4D-STEM DPC electric field direction calibration method

Abstract

The application discloses a 4D-STEM DPC electric field direction calibration method, and relates to the technical field of electron microscopy. The method comprises the following steps: obtaining original 4D-STEM data, generating an ADF image and calculating an electric field vector field; extracting a set of atomic column center positions from the ADF image; constructing a candidate set of discrete coordinate transformations, selecting an optimal flip transformation based on the radial consistency of the electric field near the atoms; performing a coarse-to-fine search on the rotation angle through a target function to determine an optimal rotation angle; eliminating the 180-degree direction ambiguity by calculating the sign of the weighted charge density indicator in the atomic neighborhood; and calibrating the original centroid shift field by applying the optimal flip transformation and the final rotation angle, and outputting the calibrated electric field, charge density and potential distribution. The application can automatically identify and correct the coordinate axis flip and mirror transformation, eliminate the 180-degree direction ambiguity, output the unique and correct electric field direction, and be suitable for the general 4D-STEM DPC analysis process.

Description

Technical Field

[0001] This invention relates to the field of electron microscopy, specifically to a 4D-STEM DPC electric field direction calibration method. Background Technology

[0002] 4D-STEM DPC (including CoM / iCoM) technology calculates the two-dimensional projected electric field vector field inside the sample by analyzing the centroid shift of the diffraction pattern formed after the electron beam passes through the sample. This method has been widely used in the characterization of polarization direction of ferroelectric materials, measurement of interface charge distribution, analysis of domain wall charge, and quantitative study of local electric field distribution.

[0003] In actual data processing workflows, it is usually necessary to precisely align the scanning coordinate system (fast scan direction and slow scan direction) with the detector coordinate system (pixel coordinates on the camera plane). This alignment process includes compensating for overall rotation, pixel axis orientation, and possible coordinate transformations.

[0004] However, existing technologies have the following defects and shortcomings: First, many existing processes assume only a global rotational deviation between the scanning and detector coordinate systems and achieve alignment through manual observation, empirical rotation, or angle searches based on electric field strength statistics. The evaluation metrics used in these methods are insensitive to the overall inversion of the electric field, which can easily lead to two equivalent solutions that differ by 180 degrees in the search results, making it impossible to uniquely determine the final direction of the electric field.

[0005] Secondly, in actual experiments and data processing, common issues such as coordinate axis reversal, image mirroring, or component exchange (e.g., transposition during data reading, differences in the definition of fast scan direction, differences in detector coordinate conventions, etc.) are geometric relationships that alter the chirality of the coordinate system. Simple rotation models cannot cover and correct for these chiral changes. In such cases, the evaluation quantities related to the electric field rotation direction mathematically change sign with the change in chirality. Without sign unification, the scores between different candidate results will be incomparable, leading to the failure of automated calibration processes or obtaining incorrect electric field direction chirality.

[0006] Therefore, there is an urgent need for an electric field direction calibration method that can automatically identify and correct coordinate axis flips and mirror transformations, while eliminating ambiguity in the 180-degree direction, to ensure the accuracy and uniqueness of 4D-STEM DPC electric field analysis. Summary of the Invention

[0007] In view of this, the present invention provides a 4D-STEM DPC electric field direction calibration method, which can solve the overall geometric correspondence between the scanning coordinate system and the detector coordinate system under conditions of no standard and little human intervention, and output a unique and directional electric field result and calibration parameters.

[0008] This invention provides a 4D-STEM DPC electric field direction calibration method, comprising the following steps: S1. Obtain the raw 4D-STEM data cube. ;in, For scanning spatial coordinates; These are the coordinates in the diffraction space. S2. Generate an ADF image from the original 4D-STEM data cube. The centroid offset is calculated for each scanning position, thereby obtaining the electric field vector field. ; S3, from the ADF image Extract the set of atomic column center positions ;in, ; The total number of atomic columns; S4. Construct a candidate set for discrete coordinate transformation. For each candidate transformation The global score is calculated based on the radial consistency of the electric field near the center of the atomic pillar. And select the candidate transformation with the highest score. As the optimal flip transformation, for the electric field vector field Perform a flip correction to obtain the corrected electric field vector field. ; S5, Rotation angle of candidate The electric field vector field after flipping and correction Rotation to obtain Calculate its divergence Curl and in the area of ​​interest The standard deviations of the two are calculated internally, and the optimal rotation angle is obtained by optimizing the objective function. ; S6. Based on the center position of the atomic column and the ADF image Calculate the sign decision factor ; Based on the sign determination quantity The sign determines the final rotation angle. ; S7. Perform the optimal flip transformation. and the final rotation angle It is applied to the original centroid offset field to obtain the calibrated centroid offset field, and outputs the calibrated electric field, charge density and potential distribution based on the calibrated centroid offset field.

[0009] Compared with the prior art, the present invention has the following beneficial effects: (1) It can automatically identify and correct the mirror flip, coordinate axis reversal and component exchange between the scanning coordinates and the detector coordinates, and avoid misjudgment of electric field direction caused by chiral error.

[0010] (2) It can eliminate equivalent solutions that are 180 degrees out of phase and output a uniquely determined electric field direction.

[0011] (3) It is applicable to the general DPC / iCoM electric field analysis process and is easy to integrate into the existing 4D-STEM data processing software pipeline.

[0012] (4) No standard samples or manual intervention are required, achieving fully automated calibration. Attached Figure Description

[0013] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained from these drawings without creative effort.

[0014] Figure 1 This is a flowchart illustrating a 4D-STEM DPC electric field direction calibration method according to an embodiment of the present invention. Detailed Implementation

[0015] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0016] like Figure 1 As shown, the present invention provides a 4D-STEM DPC electric field direction calibration method, characterized by comprising the following steps: Step S1: Obtain the raw 4D-STEM data cube ;in, For scanning spatial coordinates; These are the coordinates in the diffraction space.

[0017] Specifically, this invention obtains raw 4D-STEM data through 4D-STEM (four-dimensional scanning transmission electron microscopy) experiments. The raw 4D-STEM data is a four-dimensional intensity tensor (or a four-dimensional data cube), which can be expressed as: in, These are real-space (scanning space) coordinates. The electron probe scans the sample surface, forming a... A two-dimensional grid. Each scan point corresponds to a specific location on the sample. These are reciprocal space (diffraction space) coordinates, corresponding to the pixel positions of the detector. The detector includes... Each pixel records the intensity of electrons scattered in that direction.

[0018] Therefore, the original 4D-STEM data is a dimensional [database name missing]. The four-dimensional data records the complete intensity distribution of the diffraction pattern at each scanning point. As an example, assuming 10×10 points (100 positions) are scanned on the sample and the detector is 256×256 pixels, then the raw 4D-STEM data is a four-dimensional array with a shape of 10×10×256×256.

[0019] Furthermore, the raw 4D-STEM data can be stored in a computer in the following form: Among them, index Corresponding scanning position ;index Corresponding detector pixels ; ; ; ; .

[0020] Step S2: Generate ADF image from the original 4D-STEM data cube The centroid offset is calculated for each scanning position, thereby obtaining the electric field vector field. .

[0021] In one alternative implementation, in step S2, an ADF image is generated. Specifically: Select a ring-shaped area on the detector. For each scan position The diffraction intensity in the annular region Integrating within: .

[0022] Specifically, an ADF (Annular Dark Field) image is generated from the original 4D-STEM data as a reference for subsequent atomic position localization. (Annular region) Typically, a mid-to-high scattering angle range is selected to clearly image the atomic column contrast. The generated... It is a two-dimensional image, in which bright spots roughly correspond to the positions of atomic pillars in the sample.

[0023] In one optional implementation, step S2, the calculation of the centroid offset, includes: S21. Calculate the average diffraction pattern .

[0024] Specifically, regarding the scanning dimensions Averaging yields the average diffraction pattern PACBED. .

[0025] The formula for calculating the average diffraction pattern is as follows: S22, according to The intensity-weighted centroid positioning of the BF disk center And fit the disk radius Constructing a BF mask .

[0026] S23. For each scan position Calculate the zeroth moment and first moment , : in, , ; The detector pixel coordinate grid; , These are the displacement coordinates relative to the center of the BF disk; S24. Based on S23, obtain the centroid offset of each scan position: S25. Based on S24, the electric field vector field is obtained. : in, k This is a proportionality coefficient related to the experimental parameters.

[0027] Specifically, using the average diffraction pattern PACBED Intensity-weighted centroid estimation of bright field (BF) disk center .

[0028] Among them, the center of the Bright Field (BF) disc The calculation expression is: in, For detector pixel coordinate grid, ; Based on PACBED The gradient magnitude is used to locate the boundary of the BF disk, and the radius of the BF disk is obtained by fitting a circular model. ; Constructing a BF mask : For each scan position iCoM is calculated only within the BF mask (denominator is the sum of BF intensities, numerator is the first moment).

[0029] Define the displacement coordinates relative to the center of the BF disk as: Calculate the zeroth moment and first moment , : And in the numerical implementation, for those that are too small join in Prevent division by zero.

[0030] The final centroid offset (in detector pixels) for each scan point is: The resulting centroid offset Subsequently, because the electric field and the centroid shift direction are opposite, their values ​​differ by a coefficient. k Thus, the electric field vector field is obtained. : .

[0031] Step S3: From the ADF image Extract the set of atomic column center positions ;in, ; This represents the total number of atomic columns.

[0032] Step S4: Construct a candidate set of discrete coordinate transformations For each candidate transformation The global score is calculated based on the radial consistency of the electric field near the center of the atomic pillar. And select the candidate transformation with the highest score. As the optimal flip transformation, for the electric field vector field Perform a flip correction to obtain the corrected electric field vector field. .

[0033] In an alternative implementation, in step S4, the discrete coordinate transformation candidate set... Including but not limited to: identity transformations , x Axis Reverse Transformation and y Axial direction ; The representation of each candidate transformation is as follows: .

[0034] In an alternative implementation, in step S4, for each candidate transformation Calculate the global score Specifically, it includes: S41, Regarding the electric field vector field Applying a candidate transformation yields ; The transformation rules are as follows: S42, For each atomic center Multiple sampling points are selected within a surrounding annular region (avoiding the central pixel and limiting the radius to reduce interference from neighboring atoms). Calculate the projection of the electric field in the radial direction. : in, The electric field vector field after candidate transformation at the sampling point The value at; It is the center of the ring-shaped area.

[0035] S43. Based on S42, calculate the consistency score for all atomic columns. : in, The number of sampling points in the target atomic column region; Specifically, when the electric field exhibits a uniform radial pattern near the atomic column, A higher degree of consistency should be exhibited on this ring. Based on this, the consistency score of a single-atom column is defined as follows: in, The number of sampling points in the neighborhood of the target atomic column. .

[0036] S44. The global score is obtained by weighted summation of the consistency scores of all atomic pillars. : Among them, weight Intensity at the atom center from the ADF image It may be a constant.

[0037] In an optional implementation, in step S4, the corrected electric field vector field The calculation steps include: S45. Select the candidate transform with the highest global score. As the optimal flip transformation: This step determines whether there is an exchange of orientation or components on the coordinate axes. If the confirmed optimal flip transformation corresponds to a change in chirality ( ,For example x shaft or y If the axis is reversed, the quantities related to the direction of electric field rotation will mathematically undergo a sign reversal. Subsequent calculations will require corresponding reversal processing.

[0038] S46, Regarding the electric field vector field The corrected electric field vector field is obtained by using the optimal flip transformation. : .

[0039] Step S5: Select candidate rotation angles The electric field vector field after flipping and correction Rotation to obtain Calculate its divergence Curl and in the area of ​​interest The standard deviations of the two are calculated internally, and the optimal rotation angle is obtained by optimizing the objective function. .

[0040] In one optional implementation, step S5 specifically includes: S51, regarding the candidate rotation angle For the electric field vector field after flip correction Perform a complete rotation to obtain : .

[0041] S52. Based on step S51, calculate divergence Curl ; in, and They are respectively exist x direction and y The directional component.

[0042] Specifically, the definition is based on the rotation angle. The electric field vector field after gradient tensor : in, and They are respectively exist x direction and y Component of direction; Subsequently Construct two types of key scalars: Divergence: Curl (2D curl) Quantity / vorticity): .

[0043] S53. In the Region of Interest (ROI) Calculate the divergence separately. Curl Standard deviation: in, Indicates the region of interest Internal standard deviation; The standard deviation of curl; The standard deviation of the divergence; S54. The optimal rotation angle is obtained by optimizing the objective function. ; The expression for the objective function is: in, Pick .

[0044] Specifically, when the rotation angle is correct, the electric field is closer to a conservative field. Smaller and smoother), while the charge structure is clearer ( The contrast is stronger, that is, the changes are more "sharp", therefore a joint objective function is adopted: in, Pick .

[0045] In an alternative implementation, in step S5, the optimal rotation angle is searched. Employ a multi-round search strategy, progressing from coarse to fine: exist Within the interval, a coarse search is performed with a preset first step length to locate the minimum region of the objective function; Within the minimum region, a fine search is performed with a preset second step size to obtain the optimal rotation angle. The second step length is less than the first step length.

[0046] Step S6: Based on the center position of the atomic pillar and ADF image Calculate the sign decision factor ; Based on the sign determination quantity The sign determines the final rotation angle. ; In one optional implementation, step S6 specifically includes: S61, Based on the center position of the atomic column and ADF image Calculate the conformity judgment quantity : in, At the atomic center The local neighborhood; ; It is the vacuum permittivity; S62. Determining the sign based on the quantity The sign determines the final rotation angle. : .

[0047] Specifically, in solving rotation angles based on curl / divergence, there are... Ambiguity. For any angle ,have Therefore, the charge indication (projection) is: satisfy: Since the aforementioned angle evaluation index is based on standard deviation (std) and is not sensitive to overall sign reversal, additional criteria are needed to uniquely determine the final direction.

[0048] Therefore, by utilizing the atomic column position information provided by the ADF image, the symbol is determined only at the center of the atomic column and within its defined local neighborhood. The specific procedure is as follows: Extracting the center of the atomic column from the ADF image And define a local neighborhood mask around each center. (For example, with) Center and radius (small circular area).

[0049] Constructing sign decision variables: when When the value is 0, it means that the charge density indicator is generally positive within the ADF atomic pillar mask region; otherwise, it is generally negative. Because ,have: Therefore, this can be used to eliminate ambiguity and determine the final perspective: .

[0050] This step relies solely on the atomic pillar centers and their local neighborhoods provided by the ADF, along with the corresponding ADF intensity weights, thus simplifying the implementation while maintaining physical consistency.

[0051] Step S7: Perform the optimal flip transformation. and final rotation angle It is applied to the original centroid offset field to obtain the calibrated centroid offset field, and outputs the calibrated electric field, charge density and potential distribution based on the calibrated centroid offset field.

[0052] Specifically, in determining the optimal flip transformation With final rotation angle Then, the original centroid offset (CoM) field is... Mapping to a unified coordinate system yields the calibrated centroid offset field. .

[0053] .

[0054] in, The final CoM vector field used for subsequent calculations of the projected electric field, potential distribution, and charge density.

[0055] 1. Output calibrated electric field: Given the calibrated CoM field The electric field components are defined in code as follows: The electric field amplitude is: in: For reciprocal space calibration ( / pixel), For electron velocity, The thickness of the sample is in meters (m). is Planck's constant. It represents the elementary charge.

[0056] 2. Output calibrated charge density: Given the corrected CoM field The code first calculates the combination of first derivatives: The projected charge density (expressed as electron number density, unit e / A) is obtained according to Gauss's law. 2 ): in, For real space calibration (m / pixel); For reciprocal space calibration ( / pixel); It is Planck's constant; It is the elementary charge; For electron velocity.

[0057] 3. Output potential distribution after calibration: Given the corrected CoM field and according to Solving the Poisson equation in Fourier space is therefore equivalent to performing the inverse operation on the gradient.

[0058] Let the discrete frequency be: And define: The intermediate potential is: Finally, multiply by the potential prefactor. The potential distribution is obtained (in V): in, For reciprocal space calibration ( / pixel), For real space calibration (m / pixel). For electron velocity, Thickness (m) is Planck's constant. It represents the elementary charge.

[0059] Although embodiments of the invention have been described in conjunction with the accompanying drawings, those skilled in the art can make various modifications and variations without departing from the spirit and scope of the invention, and such modifications and variations all fall within the scope defined by the appended claims.

Claims

1. A method for calibrating the electric field direction of a 4D-STEM DPC, characterized in that, Includes the following steps: S1. Obtain the raw 4D-STEM data cube. ;in, For scanning spatial coordinates; These are the coordinates in the diffraction space. S2. Generate an ADF image from the original 4D-STEM data cube. The centroid offset is calculated for each scanning position, thereby obtaining the electric field vector field. ; S3, from the ADF image Extract the set of atomic column center positions ;in, ; The total number of atomic columns; S4. Construct a candidate set for discrete coordinate transformation. For each candidate transformation The global score is calculated based on the radial consistency of the electric field near the center of the atomic pillar. And select the candidate transformation with the highest score. As the optimal flip transformation, for the electric field vector field Perform a flip correction to obtain the corrected electric field vector field. ; S5, Rotation angle of candidate The electric field vector field after flipping and correction Rotation to obtain Calculate its divergence Curl and in the area of ​​interest The standard deviations of the two are calculated internally, and the optimal rotation angle is obtained by optimizing the objective function. ; S6. Based on the center position of the atomic column and the ADF image Calculate the sign decision factor ; Based on the sign determination quantity The sign determines the final rotation angle. ; S7. Perform the optimal flip transformation. and the final rotation angle It is applied to the original centroid offset field to obtain the calibrated centroid offset field, and outputs the calibrated electric field, charge density and potential distribution based on the calibrated centroid offset field.

2. The method according to claim 1, characterized in that, In step S2, an ADF image is generated. Specifically: Select a ring-shaped area on the detector. For each scan position The diffraction intensity in the annular region Integrating within: 。 3. The method according to claim 1, characterized in that, In S2, the calculation steps for the centroid offset include: S21. Calculate the average diffraction pattern ; S22, according to The intensity-weighted centroid positioning of the BF disk center And fit the disk radius Constructing a BF mask ; S23. For each scan position Calculate the zeroth moment and first moment , : in, , ; The detector pixel coordinate grid; , These are the displacement coordinates relative to the center of the BF disk; S24. Based on S23, obtain the centroid offset of each scan position: S25. Based on S24, the electric field vector field is obtained. : in, k This is a proportionality coefficient related to the experimental parameters.

4. The method according to claim 1, characterized in that, In S4, the discrete coordinate transformation candidate set Including but not limited to: identity transformations , x Axis Reverse Transformation and y Axial direction ; The representation of each candidate transformation is as follows: 。 5. The method according to claim 1, characterized in that, In S4, for each candidate transformation Calculate the global score Specifically, it includes: S41, Regarding the electric field vector field Applying a candidate transformation yields ; The transformation rules are as follows: ； S42, For each atomic center Multiple sampling points were selected within the surrounding annular area. (by Center and radius (Uniform sampling on the ring, or sampling within a small window) to calculate the projection of the electric field in the radial direction. : in, The electric field vector field after candidate transformation at the sampling point The value at; The center of the ring-shaped area; S43. Based on S42, calculate the consistency score for all atomic columns. : in, The number of sampling points in the target atomic column region; S44. The global score is obtained by weighted summation of the consistency scores of all atomic pillars. : Among them, weight Intensity at the atom center from the ADF image It may be a constant.

6. The method according to claim 5, characterized in that, In S4, the corrected electric field vector field The calculation steps include: S45. Select the candidate transform with the highest global score. As the optimal flip transformation: ； S46, Regarding the electric field vector field The corrected electric field vector field is obtained by using the optimal flip transformation. : 。 7. The method according to claim 1, characterized in that, S5 specifically includes the following steps: S51, regarding the candidate rotation angle For the electric field vector field after flip correction Perform a complete rotation to obtain : ； S52. Based on S51, calculate divergence Curl ; in, and They are respectively exist x direction and y Component of direction; S53, In the region of interest Calculate the divergence separately. Curl Standard deviation: in, Indicates the region of interest Internal standard deviation; The standard deviation of curl; The standard deviation of the divergence; S54. The optimal rotation angle is obtained by optimizing the objective function. ; The expression for the objective function is: in, Pick .

8. The method according to claim 7, characterized in that, In step S5, the optimal rotation angle is searched. Employ a multi-round search strategy, progressing from coarse to fine: exist Within the interval, a coarse search is performed with a preset first step length to locate the minimum region of the objective function; Within the minimum region, a fine search is performed with a preset second step size to obtain the optimal rotation angle. The second step length is less than the first step length.

9. The method according to claim 1, characterized in that, S6 specifically includes: S61, Based on the center position of the atomic column and the ADF image Calculate the conformity judgment quantity : in, At the atomic center The local neighborhood; ; It is the vacuum permittivity; S62. Determining the sign based on the quantity The sign determines the final rotation angle. : 。

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