Novel component measurement method based on Raman peak intensity

Through a new component measurement method based on Raman peak strength, combined with molecular dynamics simulation and Raman spectral peak strength analysis, the accuracy problem of quantitative analysis of perovskite materials is solved, and high-precision component measurement is achieved, which is suitable for a variety of perovskite materials.

CN120404690APending Publication Date: 2025-08-01NORTHWEST UNIV
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Patent Information

Application Number
CN202510280037.2
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2025-03-11
Publication Date
2025-08-01

AI Technical Summary

Technical Problem

The prior art cannot achieve high-precision quantitative analysis of perovskite material components, and micro component deviations lead to significant changes in performance, and traditional analysis methods cannot meet the needs of high-precision quantitative.

Method used

A new component measurement method based on Raman peak strength was adopted, and a linear relationship with the ratio of cations of perovskite materials was identified through molecular dynamics simulation and Raman spectral peak strength analysis.

Benefits of technology

High-precision quantitative analysis of perovskite material components is achieved, with an accuracy of better than 1%, no complex sample preparation process is required, and material structural damage is avoided. It is suitable for organic-inorganic hybrid and all-inorganic perovskite systems.

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Abstract

The invention discloses a novel component measurement method based on Raman peak intensity, and relates to the technical field of material science, and the method comprises molecular dynamics simulation and Raman spectrum peak intensity analysis, construction of a structural model of a target perovskite system, molecular dynamics simulation of the structural model, and acquisition of a stable dynamics track. The polarizability and Raman spectrum of the material are calculated based on a dynamic trajectory, a characteristic peak corresponding to a specific vibration mode is identified, and a linear relationship between the intensity of the characteristic peak and the cation ratio of the perovskite material is established, so that quantitative analysis of components is realized. The method has the advantages that molecular dynamics simulation and experimental data are deeply combined through core steps of constructing a target perovskite system structure model, carrying out molecular dynamics simulation, calculating a Raman spectrum, establishing a linear relation between the characteristic peak intensity and the cation ratio and the like, and the limitation that a traditional analysis method can only be qualitative is broken through; the high-precision quantitative analysis on the perovskite material components is realized.
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Description

Technical Field

[0001] The present invention relates to the field of material science and technology, and in particular to a novel component measurement method based on Raman peak intensity. Background Art

[0002] Perovskite materials, with their unique optoelectronic properties, have demonstrated significant application value in photovoltaics, semiconductor devices, and other fields. Their performance is highly sensitive to the proportions of their internal components, and accurate component determination is crucial for material research and development and performance optimization. Currently, traditional analytical methods such as XRD and XPS can only achieve qualitative or low-precision quantitative analysis in perovskite material analysis.

[0003] However, in perovskite material systems, slight deviations in composition can lead to significant changes in performance. Existing analytical methods are unable to break through the limitations of "only qualitative or low-precision quantitative analysis" and are therefore unable to meet the needs of high-precision quantitative analysis. How to achieve high-precision quantitative analysis of perovskite material components has become a technical problem that needs to be urgently solved in this field. To this end, we propose a new component measurement method based on Raman peak intensity. Summary of the Invention

[0004] The purpose of the present invention is to provide a new component measurement method based on Raman peak intensity.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a new component measurement method based on Raman peak intensity, comprising molecular dynamics simulation and Raman spectrum peak intensity analysis. The specific operating steps of the new component measurement method are as follows:

[0006] Step 1: Construct a structural model of the target perovskite system;

[0007] Step 2: performing molecular dynamics simulation on the structural model to obtain a stable dynamic trajectory;

[0008] Step 3: Calculate the polarizability and Raman spectrum of the material based on the kinetic trajectory, and identify the characteristic peak corresponding to the specific vibration mode;

[0009] Step 4: Establish a linear relationship between the characteristic peak intensity and the cation ratio of the perovskite material to achieve quantitative component analysis.

[0010] As a further solution of the present invention: the structural model is a 1×2×2 cubic supercell, which is used to construct FA X (MA) 1-X PbI3 and MA X (FA) 1-X PbI3 perovskite system, in which it is used to adjust FA + and MA + The composition ratio of these two cations.

[0011] As a further solution of the present invention: in the second step, the molecular dynamics simulation adopts the density functional theory combined with the BLYP exchange-correlation functional and the Grimme D3 dispersion correction, uses the MOLOPT-DZVP-SR-GTH double-zeta basis set and the Goedecker-Teter-Hutter (GTH) pseudopotential, the time step is 0.5 femtoseconds, the temperature is controlled at 300 K, and the number of simulation steps is 60,000 steps.

[0012] As a further solution of the present invention: in the third step, the calculation of the Raman spectrum is based on the dynamic change of the polarizability, by identifying the FA + formamidinium cation (FA + cation) at the NH2-C-NH2 bending vibration peak at 520 - 530 cm -1 and the MA + methylammonium cation (MA + cation) at the bending vibration peak at 1480 - 1490 cm -1 for the characteristic peak attribution.

[0013] As a further solution of the present invention: in the fourth step, a linear regression analysis is used to establish the linear relationship between the peak intensity of the FA + cation at 520 - 530 cm -1 and the X value in the FA X (MA) 1-X PbI3 system, and the linear relationship between the peak intensity of the MA + cation at 1480 - 1490 cm -1 and the X value in the FA X (FA) 1-X PbI3 system.

[0014] As a further solution of the present invention: the perovskite material includes organic-inorganic hybrid perovskite and all-inorganic perovskite, and the specific systems are FA X (MA) 1-X PbI3 and FA X (FA) 1-X PbI3.

[0015] As a further solution of the present invention: the characteristic peaks include the C-N bond stretching vibration peak of the FA + cation at 1090 cm -1 , the N-H stretching vibration peak at 3400 cm -1 and the C-H stretching vibration peak of the MA + cation at 3040 cm -1 .

[0016] As a further solution of the present invention: The molecular dynamics simulation is completed using the open-source software package CP2K: OpenSource Molecular Dynamics.

[0017] As a further solution of the present invention: By combining molecular dynamics simulation with Raman spectroscopy peak intensity analysis, and based on the linear relationship between the characteristic peak intensity and the cation ratio, the component optimization and performance evaluation of perovskite materials are completed with high precision.

[0018] Adopting the above technical solutions, compared with the prior art, the beneficial effects of the present invention are as follows:

[0019] 1. Through core steps such as constructing the target perovskite system structure model, conducting molecular dynamics simulation, calculating Raman spectroscopy, and establishing the linear relationship between the characteristic peak intensity and the cation ratio, the present invention deeply combines molecular dynamics simulation with experimental data, breaks through the limitation of traditional analysis methods that can only be qualitative, and realizes the high-precision quantitative analysis of the components of perovskite materials.

[0020] 2. By combining molecular dynamics simulation with Raman spectroscopy peak intensity analysis, the present invention does not require a complex sample preparation process during measurement, directly uses Raman spectroscopy analysis to detect perovskite materials, avoids damage to the material structure, and achieves the effect of efficient and non-destructive analysis of perovskite materials.

[0021] 3. The method described in the present invention is clearly applicable to organic-inorganic hybrid perovskite and all-inorganic perovskite systems, and can be applied to the component optimization and performance evaluation of perovskite materials in fields such as photovoltaic materials and semiconductor devices, demonstrating the wide universality advantage for different types of perovskite materials. Description of the Drawings

[0022] Figure 1 Schematic diagram of the process of the new component measurement method based on Raman peak intensity in the embodiment of the present invention;

[0023] Figure 2 Schematic diagram of the crystal structures of FAPI3 and MAPBI3 in the embodiment of the present invention;

[0024] Figure 3 For FA in the embodiment of the present invention X (MA) 1-X Schematic diagram of the Raman spectrum of the PbI3 system;

[0025] Figure 4 For FA in the embodiment of the present invention + The schematic diagram of the linear relationship between the intensity of the NH2-C-NH2 bending vibration peak of the cation at 520 - 530 cm -1 and the X value;

[0026] Figure 5For MA in the embodiments of the present invention + The schematic diagram of the linear relationship between the intensity of the bending vibration peak of NH3 molecules at 1460 cm -1 and the value of X;

[0027] Figure 6 For MA in the embodiments of the present invention X (FA) 1-X The schematic diagram of the Raman spectrum of the PbI3 system;

[0028] Figure 7 For FA in the embodiments of the present invention + The schematic diagram of the linear relationship between the intensity of the bending vibration peak of NH2-C-NH2 at 519 cm -1 and the value of X;

[0029] Figure 8 For FA in the embodiments of the present invention + The schematic diagram of the linear relationship between the intensity of the stretching vibration peak of C-H at 1090 cm -1 and the value of X;

[0030] Figure 9 For FA in the embodiments of the present invention + The schematic diagram of the linear relationship between the intensity of the stretching vibration peak of N-H at 3400 cm -1 and the value of X;

[0031] Figure 10 For MA in the embodiments of the present invention + The schematic diagram of the linear relationship between the intensity of the bending vibration peak of NH3 molecules at 1460 cm -1 and the value of X;

[0032] Figure 11 For MA in the embodiments of the present invention + The schematic diagram of the linear relationship between the intensity of the stretching vibration peak of C-H at 3040 cm -1 and the value of X. Detailed implementation manners

[0033] The following further describes the detailed implementation manners of the present invention with reference to the accompanying drawings. It should be noted here that the description of these implementation manners is used to help understand the present invention, but does not limit the present invention.

[0034] In addition, the technical features involved in the various implementation manners of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0035] Please refer to the attached Figure 1 - attached Figure 11, a novel component measurement method based on Raman peak intensity of the present invention, includes molecular dynamics simulation and Raman spectrum peak intensity analysis. The specific operation steps of the novel component measurement method are as follows:

[0036] Step 1: Construct a structural model of the target perovskite system;

[0037] Step 2: Perform molecular dynamics simulation on the structural model to obtain a stable kinetic trajectory;

[0038] Step 3: Calculate the polarizability and Raman spectrum of the material based on the kinetic trajectory, and identify the characteristic peaks corresponding to specific vibration modes;

[0039] Step 4: Establish a linear relationship between the intensity of the characteristic peaks and the cation ratio of the perovskite material to achieve quantitative component analysis.

[0040] In an embodiment of the present invention: The structural model is a 1×2×2 cubic supercell, and this supercell is used to construct FA X (MA) 1-X PbI3 and MA X (FA) 1-X PbI3 perovskite system, which is used to adjust the component ratio of FA + and MA + these two cations.

[0041] In an embodiment of the present invention: In the step 2, the molecular dynamics simulation adopts density functional theory combined with BLYP exchange-correlation functional and Grimme D3 dispersion correction, uses the MOLOPT-DZVP-SR-GTH double zeta basis set and Goedecker-Teter-Hutter (GTH) pseudopotential, the time step is 0.5 femtoseconds, the temperature is controlled at 300K, and the number of simulation steps is 60000 steps.

[0042] In an embodiment of the present invention: In the step 3, the calculation of the Raman spectrum is based on the dynamic change of the polarizability. By identifying the NH2-C-NH2 bending vibration peak of the formamidinium cation (FA + cation) at 520 - 530 cm + and the bending vibration peak of the methylammonium cation (MA -1 cation) at 1480 - 1490 cm + for the attribution of the characteristic peaks. + cation) at 1480 - 1490 cm -1 to perform the characteristic peak attribution.

[0043] In an embodiment of the present invention: In the step 4, a linear regression analysis is used to establish the formamidinium cation (FA + cation) at 520 - 530 cm -1Peak intensity and FA X (MA) 1-X The linear relationship of the X value in the PbI3 system, and MA + Cation at 1480 - 1490 cm -1 Peak intensity and FA X (FA) 1-X The linear relationship of the X value in the PbI3 system.

[0044] In one embodiment of the present invention: the perovskite material includes organic - inorganic hybrid perovskite and all - inorganic perovskite, and the specific systems are FA X (MA) 1-X PbI3 and FA X (FA) 1-X PbI3.

[0045] In one embodiment of the present invention: the characteristic peaks include the C - N bond stretching vibration peak of FA + cation at 1090 cm -1 , the N - H stretching vibration peak at 3400 cm -1 and MA + cation's C - H stretching vibration peak at 3040 cm -1 .

[0046] In one embodiment of the present invention: the molecular dynamics simulation is completed using the open - source software package CP2K: OpenSource Molecular Dynamics.

[0047] In one embodiment of the present invention: through the combination of molecular dynamics simulation and Raman spectrum peak intensity analysis, and based on the linear relationship between the characteristic peak intensity and the cation ratio, the component optimization and performance evaluation of the perovskite material are completed with high precision.

[0048] Example 1. Please refer to the attached Figure 1 - attached Figure 5 , in this example, the FA X (MA) 1-X PbI3 perovskite system is taken as the research object, and a 1×2×2 cubic supercell structure model is adopted. By gradually replacing the A - site cations (formamidinium FA + and methylammonium MA +) Five models with X values of 0 (pure MAPbI3), 0.25, 0.5, 0.75, and 1 (pure FAPbI3) were constructed. The molecular dynamics simulations were strictly carried out using the open-source software CP2K, based on the BLYP exchange-correlation functional of density functional theory combined with the Grimme D3 dispersion correction, using the double-zeta basis set MOLOPT-DZVP-SR-GTH and the Goedecker-Teter-Hutter (GTH) pseudopotential. The simulations were performed in the NVT system. The temperature was controlled at 300 K through a Nosé-Hoover chain thermostat, and the time step was set to 0.5 femtoseconds. The total number of simulation steps reached 60,000 steps to ensure that the system reached thermodynamic equilibrium. Based on the obtained kinetic trajectories, the dynamic change of the polarizability of the material was calculated and the Raman spectrum was generated ( Figure 2 ), by comparing the spectral characteristics of systems with different X values, it was found that the intensity of the NH2-C-NH2 bending vibration peak (δ(NCN)) in the range of 520 - 530 cm -1 of the FA+ cation showed a significant linear enhancement trend with the increase of the FA + ratio. Experimental data showed that when X increased from 0 to 1, the intensity of this peak gradually increased from the baseline value to the maximum value, indicating a direct correlation with the FA+ content. At the same time, the intensity of the NH3 molecular bending vibration peak (δ(NH3)) at 1460 cm -1 of the MA+ cation increased linearly with the increase of the MA+ ratio, further verifying the quantitative relationship between the characteristic peak intensity and the cation ratio. Through the linear regression analysis of claim 5, a linear equation of the peak intensity of FA + in the range of 520 - 530 cm -1 and the X value, and a linear equation of the peak intensity of MA + in the range of 1460 cm -1 and the X value were successfully established. After experimental verification, the measurement accuracy of the components of the FA X (MA) 1-X PbI3 system by this method is better than 1%, significantly better than the accuracy level of traditional analysis methods.

[0049] Example two: Please refer to the appendix Figure 6 -appendix Figure 11 , in this example, the MA X (FA) 1-X PbI3 system was taken as the research object. A 1×2×2 cubic supercell was also used. By adjusting the ratio of FA + and MA + (X values were 0, 0.25, 0.5, 0.75, 1), models were constructed. The simulation parameters were exactly the same as those in Example one to ensure the repeatability of the experimental conditions. Through Raman spectrum analysis, it was found that the FA + cation had a peak at 519 cm -1The intensity of the NH2-C-NH2 bending vibration peak (δ(NCN)) at [location] increases linearly with the increase in the FA + proportion. In addition, a good linear relationship is also shown in the newly added characteristic peaks: FA + The intensity of the stretching vibration peak (1090 cm -1 of the C-N bond, ν(CN)) is positively correlated with the FA + proportion. The stretching vibration peak of N-H (3400 cm -1 , ν(NH)) also shows a significant linear trend. For the MA + cation, except for the bending vibration peak of NH3 at 1060 cm -1 , the intensity of its C-H stretching vibration peak (3040 cm -1 , ν(CH)) increases with the increase in the MA + proportion, further expanding the selection range of characteristic peaks. The experimental data show that the peak intensity of FA + at 519 cm -1 is highly linearly correlated with the X value (the proportion of FA in MA x (FA) 1-X ), and the correlation coefficient exceeds 0.99. Similarly, the linear relationship between the peak intensity of MA + at 3040 cm -1 and the X value is also verified by experiments. This example not only verifies the overall system applicability of claim 6 but also improves the detection accuracy of the cation proportion in complex systems through the newly added characteristic peaks, with the error controlled within 1.5%.

[0050] Example 3. Please refer to the appendix Figure 1 - appendix Figure 2 . In this example, the all-inorganic perovskite system is taken as the research object, and a model is constructed using a 1×2×2 cubic supercell. By adjusting the GTH pseudopotential to adapt to the all-inorganic cations, other simulation conditions (claim 3) are kept consistent with the previous two examples. It is experimentally found that the vibration peak intensities of cations and halide ions in the all-inorganic perovskite system are linearly related to the concentration (such as the bond vibration peak of cations at 120 cm -1 ), verifying the universality of the method (claim 9).

[0051] Specifically, during the measurement process, there is no need for complex sample preparation processes (such as grinding, coating, etc.). The perovskite material is directly detected using a Raman spectrometer. By comparing the experimental Raman spectrum with the theoretical calculation spectrum (such as the 520 - 530 cm X peak of the FA 1-X (MA) -1 PbI3 system matching the simulated spectrum), the attribution of characteristic peaks can be quickly determined (such as the NH2-C-NH2 bending vibration peak of FA + ), avoiding damage to the material structure.

[0052] Specifically, by constructing a 1×2×2 cubic supercell model of the target perovskite system (such as FA X (MA) 1-X PbI3 or MA X (FA) 1-X PbI3), molecular dynamics simulations were performed using CP2K software (density functional theory + BLYP functional + Grimme D3 dispersion correction, 300K, 0.5 femtosecond step size, 60,000 steps), to calculate the dynamic changes in the polarizability and Raman spectrum of the material, and to identify FA + At 520-530cm -1 (δ(NCN)), MA + At 1460cm -1 (δ(NH3)) and other characteristic peaks, and the linear relationship between peak intensity and cation ratio was established by linear regression analysis (such as FA X (MA) 1-X The X value in PbI3 and 520-530cm -1 Linear equation for peak intensity, R 2 >0.99), achieving high-precision quantitative analysis of components (accuracy better than 1%).

[0053] Specifically, by adjusting the cation combination in the supercell (such as FA X (MA) 1-X PbI3 or MA X (FA) 1-X PbI3), verify the applicability of different organic-inorganic hybrid systems (such as FA X (MA) 1-X FA in PbI3 + The δ(NCN) peak intensity is linearly related to the X value, and MA X (FA) 1-X New FA in PbI3 + ν(CN) and MA + For the all-inorganic perovskite system, by adapting the GTH pseudopotential and verifying the cation and halide ion vibration peaks (such as 120cm -1 The linear relationship between the peak of the bond vibration and the concentration was confirmed, confirming the universality of the method (the error was controlled within 1.5%).

[0054] Working principle:

[0055] First, a 1×2×2 cubic supercell structure model of the target perovskite system is constructed, specifically FA X (MA) 1-X PbI3 or MA X (FA) 1-XPbI3 system, by gradually replacing the A-site cation (formamidine FA + With methylammonium MA + ), five models with X values (cation ratio) of 0, 0.25, 0.5, 0.75, and 1 were constructed to ensure coverage of the full component range. Subsequently, molecular dynamics simulations were carried out using the open source software package CP2K. The calculations were based on density functional theory, and the BLYP exchange-correlation functional combined with GrimmeD3 dispersion correction was used to accurately describe van der Waals interactions. A double zeta basis set (MOLOPT-DZVP-SR-GTH) and GTH pseudopotential were used to treat electron-ion interactions. The simulations were performed under an NVT system, maintained at a constant temperature of 300K by a Nosé-Hoover chain heat bath, with a time step of 0.5 femtoseconds and a total of 60,000 steps to ensure that the system reached thermodynamic equilibrium and obtained a stable kinetic trajectory. Based on the simulated trajectory, the dynamic changes in the polarizability of the material were calculated, and the theoretical Raman spectrum was generated by Fourier transform. By comparing the experimental and simulated spectra, FA was identified. + Cation at 520-530cm -1 NH2-C-NH2 bending vibration peak (δ(NCN)), MA + Cation at 1460 cm -1 The NH3 molecular bending vibration peak (δ(NH3)) at + The CN bond stretching vibration peak (1090cm -1 ), NH stretching vibration peak (3400cm -1 ) and MA + The CH stretching vibration peak (3040cm -1 ), finally, a quantitative relationship between the characteristic peak intensity and the cation ratio was established by linear regression analysis, e.g., FA + At 520-530cm -1 The peak intensity of X(FA X (MA) 1- X FA ratio in PbI3) is highly linearly correlated (R 2 >0.99), MA + At 1460cm -1 The peak intensity and X value (MA X (FA) 1-X The MA ratio in PbI3) also shows a linear relationship. Experimental verification shows that this method has a component measurement accuracy of better than 1% for FA / MA mixed systems, and is also applicable to all-inorganic perovskite systems (such as by adjusting the GTH pseudopotential to adapt cations). At this point, the entire workflow is completed.

[0056] Although the present invention is disclosed above in preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications without departing from the spirit and scope of the present invention. Therefore, any modification, equivalent change, and modification made to the above embodiments based on the technical essence of the present invention without departing from the technical solution of the present invention shall fall within the protection scope defined by the claims of the present invention.

Claims

1. A novel component measurement method based on Raman peak intensity, including molecular dynamics simulation and Raman spectrum peak intensity analysis, is characterized in that, The specific operation steps of the novel component measurement method are as follows: Step 1: Construct a structural model of the target perovskite system; Step 2: Perform molecular dynamics simulations on the structural model to obtain stable kinetic trajectories; Step 3: Calculate the polarizability and Raman spectra of the material based on the kinetic trajectories, and identify the characteristic peaks corresponding to specific vibration modes; Step 4: Establish a linear relationship between the intensity of the characteristic peaks and the cation ratio of the perovskite material to achieve quantitative component analysis.

2. The novel component measurement method based on Raman peak intensity according to claim 1, characterized in that: The structural model is a 1×2×2 cubic supercell, which is used to construct FA X (MA) 1-X PbI3 and MA X (FA) 1-X PbI3 perovskite system, in which it is used to adjust FA + and MA + the component ratios of these two cations.

3. A novel component measurement method based on Raman peak intensity according to claim 1, characterized in that: In Step 2, the molecular dynamics simulation uses density functional theory combined with the BLYP exchange-correlation functional and Grimme D3 dispersion correction, uses the MOLOPT-DZVP-SR-GTH double-zeta basis set and the Goedecker-Teter-Hutter (GTH) pseudopotential, the time step is 0.5 femtoseconds, the temperature is controlled at 300 K, and the number of simulation steps is 60,000 steps.

4. A novel component measurement method based on Raman peak intensity according to claim 2, characterized in that: In the third step, the calculation of the Raman spectrum is based on the dynamic change of the polarizability, by identifying the FA + formamidinium cation (FA + cation) at the NH2-C-NH2 bending vibration peak at 520-530 cm -1 and the MA + methylammonium cation (MA + cation) at the bending vibration peak at 1480-1490 cm -1 to assign the characteristic peaks.

5. A novel component measurement method based on Raman peak intensity according to claim 4, characterized in that: In the fourth step, a linear regression analysis is used to establish the linear relationship between the peak intensity of FA + cations at 520 - 530 cm -1 and the X value in the FA X (MA) 1-X PbI3 system, and the linear relationship between the peak intensity of MA + cations at 1480 - 1490 cm -1 and the X value in the FA X (FA) 1-X PbI3 system.

6. A novel component measurement method based on Raman peak intensity according to claim 5, characterized in that: The perovskite materials include organic-inorganic hybrid perovskites and all-inorganic perovskites, and the specific systems are FA X (MA) 1-X PbI3 and FA X (FA) 1-X PbI3.

7. A novel component measurement method based on Raman peak intensity according to claim 2, characterized in that: The characteristic peaks include FA + The stretching vibration peak of the C-N bond of the cation at 1090 cm -1 , the stretching vibration peak of N-H at 3400 cm -1 and MA + The stretching vibration peak of C-H of the cation at 3040 cm -1 .

8. A novel component measurement method based on Raman peak intensity according to claim 3, characterized in that: The molecular dynamics simulation is completed using the open-source software package CP2K: Open Source Molecular Dynamics.

9. A novel component measurement method based on Raman peak intensity according to claim 1, characterized in that: Through the combination of molecular dynamics simulation and Raman spectrum peak intensity analysis, and based on the linear relationship between the characteristic peak intensity and the cation ratio, the component optimization and performance evaluation of the perovskite material are completed with high precision.