Measurement and decoupling method for rotation angle and inclination angle of crystal interface

Abstract

The invention relates to a method for measuring and decoupling a rotation angle and an inclination angle of a crystal interface, and the method comprises the steps: S1, obtaining a crystal sample containing a top crystal, a bottom crystal and the interface, building a Cartesian rectangular coordinate system with the offset tangent axis of the crystal sample as the x axis, recording the offset tangent angle of the crystal sample as alpha, and calculating the crystallographic directions n z and n y of the crystal sample; s2, fixing the bottom crystal, rotating the top crystal relative to the interface psi, and calculating to obtain crystallographic directions n z'and n y 'of the rotated top crystal according to n z, n y and psi; and S3, according to n z, n z ', n y and n y', calculating to obtain a crystallographic rotation angle psi 'and a crystallographic inclination angle theta' of the crystal sample. According to the method, the known offset angle alpha and the initial crystallographic direction are used as references, the mathematical relationship between Psi'and Theta 'of crystallography and the direction vector after rotation are deduced through the controllable rotation Psi of the top crystal relative to the interface, then the interface rotation angle Psi of the actual crystal is calculated through inversion, and the measurement precision and repeatability are improved.

Description

Technical Field

[0001] This invention relates to the field of semiconductor technology, and specifically to a method for measuring and decoupling the crystal interface rotation angle and tilt angle. Background Technology

[0002] The atomic arrangement at the interface of heterojunctions (or homojunctions) is a core factor determining the physicochemical properties of crystalline composite materials. Aberration-corrected transmission electron microscopy (Ac-TEM), as an important high-resolution characterization technique, can directly observe and finely analyze the crystallographic features of heterojunction materials (or homojunctions), including the spatial arrangement of bulk atoms, precise identification of atom types, and key information such as the atomic configuration of the interface region.

[0003] However, the influence of the relative tilt angle and rotation angle of the two layers in a heterojunction on the physicochemical properties of the heterojunction interface often lacks scientific research. This is due to the lack of effective characterization methods for the tilt angle of heterojunctions. Specifically, the rotation mismatch (Twist) of the interface during crystal and interface fabrication processes affects the tilt mismatch (Tilt), and vice versa. The coupling between the Twist angle and the Tilt angle is difficult to measure and analyze directly, necessitating the development of new methods to decouple and accurately characterize the atomic arrangement of the interface during the process. Summary of the Invention

[0004] Based on the above description, the present invention provides a method for measuring and decoupling the rotation angle and tilt angle of a crystal interface, aiming to improve the accuracy of the measurement of the rotation angle and tilt angle and decouple their relationship.

[0005] The technical solution of the present invention to solve the above-mentioned technical problems is as follows: This invention provides a method for measuring the rotation angle and tilt angle of a crystal, comprising: S1. Obtain a crystal sample containing the top crystal, bottom crystal, and interface. Establish a Cartesian coordinate system with the tangent axis of the crystal sample as the x-axis. Denote the tangent angle of the crystal sample as α. Calculate the crystallographic orientation of the crystal sample. n z and n y ; S2. Fix the bottom crystal, rotate the top crystal relative to the interface by ψ, according to... n z , n y The crystallographic orientation of the rotated top crystal is calculated using ψ. n z ’ andn y ’ ; S3. According to n z , n z ’ , n y and n y ’ The crystallographic rotation angle ψ' and crystallographic tilt angle θ' of the crystal sample were calculated.

[0006] Furthermore, in step S1, n z The calculation method is as follows: Let the unit vector along the z-axis be denoted as n 1. As shown in equation (1), (1), Will n 1. Rotate α around the x-axis to obtain n z As shown in equation (2), (2).

[0007] Furthermore, in step S2, n z ’ The calculation method is as follows: Will n z Rotate ψ about the z-axis to obtain n z ’ As shown in equation (3), (3).

[0008] Furthermore, in step S3, the method for calculating the crystallographic tilt angle θ' is shown in equation (4). (4).

[0009] Furthermore, in step S1, n y The calculation method is as follows: Let the unit vector of the y-axis be denoted as n 2. As shown in equation (5), (5); Will n 2. Rotate α around the x-axis to obtain n y As shown in equation (6), (6).

[0010] Furthermore, in step S2, n y ’ The calculation method is as follows: Will n y Rotate ψ about the z-axis to obtain n y ’ As shown in equation (7), (7).

[0011] Furthermore, in step S3, the method for calculating the crystallographic rotation angle ψ' is shown in equation (8). (8).

[0012] Furthermore, when α > 5°, the crystallographic rotation angle ψ' is calculated as shown in equation (9). (9).

[0013] Furthermore, the magnitude of the shear angle α is detected by any one of the following methods: TEM, STEM, EBSD, CBED, Kikuchi Pattern, NBD, PED, Moiré Pattern, and 4D-STEM.

[0014] The present invention also proposes a decoupling method for crystal rotation angle and tilt angle, including the aforementioned measurement methods for crystal rotation angle and tilt angle; Step S3 is followed by: S4. The crystallographic rotation angle ψ' is obtained by PVTEM, and the crystallographic tilt angle θ' is obtained by XS-TEM. The relationship between the angles is calculated based on ψ, α, ψ' and θ'.

[0015] Compared with the prior art, the technical solution of this application has the following beneficial technical effects: (1) In the technical solution of the present invention, the known offcut angle α and the initial crystallographic direction are utilized ( n z , n y Using ψ as a reference, the mathematical relationship between ψ' and θ' in crystallography and the direction vector after rotation are derived through the controllable rotation ψ of the top crystal relative interface. n z '、 n yThe interface rotation angle ψ of the actual crystal is then calculated by inversion, avoiding subjective errors caused by reliance on electron diffraction or image fitting, and significantly improving measurement accuracy and repeatability. (2) The crystallographic rotation and tilt angles are measured using TEM (transmission electron microscopy). Then, modeling and mathematical calculations are used to analyze the coupling relationship between the crystal and interface fabrication processes: the Twist angle, Tilt angle, and the crystal surface Off angle. First, the crystallographic Twist angle is measured using PVTEM (planar view transmission electron microscopy) sample preparation and TEM analysis. Then, two TEM slices are prepared at two different locations and orientations (dual-point positions), and the corresponding rotation angles 1 and 2 are measured using TEM imaging. For cubic crystals, these two orientations are generally 100° and 010°; for hexagonal crystals, these two orientations are generally 1-100° and 11-20°. Then, a geometric model is established, and the accurate crystallographic tilt angle is calculated. Finally, the mathematical relationship between the Twist angle, Tilt angle, and crystal surface Off angle between the interface and the crystal is calculated using geometric modeling and mathematical calculations. Attached Figure Description

[0016] Figure 1 This is a schematic diagram of the tilt angle and rotation angle of the crystal interface in the existing crystal and interface preparation process; Figure 2 This is a schematic diagram of the existing technology for measuring crystallographic tilt angle and rotation angle based on TEM sample rod rotation; Figure 3 This is a schematic diagram of the existing crystal interface Twist and Tilt angle measurement methods based on small-angle grain boundaries and PVTEM technology; Figure 4 This is a schematic flowchart illustrating the method for measuring the crystal rotation angle and tilt angle in Embodiment 4 provided in this application. Detailed Implementation

[0017] To facilitate understanding of this application, a more complete description will be provided below with reference to the accompanying drawings, which illustrate embodiments of the present application. However, the present application can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided so that the disclosure of this application will be thorough and complete.

[0018] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the application.

[0019] The atomic arrangement at the interface of heterojunctions (or homojunctions) is a core factor determining the physicochemical properties of crystalline composite materials. Aberration-corrected transmission electron microscopy (Ac-TEM), as an important high-resolution characterization technique, can directly observe and finely analyze the crystallographic features of heterojunction materials (or homojunctions), including the spatial arrangement of bulk atoms, precise identification of atom types, and key information such as the atomic configuration of the interface region.

[0020] However, the influence of the relative tilt angle and rotation angle of the two layers in a heterojunction on the physicochemical properties of the heterojunction interface often lacks scientific research. This is due to the lack of effective characterization methods for the tilt angle of heterojunctions. Specifically, the rotational mismatch (Twist) at the interface during crystal and interface fabrication processes affects the tilt angle (Tilt) of the crystal, and vice versa. Figure 1 As shown, the coupling between the Twist angle and the Tilt angle is difficult to measure and analyze directly, and new methods need to be developed to decouple them and accurately characterize the atomic arrangement of the interface in the process.

[0021] like Figure 1 As shown, the rotation angle ψ and tilt angle between crystals are coupled during the crystal and interface fabrication process. That is, the interface rotation misalignment operation, while causing a crystallographic rotation angle, also spontaneously introduces a crystallographic tilt angle. This figure illustrates the tilt and rotation angle processes between the top and bottom crystals caused by rotational misalignment during the crystal and interface fabrication process. Figure 1 In (a), the off-cut angle represents a common optimized structure in crystal preparation, including a schematic diagram of the crystallographic structure. Because... n 1 is the surface normal vector of the crystal, perpendicular to the surface of the crystal. n z The normal vector to the crystallographic surface caused by the off-cut is perpendicular to the crystallographic surface, as shown in yellow in the figure. It can be seen that... n 1≠ n z In fact, the angle between the two is also an off-cut angle. Figure 1 (b) is a schematic diagram of the structure after two crystals with an off-cut angle of α are fabricated together. This is the ideal case, where the rotation angle ψ between the top and bottom crystals is 0. The surfaces of the top and bottom crystals are parallel to each other, with an inclination angle of 0. However, during the crystal and interface fabrication process, misalignment will cause a rotation between the top and bottom crystals, with the axis of rotation being the normal to the crystal surface. n 1. Figure 1(c) is a schematic diagram of the actual structure of the crystal interface. We fix the bottom crystal A in place and move the top crystal B around... n 1. The structure is established by rotating counterclockwise (ψ≠0). At this point, the crystallographic surfaces of the top and bottom crystals are no longer parallel but have a tilt angle equal to... n z and n’ z Angle. Additionally, there exists a crystallographic rotation angle ψ', the magnitude of which can be considered as... n y and n’ y Angle. To easily confirm that this tilt angle exists, we can compare it with the rotation axis of crystal B. n z In this case: the crystallographic surfaces of the top and bottom crystals remain parallel, with an inclination angle of 0. Figure 1 In (d), by translating the crystallographic surfaces of the top and bottom crystals, the tilting and rotation phenomena between the surfaces can be observed directly.

[0022] Existing methods for measuring tilt and rotation angles include TEM measurement and PVTEM (Plane View TEM) measurement.

[0023] The method for measuring the crystallographic twist and tilt angles based on TEM sample rod rotation is described in [reference needed]. Figure 2 : Sample preparation cutting (FIB Cut) direction is parallel to the interface, such as Figure 2 As shown in (a), the TEM slice is lifted with its orientation perpendicular to the screen. The resulting TEM slice retains the top crystal, interface, and bottom crystal. The crystallographic rotation angle and tilt angle are as follows: Figure 2 As shown in (b). The relationship between the direction of the electron beam (E-beam) and the direction of the TEM sheet during TEM testing is as follows: Figure 2 As shown in (c), crystals A and B are positioned on the positive band axis by horizontally rotating the TEM sample rod. The angle of rotation of the TEM sample rod is the Twist angle. Similarly, crystals A and B are positioned on the positive band axis by vertically rotating the TEM sample rod. The angle of rotation of the TEM sample rod is the Tilt angle.

[0024] Sample preparation, such as Figure 2 As shown in (a), TEM thin films are prepared using mechanical methods, such as Ion Milling, Dimple, or FIB. The thin films need to include a top crystal, an interface, and a bottom crystal. The top and bottom crystals are brought into a axial position by rotating the TEM sample rod in the horizontal and vertical directions, and the rotation angle of the TEM sample rod is recorded. The rotation twist and tilt angle between the crystals are then calculated.

[0025] The above method has the following drawbacks: 1. Twist angle and Tilt angle are calculated and indirectly measured, and their accuracy is insufficient; 2. To verify the Twist and Tilt angles by recording the tilt angle of the TEM sample rod, it is necessary to carefully design the orientation and welding position of the FIB Cut sample, the placement of the TEM sheet on the sample rod, and to consider the calibration and accuracy of the TEM goniometer. 3. This method is a test after interface preparation, and it does not resolve the coupling relationship between the Twist angle and the Tilt angle, nor does it provide any guidance for improving the interface preparation process; 4. The experimental principle requires prior knowledge of the spatial positions of the Twist and Tilt axes and the design of the TEM sample preparation direction accordingly. However, the spatial positions of the Twist and Tilt axes are often unknown.

[0026] For methods of measuring the crystal interface tit angle based on small-angle grain boundaries and PVTEM technology, please refer to [link / reference]. Figure 3 : Sample preparation, such as Figure 3 As shown in (a), PVTEM thin sections are prepared using mechanical methods, Ion Milling, Dimple, or FIB techniques. The thin sections must contain a top crystal, an interface, and a bottom crystal. The thin section sample is imaged using TEM to resolve the moiré fringe pattern formed by periodic dislocations, and the average distance between adjacent TIDs (Time Dislocations) (DTIDs) is measured. The interface tilt angle θ is calculated using the Frank relation formula, combined with the material's lattice parameters. During the analysis, it is essential to ensure that the TEM image clearly shows the continuous distribution of TIDs to guarantee measurement accuracy.

[0027] This technology is based on small-angle grain boundary theory and TEM imaging. In small-angle grain boundary theory, crystal interfaces consist of periodic dislocation distributions, including Tilt Interfacial Dislocations (TIDs). In TEM imaging, TIDs form banded moiré patterns, such as... Figure 3 As shown in (b).

[0028] The above method has the following drawbacks: 1. This technique is no longer applicable when the crystal interface is not a periodic dislocation structure; 2. The measurement of spacing in PVTEM images has an inherent error of ±30%. A large error range will be directly passed to the calculation results of ψ and θ through the Frank relation, resulting in a decrease in accuracy. 3. The spacing between TWIDs and TIDs in actual samples will be uneven. This is because they are affected by factors such as the dislocation network caused by the rotation angle and the interaction between dislocations, which deviate from the ideal model assumption of the Frank relation and increase the measurement error. 4. This method is a test after interface preparation. It does not resolve the coupling relationship between the Twist angle and the Tilt angle, and has no guiding role in improving the interface preparation process.

[0029] In view of this, see Figure 4 This invention provides a method for measuring the rotation angle and tilt angle of a crystal, comprising: S1. Obtain a crystal sample containing the top crystal, bottom crystal, and interface. Establish a Cartesian coordinate system with the tangent axis of the crystal sample as the x-axis. Denote the tangent angle of the crystal sample as α. Calculate the crystallographic orientation of the crystal sample. n z and n y ; S2. Fix the bottom crystal, rotate the top crystal relative to the interface by ψ, according to... n z , n y The crystallographic orientation of the rotated top crystal is calculated using ψ. n z ’ and n y ’ ; S3. According to n z , n z ’ , n y and n y ’ The crystallographic rotation angle ψ' and crystallographic tilt angle θ' of the crystal sample are calculated. In the technical solution of this invention, the known off-cut angle α and initial crystallographic direction (…) are utilized. n z , n y Using ψ as a reference, the mathematical relationship between ψ' and θ' in crystallography and the direction vector after rotation are derived through the controllable rotation ψ of the top crystal relative interface. n z '、 n yThe actual interface rotation angle ψ is then calculated by inversion, avoiding subjective errors caused by reliance on electron diffraction or image fitting, and significantly improving the quantitative accuracy and repeatability of the measurement. The crystallographic rotation and tilt angles are measured using TEM (transmission electron microscopy), and then the coupling relationship between the crystal and interface preparation process's Twist angle, Tilt angle, and crystal surface Off angle is analyzed using modeling and mathematical calculations. First, the crystallographic Twist angle is measured through PVTEM (planar view transmission electron microscopy) sample preparation and TEM analysis. Then, two TEM slices are prepared at two different positions and directions (dual-point positions), and the corresponding rotation angles 1 and 2 are measured by TEM imaging. For cubic crystals, these two directions are generally 100° and 010°; for hexagonal crystals, these two directions are generally 1-100° and 11-20°. Then, the accurate crystallographic tilt angle is obtained by establishing a geometric model and calculating. Finally, the mathematical relationship between the interface and crystal Twist angle, Tilt angle, and crystal surface Off angle is calculated through geometric modeling and mathematical calculations.

[0030] Specifically, in some embodiments of the present invention, by preparing a TEM thin film of the crystal sample to detect the off-cut angle of the crystal sample, local off-cut angle information that completely corresponds to the subsequent rotation angle / tilt angle analysis is obtained, avoiding errors caused by macroscopic averaging measurements.

[0031] Furthermore, in step S1, n z The calculation method is as follows: Let the unit vector along the z-axis be denoted as n 1. As shown in equation (1), (1), Will n 1. Rotate α around the x-axis to obtain n z As shown in equation (2), (2).

[0032] In the technical solution of this invention, a Cartesian coordinate system is constructed with the tangent axis as the x-axis, and the rotation matrix is ​​explicitly calculated. n z This ensures the consistency between the crystallographic reference frame and the actual processing orientation of the sample; n z As a key reference direction, its analytical expression directly relates to the orientation after the top crystal is rotated. n z ’ The derivation of ψ' and the final inversion calculation of θ'.

[0033] Furthermore, in step S2, nz ’ The calculation method is as follows: Will n z Rotate ψ about the z-axis to obtain n z ’ As shown in equation (3), (3).

[0034] In the technical solution of this invention, a standard three-dimensional rotation matrix is ​​used to... n z A rigid body rotation transformation about the z-axis accurately reproduces the orientation changes of the crystal in space, avoiding orientation errors caused by simplification assumptions or projection approximations; because n z The substrate itself already contains the influence of the offset angle α (non-ideal (001) orientation), and further rotating it around the z-axis ψ can simultaneously reflect the combined geometric effect of the offset substrate and the rotation of the top crystal.

[0035] Furthermore, in step S3, the method for calculating the crystallographic tilt angle θ' is shown in equation (4). (4).

[0036] In the technical solution of the present invention, by adopting formula (4), it is clearly expressed that θ' is a nonlinear function of the shear angle α and the rotation angle ψ: when ψ = 0, θ' = 0 (no additional tilt); when ψ ≠ 0, even if the original interface is a pure shear structure, an equivalent tilt component will be induced due to the rotation of the top crystal.

[0037] In typical applications such as crystal and interface fabrication processes and two-dimensional material rotation stacking, α and ψ are usually small angles (<5°). At this time, traditional TEM or EBSD methods are difficult to distinguish between tilt and twist components. However, this invention directly calculates θ' through analytical formulas, avoiding the noise and resolution limitations in experimental characterization and achieving tilt angle evaluation with sub-degree accuracy.

[0038] Furthermore, in step S1, n y The calculation method is as follows: Let the unit vector of the y-axis be denoted as n 2. As shown in equation (5), (5); Will n 2. Rotate α around the x-axis to obtain n y As shown in equation (6), (6).

[0039] In the technical solution of this invention, after obtaining the normal direction n z Based on this, further precise calculations are made of the horizontal direction orthogonal to it. n y , making n x 、n y 、n z This constructs a right-handed orthogonal crystallographic coordinate system that conforms to the actual geometric orientation of the sample. This coordinate system accurately reflects the lattice orientation of the offset substrate, providing a complete vector basis for subsequent rotation operations and orientation difference decomposition.

[0040] Furthermore, in step S2, n y ’ The calculation method is as follows: Will n y Rotate ψ about the z-axis to obtain n y ’ As shown in equation (7), (7).

[0041] In the technical solution of this invention, after obtaining the rotated normal... n z ’ Based on this, the lateral direction is calculated simultaneously and accurately. n y ’ This ensures the consistency and integrity of rotational operations in three-dimensional space. n y ’ and n z ’ Together, they form a local orthogonal basis for the rotated crystal, providing complete directional information for subsequent orientation difference analysis; combined with known... n y 、n z 、n y ’ and n z ’ The present invention can extract the tilt component θ' and the rotation component ψ' generated by the combined effect of shearing and rotation.

[0042] Furthermore, in step S3, the method for calculating the crystallographic rotation angle ψ' is shown in equation (8). (8).

[0043] In the technical solution of this invention, ψ' is directly calculated through rigorous vector operations, eliminating the need for experimental image inversion and significantly improving measurement accuracy and repeatability. It should be noted that the crystallographic rotation angle ψ' actually observed using physical observation methods is not equal to the physical rotation angle ψ of the top crystal, but rather is modulated by the deflection angle α. When α = 0 (no deflection), ψ' = ψ; however, when α > 0, even if the top crystal rotates ψ, its effective twist angle ψ' in the crystallographic coordinate system will be either "compressed" or "enhanced." This invention introduces a method based on... n y and n y ’ The analytical formula (8) of the dot product is used to calculate the crystallographic rotation angle ψ'. This invention not only achieves high-fidelity, non-destructive, and quantitative characterization of twist behavior, but also profoundly reveals the modulation mechanism of the tangent geometry on the rotation response, laying a solid theoretical and methodological foundation for the precise control, performance prediction, and device application of advanced crystal interfaces.

[0044] Furthermore, when α > 5°, the crystallographic rotation angle ψ' is calculated as shown in equation (9). (9).

[0045] In the technical solution of this invention, when the off-cut angle α of the crystal sample is greater than 5°, which exceeds the applicable range of the small angle approximation, a more accurate correction formula is adopted, as shown in equation (9). In some advanced applications (such as epitaxial growth on a substrate with a large angle tilt, special crystal plane bonding, or artificially designed large-angle two-dimensional stacking), the off-cut angle may be as high as 15°-30°. If the small angle approximation formula is still used at this time, the ψ' error will reach several degrees, which will seriously affect the judgment of the moiré period, interface dislocation density, or band structure.

[0046] Furthermore, the magnitude of the shear angle α is detected by any one of the following methods: TEM, STEM, EBSD, CBED, Kikuchi Pattern, NBD, PED, Moiré Pattern, and 4D-STEM.

[0047] In the technical solution of this invention, through a variety of mainstream crystal orientation characterization techniques from the nanoscale (such as TEM, STEM, NBD) to the micrometer / macroscale (such as EBSD), users can flexibly choose the most suitable method according to the sample type, equipment conditions and accuracy requirements, ensuring that the acquisition of α value is both accurate and practical.

[0048] The present invention also proposes a decoupling method for crystal rotation angle and tilt angle, including the aforementioned measurement methods for crystal rotation angle and tilt angle; Step S3 is followed by: S4. The crystallographic rotation angle ψ' is obtained by PVTEM, and the crystallographic tilt angle θ' is obtained by XS-TEM. The relationship between the angles is calculated based on ψ, α, ψ' and θ'.

[0049] In the technical solution of this invention, the experimental rotation angle ψ' is obtained by using the phase contrast response of PVTEM to the twist, and the experimental tilt angle θ' is obtained by using the high-sensitivity diffraction / imaging characteristics of XS-TEM to the lattice tilt. The results are compared with the analytically calculated ψ' and θ' in step S3, forming a complete technical closed loop of "modeling-prediction-measurement-correction", which significantly improves the credibility and physical authenticity of the entire decoupling method.

[0050] The technical solution of the present invention will be further described in detail below with reference to specific embodiments. It should be understood that the following embodiments are only used to explain the present invention and are not intended to limit the present invention.

[0051] Unless otherwise specified, all materials and reagents used in the following examples are commercially available.

[0052] Example 1 This embodiment provides a method for measuring and decoupling the crystal rotation angle and tilt angle; the principle is described in [reference needed]. Figure 4 The specific steps are as follows: Establish a Cartesian coordinate system with the crystal's off-cut axis as the x-axis and the off-cut angle as α. Under ideal process conditions, ψ=0, and the crystallographic direction can be calculated. n z and n y ,like Figure 4 As shown in (a). The bottom crystal A remains stationary, while the top crystal rotates counterclockwise by an angle ψ about the z-axis

[001] . The crystallographic orientation of the top crystal changes as follows: n z ’ and n y ’ ,like Figure 4 As shown in (b), the crystallographic orientation of the bottom crystal A remains unchanged. n z , n y Unchanged. By normalizing the crystallographic orientations of the top and bottom crystals to the same coordinate system and performing analysis, the tilt angles of crystal A and crystal B are equivalent to the angle between the interface and the off-cut crystallographic surface, and also equivalent to the vector... n z and n z’ The included angle between them, such as Figure 4 As shown in (c). Similarly, the crystallographic rotation angle is equivalent to... n y and n y ’ The angle between them. The crystallographic rotation angle ψ' and tilt angle θ′ were obtained by PVTEM and XS-TEM techniques, respectively. The vectors were calculated. n z ’ and n y ’ The values ​​can decouple the relationship between the rotation angle, tilt angle, and off-cut angle.

[0053] The tilt angle between crystals A and B is equivalent to the angle between the interface plane and the off-cut crystallographic plane, and also equivalent to the vector... n z and n z ’ The angle between them. Similarly, the rotation angle between crystal A and crystal B is equivalent to the vector. n y and n y ’ The angle between the two points corresponds to the rotation angle measured by TEM. Measuring the rotation and tilt angles of crystals and interface fabrication processes using TEM technology is actually achieved by measuring crystallographic angles (crystal plane or crystal direction angles). This technique offers high accuracy and quantitative precision, but it doesn't necessarily reflect the rotation mismatch at the interface. This is because crystals have off-cut angles. This technique can decouple the correlation between rotation and tilt angles, and also decouple the correlation between rotation mismatch in the interface fabrication process and TEM detection results.

[0054] The calculations and formula derivations are as follows: (1), Rotate about the x-axis by α (the off-cut angle of the crystal). (2), Rotate the interface about the z-axis by ψ (the rotation angle of the interface). (3), Rotate about the x-axis (4), (5), Rotate ψ around the z-axis (6), The crystallographic Tilt angle θ′ satisfies (7), Semiconductor processes can achieve a crystal titer angle θ = 0. The case where the interface titer angle θ ≠ 0 due to process failures will be discussed later.

[0055] When α ≤ 5°, the crystallographic angle ψ' satisfies (8), When α > 5°, considering the rotation angle TEM test as a projected image, a correction term is added. (9).

[0056] The rotation angle ψ' and tilt angle θ' were obtained by PVTEM and XS-TEM techniques, respectively, and were calculated using formula (9). (11) Angular relationship in the decoupling interface preparation process.

[0057] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

[0058] In summary, the technical solution of this application has the following beneficial technical effects: (1) In the technical solution of the present invention, the known offcut angle α and the initial crystallographic direction are utilized ( n z , n y Using ψ as a reference, the mathematical relationship between ψ' and θ' in crystallography and the direction vector after rotation are derived through the controllable rotation ψ of the top crystal relative interface. n z '、 n y The interface rotation angle ψ of the actual crystal is then calculated by inversion, avoiding subjective errors caused by reliance on electron diffraction or image fitting, and significantly improving measurement accuracy and repeatability. (2) The crystallographic rotation and tilt angles are measured using TEM (transmission electron microscopy). Then, modeling and mathematical calculations are used to analyze the coupling relationship between the crystal and interface fabrication processes: the Twist angle, Tilt angle, and the crystal surface Off angle. First, the crystallographic Twist angle is measured using PVTEM (planar view transmission electron microscopy) sample preparation and TEM analysis. Then, two TEM slices are prepared at two different locations and orientations (dual-point positions), and the corresponding rotation angles 1 and 2 are measured using TEM imaging. For cubic crystals, these two orientations are generally 100° and 010°; for hexagonal crystals, these two orientations are generally 1-100° and 11-20°. Then, a geometric model is established, and the accurate crystallographic tilt angle is calculated. Finally, the mathematical relationship between the Twist angle, Tilt angle, and crystal surface Off angle between the interface and the crystal is calculated using geometric modeling and mathematical calculations.

Claims

1. A method for measuring the rotation angle and tilt angle of a crystal, characterized in that, include: S1. Obtain a crystal sample containing the top crystal, bottom crystal, and interface. Establish a Cartesian coordinate system with the tangent axis of the crystal sample as the x-axis. Denote the tangent angle of the crystal sample as α. Calculate the crystallographic orientation of the crystal sample. n z and n y ; S2. Fix the bottom crystal, rotate the top crystal relative to the interface by ψ, according to... n z , n y The crystallographic orientation of the rotated top crystal is calculated using ψ. n z ’ and n y ’ ; S3. According to n z , n z ’ , n y and n y ’ The crystallographic rotation angle ψ' and crystallographic tilt angle θ' of the crystal sample were calculated.

2. The method for measuring the crystal rotation angle and tilt angle according to claim 1, characterized in that, In step S1, n z The calculation method is as follows: Let the unit vector along the z-axis be denoted as n 1. As shown in equation (1), （1）， Will n 1. Rotate α around the x-axis to obtain n z As shown in equation (2), （2）。 3. The method for measuring the crystal rotation angle and tilt angle according to claim 2, characterized in that, In step S2, n z ’ The calculation method is as follows: Will n z Rotate ψ about the z-axis to obtain n z ’ As shown in equation (3), （3）。 4. The method for measuring the crystal rotation angle and tilt angle according to claim 3, characterized in that, In step S3, the crystallographic tilt angle θ' is calculated using equation (4). （4）。 5. The method for measuring the crystal rotation angle and tilt angle according to claim 1, characterized in that, In step S1, n y The calculation method is as follows: Let the unit vector of the y-axis be denoted as n 2. As shown in equation (5), （5）； Will n 2. Rotate α around the x-axis to obtain n y As shown in equation (6), （6）。 6. The method for measuring the crystal rotation angle and tilt angle according to claim 5, characterized in that, In step S2, n y ’ The calculation method is as follows: Will n y Rotate ψ about the z-axis to obtain n y ’ As shown in equation (7), （7）。 7. The method for measuring the crystal rotation angle and tilt angle according to claim 6, characterized in that, In step S3, the crystallographic rotation angle ψ' is calculated using equation (8). （8）。 8. The method for measuring the crystal rotation angle and tilt angle according to claim 7, characterized in that, When α > 5°, the crystallographic rotation angle ψ' is calculated as shown in equation (9). （9）。 9. The method for measuring the crystal rotation angle and tilt angle according to claim 1, characterized in that, The magnitude of the shear angle α is detected by any one of the following methods: TEM, STEM, EBSD, CBED, Kikuchi Pattern, NBD, PED, Moiré Pattern, and 4D-STEM.

10. A method for decoupling crystal rotation angle and tilt angle, characterized in that, Including the crystal rotation angle and tilt angle measurement method as described in any one of claims 1 to 9; Step S3 is followed by: S4. The crystallographic rotation angle ψ' is obtained by PVTEM, and the crystallographic tilt angle θ' is obtained by XS-TEM. The relationship between the angles is calculated based on ψ, α, ψ' and θ'.

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