings of existing technologies, the present invention proposes a method for assessing the feasibility of simultaneously measuring multiple thermophysical parameters using frequency-domain thermoreflectance. This method provides a systematic analytical framework that accurately assesses the feasibility of simultaneously measuring multiple thermophysical parameters in a single experiment. This method addresses the issues of low data analysis efficiency and poor accuracy during multivariate thermophysical parameter measurements using frequency-domain thermoreflectance technology. Furthermore, the method is simple to operate and does not require additional parameter measurements or complex mathematical calculations, thereby improving the efficiency and accuracy of data analysis during the simultaneous measurement of multiple thermophysical parameters.

[0004] The purpose of the present invention is achieved through the following technical solutions:

[0005] A method for evaluating the feasibility of simultaneously measuring multiple thermophysical parameters based on a frequency domain thermoreflectance method, the method comprising:

[0006] Step 1: Change the heating frequency of the frequency domain thermoreflection method experiment ω=(ω1,ω2,…,ω m ), collect the phase difference signal φ=(φ1,φ2,…,φ m ); The heating frequency ω and the phase difference signal φ are fitted by the multi-layer heat conduction model φ=F(ω,θ), and the thermal physical property parameters to be measured θ=(A1,A2,…,A n )’s nominal value;

[0007] Step 2: Calculate the phase difference signal φ with respect to any thermophysical parameter A to be measured by the differential method i Sensitivity parameters within the experimental frequency range

[0008]

[0009] Step 3: Calculate any two thermophysical parameters A to be measured i ,A j Logarithmic relative sensitivity parameters within the experimental heating frequency range

[0010]

[0011] Step 4: Calculate any two thermophysical parameters A to be measured i ,A j Logarithmic relative sensitivity parameters within the experimental frequency range Whether the difference between the maximum value and the minimum value of exceeds the preset threshold value, if it exceeds, the two thermophysical parameters A to be measured i ,A j Can be accurately measured simultaneously within the experimental frequency domain in one experiment; otherwise, the thermophysical parameter A to be measured i ,A j It cannot be accurately measured simultaneously in one experiment, and only the thermophysical parameter A to be measured can be accurately measured. i ,A j A combination of parameters.

[0012] Furthermore, the preset threshold is 0.5.

[0013] The beneficial effects of the present invention are as follows:

[0014] The feasibility evaluation method of the present invention for simultaneously measuring multiple thermophysical parameters based on the frequency domain thermoreflection method calculates the difference between the maximum and minimum values of the logarithmic relative sensitivity parameters of the parameters to be measured in the experimental frequency domain and compares it with a preset threshold value, thereby directly evaluating whether the parameters to be measured can be accurately measured simultaneously, avoiding complex mathematical calculations, greatly reducing the time cost of data analysis in the frequency domain thermoreflection method multivariate thermophysical parameter measurement experiment, and effectively improving the efficiency and accuracy of experimental data analysis and measurement in heat transfer parameter measurement. BRIEF DESCRIPTION OF THE DRAWINGS

[0015] Figure 1 This is a flow chart of a method for evaluating the feasibility of simultaneously measuring multiple thermophysical parameters based on the frequency domain thermoreflectance method provided in Example 1 of the present application.

[0016] Figure 2 Schematic diagram of the phase difference data and the optimal fitting curve obtained by experiment in Example 1 of the present application.

[0017] Figure 3 This is a schematic diagram of a sensitivity curve drawn with the experimental heating frequency as the independent variable and the sensitivity parameter of the thermophysical property parameter Au / SiO2 to be measured obtained by calculation as the dependent variable in Example 1.

[0018] Figure 4 This is a schematic diagram of a sensitivity curve drawn with the experimental heating frequency as the independent variable and the sensitivity parameter of the thermophysical property parameter Au / Al2O3 to be measured obtained by calculation as the dependent variable in Example 1.

[0019] Figure 5 Schematic diagram of the logarithmic relative sensitivity curve of Au / SiO2 in Example 1.

[0020] Figure 6 Schematic diagram of the logarithmic relative sensitivity curve of Au / Al2O3 in Example 1.

[0021] Figure 7 Schematic diagram of the sensitivity curve and relative sensitivity curve of the double-layer material sample to be tested in Example 2; wherein, Figure (a) is the sensitivity curve, and Figure (b) is the relative sensitivity curve.

[0022] Figure 8 Schematic diagram of the change of the normalized results of the fitting of the parameters in the low-frequency range and the high-frequency range in Example 2 with the number of fitting parameters. DETAILED DESCRIPTION

[0023] The present invention will be described in detail below with reference to the accompanying drawings and preferred embodiments, and the purpose and effects of the present invention will become more apparent. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0024] The assessment of the feasibility of simultaneously measuring multiple thermophysical parameters in the frequency domain thermoreflectance method refers to the use of reasonable data analysis methods to evaluate or determine whether multiple unknown thermophysical parameters to be measured can be accurately fitted and measured simultaneously in a single frequency domain thermoreflectance experiment. Ideally, a single experiment can only measure and determine one unknown thermophysical parameter, and fitting multiple parameters simultaneously often leads to unreliable results. Existing analysis and assessment methods for the feasibility of multivariate thermophysical parameter measurements generally attempt to determine the feasibility of simultaneously measuring multiple parameters in a single experiment by comparing the differences in the sensitivity curves of the parameters to be measured. However, such assessments are not rigorous and may even fail in simple cases, or require complex mathematical calculations for multi-parameter measurement feasibility analysis. Such methods can easily reduce the efficiency and accuracy of experimental data analysis and measurement during the multi-parameter measurement process.

[0025] To solve the above problems, an embodiment of the present application provides an evaluation method for the feasibility of simultaneously measuring multiple thermal properties based on the frequency domain thermal reflection method. It does not require additional parameter measurements or complex mathematical calculations. By calculating the sensitivity of several thermal property parameters to be measured, the logarithmic relative sensitivity parameters of two of the thermal property parameters can be obtained based on the logarithmic relative sensitivity analysis. Based on whether the difference between the maximum and minimum values of the logarithmic relative sensitivity parameters of the thermal property parameters to be measured within the experimental frequency range exceeds a certain threshold, the feasibility of measuring multiple variable thermal property parameters can be directly evaluated.

[0026] Optionally, the method for evaluating the feasibility of simultaneously measuring multiple thermal properties based on the frequency domain thermal reflection method in the embodiment of the present application can be applied to any metrology technology to evaluate the feasibility of accurately measuring multiple parameters to be measured simultaneously. The following is further explained using an embodiment of experimentally measuring the thermal properties of materials using the frequency domain thermal reflection method. Through this embodiment, the method for evaluating the feasibility of simultaneously measuring multiple thermal property parameters based on logarithmic relative sensitivity parameter LSR analysis is described in detail.

[0027] Example 1

[0028] Table 1 shows the parameters of the materials to be tested in Example 1. There are two materials to be tested, one is gold (Au)-plated heat-resistant glass 7740 (SiO2), and the other is gold (Au)-plated sapphire (Al2O3). The thermal physical parameters to be tested are the thermal conductivity of the two materials SiO2 and Al2O3. and And the interface thermal conductivity h between the two materials and the gold-plated layer Au / 2 and The other material parameters are known. The experiment uses a test bench with an effective spot radius of 2.85μm for measurement, and the effective spot radius obtained by fitting the two materials is measured simultaneously during the experimental data fitting process. To verify the accuracy of other thermophysical parameters of the two materials measured.

[0029] Table 1 Parameters of the materials to be tested according to the specific embodiment

[0030]

[0031] like Figure 1 As shown, an embodiment of the present application provides a method for evaluating the feasibility of simultaneously measuring multiple thermophysical parameters based on a frequency domain thermoreflectance method, including:

[0032] Step S101: Perform a frequency domain thermoreflection method experiment and change the heating frequency ω=(ω1,ω2,…,ω m ), collect the temperature rise ΔT and heat flow phase difference signal φ=(φ1,φ2,…,φ m ); The heating frequency ω and the phase difference signal φ are fitted by the multi-layer heat conduction model φ=F(ω,θ), and the thermal physical property parameters to be measured θ=(A1,A2,…,A n )’s nominal value.

[0033] In the multilayer heat diffusion model, the relationship between ΔT and the corresponding phase difference φ in the system frequency domain is:

[0034] ΔT=Ae iφ (1)

[0035] The temperature rise ΔT is determined by the following experimental parameter relationship:

[0036]

[0037] Where P0 is the power of the experimental heating laser, x is the Hankel transformation parameter, ω is the experimental heating frequency, θ t and f t are the surface temperature and heat flux density of the sample respectively, and r is the effective spot radius of the heating laser.

[0038] Upper surface temperature θ t and heat flux f t It can be derived and calculated based on the multi-layer stacking structure of the experimental sample. In the multi-layer stacking structure, the temperature of the upper surface of the i-th layer is and heat flux and the temperature of the lower surface of this layer and heat flux The relationship between

[0039]

[0040] Among them, M i Calculated by the following formula:

[0041]

[0042]

[0043] In the above formula, d i , ρ i ,c i , are the thickness, in-plane thermal conductivity, density, specific heat and normal thermal conductivity of the i-th layer of material in the sample. Multiplying all layers in the sample in sequence can give the temperature of the bottom layer of the sample θ b and heat flux f b The surface temperature of the sample and heat flux The relationship

[0044]

[0045] The adiabatic boundary condition at the bottom of the sample Calculate the sample surface temperature and heat flux The relationship:

[0046]

[0047] From the above equations (1)-(7), it can be seen that the heating frequency ω i Phase difference φ under i It is determined only by the physical properties of the sample and the power P0 and effective spot radius r of the experimental heating laser.

[0048] Therefore, the physical properties of the multilayer sample in equation (7) can be determined by the phase difference signal measured in equation (1). This determination usually adopts a nonlinear fitting process based on least squares.

[0049] Step S102: Calculate the phase difference signal φ with respect to any thermophysical parameter A to be measured by differential method i Sensitivity parameters within the experimental frequency range

[0050]

[0051] This embodiment also uses the experimental heating frequency as the independent variable, and the corresponding thermal physical parameter A to be measured at the frequency i Sensitivity The sensitivity curve of the thermophysical property parameter to be measured is drawn for the dependent variable.

[0052] like Figure 2As shown, in a specific embodiment, we take 20 heating frequencies in the experimental frequency domain of the frequency domain thermal reflection method to measure the two materials to be tested respectively, and each material obtains 20 phase difference signals at the corresponding frequencies, and the phase difference signals measured at the corresponding experimental frequencies are converted into Fitting with the multilayer heat conduction model φ=F(ω,θ) respectively, the thermal physical parameters to be measured in the two materials are obtained

[0053] Nominal value.

[0054] According to the nominal values of the thermophysical parameters of the two materials to be measured and the heat conduction model φ=F(ω,θ), the definition of the sensitivity parameter is The phase difference signal of the two materials is calculated by the difference method and In a certain frequency domain, the thermal physical parameters to be measured are respectively Sensitivity parameters:

[0055]

[0056]

[0057] The sensitivity of the two materials' thermophysical parameters to be measured is calculated using the experimental frequency as the independent variable.

[0058]

[0059] The sensitivity curves of the thermophysical parameters of the two materials to be tested are drawn as dependent variables, including three sensitivity curves of Au / SiO2 like Figure 3 As shown, the three sensitivity curves of Au / Al2O3 like Figure 4 shown.

[0060] Step S103: Calculate any two thermophysical parameters A to be measured i , A j Logarithmic relative sensitivity parameters within the experimental heating frequency range

[0061]

[0062] Taking the experimental frequency as the independent variable, two thermophysical parameters to be measured A i and A j Logarithmic relative sensitivity As the dependent variable, the logarithmic relative sensitivity change curve can be drawn.

[0063] like Figure 5 and Figure 6 As shown in the figure, the two materials to be tested in the embodiment of this application are Au / SiO2 and Au / Al2O3, and the sensitivity curves of the thermophysical parameters to be tested are shown in the figure. and Get the corresponding logarithmic relative sensitivity curve and Schematic diagram of .

[0064] Among them, according to the definition of logarithmic relative sensitivity The calculated logarithmic relative sensitivity values between the parameters are:

[0065]

[0066]

[0067]

[0068] With the experimental frequency as the independent variable, the logarithmic relative sensitivity parameter and As the dependent variable, we can draw Figure 5 and Figure 6 The logarithmic relative sensitivity change curve of any two measured thermophysical property parameters of the two materials shown.

[0069] Step S104: Calculate the logarithmic relative sensitivity parameter within the experimental frequency range Whether the difference between the maximum value and the minimum value of exceeds the preset threshold value, if it exceeds, the two thermophysical parameters A to be measured i , A j Can be accurately measured simultaneously within the experimental frequency domain in one experiment; otherwise, the thermophysical parameter A to be measured i , A j It cannot be accurately measured simultaneously in one experiment, and only the thermophysical parameter A to be measured can be accurately measured. i , A j A combination of parameters.

[0070] In this embodiment, the preset threshold is 0.5, such as Figure 5 and 6 As shown, the three logarithmic relative sensitivity curves of Au / SiO2 The difference between the maximum and minimum values in the experimental frequency domain is greater than 0.5, indicating that the thermal physical parameters to be measured in the experimental frequency domain The three logarithmic relative sensitivity curves of Au / Al2O3 are unrelated and can be accurately measured simultaneously in one experiment. middle, The difference between the maximum and minimum values in the experimental frequency domain is less than 0.5, indicating that and The two parameters are related and cannot be accurately measured simultaneously in all frequency domains through a single experiment.

[0071] The effective spot radius of the two materials measured experimentally Compared with the effective spot radius of 2.85μm set by the experimental bench, it can be seen that the experimental measurement value of Au / SiO2 is more reliable than that of Au / Al2O3, which is the same conclusion obtained by the logarithmic relative sensitivity parameter LSR analysis. Figure 3 and 4 The feasibility of multi-parameter measurement can be evaluated by the sensitivity of the sensitivity parameters. Figure 4 The sensitivity curve of Au / Al2O3 sample is compared with Figure 3 The sensitivity curve for the Au / SiO2 sample is more sensitive, leading to the conclusion that the multivariate fitting accuracy for the Au / Al2O3 sample is higher than that for the Au / SiO2 sample. This is contrary to the actual situation and will lead to incorrect evaluation results. This also confirms that directly using sensitivity curves is not sufficient to evaluate the feasibility of multivariate fitting.

[0072] Example 2

[0073] This embodiment studies the thermal conductivity k2, specific heat capacity C2 of the second layer material in a double-layer material sample and the interface thermal conductivity h between the second layer material and the first layer material. 12 A total of three thermophysical parameters were measured. In addition to these three physical quantities, the other physical quantities of the double-layer material are known, as shown in Table 2.

[0074] Table 2 Physical parameters of the double-layer material samples to be tested

[0075]

[0076] Similar to the first embodiment, in the frequency domain thermal reflection method experiment, the phase difference signal φ between the temperature rise and heat flow of the material to be tested is collected at different heating frequencies ω, and the phase difference is fitted by the multi-layer heat conduction model φ = F(ω, θ), and the three thermal physical parameters to be tested θ = (k2, C2, h 12 )’s nominal value.

[0077] By the definition of sensitivity parameter The phase difference signal φ is obtained with respect to the three thermophysical parameters to be measured θ=(k 2，2 , h 12 ) sensitivity parameter The experimental heating frequency ω is used as the independent variable, and the sensitivity parameters of the thermophysical properties at the corresponding frequency are used as the dependent variables. The sensitivity curves of the three thermophysical parameters to be measured are plotted, as shown in the figure: Figure 7 As shown in (a).

[0078] According to the definition of logarithmic relative sensitivity The logarithmic relative sensitivity parameters of the three thermophysical parameters to be measured can be calculated The experimental frequency is used as the independent variable, and the logarithmic relative sensitivity of the thermophysical parameters at the corresponding frequency is The logarithmic relative sensitivity curves of the three thermophysical parameters to be measured are drawn for the dependent variable, such as Figure 7 As shown in (b).

[0079] The feasibility of using the logarithmic relative sensitivity LSR curve of the present invention to simultaneously measure multiple thermophysical parameters is evaluated, such as Figure 7 As shown in (b), and The difference between the maximum and minimum values of the curve in the frequency domain of this experiment is greater than 0.5, indicating that the pair of thermophysical parameters k2 and h 12 , C2 and h 12 Can be measured accurately and simultaneously in the experimental frequency domain; and the curve Although the difference between the maximum and minimum values is greater than 0.5 in the low frequency range less than 5MHz, in the high frequency range greater than 5MHz, The difference between the maximum and minimum values of the curve is less than 0.5, indicating that k2 and C2 can only be accurately measured simultaneously in the low-frequency range, while in the high-frequency range, k2 and C2 are correlated. Through a single experiment, it is impossible to accurately measure the values of k2 and C2 at the same time. Only the combined parameter of k2 and C2, the thermal diffusivity, can be accurately measured. value.

[0080] In order to verify the accuracy of the feasibility assessment of multi-parameter measurement using the logarithmic relative sensitivity (LSR) curve of the present invention, a Monte Carlo simulation method was used for verification. Monte Carlo simulation is a statistical simulation method based on the theory of probability statistics. Monte Carlo simulation is used to statistically simulate the multi-parameter fitting process in the frequency domain thermoreflectance method experiment. First, the theoretical phase difference signal φ of the material's temperature rise and heat flow is calculated using the multilayer heat diffusion model φ = F(ω,θ). Then, random noise sampled from a normal distribution with a mean of zero and a variance of 0.2° is added to the simulation process. This noise is used to simulate the various noise signals in the actual frequency domain thermoreflectance method experiment process to obtain a simulated phase difference signal φ affected by random noise. After fitting this simulated phase difference signal, the various thermophysical parameters to be measured can be obtained. Different random noises are obtained from the above normal distribution, and the simulation is repeated 5000 times to ensure the statistical reliability of the Monte Carlo simulation. The simulation results are obtained through the above repeated simulation process, and the distribution, mean, and uncertainty of each thermophysical parameter to be measured are analyzed.

[0081] In this embodiment, 5000 sets of simulated phase difference signals at 20 frequency points are generated in the low frequency range of 50kHz-5MHz and the high frequency range of 5MHz-50MHz, and single parameter and double parameter fitting are performed on the two parameters to be measured, k2 and C2, in the high frequency range and low frequency range, respectively. 12 ) three-parameter fitting. All fitting results are normalized with the set values and displayed in Figure 8 In the figure, the horizontal dotted line represents the reference line where the fitting result is equal to the set value. The closer the fitting result is to the horizontal dotted line, the closer it is to the set value. Figure 8 As shown in (a), no matter how the number of fitting parameters changes, the measured parameters k2, C2 and h are obtained in the low frequency range. 12 The fitting mean of is consistent with the set value in Table 2. Although the measurement uncertainty increases with the increase of the number of simultaneous fitting parameters, the maximum fitting uncertainty is only 15%, which also proves the feasibility of fitting any multiple parameters in the low frequency range; in the high frequency range, such as Figure 8 As shown in (b), only the mean values of k2 and C2 from the single parameter fitting are consistent with the experimental measurements in Table 2. The k2 and C2 results from the two parameter fitting deviate from the set values and have large uncertainties. However, the combined parameters calculated from the fitting results are consistent with the experimental measurements in Table 2. The results are consistent with the set values, which proves that the combined parameter e2 can be accurately measured within this high-frequency range, while k2 and C2 are correlated and cannot be accurately measured simultaneously. In summary, the Monte Carlo simulation results are consistent with the analytical conclusions of the logarithmic relative sensitivity (LSR) of the present invention. Therefore, the Monte Carlo simulation analysis of this embodiment shows that it is feasible to fit unrelated parameters simultaneously, while fitting related parameters will produce erroneous results. This further confirms the effectiveness and accuracy of the logarithmic relative sensitivity (LSR) analysis method as an analytical method for the feasibility of measuring multivariate thermophysical parameters.

[0082] Those skilled in the art will understand that the foregoing descriptions are merely preferred embodiments of the invention and are not intended to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, those skilled in the art will still be able to modify the technical solutions described in the foregoing examples or substitute equivalents for some of the technical features therein. Any modifications, equivalent substitutions, etc. made within the spirit and principles of the invention shall be included within the scope of protection of the invention.

Claims

1. A method for evaluating the feasibility of simultaneously measuring multiple thermophysical parameters based on frequency domain thermoreflectance method, characterized in that: The method includes: Step 1: Change the heating frequency of the frequency domain thermoreflectance method experiment , collect the temperature rise of the material in the frequency domain Phase difference signal with heat flow ;Through the multi-layer heat conduction model Heating frequency and phase difference signal Perform fitting to obtain the thermal physical parameters to be measured The nominal value of Step 2: Calculate the phase difference signal by differential method Regarding any thermophysical parameter to be measured Sensitivity parameters within the experimental frequency range : ； Step 3: Calculate any two thermophysical parameters to be measured Logarithmic relative sensitivity parameters within the experimental heating frequency range : ， ； in, Indicates the thermal physical parameters to be measured Sensitivity parameters within the experimental heating frequency range; Step 4: Calculate any two thermophysical parameters to be measured Logarithmic relative sensitivity parameters within the experimental frequency range Whether the difference between the maximum value and the minimum value of exceeds the preset threshold, if so, the two thermophysical parameters to be measured Can be accurately measured simultaneously within the experimental frequency domain in one experiment; otherwise, the thermal physical parameters to be measured It cannot be accurately measured simultaneously in one experiment, and only the thermophysical parameters to be measured can be accurately measured. A combination of parameters.

2. The feasibility evaluation method for simultaneously measuring multiple thermophysical parameters based on frequency domain thermoreflectance method according to claim 1 is characterized in that: The preset threshold is 0.5.

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