A method for testing the mismatch stress change rule of a thin film

Abstract

The application relates to a test method for the change rule of film mismatch stress, and belongs to the technical field of film stress testing. The method uses a standard sample with a known lattice constant, combines a diffraction principle to perform reverse calibration, obtains an accurate working distance on any device, and eliminates the dependence on fixed mechanical size. The method clearly adopts a calculation of an electron wavelength that is corrected by relativity, ensures that the calculation of the wavelength still maintains theoretical accuracy under high-voltage conditions, realizes accurate determination of the size and type of film stress, and makes the measurement results have reliable comparability and reproducibility between different laboratories or devices. The method changes the electron beam incidence angle by rotating a sample table, collects RHEED images of a substrate and a grown film, determines the greatest common divisor of the lattice constants of the substrate and the film at the interface, clearly determines the interface matching mode, corrects the actual lattice constant of the substrate under the growth temperature by considering the influence of a thermal expansion coefficient on the lattice constant, and calculates the actual mismatch degree after the thermal expansion correction.

Description

Technical Field

[0001] This invention relates to the field of thin film stress testing technology, and in particular to a method for testing the variation law of mismatch stress in thin films. Background Technology

[0002] The in-plane stress state of thin films is a fundamental and crucial piece of information in condensed matter physics for studying surface and interface physics. In-plane stress refers to the internal stress of a thin film along its planar direction. This stress can be caused by factors such as lattice mismatches, differences in thermal expansion coefficients, or deposition conditions during film growth. For ultrathin films, in-plane stress can affect their crystal structure, electron transport properties, thermal properties, and mechanical stability. For example, stress can cause lattice distortion, altering the band structure and thus affecting conductivity. In positive applications, the in-plane stress state of ultrathin films can be used to improve thermoelectric efficiency, quantum effects, or sensor sensitivity through stress design. Simultaneously, residual stress must be controlled to avoid mechanical failure or electrical performance degradation, preventing it from affecting device reliability and lifespan. Therefore, stress and its variation are among the most important parameters of thin films.

[0003] In existing thin-film stress detection methods, wavelength and sample-to-screen distance are often used as quantitative parameters, while fringe spacing is used as a variable to calculate stress magnitude. However, in actual testing, when the sample height is changed to obtain a clear RHEED diffraction pattern or to accommodate Bragg diffraction, the distance L from the sample center to the screen changes according to the Pythagorean theorem, leading to calculation errors. Furthermore, when the working distance L changes to obtain a clear RHEED diffraction pattern or to accommodate Bragg diffraction, it also changes, causing errors in the calculation results. In practical scientific and industrial scenarios, the working distance of RHEED systems from different manufacturers, different models, and even the same equipment may vary after maintenance or adjustment, leading to errors in the calculation of the lattice constant and consequently, an inability to accurately determine the stress magnitude. Moreover, when electron energy reaches above 10 keV, the electron velocity is significantly close to the speed of light, and the mass increase effect (relativistic effect) cannot be ignored; wavelength errors will affect data reliability. Because the electron beam wavelength was ignored as a key parameter in the error control process, the accuracy of the electron beam wavelength in measuring the stress state was affected.

[0004] Furthermore, existing technologies directly use the lattice constants of the thin film and substrate for calculations, assuming single-atom matching. However, in actual thin film growth, several substrate atoms may correspond to several substrate atoms for matching, leading to significant errors in the mismatch calculations based on single-atom matching. Moreover, directly using the lattice constants at room temperature fails to consider the thermal strain introduced by the difference in thermal expansion coefficients during cooling from the growth temperature to room temperature. First-principles calculations confirm that neglecting the thermal expansion matching issue leads to qualitative errors in predicting the phase transition temperature and structural evolution trend of the thin film, thereby affecting the testing of the stress variation law of the film mismatch. Summary of the Invention

[0005] In view of the shortcomings of the prior art, the purpose of this invention is to provide a test method for the variation law of mismatch stress in thin films. In the existing practice of using RHEED for lattice analysis, the wavelength and sample-fluorescent screen distance are often regarded as fixed but not necessarily precisely known mechanical dimensions. The error in their values ​​will directly lead to the calculation error of the lattice constant. Moreover, the mismatch degree is assumed to be a single atom match, without considering the influence of thermal strain, which leads to the problem of low stress test accuracy of thin films.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: A method for testing the variation law of mismatch stress in thin films, the method comprising: Step 1: First, a standard sample is used and irradiated with RHEED to obtain the parameter information of the standard sample. Then, the wavelength of the incident electron beam is corrected according to the working voltage of RHEED to obtain the corrected electron beam wavelength information and further obtain the precise working distance. Step 2: Obtain the sample to be tested, preprocess the substrate of the sample to be tested, obtain the substrate diffraction images of the substrate in multiple angle directions, and obtain the substrate diffraction fringe spacing based on the substrate diffraction images. Combined with the precise working distance, the surface lattice constant of the substrate is obtained. Step 3: Start epitaxial growth on the sample to be tested, obtain the real-time diffraction fringe spacing and working voltage during the epitaxial growth process of the sample to be tested, and obtain the growth lattice constant of the sample to be tested by combining the precise working distance; after the growth is completed, obtain the diffraction images of the finished product in multiple angle directions, and obtain the diffraction fringe spacing of the finished product based on the diffraction images of the finished product, and obtain the lattice constant of the finished product by combining the precise working distance. Step 4: Compare the substrate diffraction image and the finished product diffraction image to analyze the matching mode between the finished product and the substrate. Correct the thermal expansion coefficient of all surface lattice constants in Steps 2 and 3 to determine the periodic lattice constant in the matching mode between the finished product and the substrate. Based on the matching mode and periodic lattice constant determined by rotating the sample stage, obtain the mismatch degree of the sample to be tested. Based on the corrected growth lattice constant, determine the stress and stress variation law of the sample to be tested.

[0007] Furthermore, in step 1, the parameter information of the standard sample includes lattice constant, fringe spacing and diffraction index. The fringe spacing of the standard sample is identified by the average value. By using the known lattice constant of the diffraction fringes of the standard substrate, combined with the diffraction fringe spacing, the actual lattice constant corresponding to each pixel in the diffraction pattern is calibrated.

[0008] Furthermore, in step 1, the operating voltage of the RHEED is 10-50 keV. When the operating voltage of the RHEED is greater than 10 keV, the speed of the electrons is close to the speed of light, and the mass increase effect cannot be ignored. Wavelength error will affect the reliability of the data. Therefore, the wavelength of the incident electron beam is corrected based on relativistic effects to obtain the corrected electron beam wavelength information, which is expressed as: ; in, Let be Planck's constant. For electronic quality, The number of elementary charges, V This is the real-time operating voltage of RHEED. The speed of light; Based on the corrected electron beam wavelength information and standard sample parameter information, the accurate working distance is expressed as follows: ; in a is the lattice constant. d This represents the average stripe spacing of the standard sample. λ The wavelength of the incident electron beam. h , k The diffraction index is denoted as .

[0009] Further, in step 2, the sample to be tested is acquired, and the substrate of the sample to be tested is subjected to ultrasonic cleaning and thermal cleaning treatment. The RHEED equipment is turned on, the electron beam energy is set to 30 keV, the sample stage is rotated, and the incident direction of the electron beam relative to the crystal orientation of the substrate is changed. An RHEED image is acquired every 15° in the range of 0° to 360° to obtain the substrate diffraction images of the substrate in multiple angle directions. The substrate diffraction fringe spacing is obtained according to the substrate diffraction images. Combined with the precise working distance, the surface lattice constant of the substrate is obtained.

[0010] Furthermore, in step 3, after replacing the standard sample with the sample to be tested, an epitaxial growth operation is performed on the sample to be tested, and the real-time diffraction fringe spacing and working voltage during the epitaxial growth process are obtained, with the average value of the diffraction fringe spacing being taken; and combined with the precise working distance, the lattice constant of the sample to be tested is obtained, expressed as: ; in, The average value of the diffraction fringe spacing. L The precise working distance of the RHEED system used. Let be Planck's constant. For electronic quality, The number of elementary charges, V This is the real-time operating voltage of RHEED. At the speed of light, h , k The diffraction index is denoted as .

[0011] Furthermore, in step 3, after growth is completed, the sample stage is rotated, and RHEED images of the finished product are acquired every 15° within the range of 0° to 360°. The diffraction images at the same angular position as the substrate during calibration are recorded in particular. The diffraction images of the finished product in multiple angular directions are obtained, and the diffraction fringe spacing of the finished product is obtained based on the diffraction images. Combined with the precise working distance, the lattice constants at room temperature along different angular directions are calculated.

[0012] Furthermore, in step 4, the substrate diffraction image and the finished product diffraction image are compared along different rotation angles. When the diffraction fringes of the substrate and the diffraction fringes of the finished product periodically overlap, the result is recorded. The continuous fraction method is used to determine the integer pair with the smallest difference between the substrate lattice constant and the finished product lattice constant, thus determining the quantitative description of the matching between the finished product and the substrate.

[0013] Furthermore, thermal expansion coefficient correction is performed on all surface lattice constants in steps 2 and 3, specifically including: Obtain the thermal expansion coefficients of the substrate and thin film materials within the growth temperature range, and use the following formula to correct the lattice constant measured at room temperature to the actual lattice constant at the growth temperature: ; in, a T Growth temperature T growth The lattice constant of the crystal is as follows. a 室温 α is the lattice constant measured at room temperature, and α is the linear thermal expansion coefficient of the material within the temperature range. When the coefficient of thermal expansion of a material changes significantly with temperature, the integral form is used: ; T growth The substrate temperature during thin film growth. T room Room temperature; The lattice constants of the substrate and the thin film were corrected as described above to obtain the actual lattice constant of the substrate at the growth temperature. a s (T growth ) and the stress-free lattice constant of the thin film a f (T growth ) .

[0014] Based on quantitative description, and combined with the finished lattice constant of the test sample while maintaining substrate temperature at the end of growth and the lattice constant of the corrected substrate, the periodic lattice constant is determined. The misfit degree of the test sample is then obtained from the periodic lattice constant, expressed as: ; in a f It is the periodic lattice constant of the sample under test. a s It is the periodic lattice constant of the substrate after correction for the coefficient of thermal expansion.

[0015] Furthermore, in step 4, based on the corrected growth lattice constant of the sample to be tested, the Young's modulus of the sample is obtained in conjunction with the sample material, and the stress value of the sample is obtained in conjunction with the lattice constant, expressed as: ; in, Let be the strain value of the sample to be tested, and E be the Young's modulus of the sample material.

[0016] Furthermore, in step 4, the strain state of the sample under test is obtained based on the lattice constant. The corrected growth lattice constant is compared with the lattice constant of the stress-free sample under room temperature to obtain the strain of the thin film, expressed as: ; Where a is the growth lattice constant of the sample to be tested, and a0 is the lattice constant of the stress-free sample to be tested at room temperature. The stress variation law is determined based on the strain value.

[0017] The technical solution adopted in this invention has the following beneficial effects: In this invention, standard samples are first used for RHEED quantitative testing. The actual precise working distance of the current device, i.e., the distance from the sample to the fluorescent screen, is calculated, reducing the impact of working distance errors. Simultaneously, the wavelength variation of the electron beam under different operating voltages is considered, and the wavelength acquisition is corrected based on relativistic effects, further reducing the impact of wavelength errors and thus improving the accuracy in thin film stress testing. Furthermore, different RHEED devices have different electron beams and working distances, while this invention can use any RHEED device to perform relevant stress testing on thin films, and the testing accuracy is not affected by the device, improving the universality of the testing method.

[0018] This invention, through precise physical formulas, not only ensures the absolute accuracy of a core input parameter, but also conveys a clear message to the field: because the strain caused by mismatch stress is on the order of Å, precise control of the working distance and electronic wavelength is necessary to achieve accurate quantitative stress measurement.

[0019] This invention does not presuppose a 1:1 lattice match between the thin film and the substrate. Instead, it uses a rotating sample stage to change the electron beam incident angle, and the system acquires RHEED images of the substrate and thin film under different crystal orientations. By comparing the periodic overlap of the diffraction patterns, it identifies the actual multiple matching modes at the interface—that is, determining how many substrate atoms and how many thin film atoms constitute a matching repeating unit. This method fully utilizes the advantages of in-situ, real-time RHEED characterization, accurately capturing the true matching relationship at the interface and avoiding misjudgments of matching modes caused by simply applying lattice constant ratios. Furthermore, this invention reflects the actual strain state of the interface better than traditional unit cell-scale mismatch calculations, providing more accurate physical parameters for optimizing epitaxial growth processes.

[0020] After obtaining the RHEED diffraction pattern, this invention proposes to quantitatively determine the matching period by calculating the greatest common divisor of the lattice constants of the substrate and the thin film. Specifically, the ratio of the two lattice constants is determined using the continued fraction method, finding the integer pair that minimizes the difference between the substrate and thin film lattice constants. This mathematical processing method transforms the geometric relationship of the diffraction pattern into a precise integer matching relationship, providing an accurate basis for subsequent mismatch calculations of the period length.

[0021] This invention introduces a thermal expansion correction mechanism into the mismatch calculation, which corrects the lattice constant measured at room temperature using the material's coefficient of thermal expansion to the value at the actual growth temperature. This mechanism effectively eliminates the influence of thermal strain introduced by the difference in coefficients of thermal expansion during the cooling process from the growth temperature to room temperature, ensuring that the calculated mismatch accurately reflects the interface matching state during film growth. This correction mechanism is particularly important for heterogeneous systems grown at high temperatures or with significant differences in coefficients of thermal expansion. Attached Figure Description

[0022] Figure 1 A schematic flowchart of a method for testing the variation law of mismatch stress in thin films provided by the present invention; Figure 2 The above are RHEED diffraction patterns of standard sample Si(110)

[100] and Bi(00l) films with different growth times in the embodiments of the present invention. Figure 3 This is a graph showing the lattice constant variation of Bi thin films of different thicknesses provided in the embodiments of the present invention; Figure 4 This is a schematic diagram illustrating the matching of the MgO substrate and the Bi thin film provided in an embodiment of the present invention. Detailed Implementation

[0023] In current practices of using RHEED for lattice analysis, the distance L is often treated as a fixed but not necessarily precisely known mechanical dimension, and errors in its value are directly and linearly transmitted to the calculated lattice constant. The electron beam wavelength λ is often approximated, especially at higher accelerating voltages; neglecting relativistic corrections introduces non-negligible systematic errors. The imprecise determination of these two parameters is one of the key bottlenecks preventing RHEED from transitioning from a qualitative observation tool to a quantitative analysis instrument.

[0024] To address this issue, this invention offers a systematic and innovative solution: First, it proposes using a standard sample with a known lattice constant for reverse calibration based on diffraction principles. Specifically, the working distance L is determined using a standard sample with a known lattice constant a. Given the lattice constant a, the fringe spacing d is obtained through RHEED, and the working distance L can be calculated using a formula. Reverse calibration using the diffraction principle (lattice constant formula) allows for the derivation of the working distance L, thus obtaining a precise value for L on any device. This eliminates dependence on fixed mechanical dimensions and achieves cross-device universality. Second, it explicitly employs a relativistically modified formula for calculating electron wavelength, linking the accelerating voltage and wavelength through rigorous physical laws. This ensures that the calculation of λ maintains theoretical accuracy even under high-voltage conditions, eliminating fundamental errors at the source.

[0025] By precisely locking the two most fundamental input parameters, L and λ, this invention constructs a complete quantitative measurement chain that is fully traceable and error-controllable, from diffraction images to stress values. This not only enables accurate determination of the magnitude and type of thin film stress but also ensures reliable comparability and reproducibility of measurement results across different laboratories and equipment, thereby elevating RHEED technology to a new quantitative stage that can serve precision materials research and process control.

[0026] A method for testing the variation law of mismatch stress in thin films, combined with Figure 1 The testing methods include: Step 1: First, a standard sample is used and irradiated with RHEED to obtain the parameter information of the standard sample. Then, the wavelength of the incident electron beam is corrected according to the working voltage of RHEED to obtain the corrected electron beam wavelength information and further obtain the precise working distance. The parameters of the standard sample include lattice constant, fringe spacing, and diffraction index. In the RHEED experiment, when an electron beam irradiates the sample surface at a grazing incidence angle, the diffraction pattern is generated by the intersection of the surface's two-dimensional reciprocal lattice with the Ewald sphere. Since the surface loses its periodicity in the vertical direction, the reciprocal lattice points elongate along the vertical direction to form reciprocal bars (diffraction fringes). h and k are used to calibrate these diffraction fringes: each diffraction fringe is uniquely identified by its corresponding two-dimensional Miller index (hk).

[0027] Different (hk) values ​​correspond to different diffraction fringes: Each diffraction spot or fringe observed on the fluorescent screen corresponds to a specific set of (hk) values. For example, (00) is a specular reflection spot (principal axis), while (10), (01), (11), etc. are diffraction spots of different orders (secondary axes).

[0028] The fringe spacing of the standard sample is calculated using the average value. By combining the known lattice constant of the diffraction fringes on the standard substrate with the fringe spacing, the actual lattice constant corresponding to each pixel in the diffraction pattern is calibrated. A collaborative extraction method of "DigitalMicrograph physical calibration + ImageJ multi-peak fitting" is employed. This method establishes the correspondence between pixels and lattice constants through the calibration pattern, and uses multi-peak fitting to accurately locate the peak values ​​of the diffraction fringes, thus obtaining the actual distance of the diffraction fringes, i.e., the characteristic parameter d.

[0029] In this embodiment, the diffraction pattern is imported into ImageJ software and DigitalMicrograph software to extract precise feature parameters d, DigitalMicrograph software is used for precise calibration, and ImageJ software is used for multi-peak fitting to obtain the average value.

[0030] In DigitalMicrograph software, the pixel distance of diffraction fringes in the diffraction pattern is measured using the "Analysis > Calibrate > Spatial Scale" function. Combined with the lattice constant and pixel distance of the standard sample, the true physical unit is assigned to each pixel in the diffraction pattern. For example... .

[0031] Import the calibration pattern into ImageJ software, select all diffraction fringes in the image, draw a straight line along the horizontal direction of the diffraction fringes, and use the "Analyze>PlotProfile" function to generate an intensity distribution curve, where each peak represents the precise location of the diffraction fringes.

[0032] Using multi-peak fitting PeakFitter, the average pixel distance between two peaks is accurately determined. Based on the calibrated diffraction pattern, the actual distance of the diffraction fringes is obtained, which is the characteristic parameter d of the diffraction pattern.

[0033] The equation typically used in wavelength calculation methods (the non-relativistic modified electron beam wavelength formula) is: ; When electron energies reach 10 keV or higher, the electron velocity approaches a significant proportion of the speed of light, and the mass increase effect (relativistic effect) becomes non-negligible, causing wavelength errors to affect data reliability. However, compared to the relativistic formulas calculating wavelengths of 0.7% at 19 keV and 1.5% at 30 keV, this directly impacts the calculation of the lattice constant 'a', leading to increased systematic bias. Therefore, this invention employs a relativistically corrected electron beam wavelength formula for calculation, using the following equation to relativistically correct the wavelength λ: ; in Let be Planck's constant. For electronic quality, The number of elementary charges, V For real-time operating voltage, The speed of light; The RHEED electron gun is designed to operate within the 10-50 keV range, with 10 keV being a physically significant threshold. 10 keV is the lower limit for RHEED to be considered "high-energy electron diffraction." When the voltage is below 10 keV, this method is no longer classified as RHEED but rather as "low-energy electron diffraction" (LEED), and therefore cannot be called high-energy electron diffraction. Therefore, relativistic corrections to the electron beam wavelength are necessary for RHEED characterization.

[0034] According to the Bragg diffraction equation, the working distance L can be expressed as: ; in a is the lattice constant. d This represents the average stripe spacing of the standard sample. λ The wavelength of the incident electron beam. h , k The diffraction index is the working distance of RHEED. L (Distance from sample to fluorescent screen).

[0035] To obtain precise working distance LBy positioning a standard sample with a known lattice constant at the location set for the planned thin film preparation, the spacing of the diffraction fringes of the standard sample is obtained. Since the lattice constant is known, the correction wavelength has been obtained, and thus the precise working distance can be determined.

[0036] Step 2: Obtain the sample to be tested. Perform ultrasonic cleaning and thermal cleaning on the sample substrate to make the surface of the sample substrate clean and flat. Turn on the RHEED equipment, set the electron beam energy to 30 keV, rotate the sample stage, change the incident direction of the electron beam relative to the crystal orientation of the substrate, and acquire an RHEED image every 15° in the range of 0° to 360° to obtain substrate diffraction images of the substrate in multiple angle directions. Obtain the substrate diffraction fringe spacing based on the substrate diffraction images, and combine it with the precise working distance to obtain the surface lattice constant of the substrate.

[0037] Step 3: Start epitaxial growth on the sample to be tested, obtain the real-time diffraction fringe spacing and working voltage during the epitaxial growth process of the sample to be tested, and obtain the growth lattice constant of the sample to be tested by combining the precise working distance; after the growth is completed, obtain the diffraction images of the finished product in multiple angle directions, and obtain the diffraction fringe spacing of the finished product based on the diffraction images of the finished product, and obtain the lattice constant of the finished product by combining the precise working distance. In this process, after replacing the standard sample with the test sample, the substrate temperature was kept at room temperature, the source furnace was turned on, the growth rate was 1 Å / min, and a thin film was grown. The real-time diffraction fringe spacing and working voltage during the epitaxial growth of the test sample were obtained, and the average value of the diffraction fringe spacing was taken. Combined with the precise working distance, the lattice constant of the test sample was obtained, expressed as: ; in, The average value of the diffraction fringe spacing. L The precise working distance of the RHEED system used. Let be Planck's constant. For electronic quality, The number of elementary charges, V This is the real-time operating voltage of RHEED. At the speed of light, h , k The diffraction index is denoted as .

[0038] In the embodiments of the present invention, all the above calculations of lattice constants are performed using the above calculation formulas, thereby obtaining more accurate lattice constant values.

[0039] After growth, the sample stage was rotated, and RHEED images of the finished product were acquired every 15° within the range of 0° to 360°. Special attention was paid to recording diffraction images at the same angular positions as those calibrated on the substrate. Multi-angle diffraction images of the finished product were obtained, revealing a clear striped diffraction pattern, indicating a smooth surface and good crystallinity. The spacing of the RHEED diffraction fringes was measured, and combined with the precise working distance, the lattice constants along different directions at room temperature were calculated.

[0040] Step 4: Compare the substrate diffraction image and the finished product diffraction image, analyze the matching mode between the finished product and the substrate, correct the thermal expansion coefficient of all surface lattice constants in Step 2 and Step 3, determine the periodic lattice constant in the matching mode between the finished product and the substrate, obtain the mismatch degree of the sample under test based on the matching mode and the periodic lattice constant, and determine the stress and stress change law of the sample under test according to the corrected growth lattice constant.

[0041] In this embodiment, the substrate diffraction image and the finished product diffraction image are compared along different rotation angles. When the diffraction fringes of the substrate and the diffraction fringes of the finished product periodically overlap, the result is recorded. The integer pair with the smallest difference between the lattice constant of the substrate and the lattice constant of the finished product is determined by the continued fraction method, which determines the quantitative description of the matching between the finished product and the substrate.

[0042] The RHEED images of the substrate and the thin film were compared and analyzed. By superimposing diffraction patterns at different rotation angles, the overlap relationship of diffraction spots or fringes was sought. Specifically, the RHEED images of the substrate and the thin film were compared along different rotation angles. It was found that when the sample stage was rotated to a specific angle, the diffraction fringes of the thin film and the substrate periodically overlapped. This phenomenon indicates that the lattice and substrate form a certain multiple matching relationship at this orientation. If clear diffraction fringes of the thin film were directly obtained at a specific substrate orientation, the lattice constants of the substrate and the thin film were directly calculated to determine the matching mode.

[0043] Thermal expansion coefficient correction is performed on all surface lattice constants in steps 2 and 3. This invention introduces a thermal expansion correction mechanism in the mismatch calculation, with the specific steps as follows: First, based on the material property manuals or experimental measurements of the substrate and thin film materials, obtain their linear coefficient of thermal expansion α within the growth temperature range. If the change in the coefficient of thermal expansion with temperature is not negligible, then the temperature-dependent coefficient of thermal expansion function α(T) is used. Second, record the substrate temperature T during thin film growth. growth (Typically read directly by the temperature-controlled thermocouple of the MBE or sputtering equipment) and room temperature T room (Take 298 K or the actual test environment temperature).

[0044] Then, the substrate lattice constant measured in step 2 at room temperature... a s 室温 After correction, the actual lattice constant of the substrate at the growth temperature is obtained: ; Similarly, the lattice constant of the thin film at room temperature measured in step 3... a f 室温 After correction, the stress-free lattice constant of the thin film at the growth temperature is obtained: ; a f (T growth ) It reflects the actual lattice constant of the thin film after it is subjected to stress at the growth temperature, while a f 室温 This is then used for comparison with the stress-free state in subsequent strain analysis. Finally, the corrected... a s (T growth By combining the period integer multiples in the matching mode with the period scale mismatch calculation formula, we can obtain the mismatch degree that truly reflects the matching state of the growth interface.

[0045] Based on quantitative description, and combined with the finished lattice constant of the test sample while maintaining substrate temperature at the end of growth and the lattice constant of the corrected substrate, the periodic lattice constant is determined. The misfit degree of the test sample is then obtained from the periodic lattice constant, expressed as: ; in, a f It is the periodic lattice constant of the sample under test. a s It is the periodic lattice constant of the substrate after correction for the coefficient of thermal expansion.

[0046] The periodic lattice constant is the lattice constant of several thin films that maintain the substrate temperature and the lattice constant of several corrected substrates. For example, 8 Bi corresponds to 15 MgO, which is a matching method determined by rotating the substrate. 8 Bi is one periodic lattice constant, and 15 MgO is another periodic lattice constant. The mismatch degree is calculated using the periodic lattice constant, which replaces the traditional method of simply calculating the mismatch degree by matching the lattice constant of a single unit cell of the thin film with the lattice constant of a substrate, thus improving the accuracy of the mismatch degree calculation.

[0047] In this embodiment, based on the growth lattice constant of the sample to be tested, the Young's modulus of the sample is obtained according to the material of the sample to be tested, and the stress value of the sample to be tested is obtained in combination with the lattice constant, expressed as: ; in, Here, E represents the strain value of the sample to be tested, and E represents the Young's modulus of the sample material. The obtained lattice constant is compared with the lattice constant of the stress-free sample under room temperature to obtain the strain of the thin film, which is expressed as: ; Where a is the lattice constant of the sample under test, and a0 is the lattice constant of the sample under test without stress at room temperature.

[0048] Based on the above stress values, the relationship between stress and film growth time was determined, the stress-growth time relationship curve was plotted, and the stress variation law was obtained.

[0049] Based on the above-mentioned positive and negative mismatch values, and by corroborating them with the stress states, the variation law of thin film growth can be further refined.

[0050] This method is applicable to thin film material systems that can be epitaxially grown on single-crystal substrates and produce clear RHEED diffraction patterns, such as semiconductor thin films, oxide thin films, and metal thin films. The thin film must be single-crystal or highly oriented polycrystalline, capable of producing clear RHEED diffraction spots or stripes.

[0051] Based on the above method, the present invention also conducts a control experiment, as follows: First, a stress calculation experiment was conducted: Experimental conditions and methods: Bismuth (Bi) films were grown on Si(100) single crystal substrates in a molecular beam epitaxy (MBE) apparatus. The growth conditions were: substrate temperature 70°C, deposition rate 1 Å / min. Using the method of this invention, the RHEED accelerating voltage was set to 30 keV and the incident angle was 3°. Bi films of different thicknesses were grown, namely 100 Å, 150 Å, 180 Å, 210 Å, 300 Å and 400 Å. By controlling the deposition time, RHEED diffraction images of different film thicknesses were obtained by CCD. Using Si(100)

[100] as the standard sample, the electron beam wavelength was relativistically corrected to 0.086 Å. The diffraction patterns were imported into Imagej and DigitalMicrograph software to accurately extract the average distance between the principal and secondary axes of the diffraction fringes. .Will Substituting the lattice constant and the functional relationship, we obtain the working distance. L =527820μm. Applying the complete process of this invention, precise extraction is achieved from diffraction images. Substituting into the formula, the real-time lattice constant of the 90Å thick thin film is calculated. a f =4.511Å, such as Figure 2As shown, Figure 2 The upper center shows the diffraction fringes of the standard sample, and the lower center shows the diffraction fringes at different times: t1 is 100 min, t2 is 180 min, and t3 is 300 min. Plotting the lattice constants calculated from films of different thicknesses deposited at different times against the film thickness clearly shows the changes in the lattice constant. Figure 3 As shown.

[0052] The bulk lattice constant of Bi is known. a O = 4.54 Å, biaxial Young's modulus E = 32.3 GPa. The strain was calculated. ε =(4.511-4.54) / 4.54≈-0.639%, and the stress σ is quantitatively calculated to be 32.3GPa×-0.639%=-0.206GPa.

[0053] This invention elevates RHEED stress measurement to "high-precision quantitative measurement" by precisely calibrating the L value and employing relativistically corrected wavelengths. In an example with a 90 Å thick Bi thin film, this invention obtained a precise stress value of -0.206 GPa, and every formula symbol can be accurately derived.

[0054] Control group 1: Only the effect of inaccurate L calibration. It cannot be calculated for devices with unknown working distances. The working distance of this invention is calculated to be 527820 μm, which not only applies to devices with unknown working distances but also improves accuracy by several orders of magnitude.

[0055] Control group 2: The effect of λ being uncorrected only. L was precisely calibrated to 527820 μm, and λ was approximated to 0.0873 Å (error +1.5%) using a non-relativistic approximation. This error was propagated to the lattice constant and further amplified by Young's modulus, ultimately leading to an order-of-magnitude deviation in stress calculation.

[0056] The calculated lattice constants are a1 = 4.511 Å after relativistic correction and a2 = 4.579 Å without relativistic correction. The calculated stress is -0.206 GPa after relativistic correction and 0.277 GPa without relativistic correction, with an error rate of 234%. Furthermore, the stress states are incorrect: the relativistically corrected stress is compressive stress, while the uncorrected stress is tensile stress.

[0057] Next, an experiment was conducted to calculate the film mismatch degree. Experimental conditions and methods: A Si(100) single crystal substrate with dimensions of 10 mm × 10 mm × 0.5 mm was selected and sequentially ultrasonically cleaned with acetone, ethanol, and deionized water for 10 minutes each. After drying with nitrogen, it was placed into a molecular beam epitaxy (MBE) growth chamber. The background vacuum of the growth chamber was better than 1 × 10⁻⁶ mm. -7Pa. The substrate was heated to 600°C for 30 minutes for thermal cleaning to remove adsorbed carbon contaminants and obtain an atomically flat surface.

[0058] After the substrate temperature dropped to room temperature, a reflective high-energy electron diffractometer (RHEED) was turned on, and the electron beam energy was set to 30 keV. The incident direction of the electron beam relative to the crystal orientation of the substrate was changed by rotating the sample stage. An RHEED image was acquired every 15° in the range of 0° to 360°, focusing on recording the diffraction fringe images along the Si

[100] and

[110] directions (i.e., the sample stage rotation angles of 0° and 45°). The surface lattice constant of the Si substrate was calculated based on the diffraction fringe spacing. The calibration result was: the diffraction fringe spacing along the

[100] direction corresponds to the reciprocal lattice constant, and the lattice constant of Si at room temperature was calculated to be a_Si_room temperature = 5.435 Å (which is consistent with the standard value of 5.43 Å), confirming that the substrate surface is clean and ordered.

[0059] During the growth process, the changes in the RHEED image were monitored in real time, and the diffraction stripes gradually changed from Si characteristics to Bi characteristics, indicating that the film grew continuously.

[0060] After growth, RHEED images of the Bi thin film were acquired again at 15° intervals within the range of 0° to 360° by rotating the sample stage. Special attention was paid to recording diffraction images at the same angular positions as those calibrated with Si (e.g., 0° and 45°) for subsequent comparative analysis. The Bi thin film exhibited a clear striped diffraction pattern, indicating a smooth surface and good crystallinity. The spacing of the Bi thin film's RHEED diffraction fringe was measured: the interplanar spacing corresponding to the Bi thin film diffraction fringe was d = 4.54 Å, therefore the longitudinal lattice constant was √3d = 7.86 Å. The RHEED images of the Si substrate and the Bi thin film were compared and analyzed. By superimposing diffraction patterns at different rotation angles, the overlap relationship between the two sets of diffraction spots or fringes was sought.

[0061] The RHEED image of Si along the

[100] direction was compared with the images of the Bi film along different rotation angles. It was found that when the sample stage was rotated to 60°, the diffraction fringes of the Bi film and the Si substrate periodically overlapped. This phenomenon indicates that the Bi lattice and the Si lattice form a certain multiple matching relationship at this orientation. Based on the unit cell data of Bi and Si, the matching direction was determined, and a schematic diagram was drawn. The specific values ​​of both were calculated: a_Si_room temperature = 5.435 Å; a_Bi_room temperature single atom longitudinal direction = 7.86 Å; The greatest common divisor of the lattice constants of Si and Bi is calculated using the continued fraction method: a_Si_Room / a_Bi_Room= 5.435 / 7.86= 0.691; Its continued fraction expansion is 0.691 = 1 / (1.447) = 1 / (1 + 0.447) ≈ 1 / (1 + 1 / 2.237) ≈ 10 / 7; Therefore, a_Si_room temperature / a_Bi_room temperature ≈ 10 / 7, meaning the periodic lattice constant of Si is 10 × a_Si = 10 × 5.435 = 54.35 Å, and the periodic lattice constant of Bi is 7 × a_Bi = 7 × 7.863 = 55.02 Å. This indicates that Bi and Si have a matching mode of 10 Si lattices matching 7 Bi lattices, as shown below. Figure 4 As shown.

[0062] The periodic lattice constant of 10 Si atoms matches that of 7 Bi atoms. The mismatch is calculated as follows: δ = |(10× a_Si - 8 × a_Bi_)| / (10 × a_Si) × 100%=-1.23%.

[0063] Control group: The lattice constants of Si (5.435 Å) and Bi thin film (4.54 Å) were directly substituted into δ = (a f - a s ) / a s In the formula, the unit cells are assumed to be matched 1:1, and the calculated value is 16.467%. This is not only inaccurate, but the stress state is also the opposite of the actual situation. The mismatch is positive and indicates tensile stress, while the data we measured is compressive stress.

[0064] This invention determines the lattice constant. a and accurate working distance L and wavelength λ The calculation method was proposed, and a formula for calculating the lattice constant was put forward, enabling accurate acquisition of the working distance based on standard samples and corrected wavelengths. L In order to achieve a level of technology that the background technology could not reach.

[0065] This invention solves the problem of how to make the RHEED quantitative stress measurement method free from equipment selectivity, thus enabling its application on any RHEED device (regardless of whether its L value is known in advance). Based on the lattice constant calculation formula of this application, users can measure a standard sample with a known lattice constant to accurately calibrate the actual L value of the current device, and the accuracy of the L value can be improved by several orders of magnitude. Therefore, the lattice constant calculation formula of this invention has universality. Regardless of the device (with an unknown L value), it can be accurately calibrated using a standard sample and then used for the direct measurement of any thin film mismatch stress.

[0066] This invention aims to provide a systematic approach that transforms Reflection High Energy Electron Diffraction (RHEED) technology from a qualitative measurement technique combining high-resolution transmission electron microscopy (HRTEM) image observation and computational simulation into a standardized method for in-situ, quantitative, and high-precision measurement of the mismatch stress state of thin films. This method utilizes standard samples to calibrate the key geometric parameter L on any device. This enables accurate measurement on any RHEED device, greatly enhancing the method's practical value. The invention explicitly employs a relativistically corrected formula for calculating electron wavelength, ensuring that the calculation of λ maintains theoretical accuracy under high-voltage conditions, thus eliminating fundamental errors at the source.

[0067] The key difference between this invention and traditional methods lies in that it does not presuppose a 1:1 lattice match between the thin film and the substrate. Instead, by rotating the sample stage to change the electron beam incident angle, the system acquires RHEED images of the substrate and thin film under different crystal orientations. By comparing the periodic overlap of the diffraction patterns, the system identifies the actual multiple matching modes at the interface—that is, determining how many substrate atoms and how many thin film atoms constitute a matching repeating unit. This method fully utilizes the advantages of in-situ, real-time RHEED characterization, accurately capturing the true matching relationships at the interface and avoiding misjudgments of matching modes caused by simply applying lattice constant ratios.

[0068] After obtaining the RHEED diffraction pattern, this invention proposes to quantitatively determine the matching period by calculating the greatest common divisor of the lattice constants of the substrate and the thin film. Specifically, the ratio of the two lattice constants is determined using the continued fraction method, and the integer pair that minimizes the difference between the substrate and thin film lattice constants is found. This mathematical processing method transforms the geometric relationship of the diffraction pattern into a precise integer matching relationship, providing an accurate basis for the period length in subsequent mismatch calculations.

[0069] This invention introduces a thermal expansion correction mechanism into the mismatch calculation, which corrects the lattice constant measured at room temperature using the material's coefficient of thermal expansion to the value at the actual growth temperature. This mechanism effectively eliminates the influence of thermal strain introduced by the difference in coefficients of thermal expansion during the cooling process from the growth temperature to room temperature, ensuring that the calculated mismatch accurately reflects the interface matching state during film growth. This correction mechanism is particularly important for heterogeneous systems grown at high temperatures or with significant differences in coefficients of thermal expansion.

[0070] Based on a defined matching mode (m substrate atoms matching n thin film atoms) and a thermally corrected lattice constant, this invention proposes to calculate the mismatch using the period length ratio: δ = |m×af - n×as / (n×as) × 100%. This formula defines the mismatch at the scale of matching repeating units, which better reflects the actual strain state of the interface than the traditional unit cell-scale mismatch calculation, providing more accurate physical parameters for optimizing epitaxial growth processes.

[0071] The method established in this invention is not limited to the Bi / Si system and can be extended to the study of matching relationships and the calculation of mismatch degree in various heteroepitaxial materials. Its core process—rotating RHEED experiments to determine the matching mode, the greatest common divisor method to determine the matching period, thermal expansion correction, and periodic scale mismatch degree calculation—forms a complete technical solution with systematicity and portability.

[0072] The core difference between the technical means of this invention and existing research lies in the systematic upgrading of Reflection High Energy Electron Diffraction (RHEED) from a qualitative or semi-quantitative surface analysis tool to a cross-device, high-precision, traceable in-situ quantitative measurement method. Its non-obviousness and unexpected effects are mainly reflected in the following four aspects: Existing techniques assume the distance L from the sample to the fluorescent screen is a fixed mechanical dimension during device calibration, directly using nominal values ​​or ignoring them, resulting in unavoidable systematic errors. This invention utilizes standard samples with known lattice constants and combines the diffraction principle to reverse-calibrate the precise L value for any device, eliminating mechanical dimension dependence. Regarding wavelength calculation, existing techniques use non-relativistic approximation formulas, especially at high voltages ≥10keV, ignoring the electron mass increase effect and introducing fundamental errors. This invention explicitly uses a relativistically corrected electron wavelength formula, ensuring the theoretical accuracy of wavelength at high voltages and eliminating wavelength errors at the source. Existing techniques assume a 1:1 unit cell matching between the thin film and substrate, directly applying lattice constant ratios to calculate mismatch, easily leading to mode misjudgment. This invention acquires multi-angle RHEED images by rotating the sample stage, identifies the periodic overlap of diffraction fringes, and uses the continued fraction method to determine the m:n integer matching pattern (e.g., 10 Si matching 7 Bi). Existing techniques directly use the lattice constant measured at room temperature, neglecting the thermal strain introduced by the difference in thermal expansion coefficients during cooling from the growth temperature to room temperature. This invention introduces a thermal expansion coefficient correction mechanism to correct the room temperature lattice constant to the value at the actual growth temperature, eliminating thermal strain interference.

[0073] For a long time, researchers in the RHEED field have generally accepted it as a "semi-quantitative" tool, believing that... L and λ Tiny errors may have little impact on the final result, or it may be assumed that differences between devices cannot be standardized. This invention takes the opposite approach, actively treating these two seemingly fixed parameters as variables requiring precise calibration. Traditional thin-film physics assumes a 1:1 interface match, as this is the simplest assumption for theoretical calculations. This invention, through experimental observation, discovers that complex multiple matches (such as 7:10) often exist in actual epitaxial growth, and uses the continued fraction method to deduce integer matching pairs from diffraction geometry.

[0074] Improved accuracy and reliability: Comparisons in the examples show that ignoring only the relativistic correction (1.5% wavelength error at 30keV) causes the calculated stress to incorrectly change from the true -0.206GPa (compressive stress) to +0.277GPa (tensile stress), an error rate as high as 234%, and the stress states are completely reversed. This result completely overturns the traditional understanding that "small errors are negligible." In the Bi / Si system, the traditional 1:1 default mode calculates a mismatch of 16.467% (tensile stress), while the 10:7 actual matching mode determined in this invention calculates a mismatch of only -1.23% (compressive stress). The two not only differ by an order of magnitude in value, but also have opposite stress properties, directly affecting the judgment of the thin film growth mechanism.

[0075] Cross-device universality: This invention reverse-calibrates the L value using standard samples, enabling any RHEED device, regardless of whether its mechanical dimensions are known or whether it has been maintained and adjusted, to output comparable and accurate stress data. This is equivalent to establishing a "metrological standard" for RHEED technology, enabling it to serve precision materials research and industrial process control.

[0076] This invention reveals the true physical picture of the interface: by rotating the sample stage and using the continued fraction method, this invention clearly reveals, for the first time, non-1:1 multiple matching modes under RHEED in-situ conditions. This result proves that many mismatch degrees and stress evolution laws calculated based on the 1:1 assumption may be incorrect. This discovery has important implications for correcting heteroepitaxial growth theories (such as the lattice mismatch dislocation generation threshold and critical thickness model).

[0077] The invention eliminates the interference of thermal strain: By correcting for the coefficient of thermal expansion, it can distinguish between growth stress caused by lattice mismatch and thermal stress caused by the cooling process. For heterogeneous systems grown at high temperatures or with large differences in the coefficient of thermal expansion, this correction can avoid misjudging thermal stress as mismatch stress, thereby providing accurate physical parameters for process optimization.

Claims

1. A method for testing the variation law of mismatch stress in thin films, characterized in that, The testing method includes: Step 1: First, a standard sample is used and irradiated with RHEED to obtain the parameter information of the standard sample. Then, the wavelength of the incident electron beam is corrected according to the working voltage of RHEED to obtain the corrected electron beam wavelength information and further obtain the precise working distance. Step 2: Obtain the sample to be tested, preprocess the substrate of the sample to be tested, obtain the substrate diffraction images of the substrate in multiple angle directions, and obtain the substrate diffraction fringe spacing based on the substrate diffraction images. Combined with the precise working distance, the surface lattice constant of the substrate is obtained. Step 3: Start epitaxial growth on the sample to be tested, obtain the real-time diffraction fringe spacing during the epitaxial growth process of the sample to be tested, and obtain the growth lattice constant of the sample to be tested by combining the precise wavelength and the precise working distance; after the growth is completed, obtain the diffraction images of the finished product in multiple angle directions, and obtain the diffraction fringe spacing of the finished product based on the diffraction images of the finished product, and obtain the lattice constant of the finished product by combining the precise working distance. Step 4: Compare the substrate diffraction image and the finished product diffraction image, analyze the matching mode between the finished product and the substrate, correct the thermal expansion coefficient of all surface lattice constants in Step 2 and Step 3, determine the periodic lattice constant in the matching mode between the finished product and the substrate, obtain the mismatch degree of the sample under test based on the matching mode and the periodic lattice constant, and determine the stress and stress change law of the sample under test according to the corrected growth lattice constant.

2. The method for testing the variation law of thin film mismatch stress according to claim 1, characterized in that, In step 1, the parameter information of the standard sample includes lattice constant, fringe spacing and diffraction index. The fringe spacing of the standard sample is identified by the average value. By using the known lattice constant of the diffraction fringes of the standard substrate, combined with the fringe spacing, the actual lattice constant corresponding to each pixel in the diffraction pattern is calibrated.

3. The method for testing the variation law of thin film mismatch stress according to claim 1, characterized in that, In step 1, the RHEED operating voltage is 10-50 keV. When the RHEED operating voltage is greater than 10 keV, the electron velocity is close to the speed of light, and its mass increase effect cannot be ignored. Wavelength error will affect the reliability of the data. Therefore, the wavelength of the incident electron beam is corrected based on relativistic effects to obtain the corrected electron beam wavelength information, which is expressed as: ； in, is Planck's constant. For electronic quality, The number of elementary charges, V This is the real-time operating voltage of RHEED. The speed of light; Based on the corrected electron beam wavelength information and standard sample parameter information, the accurate working distance is expressed as follows: ； in a is the lattice constant. d This represents the average stripe spacing of the standard sample. λ The wavelength of the incident electron beam. h , k The diffraction index is denoted as .

4. The method for testing the variation law of thin film mismatch stress according to claim 1, characterized in that, In step 2, the sample to be tested is acquired, and the substrate of the sample to be tested is subjected to ultrasonic cleaning and thermal cleaning treatment. The RHEED equipment is turned on, the electron beam energy is set to 30 keV, the sample stage is rotated, and the incident direction of the electron beam relative to the crystal orientation of the substrate is changed. An RHEED image is acquired every 15° in the range of 0° to 360° to obtain the substrate diffraction images of the substrate in multiple angle directions. The substrate diffraction fringe spacing is obtained from the substrate diffraction images. Combined with the precise working distance, the surface lattice constant of the substrate is obtained.

5. The method for testing the variation law of thin film mismatch stress according to claim 1, characterized in that, In step 3, after replacing the standard sample with the sample to be tested, an epitaxial growth operation is performed on the sample to be tested. The real-time diffraction fringe spacing and working voltage during the epitaxial growth process are obtained, and the average value of the diffraction fringe spacing is taken. Combined with the precise working distance, the lattice constant of the sample to be tested is obtained, expressed as: ； in, The average value of the diffraction fringe spacing. L The precise working distance of the RHEED used. is Planck's constant. For electronic quality, The number of elementary charges, V This is the real-time operating voltage of RHEED. At the speed of light, h , k The diffraction index is denoted as .

6. The method for testing the variation law of thin film mismatch stress according to claim 1, characterized in that, In step 3, after growth is completed, the sample stage is rotated, and RHEED images of the finished product are acquired every 15° in the range of 0° to 360°. The diffraction images at the same angle position as the substrate are recorded in particular. The diffraction images of the finished product in multiple angle directions are obtained, and the diffraction fringe spacing of the finished product is obtained based on the diffraction images. Combined with the precise working distance, the lattice constants at room temperature along different angle directions are calculated.

7. The method for testing the variation law of thin film mismatch stress according to claim 1, characterized in that, In step 4, the substrate diffraction image and the finished product diffraction image are compared along different rotation angles. When the diffraction fringes of the substrate and the diffraction fringes of the finished product periodically overlap, the result is recorded. The ratio of the lattice constants at the point of overlap is determined using the continued fraction method. The integer pair that minimizes the difference between the substrate lattice constant and the finished product lattice constant is obtained, and a quantitative description of the matching between the finished product and the substrate is determined.

8. The method for testing the variation law of thin film mismatch stress according to claim 7, characterized in that, Thermal expansion coefficient correction is performed on all surface lattice constants in steps 2 and 3, specifically including: Obtain the thermal expansion coefficients of the substrate and thin film materials within the growth temperature range, and use the following formula to correct the lattice constant measured at room temperature to the actual lattice constant at the growth temperature: ； in, a T Growth temperature T growth The lattice constant of the crystal is as follows. a 室温 α is the lattice constant measured at room temperature, and α is the linear thermal expansion coefficient of the material within the temperature range. When the coefficient of thermal expansion of a material changes significantly with temperature, the integral form is used: ； T growth The substrate temperature during thin film growth. T room Room temperature; The lattice constants of the substrate and the thin film were corrected as described above to obtain the actual lattice constant of the substrate at the growth temperature. a s (T growth ) and the stress-free lattice constant of the thin film a f (T growth ) .

9. The method for testing the variation law of thin film mismatch stress according to claim 1, characterized in that, In step 4, based on the corrected growth lattice constant of the sample under test, the Young's modulus of the sample is obtained in conjunction with the sample material, and the stress value of the sample is obtained in conjunction with the lattice constant, expressed as: ； in, Let be the strain value of the sample to be tested, and E be the Young's modulus of the sample material.

10. The method for testing the variation law of thin film mismatch stress according to claim 1, characterized in that, In step 4, the strain state of the sample under test is obtained based on the lattice constant. The corrected growth lattice constant is compared with the lattice constant of the stress-free sample under room temperature to obtain the strain of the thin film, expressed as: ； Where a is the growth lattice constant of the sample to be tested, and a0 is the lattice constant of the stress-free sample to be tested at room temperature. The stress variation law is determined based on the strain value.

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