Solid-state spin coherent simulation method considering intragranular magnetic field

Abstract

The invention relates to a solid-state spin coherent simulation method considering an intragranular magnetic field, and belongs to the field of quantum information science. Comprising the following steps: constructing super-unit cells, performing natural isotope replacement on the super-unit cells, and introducing defects; performing structure optimization on the super-unit cells with the introduced defects, obtaining magnetization density distribution and spin magnetic moment distribution of the super-unit cells and magnetic fields of core spin positions and center spin positions of the super-unit cells, and constructing a spin data set; obtaining a simulation spin data set according to the bath cut-off radius, obtaining a cluster set according to the calculation order and the dipole cut-off radius, and performing spin coherence property simulation based on the cluster set and the central spin to obtain Hamiltonian and coherence functions of the central spin and the cluster; and obtaining non-uniform and uniform dephasing time based on the coherence function and a corresponding fitting formula. According to the method, the complexity of the influence of the intragranular magnetic field on spin is considered, the construction of the intragranular magnetic field is realized, and the accuracy of spin coherence simulation of the solid-state spin system material with complex atom defects is improved.

Description

Technical Field

[0001] The present invention relates to the field of quantum information science, and in particular to a solid-state spin coherence simulation method taking into account the intra-crystal magnetic field. Background Art

[0002] Cluster-correlation expansion (CCE) theory is currently one of the most effective methods for simulating the coherence time of solid-state spin materials, facilitating the exploration of new quantum materials. As the performance of existing solid-state spin systems still cannot meet the demands of increasingly complex applications, color centers with complex atomic and surface defects have recently become a research hotspot.

[0003] Complex atomic defects increase the complexity of the effects of intracrystalline magnetic fields on spins, necessitating the full consideration and simulation of the impact of variations in the intracrystalline magnetic field on the spin coherence properties of solid-state spin systems. However, existing cluster correlation expansion methods have yet to implement algorithms that consider intracrystalline magnetic fields. Therefore, providing a solid-state spin coherence simulation method that considers intracrystalline magnetic fields is of great significance for improving the simulation accuracy of spin coherence properties of materials with complex atomic defect color centers and exploring quantum materials with long coherence times. This approach holds broad application prospects in quantum computing and quantum communications. Summary of the Invention

[0004] In view of the above analysis, the present invention aims to provide a solid-state spin coherence simulation method that takes into account the intracrystalline magnetic field, in order to solve the problem that the existing cluster correlation expansion method does not consider the algorithm implementation of the intracrystalline magnetic field, resulting in low simulation accuracy of the spin coherence properties of solid-state spin system materials. The present invention provides a solid-state spin coherence simulation method that takes into account the intracrystalline magnetic field, and the method comprises the following steps:

[0005] constructing an initial unit cell and expanding it into a supercell; performing natural isotope substitution in the supercell; introducing defects into the supercell;

[0006] Based on the first principles, the structure of the defect-introduced supercell is optimized; the magnetization density distribution data of the structurally optimized supercell is obtained by static electronic structure calculation; and the spin magnetic moment distribution of the structurally optimized supercell is calculated based on the magnetization density distribution data;

[0007] According to the spin magnetic moment distribution, the magnetic field induction intensity of the core spin position and the central spin position of the supercell after structural optimization is calculated to construct a spin data set;

[0008] A simulated spin data set is obtained according to the bath cutoff radius, and corresponding nuclear spins are selected from the simulated spin data set according to the calculation order and the dipole cutoff radius to obtain a cluster set. Spin coherence properties are simulated based on the cluster set and the central spin to obtain the Hamiltonian and coherence function of the central spin and the cluster; the inhomogeneous dephasing time is obtained based on the coherence function and the inhomogeneous dephasing time fitting formula, and the uniform dephasing time is obtained based on the coherence function and the uniform dephasing time fitting formula.

[0009] Furthermore, the spin coherence property simulation based on the cluster set and the central spin to obtain the Hamiltonian and coherence function of the central spin and the cluster includes:

[0010] Based on the dipole interaction between the central spin and the nuclear spin, the hyperfine tensor between the central spin and the nuclear spin and the interaction tensor between the nuclear spins are calculated;

[0011] Initialize the central spin density matrix; calculate the central spin Hamiltonian based on the central spin operator, nuclear spin operator and hyperfine tensor; obtain the central spin time evolution operator sequence based on the central spin Hamiltonian and time series; obtain the central spin density matrix evolution result based on the central spin time evolution operator sequence and the central spin density matrix; obtain the central spin coherence function based on the central spin density matrix evolution result;

[0012] The cluster density matrix is ​​obtained based on the central spin density matrix; the Hatch density of each cluster is calculated based on the central spin operator, nuclear spin operator, hyperfine tensor and interaction tensor; the time evolution operator sequence of the cluster is obtained based on the Hatch density and time series of the cluster; the density matrix evolution result of the cluster is obtained based on the time evolution operator sequence of the cluster and the cluster density matrix, and the coherence function of the cluster is obtained based on the density matrix evolution result of the cluster; the coherence function of the cluster set is obtained based on the coherence function of each cluster.

[0013] Furthermore, the interaction tensor between nuclear spins is calculated by the following formula:

[0014]

[0015] Among them, γ i is the magnetic gyroscopic ratio of nuclear spin i, γ j is the magnetic gyroscopic ratio of nuclear spin j, μ0 is the vacuum permeability, is the reduced Planck constant, E is the 3*3 unit matrix, is the distance vector between nuclear spin i and nuclear spin j, represents the outer product.

[0016] Furthermore, the hyperfine tensor between the central spin and the nuclear spin is calculated by the following formula:

[0017]

[0018] Among them, γ s is the magnetic gyro ratio of the central spin.

[0019] Furthermore, the initializing central spin density matrix includes:

[0020] generating a central spin initial state, wherein the central spin initial state includes a central spin initial |0> state probability amplitude α and a central spin initial |1> state probability amplitude β;

[0021] Constructing the central spin Pauli matrix and central spin density matrix based on the central spin initial state;

[0022] A central spin rotation matrix is ​​initialized based on the central spin Pauli matrix, and α and β values ​​and a central spin density matrix are updated based on the central spin rotation matrix.

[0023] Furthermore, the central spin Hamiltonian is calculated by the following formula:

[0024]

[0025] in, is the central spin Hamiltonian, B is the magnetic field vector including the intracrystalline magnetic field and the external magnetic field, D is the zero-field splitting constant, S = [S x ,S y ,S z ] is the central spin operator, is the nuclear spin operator, I i is the nuclear spin operator of the ith nuclear spin, γ s is the magnetic gyroscopic ratio of the central spin, A i is the hyperfine tensor between the central spin and the i-th nuclear spin.

[0026] Furthermore, the central spin time evolution operator sequence includes central spin free evolution decay and central spin Hahn echo, and the central spin free evolution decay is calculated by the following formula:

[0027]

[0028] Where i represents a complex number, h is Planck's constant, and t is a time series;

[0029] The central spin Hahn echo is calculated using the following formula:

[0030]

[0031] Among them, σ x represents the x-component of the central spin Pauli matrix.

[0032] Furthermore, the H density of the cluster is calculated by the following formula:

[0033]

[0034] Among them, γ i is the magnetic gyroscopic ratio of nuclear spin i, J ij is the interaction tensor between the ith nuclear spin and the jth nuclear spin, A i 、A a are the hyperfine tensors between the central spin and the i-th and a-th nuclear spins, respectively. i , I j , I a are the nuclear spin operators of the ith, jth, and ath nuclear spins, respectively, a > is the expected value of the cluster outer nuclear spin, and C represents the cluster.

[0035] Furthermore, the coherence function of the cluster is obtained by the following formula:

[0036]

[0037] in, represents the conjugate transpose of α, m C Represents the density matrix evolution result of the cluster.

[0038] Furthermore, the spin magnetic moment distribution of the supercell after structural optimization is calculated by the following formula:

[0039]

[0040] Among them, g e is the electron g factor, which is approximately 2.002319; μ B is the Bohr magneton; m is the magnetization density distribution; V FFT is the total number of FFT grids; is the spin angular momentum vector;

[0041] The magnetic field induction intensity of the core spin position and the central spin position of the supercell after structural optimization is calculated by the following formula:

[0042]

[0043] Where μ0 is the vacuum permeability, is the spin magnetic moment distribution, is the vector from the spin magnetic moment to the supercell core spin position or the central spin position, and r is the distance from the spin magnetic moment to the supercell core spin position or the central spin position.

[0044] Compared with the prior art, the present invention can achieve at least one of the following beneficial effects:​

[0045] 1. The present invention solves the magnetic moment distribution based on the magnetization density distribution, and then calculates the magnetic field of the supercell core spin position and the central spin position, providing a basis for improving the accuracy of simulating the spin coherence properties of materials with complex atomic defect color centers.

[0046] 2. The present invention introduces defects by performing stacking faults on the supercell and removing the corresponding atoms, and optimizes the structure through first-principles calculations, providing more accurate atomic coordinate information for cluster-related expansion simulations, and laying a foundation for improving the accuracy of simulating the spin coherence properties of materials with complex atomic defect color centers.

[0047] 3. The present invention provides a solid-state spin coherence simulation method that takes into account the intracrystalline magnetic field, takes into account the complexity of the influence of the intracrystalline magnetic field on the spin, realizes the construction of the intracrystalline magnetic field, and improves the accuracy of the simulation results of the spin coherence properties of solid-state spin system materials with complex atomic defects.

[0048] 4. The present invention implements a cluster correlation expansion algorithm that takes into account the intracrystalline magnetic field, which can realize the simulation of doublet and triplet spin coherence properties and has broad application prospects.

[0049] 5. The present invention screens the spin magnetic moment based on the preset boundary of the supercell. When calculating the magnetic field induction intensity of the supercell core spin position and the central spin position, the spin magnetic moment of electrons closer to the position is screened for calculation. While meeting the calculation accuracy, the calculation time is greatly reduced and the efficiency is improved.

[0050] In the present invention, the above-mentioned technical solutions can be combined with each other to achieve more preferred combinations. Other features and advantages of the present invention will be described in the following description, and some advantages will become apparent from the description or be learned through practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the contents particularly pointed out in the description and drawings. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] The accompanying drawings are only for the purpose of illustrating particular embodiments and are not to be considered limiting of the present invention. Like reference symbols denote like parts throughout the drawings.

[0052] Figure 1 A flow chart of a solid-state spin coherence simulation method considering intracrystalline magnetic fields provided by an embodiment of the present invention;

[0053] Figure 2 A schematic diagram of a silicon carbide stacking fault color center configuration used in simulations of an embodiment of the present invention;

[0054] Figure 3Schematic diagram of cluster sets corresponding to different calculation orders according to an embodiment of the present invention;

[0055] Figure 4 The embodiments of the present invention provide the simulated silicon carbide stacking fault color center coherence function results, as well as the uniform dephasing time and non-uniform dephasing time fitting results. DETAILED DESCRIPTION

[0056] The preferred embodiments of the present invention will be described in detail below in conjunction with the accompanying drawings, wherein the accompanying drawings constitute a part of this application and are used together with the embodiments of the present invention to illustrate the principles of the present invention, and are not used to limit the scope of the present invention.

[0057] A specific embodiment of the present invention discloses a solid-state spin coherence simulation method taking into account the intracrystalline magnetic field. Figure 1 As shown, the method includes the following steps:

[0058] Step S1, constructing an initial unit cell and expanding it into a supercell; performing natural isotope substitution in the supercell; introducing defects into the supercell;

[0059] Step S2, performing structural optimization on the defect-introduced supercell based on first principles; obtaining magnetization density distribution data of the structurally optimized supercell by static electronic structure calculation; and calculating the spin magnetic moment distribution of the structurally optimized supercell based on the magnetization density distribution data;

[0060] Step S3, calculating the magnetic field induction intensity of the core spin position and the central spin position of the supercell after structural optimization according to the spin magnetic moment distribution, and constructing a spin data set;

[0061] Step S4, obtaining a simulated spin data set according to the bath cutoff radius, selecting corresponding nuclear spins in the simulated spin data set according to the calculation order and the dipole cutoff radius to obtain a cluster set, performing spin coherence property simulation based on the cluster set and the central spin to obtain the Hamiltonian and coherence function of the central spin and the cluster; obtaining the inhomogeneous dephasing time based on the coherence function and the inhomogeneous dephasing time fitting formula, and obtaining the uniform dephasing time based on the coherence function and the uniform dephasing time fitting formula.

[0062] Specifically, in step S1, constructing an initial unit cell and expanding it into a supercell includes:

[0063] Determine the atomic types of the crystal;

[0064] Setting crystal lattice parameters and atomic coordinates based on the atomic type, and constructing an initial unit cell according to the crystal lattice parameters and atomic coordinates;

[0065] The initial unit cell is repeatedly expanded to obtain a supercell.

[0066] It should be noted that the crystal lattice parameters include the basis vectors of the unit cell and the angles between the crystal axes, which are used to describe the size and shape of a single unit cell; the basis vectors of the unit cell are the basic vectors that describe the crystal structure, usually expressed as three vectors They form the edges of the unit cell. Atomic coordinates are used to determine the positions of individual atoms within the unit cell. In simulation software, the size and shape of the initial unit cell are determined in three-dimensional space based on the set lattice parameters. Based on the atomic coordinates, the atoms are placed at corresponding positions within the initial unit cell to create the initial unit cell model. This initial unit cell is then repeatedly expanded to create a supercell model.

[0067] For example, assuming the crystal is cubic, use the variable The variables α, β, and γ represent the angles between the crystal axes b and c, a and c, and a and b, respectively. The angle between the crystal axes is set to α = β = γ = 90°. The atomic coordinates are expressed as fractional coordinates. Assuming that the fractional coordinates of an atom are (0.25, 0.25, 0.25), it means that the atom is located at a quarter position in all three coordinate axes of the unit cell.

[0068] Preferably, VESTA software is used for initial unit cell and supercell modeling.

[0069] Furthermore, performing natural isotope replacement in the supercell comprises:

[0070] determining naturally occurring isotopes of said atomic type and their concentrations;

[0071] Calculate the amount of isotopes of each type of atom based on the naturally occurring isotopes of the atom type and their concentrations;

[0072] The same number of corresponding atoms is randomly replaced in the supercell according to the isotope number of each type of atoms.

[0073] Specifically, the natural isotopes and concentrations of each atomic type are determined. For each atomic type, i.e., each class of atoms, the number of natural isotopes of each class of atoms is calculated by multiplying the number of atoms of that class in the supercell by the concentration of its natural isotope. Based on the number of natural isotopes of each class of atoms, the same number of atoms of that class in the supercell are randomly replaced with their natural isotopes.

[0074] It should be noted that when natural isotope substitution is performed, the atoms in the supercell are replaced with natural isotopes with spin.

[0075] For example, using VESTA simulation software, assuming that the crystal is silicon carbide 4H-SiC, its atomic types include carbon atoms C12 and silicon atoms Si28, and the crystal type is hexagonal. Set the basis vector of the initial unit cell to The angles between the crystal axes are set to α = β = 90°, γ = 120°, and the initial unit cell is Repeat the expansion in the direction 5, 5, and 5 times, respectively, to obtain a supercell; in which each carbon atom C12 and each silicon atom Si28 has its own number. The natural isotope of carbon atom C12 is determined to be C13, with a concentration of 1.1 at.%, and the natural isotope of silicon atom Si28 is determined to be Si29, with a concentration of 4.67 at.%. Assuming that the number of C12 and Si28 in the supercell is both 500, using 500*1.1 at.%, the number of C13 is approximately 6, and using 500*4.67 at.%, the number of Si29 is 23. Based on the number of C13, 6, Python is used to generate a first random number sequence with 6 elements, where each element has a value between 1 and 500 and each element has a different value. The value of the element represents the number of the carbon atom. In the supercell, replace the C12 with C13 for each element with the same number in the first random number sequence. Similarly, a second random number sequence with 23 elements is generated according to the number of Si29, 23, where the value range of each element is between 1 and 500 and the value of each element is different. In the supercell, Si28 with the same number corresponding to each element in the second random number sequence is replaced by Si29.

[0076] It should be noted that the unit at.% stands for atomic percentage, which represents the percentage of the number of atoms of this isotope to the total number of atoms of this type.

[0077] Furthermore, stacking faults are performed on the supercell and corresponding atoms are removed to introduce defects.

[0078] Specifically, performing stacking faults on the supercell and removing corresponding atomic defects include:

[0079] S11, selecting a first position for a stacking fault in the supercell, and translating atomic coordinates at the first position to obtain a stacking fault;

[0080] S12. Select a second location for atomic removal in the supercell after the stacking fault, and remove corresponding atoms at the second location to introduce defects.

[0081] For example, the silicon carbide stacking fault color center configuration used in the simulation of the embodiment of the present invention is as follows: Figure 2 As shown, in the supercell, the z coordinate (i.e., c coordinate) is greater than and less than The region where all atoms are located is taken as the first location for stacking fault. At this first location, the x coordinates (i.e., a coordinates) of all atoms are reduced by That is, translate to the left along the x-axis Then add the y coordinates (i.e. b coordinates) of all atoms That is, translate upward along the y-axis A stacking fault structure is obtained. In the supercell after the stacking fault, two locations with atomic coordinates of (3.08000, 7.11296, 13.21293) and (3.08000, 7.11296, 15.10500) in the absolute coordinate system are selected as the second locations for atomic removal. The corresponding atoms are removed at these second locations to form divacancy color centers, thereby introducing defects.

[0082] It can be understood that the present invention introduces defects by performing stacking faults on the supercell and removing corresponding atoms, taking into account the complexity of the influence of the intracrystalline magnetic field on the spin, and providing a basis for improving the accuracy of the calculation results of the intracrystalline magnetic field.

[0083] Specifically, in step S2, the structural optimization of the supercell with defects introduced based on first principles includes:

[0084] S21, setting optimization parameters;

[0085] S22, performing structural optimization on the supercell with defects introduced using first-principles calculation software according to the optimization parameters;

[0086] S23, judging whether the structurally optimized supercell has a stable configuration, if so, obtaining the structurally optimized supercell; otherwise, returning to step S12 to change the second position, and executing steps S22 and S23.

[0087] Preferably, the optimization parameters include the limited number of electron steps, the limited number of ion steps, the convergence accuracy of ion steps, and the convergence accuracy of electron steps.

[0088] It should be noted that, through the structural optimization, the position of atoms in space is adjusted, the atomic position distortion in the supercell is eliminated, and the total energy of the supercell is minimized, thereby obtaining a stable configuration. The electron density distribution and energy of the supercell are solved by the electron step. In each electron step, the wave function and energy of the electron are calculated according to the current position of the atomic nucleus. The movement of the atomic nucleus in the average potential field generated by the electron is obtained by the ion step. In each ion step, the force it receives is calculated according to the electronic structure at the current position of the atomic nucleus to update the speed of the atom, and the new position of the atom is calculated according to the speed of the updated atom. Therefore, it is necessary to set the maximum and minimum number of electron steps and ion steps. The ion step convergence accuracy and the electron step convergence accuracy refer to the criteria for judging whether the calculation results meet the convergence in the process of structural optimization. When the change in certain key calculation results (such as energy change, atomic displacement) is less than the corresponding convergence accuracy set, it is considered that the calculation results have converged.

[0089] Preferably, VASP software is used as the first-principles calculation software, and a corresponding input file is generated according to the optimization parameters, and the structure of the supercell with introduced defects is optimized through the input file.

[0090] Specifically, the input files include an INCAR file, a KPOINTS file, a POSCAR file, and a POTCAR file. The constructed defect-introduced supercell is exported to generate a first POSCAR file. The convergence accuracy is set to use a PBE pseudopotential, and the optimization parameters are written into the first INCAR file. A KPOINTS file containing k-points or only gamma points and a POTCAR pseudopotential file are generated. The defect-introduced supercell is structurally optimized using VASP software using these input files to generate a second POSCAR file.

[0091] It should be noted that the Brillouin zone refers to a periodically arranged region in reciprocal space. The Gamma point is the center of the Brillouin zone, and the k-point is the sampling point of the Brillouin zone. For crystal cells with high symmetry and relatively simple electronic structures, using only the Gamma point for relevant calculations can yield results that meet the required accuracy, greatly reducing the amount of computation and improving efficiency.

[0092] Preferably, whether the supercell after structural optimization has a stable configuration is determined by judging whether the absolute value of the energy difference between adjacent ion steps is less than the ion step convergence accuracy.

[0093] Specifically, the magnetization density distribution data of the structurally optimized supercell obtained by static electronic structure calculation includes:

[0094] Step S31: Keep the k-point setting and convergence accuracy unchanged, and set the spin-orbit coupling parameters;

[0095] Step S32: setting the number of ion steps to 0, using first-principles calculation software based on spin-orbit coupling parameters, obtaining the energy corresponding to the supercell through electron steps; iterating the electron steps until the electronic structure of the supercell converges;

[0096] Step S33: outputting the magnetization density distribution data corresponding to the electronic structure convergence of the supercell.

[0097] Preferably, the spin-orbit coupling parameters include: LSORBIT=TRUE, LNONCOLLINEAR=TRUE, LORBMOM=TRUE, GGA_COMPAT=FALSE.

[0098] It should be noted that, in the electronic step, the electronic wave function and energy corresponding to the supercell are obtained by constructing an effective potential and solving the Kohn-Sham equation.

[0099] Specifically, the energy corresponding to the current electron step is compared with the energy corresponding to the electron step of the previous iteration. If the absolute value of the difference between the two is less than the electron step convergence accuracy, it is determined that the electronic structure of the supercell is converged.

[0100] It should be noted that the total energy corresponding to the electronic structure of the supercell is converged to a minimum value through iterative electronic steps.

[0101] For example, VASP software is used as the first-principles calculation software. The spin-orbit coupling parameters are written into the first INCAR file to obtain a second INCAR file. Static calculations are performed based on the second INCAR file, the second POSCAR file, the KPOINTS file, and the POTCAR file to obtain the CHGCAR file and WAVECAR file corresponding to the convergence of the electronic structure of the supercell. The WAVECAR file contains the electron wave function data corresponding to the supercell, and the CHGCAR file contains the charge density distribution data and magnetization density distribution data corresponding to the supercell.

[0102] It can be understood that the present invention introduces defects by performing stacking faults on the supercell and removing the corresponding atoms, and optimizes the structure through first-principles calculations, thereby providing more accurate atomic coordinate information for cluster-related expansion simulations, and providing a basis for improving the accuracy of simulating the spin coherence properties of materials with complex atomic defect color centers.

[0103] Specifically, the spin magnetic moment distribution of the supercell after structural optimization is calculated by the following formula:

[0104]

[0105] Among them, g eis the electron g factor, which is approximately 2.002319; μ B is the Bohr magneton; m is the magnetization density distribution; V FFT is the total number of FFT grids; is the spin angular momentum vector.

[0106] Preferably, the direction of the spin angular momentum vector relative to the Cartesian coordinate system is (0, 0, 1).

[0107] It should be noted that the FFT grid refers to the grid used to discretize the real space or reciprocal space during the Fast Fourier Transform (FFT). Data reconstruction is performed on the real space or reciprocal space FFT grid based on the magnetization density distribution data, and the magnetization density distribution data on the FFT grid is converted into the spin magnetic moment distribution on the FFT grid according to formula (1). The spin magnetic moment distribution on the FFT grid means that each FFT grid point has a spin magnetic moment value.

[0108] Specifically, in step S3, the magnetic field induction intensity of the core spin position and the central spin position of the supercell after structural optimization is calculated by the following formula:

[0109]

[0110] Where μ0 is the vacuum permeability, is the spin magnetic moment distribution, is the vector from the spin magnetic moment to the supercell core spin position or the central spin position, and r is the distance from the spin magnetic moment to the supercell core spin position or the central spin position.

[0111] Specifically, the nuclear spin position is the position where natural isotope substitution occurs. The central spin position is the position corresponding to the atom with the largest mass removed from the second position. The spin dataset includes a nuclear spin magnetic field and a central spin magnetic field. The nuclear spin magnetic field is the set of magnetic fields at the nuclear spin position, and the central spin magnetic field is the magnetic field at the central spin position.

[0112] It should be noted that By changing τ to the vector from the spin magnetic moment to any point in the supercell and r to the distance from the spin magnetic moment to any point in the supercell, the magnetic field induction intensity at any point in the supercell can be calculated using formula (2).

[0113] Preferably, the magnetic field induction intensity at any point in the supercell is the sum of the magnetic field induction intensities generated by the screened spin magnetic moments at that point.

[0114] Preferably, a preset boundary of the supercell is set, and the spin magnetic moment is screened based on the spin magnetic moment distribution and the preset boundary of the supercell; the preset boundary of the supercell is a limiting length proportional to the lattice constant of the supercell.

[0115] Preferably, the method of screening the spin magnetic moment based on the preset boundary of the supercell includes:

[0116] Obtaining a calculation boundary based on the calculation point and the preset boundary of the supercell; the calculation boundary includes: a_min, a_max, b_min, b_max, c_min, c_max;

[0117] Performing virtual cell mapping according to the coordinates of the calculation boundary to obtain a calculation region;

[0118] According to the spin magnetic moment distribution, the corresponding spin magnetic moment in the calculation area is screened.

[0119] Specifically, performing virtual cell mapping according to the coordinates of the calculation boundary to obtain the calculation area includes:

[0120] Expanding the supercell with virtual cells according to the coordinates of the calculation boundary;

[0121] Within the scope of the virtual unit cell and supercell after cell expansion, a hexahedral region with the same angle as the basis vector is enclosed by the calculation boundary;

[0122] Mapping the area of ​​the hexahedral region that exceeds the supercell range to the corresponding area of ​​the supercell to obtain a first calculation area;

[0123] The area of ​​the hexahedral region that does not exceed the supercell range is used as the second calculation area;

[0124] The union of the first calculation area and the second calculation area is used as the calculation area.

[0125] For example, assuming that the area beyond the supercell in the hexahedral area falls on the lower left part of a virtual cell, the lower left part of the virtual cell is mapped to the corresponding area of ​​the supercell, that is, the lower left part of the supercell.

[0126] It should be noted that the calculation point refers to any point in the supercell where the magnetic field induction intensity is to be calculated. a_min and a_max are the minimum and maximum values ​​corresponding to the coordinates of the calculation point on the a-axis as the center and the preset boundary as the length in the a-axis direction; b_min and b_max are the minimum and maximum values ​​corresponding to the coordinates of the calculation point on the b-axis as the center and the preset boundary as the length in the b-axis direction; c_min and c_max are the minimum and maximum values ​​corresponding to the coordinates of the calculation point on the c-axis as the center and the preset boundary as the length in the c-axis direction. a_min should be less than the length of the lattice vector a, b_min should be less than the length of the lattice vector b, and c_min should be less than the length of the lattice vector c; a_max, b_max, and c_max should be greater than 0 and greater than a_min, b_min, and c_min, respectively. Formula (2) is used to obtain the magnetic field induction intensity generated by the magnetic moment at the center point on all FFT grid points in the calculation area.

[0127] Preferably, a_min, a_max, b_min, b_max, c_min, and c_max are coordinates of a coordinate system with lattice vectors as coordinate axes. When a_min>0, a_max<|a|, b_min>0, b_max<|b|, c_min>0, and c_max<|c|, virtual cell mapping is not required.

[0128] It can be understood that the present invention screens the spin magnetic moment based on the preset boundary of the supercell. When calculating the magnetic field induction intensity at any point in the supercell, the spin magnetic moment of the electrons closer to the point is screened for calculation. While meeting the calculation accuracy, the calculation time is greatly reduced and the efficiency is improved. The present invention performs virtual cell mapping based on the coordinates of the calculation boundary. When the calculation point is close to the edge of the supercell, the area in the hexahedron area that exceeds the supercell range is mapped to the corresponding area of ​​the supercell, thereby improving the accuracy of the calculation results of the magnetic field within the crystal.

[0129] It can be understood that the present invention solves the magnetic moment distribution based on the magnetization density distribution, and then calculates the magnetic field at any point in the supercell, providing a basis for improving the accuracy of simulating the spin coherence properties of materials with complex atomic defect color centers.

[0130] Specifically, in step S4, a simulated spin data set required for the spin coherence property simulation is selected from the spin data set according to the bath cutoff radius.

[0131] It should be noted that the bath cutoff radius is set as needed, and the range of nuclear spins considered in the simulation calculation is determined by the bath cutoff radius, that is, which nuclear spins will be included in the solid-state spin coherence simulation.

[0132] Specifically, the calculation order is the maximum number of nuclear spins contained in a cluster, and the dipole cutoff radius is the maximum spacing between nuclear spins in a cluster. The calculation order and dipole cutoff radius are set as needed. Clusters are generated by selecting corresponding nuclear spins from the simulated spin dataset based on the calculation order and dipole cutoff radius. All clusters form a cluster set. A single nuclear spin can belong to different clusters.

[0133] Preferably, the dipole cutoff radius is set to The bath cut-off radius is The calculation order is 2.

[0134] Furthermore, the spin coherence property simulation based on the cluster set and the central spin to obtain the Hamiltonian and coherence function of the central spin and the cluster includes:

[0135] Based on the dipole interaction between the central spin and the nuclear spin, the hyperfine tensor between the central spin and the nuclear spin and the interaction tensor between the nuclear spins are calculated;

[0136] Initialize the central spin density matrix; calculate the central spin Hamiltonian based on the central spin operator, nuclear spin operator and hyperfine tensor; obtain the central spin time evolution operator sequence based on the central spin Hamiltonian and time series; obtain the central spin density matrix evolution result based on the central spin time evolution operator sequence and the central spin density matrix; obtain the central spin coherence function based on the central spin density matrix evolution result;

[0137] The cluster density matrix is ​​obtained based on the central spin density matrix; the Hatch density of each cluster is calculated based on the central spin operator, nuclear spin operator, hyperfine tensor and interaction tensor; the time evolution operator sequence of the cluster is obtained based on the Hatch density and time series of the cluster; the density matrix evolution result of the cluster is obtained based on the time evolution operator sequence of the cluster and the cluster density matrix, and the coherence function of the cluster is obtained based on the density matrix evolution result of the cluster; the coherence function of the cluster set is obtained based on the coherence function of each cluster.

[0138] Specifically, the interaction tensor between nuclear spins is calculated using the following formula:

[0139]

[0140] Among them, γ i is the magnetic gyroscopic ratio of nuclear spin i, γ j is the magnetic gyroscopic ratio of nuclear spin j, μ0 is the vacuum permeability, is the reduced Planck constant, E is the 3*3 unit matrix, is the distance vector between nuclear spin i and nuclear spin j, represents the outer product.

[0141] Specifically, the hyperfine tensor between the central spin and the nuclear spin is calculated using the following formula:

[0142]

[0143] Among them, γ s is the magnetic gyro ratio of the central spin.

[0144] Specifically, the initialization of the central spin density matrix includes:

[0145] generating a central spin initial state, wherein the central spin initial state includes a central spin initial |0> state probability amplitude α and a central spin initial |1> state probability amplitude β;

[0146] Construct the central spin Pauli matrix and central spin density matrix based on the central spin initial state;

[0147] A central spin rotation matrix is ​​initialized based on the central spin Pauli matrix, and α and β values ​​and a central spin density matrix are updated based on the central spin rotation matrix.

[0148] Specifically, the probability amplitude α of the central spin initial |0> state is expressed as: [0+0i, 0+0i, 1+0i], and the probability amplitude β of the central spin initial |1> state is expressed as: [0+0i, 1+0i, 0+0i].

[0149] Specifically, the central spin Pauli matrix is ​​constructed using the following formula:

[0150]

[0151] Among them, σ x , σ y , σ z They represent the x-, y-, and z-direction components of the central spin Pauli matrix, and ∧ is the outer product.

[0152] Specifically, the central spin density matrix is ​​constructed using the following formula:

[0153]

[0154] Among them, ∧ represents the outer product, ρ S is the central spin density matrix.

[0155] Specifically, the central spin rotation matrix is ​​initialized using the following formula:

[0156]

[0157] Among them, E3 and E1 represent 3*3 and 1*1 unit complex matrices respectively. is the Kronecker product, and i represents a complex number.

[0158] It should be noted that when free induction decay is subsequently calculated, the central spin rotation matrix is ​​set to empty; when the Hahn echo is calculated, the central spin rotation matrix is ​​initialized using formula (7).

[0159] Specifically, the values ​​of α and β are updated using the following formula:

[0160]

[0161] Among them, R S represents the central spin rotation matrix.

[0162] Specifically, the central spin density matrix is ​​updated using the following formula:

[0163]

[0164] in, Represents R S The conjugate transpose of .

[0165] Specifically, the central spin Hamiltonian is calculated using the following formula:

[0166]

[0167] in, is the central spin Hamiltonian, B is the magnetic field vector including the intracrystalline magnetic field and the external magnetic field, D is the zero-field splitting constant, S = [S x ,S y ,S z ] is the central spin operator, is the nuclear spin operator, I i is the nuclear spin operator of the ith nuclear spin, γ S is the magnetic gyroscopic ratio of the central spin, A i is the hyperfine tensor between the central spin and the i-th nuclear spin.

[0168] It should be noted that S x 、S y 、S z The initial values ​​are as follows:

[0169]

[0170] According to the dimension of nuclear spin in cluster, S x 、S y 、S z It is expanded by the following formula:

[0171]

[0172] Where n is the product of the Haff density dimensions of the cluster core spin, En , E1 represent n*n, 1*1 unit complex matrices respectively, is the Kronecker product.

[0173] Preferably, in the simulation, the supercell is set to be in an external magnetic field environment with 0.05 T in the z-axis direction, that is, the third value in the magnetic field array is increased by 0.05 T.

[0174] Specifically, according to the total simulation time t n and time step n, define the time series t={t1,t2,…,t n The central spin time evolution operator sequence includes the central spin free evolution decay and the central spin Hahn echo. The central spin free evolution decay is calculated by the following formula:

[0175]

[0176] in, is the free evolution decay of the central spin, i represents a complex number, h is the Planck constant, and t is the time series.

[0177] Specifically, the central spin Hahn echo is calculated using the following formula:

[0178]

[0179] Specifically, the central spin density matrix evolution result is obtained by the following formula:

[0180]

[0181] in, express The conjugate transpose of .

[0182] Specifically, the central spin coherence function is obtained by the following formula:

[0183]

[0184] in, represents the conjugate transpose of α.

[0185] Specifically, the cluster density matrix is ​​obtained by the following formula:

[0186]

[0187] Among them, ρ C is the cluster density matrix, n is the product of the Haff density dimensions of the cluster core spin, is the Kronecker product.

[0188] Specifically, the H density of the cluster is calculated by the following formula:

[0189]

[0190] Among them, J ij is the interaction tensor between the ith nuclear spin and the jth nuclear spin, A i 、A a are the hyperfine tensors between the central spin and the i-th and a-th nuclear spins, respectively. i , I j , I a are the nuclear spin operators of the ith, jth, and ath nuclear spins, respectively, a > is the expected value of the cluster outer nuclear spin, and C represents the cluster.

[0191] It should be noted that I=[I x ,I y ,I z ], I x , I y , I z The initial values ​​are as follows:

[0192]

[0193] According to the dimension of nuclear spin in cluster, I x , I y , I z It is expanded by the following formula:

[0194]

[0195] Specifically, the cluster time evolution operator sequence includes cluster free evolution decay and cluster Hahn echo, and the cluster free evolution decay is calculated by the following formula:

[0196]

[0197] Where i represents a complex number, h is Planck's constant, and t is a time series.

[0198] Specifically, the cluster Hahn echo is calculated using the following formula:

[0199]

[0200] Specifically, the density matrix evolution result of the cluster is obtained by the following formula:

[0201]

[0202] in, express The conjugate transpose of .

[0203] ​Specifically, the coherence function of the cluster is obtained by the following formula:

[0204]

[0205] in, represents the conjugate transpose of α.

[0206] Specifically, the coherence function of the cluster set is obtained by the following formula:

[0207]

[0208] Among them, L i 、L j is the coherence function of the first-order cluster, is the result of phase normalization of the coherence function of the first-order cluster, L ij is the coherence function of the second-order cluster, is the result of phase normalization of the coherence function of the second-order cluster, L0 is the central spin coherence function, and L(t) is the coherence function of the cluster set.

[0209] Specifically, the expression of the uneven dephasing time fitting formula is:

[0210]

[0211] The uniform dephasing time fitting formula is expressed as:

[0212]

[0213] Specifically, the non-uniform dephasing time is obtained by fitting the coherence function of the cluster set with formula (26), and the uniform dephasing time is obtained by fitting the coherence function of the cluster set with formula (27).

[0214] It can be understood that the simulation data is obtained by simulating the coherence function of the cluster set, and the non-uniform dephasing time is obtained by minimizing the fitting error according to formula (26). The simulation data is obtained by simulating the coherence function of the cluster set, and the uniform dephasing time T2 is obtained by fitting by minimizing the fitting error according to formula (27).

[0215] For example, Figure 4 As shown in FIG. 1 , the simulation results of the color center coherence function of the silicon carbide stacking fault provided by the embodiment of the present invention, as well as the fitting results of the uniform dephasing time and the non-uniform dephasing time, are shown.

[0216] Preferably, the data of the 0-0.4 μs portion of the coherence function corresponding to the free evolution decay are used to fit the non-uniform dephasing time, and the data of the 0-1 μs portion of the coherence function corresponding to the Hahn echo are used to fit the uniform dephasing time.

[0217] As can be understood, the present invention provides a solid-state spin coherence simulation method that considers the complexity of the influence of the intracrystalline magnetic field on spins, realizes the construction of the intracrystalline magnetic field, and improves the accuracy of the simulation results of the spin coherence properties of solid-state spin systems with complex atomic defects. The present invention implements a cluster correlation expansion algorithm that considers the intracrystalline magnetic field, can realize the simulation of doublet and triplet spin coherence properties, and has broad application prospects.

[0218] Compared with the prior art, the solid-state spin coherence simulation method considering the intracrystalline magnetic field provided by the present invention has the following beneficial effects:

[0219] 1. The present invention solves the magnetic moment distribution based on the magnetization density distribution, and then calculates the magnetic field of the supercell core spin position and the central spin position, providing a basis for improving the accuracy of simulating the spin coherence properties of materials with complex atomic defect color centers.

[0220] 2. The present invention introduces defects by performing stacking faults on the supercell and removing the corresponding atoms, and optimizes the structure through first-principles calculations, providing more accurate atomic coordinate information for cluster-related expansion simulations, and laying a foundation for improving the accuracy of simulating the spin coherence properties of materials with complex atomic defect color centers.

[0221] 3. The present invention provides a solid-state spin coherence simulation method that takes into account the intracrystalline magnetic field, takes into account the complexity of the influence of the intracrystalline magnetic field on the spin, realizes the construction of the intracrystalline magnetic field, and improves the accuracy of the simulation results of the spin coherence properties of solid-state spin system materials with complex atomic defects.

[0222] 4. The present invention implements a cluster correlation expansion algorithm that takes into account the intracrystalline magnetic field, which can realize the simulation of doublet and triplet spin coherence properties and has broad application prospects.

[0223] 5. The present invention screens the spin magnetic moment based on the preset boundary of the supercell. When calculating the magnetic field induction intensity of the supercell core spin position and the central spin position, the spin magnetic moment of electrons closer to the position is screened for calculation. While meeting the calculation accuracy, the calculation time is greatly reduced and the efficiency is improved.

[0224] Those skilled in the art will appreciate that all or part of the process steps of the above-described embodiments can be implemented by instructing related hardware through a computer program, and the program can be stored in a computer-readable storage medium, such as a magnetic disk, an optical disk, a read-only memory, or a random access memory.

[0225] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by any technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. A solid-state spin coherence simulation method considering intracrystalline magnetic fields, characterized in that: The method comprises the following steps: constructing an initial unit cell and expanding it into a supercell; performing natural isotope substitution in the supercell; introducing defects into the supercell; Based on the first principles, the structure of the defect-introduced supercell is optimized; the magnetization density distribution data of the structurally optimized supercell is obtained by static electronic structure calculation; and the spin magnetic moment distribution of the structurally optimized supercell is calculated based on the magnetization density distribution data; According to the spin magnetic moment distribution, the magnetic field induction intensity of the core spin position and the central spin position of the supercell after structural optimization is calculated to construct a spin data set; A simulated spin data set is obtained according to the bath cutoff radius, and corresponding nuclear spins are selected from the simulated spin data set according to the calculation order and the dipole cutoff radius to obtain a cluster set. Spin coherence properties are simulated based on the cluster set and the central spin to obtain the Hamiltonian and coherence function of the central spin and the cluster; the inhomogeneous dephasing time is obtained based on the coherence function and the inhomogeneous dephasing time fitting formula, and the uniform dephasing time is obtained based on the coherence function and the uniform dephasing time fitting formula.

2. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 1, characterized in that: The spin coherence property simulation based on the cluster set and the central spin is performed to obtain the Hamiltonian and coherence function of the central spin and the cluster, including: Based on the dipole interaction between the central spin and the nuclear spin, the hyperfine tensor between the central spin and the nuclear spin and the interaction tensor between the nuclear spins are calculated; Initialize the central spin density matrix; calculate the central spin Hamiltonian based on the central spin operator, nuclear spin operator and hyperfine tensor; obtain the central spin time evolution operator sequence based on the central spin Hamiltonian and time series; obtain the central spin density matrix evolution result based on the central spin time evolution operator sequence and the central spin density matrix; obtain the central spin coherence function based on the central spin density matrix evolution result; The cluster density matrix is ​​obtained based on the central spin density matrix; the Hatch density of each cluster is calculated based on the central spin operator, nuclear spin operator, hyperfine tensor and interaction tensor; the time evolution operator sequence of the cluster is obtained based on the Hatch density and time series of the cluster; the density matrix evolution result of the cluster is obtained based on the time evolution operator sequence of the cluster and the cluster density matrix, and the coherence function of the cluster is obtained based on the density matrix evolution result of the cluster; the coherence function of the cluster set is obtained based on the coherence function of each cluster.

3. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 2, characterized in that: The interaction tensor between nuclear spins is calculated using the following formula: Among them, γ i is the magnetic gyroscopic ratio of nuclear spin i, γ j is the magnetic gyroscopic ratio of nuclear spin j, μ0 is the vacuum permeability, is the reduced Planck constant, E is the 3*3 unit matrix, is the distance vector between nuclear spin i and nuclear spin j, represents the outer product.

4. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 3, characterized in that: The hyperfine tensor between the central spin and the nuclear spin is calculated using the following formula: Among them, γ s is the magnetic gyro ratio of the central spin.

5. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 2, characterized in that: The initialization central spin density matrix includes: generating a central spin initial state, wherein the central spin initial state includes a central spin initial |0> state probability amplitude α and a central spin initial |1> state probability amplitude β; Construct the central spin Pauli matrix and central spin density matrix based on the central spin initial state; A central spin rotation matrix is ​​initialized based on the central spin Pauli matrix, and α and β values ​​and a central spin density matrix are updated based on the central spin rotation matrix.

6. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 2, characterized in that: The central spin Hamiltonian is calculated using the following formula: in, is the central spin Hamiltonian, B is the magnetic field vector including the intracrystalline magnetic field and the external magnetic field, D is the zero-field splitting constant, S = [S x ,S y ,S z ] is the central spin operator, is the nuclear spin operator, I i is the nuclear spin operator of the ith nuclear spin, γ s is the magnetic gyroscopic ratio of the central spin, A i is the hyperfine tensor between the central spin and the i-th nuclear spin.

7. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 6, characterized in that: The central spin time evolution operator sequence includes central spin free evolution decay and central spin Hahn echo. The central spin free evolution decay is calculated by the following formula: Where i represents a complex number, h is Planck's constant, and t is a time series; The central spin Hahn echo is calculated using the following formula: Among them, σ x represents the x-component of the central spin Pauli matrix.

8. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 6, characterized in that: The H density of the cluster is calculated by the following formula: Among them, γ i is the magnetic gyroscopic ratio of nuclear spin i, J ij is the interaction tensor between the ith nuclear spin and the jth nuclear spin, A i 、A a are the hyperfine tensors between the central spin and the i-th and a-th nuclear spins, respectively. i , I j , I a are the nuclear spin operators of the ith, jth, and ath nuclear spins, respectively, a > is the expected value of the cluster outer nuclear spin, and C represents the cluster.​ 9. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 5, characterized in that: The coherence function of the cluster is obtained by the following formula: in, represents the conjugate transpose of α, m C Represents the density matrix evolution result of the cluster.

10. The solid-state spin coherence simulation method considering the intracrystalline magnetic field according to claim 1, characterized in that: The spin magnetic moment distribution of the optimized supercell is calculated using the following formula: Among them, g e is the electron g factor, which is approximately 2.002319; μ B is the Bohr magneton; m is the magnetization density distribution; V FFT is the total number of FFT grids; is the spin angular momentum vector; The magnetic field induction intensity of the core spin position and the central spin position of the supercell after structural optimization is calculated by the following formula: Where μ0 is the vacuum permeability, is the spin magnetic moment distribution, is the vector from the spin magnetic moment to the supercell core spin position or the central spin position, and r is the distance from the spin magnetic moment to the supercell core spin position or the central spin position.

Images