Quantized static micro-magnetic model and calculation method thereof

Abstract

The invention discloses a quantized static micro-magnetic model and a calculation method thereof. The method comprises the following steps of: gridding a magnetic and multiferroic material to be researched into a plurality of small-area units or small-volume units; the angular momentum of the atom at the central position of the ith small unit is Si, the angular momentum S is a quantum mechanical operator, and the quantum number S is an integer or a half integer; the hot average value of Si is lt; sigt, Sigt; the thermal average value is equal to the thermal average value of angular momentum of all atoms in the ith small unit; the thermal average value of physical quantities such as Si is calculated according to the quantum theory, so as to be lt; sigt, Sigt; the size and the direction change along with the spatial position and the temperature; a quantum computing method is adopted in simulation; according to the quantized static micro-magnetic model and the calculation method, the Hamiltonian of the system does not comprise a complex physical expression in an LLB equation any more, and the change of the magnetic moment along with the temperature and the spatial position in the system can be described; the method not only overcomes the serious theoretical difficulty of micro-magnetism, but also greatly saves the memory of a computer, and is convenient for quickly calculating and displaying the change of microscopic magnetic and electric structures in the mesoscale range of the material at the non-zero temperature.

Description

Technical Field

[0001] The present invention relates to the technical field of computational physics and computational materials science, and specifically to a quantized static micromagnetic model and a calculation method thereof, which are used to study the macroscopic physical properties of magnetic or multiferroic materials at non-zero temperature and calculate the changes of their magnetic and electric structures within a mesoscopic scale with temperature and spatial position. Background Art

[0002] For decades, researchers at home and abroad have generally used micromagnetics and classical Monte Carlo methods to simulate the microscopic magnetic structure of magnetic and multiferroic materials and study their macroscopic physical properties. In recent years, the discovery of magnetic skyrmions, their tiny spatial scale, the topological stability of their magnetic structure, and their practical prospects as data storage and information carriers have promoted the rapid development of micromagnetism theory and methods. However, both simulation methods are based on classical or semiclassical physics and therefore have obvious limitations. In order to overcome the difficulties of the above two methods, the inventors have developed two quantum simulation methods, SCA (Self-Consistent Approach) and OQMC (Optimized Quantum Monte Carlo), both of which have been patented [21,22]. The following is a brief introduction to these numerical methods to illustrate the novelty and practical value of the present invention.

[0003] Classical micromagnetism divides the magnetic material under study into many small volume units △V. Let the atomic magnetic moment at the center point of the i-th unit be M i =M s m i , here M s is the saturation magnetization, |m i |=|m|=1, m is a unit vector. Because each small volume unit contains hundreds or thousands of atoms, it can simulate the changes in magnetic structure within the mesoscopic scale. In addition to being affected by the external magnetic field, these small volume units also interact with each other. Because the equilibrium state of the system corresponds to the state with the minimum total free energy, it can be solved using the energy minimization method based on the Brown equation [1]. However, in recent years, researchers have generally used the Landau-Lifshitz-Gilbert (LLG) equation [2-4]

[0004]

[0005] Solve the spatial stable distribution of magnetic M and its evolution over time. sIt is regarded as a constant that does not change everywhere in the whole magnet, so the LLG equation is only applicable to absolute zero temperature. It will be further discussed later that if the LLG equation is applied to low temperature regions, severe theoretical difficulties will be encountered.

[0006] In order to study the magnetic structure in the low temperature region, some scholars later proposed that according to the spin wave theory, the saturation magnetization intensity related to temperature

[0007] M s (T)=M s (0)(1-CT 3 / 2 ) (2)

[0008] Measured M s (T) changes with temperature, the value of C can be determined, and the LLG equation can be used to study the magnetic structure of the material in the low temperature region. s (T) is also just the average value of the magnetization intensity in the entire material, and it is still assumed that M s (T) is equal everywhere, which is obviously inconsistent with the actual magnetic structure.

[0009] In recent years, the use of pulsed lasers to control magnetic structures has become an important research hotspot in the field of magnetism. Experiments have shown that laser excitation can induce consistent precession of magnetic moments and spin reorientation, regulate magnetic structures, etc. In addition, the industry has also developed heat assisted magnetic recording technology for the next generation of magnetic storage technology. Unfortunately, the LLG equation does not allow radial changes in the magnetic vector. To overcome this difficulty, Garanin proposed the random Landau-Lifshitz-Bloch (LLB) equation in 1997 [5]:

[0010]

[0011] The effective magnetic field H eff Including external magnetic field, magnetocrystalline anisotropy, Heisenberg exchange and thermal coupling field, h / / and h ⊥ is the random field related to thermal effect, α / / and α ⊥ are the radial and transverse relaxation coefficients, respectively. The LLB equation can simulate the magnetic structure of many materials at non-zero temperature and describe their high-temperature magnetodynamic processes under laser excitation [6]. For ferromagnets, α / / and α ⊥ Two coefficients and h / / and h ⊥The expression is relatively simple. However, for slightly more complex ferrimagnets, the LLB equation becomes very complicated [7]. As for non-collinear complex microscopic magnetic structures, such as chiral magnetic skyrmions, the relevant expressions will be extremely complicated or even impossible to derive. This greatly limits the scope of application of the LLB equation, so it is urgent to propose new models and methods.

[0012] The Classical Monte Carlo (CMC) method is one of the most well-known and commonly used computational methods for simulating magnetic structures [8]. In each step of the calculation, a spin at a certain grid point in the system is randomly selected, randomly rotated by a spatial angle, and then the change in local energy caused by the rotation operation is calculated. The Metropolis algorithm is then used to determine whether the new orientation of the spin can be accepted or discarded. After hundreds of thousands or even millions of cycles, the total energy of the magnetic system is gradually reduced in the hope that the calculation converges to the equilibrium state of the system. However, this method regards the spin magnetic moment as a classical vector, and its limitations are very obvious. For example, it cannot simulate the antiferromagnetic skyrmion crystal in a two-dimensional square magnetic system [9,10].

[0013] In order to overcome the limitations of micromagnetism and classical Monte Carlo methods, the inventors have developed three quantum simulation methods for spin systems: SCA, QMC and OQMC [11-23]. In short, in the quantum model, the angular momentum in the system Hamiltonian is regarded as a quantum mechanical operator, and all physical quantities are calculated according to quantum theory. Strictly speaking, the microscopic magnetic structure of the material can only be accurately described by quantum theory. Therefore, the SCA and OQMC two-quantum methods proposed by the inventors can easily simulate antiferromagnetic skyrmion crystals on a two-dimensional square lattice [18, 19, 21, 22], while the CMC method cannot [9, 10].

[0014] In summary, classical micromagnetism based on the LLG equation assumes that the magnetization intensity is equal everywhere, so it can only simulate the magnetic structure of materials at zero temperature; when studying complex magnetic systems at non-zero temperature, the LLB equation also encounters unsolvable theoretical difficulties; the classical properties of CMC limit its scope of application. On the other hand, existing quantum simulation methods are suitable for simulating magnetic structure changes at the nanoscale [11-23], but the scale of magnetic skyrmions observed in experiments is often larger than tens of nanometers. Therefore, in order to simulate the changes in magnetic structure within the mesoscopic scale at any non-zero temperature and calculate its quantum effects, it is urgent to develop new quantum models and calculation methods. Summary of the invention

[0015] Purpose of the invention: In order to overcome the difficulties and limitations of micromagnetism, classical Monte Carlo and existing quantum simulation methods, the present invention proposes a quantized static micromagnetism (QSMM) model and its calculation method, which are used to study the macroscopic physical properties of magnetic and multiferroic materials at non-zero temperature, and calculate the changes of their magnetic and electric structures in the mesoscopic range with temperature and spatial position.

[0016] Technical Solution

[0017] A quantized static micromagnetic model includes the following elements and steps:

[0018] 1. From the exchange interaction intensity A and saturation magnetization intensity M S , calculate the exchange coherence length of the material

[0019]

[0020] Where μ0 is the magnetic permeability of vacuum; usually, there is a factor of 2 under the radical;

[0021] The saturation magnetization M S and the quantum number S of the atomic angular momentum (S = 1 / 2, 1, 3 / 2, 2, 5 / 2, etc. integers or half integers), calculate the lattice constant a;

[0022] The system under study is meshed into many small area or volume cells; the side length of the small cell must not be greater than the exchange coherence length L ex , and should be equal to an integer multiple of a;

[0023] By M S and the value of S, the number of atoms in the small unit can be calculated;

[0024] 2. Take the atomic angular momentum at the center point of the i-th unit as S i (i=1,2,...,N, N is the total number of small units), its thermal average value i > is equal to the thermal average of the angular momentum of all atoms in the ith cell, S i The magnetic moment m i =μ B g ii ,g is the Lande factor;

[0025] Since S is a quantum mechanical operator, its quantum number S = integers or half integers such as 1 / 2, 1, 3 / 2, 2, 5 / 2;

[0026] S i Affected by the external magnetic field, magnetocrystalline anisotropy and magneto-electric coupling, and with the atomic angular momentum S at the center of other units j ​There are interactions such as Heisenberg exchange, DM, and demagnetization field between them; the sum of the energy of these effects and interactions is S i The Hamiltonian

[0027] Each Hamiltonian They are coupled to each other, and thus the Hamiltonian of the whole system can be obtained;

[0028] 3. Hamiltonian Quantization of each item in :

[0029] For example, the energy density of the Heisenberg exchange interaction is quantized as

[0030]

[0031] Here, the exchange intensity A is defined as A = nJ / a; while in micromagnetism, A = nJS 2 / a; J represents the Heisenberg exchange constant, n is the number of magnetic atoms in the crystal unit cell; operator Discretization is required;

[0032] For example, when the DM interaction is of Bloch type, its energy density is quantized as

[0033]

[0034] Where D represents the strength of DM interaction; the operator It also needs to be discretized;

[0035] In formulas (5, 6), S is used to replace m in the micromagnetic formula, and other interaction terms also need to be treated similarly;

[0036] Under various effects and interactions, S i The sum of the energies is It is equal to the sum of the magnetic interaction energies within a single unit cell;

[0037] 4. Various actions and interactions form an effective magnetic field B at any magnetic moment eff For example, consider a two-dimensional magnetic system. Under the action of an external magnetic field H0, if the DM interaction is of Néel type, its strength is D, and the exchange interaction and anisotropy energy strengths are A and K respectively. S According to micromagnetism theory, the effective magnetic field strength acting on any atomic magnetic moment is

[28]

[0038]

[0039] Among them, H eff =B eff / μ0,H dis the demagnetization field formed by the magnetic dipole effect, which is usually ignored by researchers in calculations; take a certain atomic angular momentum S i , its energy in the effective magnetic field is

[0040]

[0041] The magnetization intensity M in the micromagnetic formula (7) S is a constant; by quantizing each term in equation (7, 8), we can also obtain the quantized Hamiltonian

[0042] 5. For multiferroic materials, the magneto-electric coupling effect must also be taken into account; there are many mechanisms for magneto-electric coupling [24,27]. For example, according to the pd orbital hybridization theory [24,25,26], the three components of the electric dipole moment P in the multiferroic material in the Cartesian coordinate system are

[0043]

[0044] Among them, S x ,S y ,S z denote the x, y, and z components of the spin S vector, respectively, so the vector P operator is Hermitian; the magneto-electric coupling energy ε ME =-P·E, where E is the electric field strength; the magneto-electric coupling term also needs to be quantized;

[0045] In this way, a quantized static micromagnetic model of magnetic and multiferroic materials was established.

[0046] 6. Solve the angular momentum S one by one during the calculation i Hamiltonian, according to the quantum mechanics formula

[0047]

[0048] Calculate the temperature T when S i and other mechanical quantities O i The thermal average value; where β = -1 / (k B T),k B is the Boltzmann constant; S calculated from this i The thermal average i >The size and direction vary with temperature and position;

[0049] In calculating S i The Hamiltonian and its thermal average i >In the process, the Heisenberg exchange constant J is reduced, and the temperature T is also reduced by the same multiple, so T is actually the reduced temperature;

[0050] ​​7. The above-mentioned quantized static micromagnetic model uses one or a combination of quantum simulation methods such as SCA, OQMC[11-22], and QMC

[23] to calculate the macroscopic properties of magnetic and multiferroic materials at non-zero temperature, and simulate and display the changes of their magnetic and electric structures (such as ferromagnetic, ferroelectric skyrmion crystals, etc.) in the mesoscopic range with position.

[0051] Beneficial Effects

[0052] In summary, compared with the prior art, the present invention combines quantum mechanics with micromagnetism theory to establish a quantized static micromagnetism (QSMM) model; the system Hamiltonian of the QSMM model does not include complex expressions such as those in the LLB equation, and can describe the changes of magnetic moments in the system with temperature and spatial position; this not only overcomes the serious theoretical difficulties of micromagnetism, but also greatly saves computer storage, and is convenient for quickly calculating and conveniently displaying the magnetic and electrical structure changes of materials within the mesoscopic scale; therefore, the present invention has opened up a new quantum approach for studying magnetic and multiferroic materials (systems). In the embodiment, a CoFeB-like multiferroic film is considered, and the spin quantum number S of the magnetic atoms is taken as 1 / 2, 1, 3 / 2, 2 and 5 / 2 respectively. The QSMM model and its calculation method are used to successfully simulate the ferromagnetic and ferroelectric skyrmion crystals on the CoFeB-like multiferroic film, and it is found that the topological number of the ferroelectric skyrmion is fractionally quantized, which may have important scientific significance. The innovative points and beneficial effects of the present invention are described as follows:

[0053] (1) Overcoming the limitations of the LLG equation: The LLG equation assumes that the magnetization intensity M in the magnetic system S The quantized static micromagnetic model established by the present invention divides the magnet into many small area or volume units, takes a representative angular momentum (magnetic moment) at the center of each small unit, and calculates its thermal average value according to quantum theory. The amplitude and direction of the thermal average value of the angular momentum (magnetic moment) calculated in this way change with temperature and position, thus overcoming the limitation of the LLG equation.

[0054] (2) Overcoming the difficulties of the LLB equation: In 1997, Garanin phenomenologically introduced the thermal mean field and established the LLB equation [5] in order to apply it to non-zero temperatures. However, the relevant mathematical expressions of the LLB equation are extremely complex and often cannot be derived, which greatly limits its scope of application. The QSMM model established in the present invention does not require the derivation of any complex expressions. It can directly solve the system Hamiltonian to calculate the macroscopic properties of the system at any non-zero temperature, and the changes of its microscopic magnetic and electrical structures with temperature and position, thus overcoming the theoretical difficulties of the LLB equation.

[0055] (3) Avoiding the theoretical difficulties of micromagnetism: In the micromagnetism formula

[0056]

[0057] In the derivation process, due to the assumption that |m|=1, the m in the exchange energy 2 is ignored as a constant; then, in the exchange magnetic field The m that appears again in the discretization expression of 2 The term is often ignored; however, at non-zero temperatures, the amplitude of the magnetic moment in the system varies with temperature and position, and the assumption of |m| = 1 does not hold; therefore, when micromagnetism is extended to non-zero temperatures, serious difficulties are encountered;

[0058] In contrast, in the quantized static micromagnetic model, according to molecular field theory, the Hamiltonian of the system can be decomposed into many individual angular momentums S i (or magnetic moment m i ) The sum of and are commutative; therefore, for any i-th angular momentum (magnetic moment), there is always:

[0059]

[0060] Where m = -S, -S+1, ..., S; therefore, regardless of temperature T and position r i how, The eigenvalue of is always a constant S(S+1) and can be ignored; therefore, the quantized micromagnetic model can accurately describe the change of magnetic moment with temperature and spatial position, overcoming the severe theoretical difficulties of micromagnetism.

[0061] (4) It is easy to calculate the microscopic quantum effects: Gilbert considered the dissipation effect and phenomenologically proposed the LLG equation, still considering the magnetic moment as a vector with constant length, and its classical properties are very obvious; the classical Monte-Carlo method is also unable to calculate the antiferromagnetic skyrmions and their crystals on a two-dimensional square magnet due to its classical properties [9,10]; the microscopic properties of materials can only be accurately described by quantum theory. Therefore, in the QSMM model proposed in the present invention, although the system is also divided into many small volume / area units, each unit is represented by a quantum angular momentum (magnetic moment), so it is easy to reveal the microscopic quantum effects of the system;

[0062] (5) Theoretical basis and characteristics of quantum simulation method: According to quantum theory, different components of angular momentum S satisfy the commutation relation α,β,γ=x,y,z,ε αβγ=0, ±1; however, the different components of the classical angular momentum vector always commute; therefore, the classical and quantum simulation methods are obviously different; it can be expected that according to the pd orbital hybridization theory, it is difficult to calculate the ferroelectric skyrmion lattice in multiferroic materials if the classical simulation method is used; in particular, the Heisenberg uncertainty relation of quantum theory makes it easier for the simulation program to tunnel through the system energy barrier and make it quickly converge to the equilibrium state of the system, which is the theoretical basis for the success of the quantum simulation method;

[0063] (6) Greatly reduces computer memory and computational complexity, making it easier to display changes in magnetic and electrical structures at the mesoscopic scale: Like classical micromagnetism, each unit considered in the quantized static micromagnetic model also contains hundreds or thousands of angular momentums (magnetic moments), and an angular momentum at its center is used to represent the average effect of the angular momentum of this unit. This greatly saves computer memory and reduces computational complexity, making it easier to quickly calculate and display changes in magnetic and electrical structures at the mesoscopic scale, and compare them with more experimental results.

[0064] (7) For the first time, a mesoscopic-scale ferroelectric skyrmion crystal in a two-dimensional multiferroic system was successfully simulated: In the examples, a CoFeB-like multiferroic film was considered, and S=1 / 2, 1, 3 / 2, 2 and 5 / 2 were set respectively. The QSMM model and its calculation method were used to calculate the periodic and symmetric mesoscopic-scale ferroelectric skyrmion lattice in the two-dimensional multiferroic system, and it was found that its topological number is fractionally quantized. This discovery may have important scientific significance. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 , one A technical solution implementation flow chart of a quantized static micromagnetic model and its calculation method;

[0066] Figure 2 , simulated Néel-type (a) ferromagnetic skyrmion crystal, (b) ferroelectric skyrmion crystal, and (c) magnetic topological number density distribution on multiferroic films in a vertical external magnetic field B = 0.09 T and an electric field E = 0.05 V / m. Here S = 1, because the Hamiltonian differs from

[28] by a factor of 2, the B value in each figure should be multiplied by 2.

[0067] Figure 3 , simulated Néel-type (a) ferromagnetic skyrmion crystal, (b) ferroelectric skyrmion crystal, and (c) magnetic topological number density distribution on multiferroic films in a vertical external magnetic field B = 0.06 T. Here S = 5 / 2, because the Hamiltonian differs from

[28] by a factor of 2, the B value in each figure should be multiplied by 2.

[0068] Figure 4, Bloch-type ferromagnetic skyrmion lattices simulated on multiferroic films in a perpendicular external magnetic field, where (a) S = 1 / 2, B = 0.04 T, (b) S = 1, B = 0.09 T, (c) S = 5 / 2, B = 0.06 T. Since the Hamiltonian differs from

[28] by a factor of 2, the B values ​​in each figure should be multiplied by 2. DETAILED DESCRIPTION

[0069] Now combine Figure 1 An embodiment of the present invention is described, which comprises the following steps:

[0070] Step 1: Mesh the system under study into many small area or volume cells, including:

[0071] (1) The exchange interaction intensity A and saturation magnetization intensity M of the material S , calculate its exchange coherence length

[0072]

[0073] where μ0 is the magnetic permeability of vacuum; usually, there is a factor of 2 under the radical of equation (4);

[0074] (2) From the saturation magnetization M S and the value of the quantum number S of the atomic angular momentum, calculate the lattice constant a of the material;

[0075] (3) The side length of a small area unit or a small volume unit shall not be greater than the coherence length L ex , and should be an integer multiple of a;

[0076] (4) Calculate the number of atoms in a small unit.

[0077] Step 2: Calculate the atomic angular momentum S i The Hamiltonian is:

[0078] (1) Take the atomic angular momentum at the center point of the i-th small area unit or small volume unit as S i (i=1,2,...,N), its thermal average value i > is equal to the thermal average of the angular momentum of all atoms in the ith unit, and its magnetic moment m i =μ B g S S i or m i =μ B g J S i ,g S and g J is the Lande factor;

[0079] (2)S i ​Affected by the external magnetic field and magnetocrystalline anisotropy field, and with the angular momentum S of other atoms j There are interactions such as Heisenberg exchange, chiral DM, and demagnetization field;

[0080] (3) For multiferroic materials, the magneto-electric coupling effect must also be taken into account. There are many mechanisms for magneto-electric coupling [24, 27]. For example, according to the pd orbital hybridization theory [24, 25, 26], the electric dipole moment at a certain lattice point in a multiferroic material is

[0081]

[0082] S x ,S y ,S z denote the x, y, and z components of the spin S vector, respectively, so the vector P operator is Hermitian; the magneto-electric coupling energy ε ME =-P·E, E is the external electric field strength;

[0083] (4) These i The sum of the energy of the action and interaction is S i The Hamiltonian Each Hamiltonian They are coupled to each other to form the Hamiltonian of the whole system.

[0084] Step 3: Quantize the Hamiltonian

[0085] (1) Let S be a quantum mechanical operator, whose quantum number S = integer or half integer such as 1 / 2, 1, 3 / 2, 2, 5 / 2;

[0086] (2) The Hamiltonian Quantization of each term in: For example, the density of the quantized Heisenberg exchange energy is expressed as

[0087]

[0088] The exchange intensity A is defined as A = nJ / a; in micromagnetism, A = nJS 2 / a; J is the Heisenberg exchange constant, n is the number of magnetic atoms in the crystal unit cell; operator It also needs to be discretized; for example, when the DM interaction is of Bloch type, its quantized energy density is

[0089]

[0090] Where D represents the strength of DM interaction, and the operator It also needs to be discretized; other actions and interaction terms also need to be quantized in a similar way;

[0091] (3)S i The Hamiltonian It is equal to the total energy of various magnetic actions and interactions within a single unit cell;

[0092] (4) Another form of expression:

[0093] Various actions and interactions form an effective magnetic field B at any magnetic moment eff For example, consider a two-dimensional magnetic system under the action of an external magnetic field H0, assuming that the DM interaction is of Néel type with a strength of D, and the exchange interaction and anisotropy energy intensities are A and K respectively. S According to micromagnetism theory, the total effective magnetic field intensity acting on the magnetic moment m is

[28]

[0094]

[0095] Among them, H eff =B eff / μ0,H d is the demagnetization field formed by the long-range magnetic dipole effect, which is usually ignored by researchers in calculations; therefore, the atomic angular momentum S i The energy in the effective magnetic field is

[0096]

[0097] The magnetization intensity m and M in the micromagnetic formula (7) S is a quantity with constant length; quantize equation (7, 8) to obtain S i The quantized Hamiltonian Step 4: Use quantum methods such as SCA and OQMC[11-22], or QMC

[23] to calculate the macroscopic physical quantities and microscopic magnetic and electrical structures of the material at non-zero temperature. The specific steps are as follows:

[0098] (1) Assume that temperature T = T0 (T0 is usually greater than the critical temperature T M );

[0099] (2) Using the above quantum simulation method, the angular momentum S i Hamiltonian and quantum mechanical formulas

[0100]

[0101] Calculate the temperature T when S i and other mechanical quantities O i The thermal average value of B T),k Bis the Boltzmann constant; when i goes through 1, 2, ..., N, the operation completes a cycle;

[0102] (3) In step (2), for infinite two-dimensional or three-dimensional systems, periodic boundary conditions need to be imposed;

[0103] (4) If the convergence condition or loop end condition [21, 22, 23] is not met, return to step (2);

[0104] (5) Calculate and output the macroscopic physical quantities and / or microscopic magnetic and electrical structures of the material at temperature T when necessary;

[0105] (6) Let T = T - ΔT (ΔT > 0);

[0106] (7) If T>T F (T F is the lowest temperature), return to step (2);

[0107] (8) Calculate and output macroscopic physical quantities such as magnetization intensity, total energy, total free energy, specific heat and entropy of the material in the temperature range under consideration.

[0108] Step 5: Calculation is complete.

[0109] Note: When solving S i The Hamiltonian and the thermal average value i >In the process of etc., the Heisenberg exchange constant J is reduced, and the temperature T is also reduced by the same multiple, so T is actually the reduced temperature.

[0110] Implementation Examples

[0111] To illustrate and prove the correctness and practicality of the present invention, the following introduces several examples of the inventor using the QSMM model and its calculation method to simulate the magnetic and electric skyrmion crystals on CoFeB-like multiferroic films. The micromagnetic parameter A of the CoFeB magnetic film is 16×10 -12 J / m,M S =1.6×10 6 A / m,D=6mJ / m 2 ,K S =2×10 6 J / m 3 , with a thickness of 0.8 nm

[28] . The effect of the demagnetization field is also considered in the system Hamiltonian in Ref.

[28] . The authors used the micromagnetic software Mumax3 to simulate the ferromagnetic skyrmion crystal on the magnetic film at zero temperature.

[0112] In the following examples, the inventors used the micromagnetic parameters A, M of the CoFeB film. S ​and D, and introduce the magneto-electric coupling, assuming its intensity α = 0.2Jm / V. S =1.6×10 6 A / m, the exchange coherence length L of the material can be calculated ex =2.23nm. For simplicity, the demagnetization field is ignored in the examples, following Ref.

[29] . At the same time, in order to simulate the ferromagnetic and ferroelectric skyrmion crystals in the multiferroic film when the spin S = 1 / 2, 1, 3 / 2, 2, 5 / 2, the periodic boundary conditions are considered, and K is used instead. S =2×10 5 J / m 3 , which is of the same order of magnitude as the corresponding parameter in Ref.

[29] . It is foreseeable that due to the difference in parameters, the results calculated in the embodiment differ from those in Ref.

[29] .

[0113] Example 1: Néel-type ferromagnetic and ferroelectric skyrmion crystals in CoFeB-like multiferroic films when S=1

[0114] When S=1, the lattice constant a can be calculated to be 0.226nm. The side length Δx=Δy=1.81nm (=8a <L ex ), a grid volume unit with Δz = 0.8nm (thickness of the film). Under the action of the vertical magnetic field B = 0.09T (Tesla) and the vertical electric field E = 0.05V / m, it is found through calculation that at the reduced temperature T S In the low temperature region of ≤10.6, ferromagnetic and ferroelectric skyrmion lattices are simultaneously formed on the multiferroic film. Figure 2 (a) and (b) show the reduced temperature T S = 0.3, and the periodic and symmetrical ferromagnetic and ferroelectric (electrically polarized) skyrmion lattices. The number of ferromagnetic and ferroelectric skyrmions in (a) and (b) is the same, and the centers of ferromagnetic and ferroelectric skyrmions coincide. Each ferroelectric skyrmion contains two symmetrical red and two blue regions, and the electric polarization vector P xy The component has an anti-skyrmion structure and has very similar characteristics to the single electrically polarized skyrmion simulated by Seki et al. (as shown in Fig. 4(b) in Ref.

[24] ). Figure 2 (c) shows the topological number density distribution of the ferromagnetic skyrmion crystal, which exhibits the same periodicity and symmetry as in (a).

[0115] The units of the vertical and horizontal axes in (a), (b) and (c) are Δx and Δy respectively, and their values ​​are related to the assumed S value. Figure 2From the number of magnetic skyrmions and the value of Δx(=Δy) in (a), we can calculate the spatial period of the skyrmion crystal to be λ=40.5nm, which is very close to the value of 43nm calculated by Kuok et al.

[28] . Since each small volume unit contains about 226 spins, when calculating the magnetization vector of a single spin, the Heisenberg exchange interaction strength J is reduced by γ=(ΔxΔy) / a. 2 times, the temperature T is also reduced by γ times, so the temperature shown in the figures of each embodiment is actually the reduced temperature T S If the true values ​​of S and a are given, the actual temperature T = γT can be calculated. S .

[0116] At the reduced temperature T S In the low temperature region of ≤10.6, the topological number Q of the magnetic skyrmion M = -1. Let the topological number of the ferroelectric skyrmion be Q E , when 0 <T S When ≤9.3, Q E =0.5; and at 9.4≤T S In the temperature range ≤10.6, Q E =2.5. Here, the topological number of the ferroelectric skyrmion is fractionally quantized, as previously discovered by the inventors [15,17]. Example 2: Néel-type ferromagnetic and ferroelectric skyrmion crystals in CoFeB-like multiferroic films when S = 5 / 2

[0117] Assume S = 5 / 2, and still use the parameter M in the first embodiment S ,A,D and K S The inventors simulated the Nelson-type ferromagnetic and ferroelectric skyrmion crystals on CoFeB-like multiferroic films, such as Figure 3 As shown in (a) and (b) in Example 1, the calculated lattice constant a, the length (Δx) and width (Δy) of the volume unit ΔV are also very different, but the height Δz is still taken as 0.8nm. Of course, Figure 3 The vertical and horizontal axes in the figure are respectively based on the newly calculated Δx and Δy. Figure 3 (a,b) show the ferromagnetic and ferroelectric skyrmion crystals, respectively. Figure 3 (c) shows the topological number density distribution of the ferromagnetic skyrmion crystal. The ferromagnetic skyrmion crystal here has the same Figure 2 Similar characteristics. In the entire low temperature region, the topological number Q of the ferromagnetic skyrmion M =-1, the topological number Q of the ferroelectric skyrmion E =2.5.

[0118] Example 3: Bloch-type ferromagnetic and ferroelectric skyrmion crystals in multiferroic films when S=1 / 2, 3 / 2, 5 / 2

[0119] Following the above method, the inventors set S = 1 / 2, 1, 3 / 2, 2, 5 / 2 respectively, and simulated Néel and Bloch-type ferromagnetic and ferroelectric skyrmion crystals on CoFeB-like two-dimensional multiferroic films, and all were successful.

[0120] Figure 4 The two-dimensional Bloch-type ferromagnetic skyrmion crystals calculated for S = 1 / 2, 1, and 5 / 2 are shown. Due to space limitations, the crystal diagram of ferroelectric skyrmions is not drawn. It is worth pointing out that when S = 1 / 2, due to P x , P y and P z are all equal to 0, so according to the pd orbital hybridization theory, there are no ferroelectric skyrmions in the system.

[0121] In summary, in the embodiment, a two-dimensional CoFeB-like multiferroic film is considered, and the quantum number of spin angular momentum S = 1 / 2, 1, 3 / 2, 2, 5 / 2 is set, and the Néel and Bloch type ferromagnetic and ferroelectric skyrmion (except S = 1 / 2) crystals on the film are calculated respectively, and the topological number Q of the ferroelectric skyrmion is found E It is fractionally quantized. This discovery may have important scientific significance.

[0122] The above descriptions are merely several embodiments of the present invention and are not intended to limit the present invention.

[0123] The present invention can be used not only to study magnetic and multiferroic systems of finite size, but also to simulate magnetic and multiferroic systems of infinite size such as two-dimensional and three-dimensional. It can be said that the present invention has explored a new quantization approach for computational magnetism. Therefore, any modification, equivalent substitution and improvement made within the spirit and principle of the present invention are within the protection scope of the present invention.

[0124] References

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Claims

1. A quantized static micromagnetic model, characterized in that: It includes the following elements: The material's exchange intensity parameter A and saturation magnetization M s , the exchange coherence length is calculated according to the following formula Where μ0 is the magnetic permeability of vacuum; Meshing the material to be studied into many small area cells or small volume cells; The side length of a small area unit or a small volume unit shall not be greater than the coherence length L ex , and should be equal to an integer multiple of the lattice constant a of the material; The interactions in the system may include Heisenberg exchange, magnetocrystalline anisotropy, chiral Dzyaloshinskii-Moriya and other interactions; as in micromagnetism, their strengths are expressed in terms of A, K and A , D and other corresponding parameters.

2. A quantized static micromagnetic model according to claim 1, characterized in that: It also includes the following elements: Take the atomic angular momentum at the center of the i-th small area unit or small volume unit as S i , its thermal average i > is equal to the thermal average of the angular momentum of all magnetic atoms in the cell;​ Angular momentum S is a quantum mechanical operator, and its quantum number S is an integer or half-integer; By M s The values ​​of and S can be used to calculate the lattice constant a and unit cell size of the material; Angular momentum S i Affected by the external magnetic field and magnetocrystalline anisotropy, it is also related to other angular momentum S j There are Heisenberg exchange, Dzyaloshinskii-Moriya, demagnetization field and other interactions; When extended to the study of multiferroic materials, the effects of magneto-electric coupling and applied electric fields also need to be taken into account: There are many magneto-electric coupling mechanisms; for example, according to the pd orbital hybridization theory, the electric dipole moment P at each magnetic atom has three components in the Cartesian coordinate system: Among them, S x ,S y ,S z denote the x, y, and z components of the spin S vector, respectively, so the P vector operator is Hermitian; the magneto-electric coupling energy ε ME =-P·E, E represents the strength of the external electric field; Under one or more of the above effects and interactions, S i The sum of the energies is S i The Hamiltonian is equal to the above total magnetic energy within a single unit cell; S i interaction, so are interrelated.

3. A quantized static micromagnetic model according to claim 2, characterized in that: One or more of the above actions and interactions form an effective magnetic field B at any atomic angular momentum. i eff ; If a two-dimensional system is considered and the DM interaction is of Néel type, then: H eff =B eff / μ0 In the formula, H0 is the external magnetic field, A is the exchange interaction strength, D, K S They represent the strength of DM and magnetocrystalline anisotropy, respectively. d is the demagnetization field formed by the magnetic dipole; So S i The Hamiltonian can be expressed as In the formula, μ B is the Bohr magneton, and g is the Lande factor.

4. A quantized static micromagnetic model according to claim 2 or 3, characterized in that: It also includes the following steps: Quantized Hamiltonian Various actions and interaction energies in the equation: For example, the energy density expression of the Heisenberg exchange interaction is quantized as For example, the energy density expression of the Bloch-type DM interaction is quantized as Here, the quantum mechanical operator S replaces the classical vector in the micromagnetic formula And other mathematical operators also need to be discretized.

5. The calculation method of a quantized static micromagnetic model according to claim 4, characterized in that: At temperature T, S i The Hamiltonian S i and other mechanical quantities O i The thermal average value is given by the quantum mechanical formula Calculate, where β = -1 / (k B T), k B is the Boltzmann constant; The angular momentum vector S calculated according to the quantum mechanics formula i The thermal average i >The size and direction vary with spatial position and temperature.​ 6. The calculation method of a quantized static micromagnetic model according to claim 5 is further characterized in that: The quantum simulation method used in the calculation is one of the SCA quantum method, the OQMC quantum method, the QMC quantum method, or a combination thereof.

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