Download MUPRO main program manual

 MUPRO main program manual Table of Content Effective property calculation 1 Purpose of the program 1.1 Elastic system 1.2 Dielectric system 1.3 Piezoelectric system 1.4 Magnetic system 1.5 Piezomagnetic system 1.6 Magnetoelectric system 1.7 Diffustion system 1.8 Thermal conduction system 2 Computation Method 3 Format of input files Ferromagnetic domain evolution 1. Purpose of the program 2. Simulation Method 2.1 Basic design 2.2 Magnetization dynamics 2.3 External field, stray field, and magnetostatic boundary condition 2.4 Magnetocrystalline anisotropy 2.5 Magneto­Elastic interaction 2.6 Magnetic exchange interaction 2.7 Dzyaloshinskii­Moriya interaction (DMI) 2.8 Thermal fluctuation 2.9 Spin torque 3. Input files MuPro­Mag examples μMag Standard Problem #1 μMag Standard Problem #2 μMag Standard Problem #3 μMag Standard Problem #4 μMag Standard Problem #5 Example #6 Example #7 Example #8 1 Effective property calculation 1 Purpose of the program The program calculates effective elastic, electric, magnetic, diffusion, and conduction, etc. properties of a given composite system with arbitrary composite structure, and the spatial distribution of mechanical, electric, and magnetic, etc. variables responsive to applied external fields. 1.1 Elastic system Calculates the effective elastic stiffness ​c​ of a composite; calculates the spatial distribution of strain ​ε​ and stress ​σ​ responsive to an applied strain/stress. The following equation(s) are solved. ∇ ∙ σ = 0, where σ = Cε0 1.2 Dielectric system Calculates the dielectric permittivity ​κ​r of a composite; calculates the spatial distribution of electric field ​E​, electric polarization ​P​, and electric displacement ​D​, responsive to an applied electric field. The following equation(s) are solved. ∇ ∙ D = 0, where D = ε0κrE 1.3 Piezoelectric system Calculates the effective elastic stiffness ​c​, dielectric permittivity ​κ​r, and piezoelectric charge coefficient ​d of a composite; calculates the spatial distribution of strain ​ε​, stress ​σ​, electric field E​, electric polarization ​P​, and electric displacement ​D​, responsive to applied strain/stress and/or electric field. The following equation(s) are solved. 1.4 Magnetic system Calculates the magnetic permeability ​μ​r of a composite; calculates the spatial distribution of magnetic field ​H​, magnetization ​M​, and magnetic induction ​B​, responsive to an applied magnetic field. The following equation(s) are solved. ∇ ∙ B = 0 , where B = μ0μrH + qσ 2 1.5 Piezomagnetic system Calculates the effective elastic stiffness ​c​, magnetic permeability ​μ​r, and piezomagnetic coefficient ​q of a composite; calculates the spatial distribution of strain ​ε​, stress ​σ​, magnetic field H​, magnetization ​M​, and magnetic induction ​B​, responsive to applied strain/stress and/or magnetic field. The following equation(s) are solved. 1.6 Magnetoelectric system Calculates the effective elastic stiffness ​c​, dielectric permittivity ​κ​r, magnetic permeability ​μ​r, piezoelectric charge coefficient ​d​, piezomagnetic coefficient ​q​, and magnetoelectric coefficient ​α of a composite; calculates the spatial distribution of strain ​ε​, stress ​σ​, electric field ​E​, electric polarization ​P​, electric displacement ​D​, magnetic field ​H​, magnetization ​M​, and magnetic induction ​B​, responsive to applied strain/stress and/or magnetic field. The following equation(s) are solved. 1.7 Diffustion system Calculates the effective diffusivity ​D of a composite; calculates the spatial profile of concentration ​c of a steady state diffusion. The following equation(s) are solved. ∇ ∙ (D∇c) = 0 1.8 Thermal conduction system Calculates the effective thermal conductivity ​k of a composite; calculates the spatial profile of temperature ​T of a steady state heat conduction. The following equation(s) are solved. ∇ ∙ (k∇T ) = 0 2 Computation Method The mechanical equilibrium, electrostatic equilibrium, magnetostatic equilibrium, steady diffusion, and/or thermal equilibrium equations are solved using the fourier spectral iterative perturbation method.​1­3 3 3 Format of input files Users need to prepare two files as input: parameter.in Declares the size of the system, the type of properties considered, properties of each phase, and external fields applied. The format is as follows: Table 1​ Format of the input file ​parameter.in Data in the file l​3 n​3 l​1 n​1 l​2 n​2 C​S N​p C​F Explanation System real size in each direction (nm) Total number of simulation grids in each direction Choice of the system: 1­elastic; 2­dielectric; 3­piezoelectric; 4­magnetic; 5­piezomagnetic; 6­magnetoelectric; 7­diffusivity; 8­thermal conductivity total # of phases Choice of the format of the input file ​struct.in: 1­order parameter array; 2­phase id array phase ID I​P c​11 c​22 c​33 c​44 c​55 c​66 c​12 c​23 c​34 c​45 c​56 c​13 c​24 c​35 c​46 c​14 c​25 c​36 c​15 c​26 c​16 ε​r11 ε​r22 ε​r33 ε​r23 ε​r13 ε​r12 (For C​S​=2,3,6) relative dielectric permittivity tensor ​ε​r (unitless) d​11 d​21 d​31 d​12 d​22 d​32 d​13 d​23 d​33 d​14 d​24 d​34 d​15 d​25 d​35 d​16 d​26 d​36 (For C​S​=3,6) piezoelectric charge coefficient tensor ​d (C/N) μ​r11 μ​r22 μ​r33 μ​r23 μ​r13 μ​r12 q​11 q​21 q​31 α​11 α​21 α​31 k​11 κ​11 q​12 q​22 q​32 α​12 α​22 α​32 k​22 κ​22 q​13 q​23 q​33 α​13 α​23 α​33 k​33 κ​33 q​14 q​24 q​34 α​14 α​24 α​34 k​23 κ​23 q​15 q​25 q​35 α​15 α​25 α​35 k​13 κ​13 q​16 q​26 q​36 α​16 α​26 α​36 k​12 κ​12 C​P F​C ε​11 (σ​11
) E​1 ε​22 (σ​22
) E​2 ε​33 (σ​33
) E​3 ε​23 (σ​23
) ε​13 (σ​13
) ε​12 (σ​12
) (For C​S​=1,3,4,6) elastic stiffness tensor ​c​ (Pa) (For C​S​=4,5,6) relative magnetic permeability tensor ​μ​r (unitless) (For C​S​=5,6) piezomagnetic coefficient tensor ​q​ (T/Pa) These lines are repeated N​p​ times, each time providin
g the propertie
s of one phase. (For C​S​=6) magnetoelectric coefficient tensor ​α​ (C/(A.m)) (For C​S​=7) diffusivity tensor ​k​ (m​2​⋅s​­1​) (For C​S​=8) thermal conductivity tensor ​κ​ (W⋅m​­1​⋅K​­1​) Choice of the problem to be solved: 1­effective properties calculation; 2­response to external field (For C​P​=2, and C​S​=1,3,4,6) flag of the mechanical boundary condition: whether to use an applied strain (or otherwise an applied stress) (For C​P​=2, and C​S​=1,3,4,6) (For C​C​=.true.) applied strain (unitless); (For C​C​=.false.) applied stress (Pa) (For C​P​=2, and C​S​=2,3,6) applied electric field (V⋅m​­1​) 4 H​1 C​1 T​1 H​2 C​2 T​2 H​3 C​3 T​3 (For C​P​=2, and C​S​=4,5,6) applied magnetic field (A⋅m​­1​) (For C​P​=2, and C​S​=7) average composition gradient (mol⋅m​­4​) (For C​P​=2, and C​S​=8) average temperature gradient (K⋅m​­1​) struct.in Contains the phase structure of the composite with order parameter arrays written in a row­major order. This file has two possible formats according to C​F​=1, and C​F​=2, respectively, as defined in parameter.in. The format is as follows: Table 2.1​ Format of the input file ​struct.in for C​F​=1 o1(1, 1, 1) Data in the file n​2 n
3 o2(1, 1, 1) … oN p(1, 1, 1) o1(1, 1, 2) o2(1, 1, 2) … oN p(1, 1, 2) Volume fraction o​k of phase k (k=1,2,…,N​p​) at grid point (1,1,1) (Similar as above) ⋮ o1(1, 1, n3) ⋮ o2(1, 1, n3) ⋮ … ⋮ oN p(1, 1, n3) o1(1, 2, 1) o2(1, 2, 1) … oN p(1, 2, 1) o1(1, 2, 2) o2(1, 2, 2) … oN p(1, 2, 2) ⋮ o1(1, 2, n3) ⋮ o2(1, 2, n3) ⋮ … ⋮ oN p(1, 2, n3) ⋮ o1(1, n2, 1) ⋮ o2(1, n2, 1) ⋮ … ⋮ oN p(1, n2, 1) o1(1, n2, 2) o2(1, n2, 2) … oN p(1, n2, 2) ⋮ o1(1, n2, n3) ⋮ o2(1, n2, n3) ⋮ ⋮ … oN p(1, n2, n3) o1(2, 1, 1) o2(2, 1, 1) … ⋮ o1(2, n2, n3) ⋮ o2(2, n2, n3) ⋮ ⋮ … oN p(2, n2, n3) ⋮ o1(n1, 1, 1) ⋮ o2(n1, 1, 1) ⋮ … n​1 ⋮ o1(n1, n2, n3) oN p(2, 1, 1) ⋮ oN p(n1, 1, 1) Explanation Total number of simulation grids ⋮ ⋮ ⋮ o2(n1, n2, n3) … oN p(n1, n2, n3) Data in the file n​1 n​2 p(1, 1, 1) p(1, 1, 2) ⋮ p(1, 1, n3) p(1, 2, 1) p(1, 2, 2) ⋮ p(1, 2, n3) Table 2.2​ Format of the input file ​struct.in for C​F​=2 n​3 Explanation Total number of simulation grids in each direction ID of the dominant phase at grid point (1,1,1) (Similar as above) 5 ⋮ p(1, n2, 1) p(1, n2, 2) ⋮ p(1, n2, n3) p(2, 1, 1) ⋮ p(2, n2, n3) ⋮ p(n1, 1, 1) ⋮ p(n1, n2, n3) A sharp interface model is adopted for C​F​=2. References [1] S.Y. Hu, L.Q. Chen, A phase­field model for evolving microstructures with strong elastic inhomogeneity, Acta Mater. 49 (2001) 1879. [2] J.J. Wang, X.Q. Ma, Q. Li, J. Britson, L.Q. Chen, Phase transitions and domain structures of ferroelectric nanoparticles: Phase field model incorporating strong elastic and dielectric inhomogeneity, Acta Mater. 61 (2013) 7591. [3] J. J. Wang, Y. Song, X. Q. Ma, L.­Q. Chen, and C.­W. Nan, Static magnetic solution in magnetic composites with arbitrary susceptibility inhomogeneity and anisotropy, J. Appl. Phys. 117 (2015) 043907. 6 Ferromagnetic domain evolution 1. Purpose of the program The program simulates the microstructure evolution of a magnet under applied external fields. 2. Simulation Method 2.1 Basic design The total size of the simulation system is l1 × l2 × l3 , which is evenly discretized into n1 × n2 × n3
cuboid grids, i.e., the size of each simulation grid is Δl1 = l1/n1 , Δl2 = l2/n2 , and Δl3 = l3/n3 .. The simulation system can be one of the following types: Bulk, 3­D; Bulk, 2­D; Bulk, 1­D; Thin film, 3­D; Thin film, 2­D; Island­on­substrate, 3­D; Island­on­substrate, 2­D; Freestanding finite­size magnet, 3­D; Freestanding finite­size magnet, 2­D; Freestanding finite­size magnet, 1­D. A set of additional parameters are used to specify the system, which may include the film thickness nf, island (or finite­size magnet) thickness ni3​, island (or finite­size magnet) lengths ni1 and ni2​, and substrate thickness ns. Schematics of all types of systems and corresponding parameters are listed below. All the parameters are in the unit of grid numbers. Table 2.1​ Types of systems System type and schematics Bulk, 3­D Parameters specifying the system n1 n2 n3 nf ni3 Value of parameters Length of the system along ​x1​ direction Length of the system along ​x2​ direction Length of the system along ​x3​ direction 0 0 7 Bulk, 2­D n1 n2 n3 nf ni3 Length of the system along ​x1​ direction 1 Length of the system along ​x3​ direction 0 0 n1 n2 n3 nf ni3 n1 n2 n3 nf ns Length of the system along ​x1​ direction 1 1 0 0 Length of the system along ​x1​ direction Length of the system along ​x2​ direction Length of the system along ​x3​ direction Thickness of the film along ​x3​ direction Thickness of the substrate along ​x3​ direction n1 n2 n3 nf ns Length of the system along ​x1​ direction 1 Length of the system along ​x3​ direction Thickness of the film along ​x3​ direction Thickness of the substrate along ​x3​ direction n1 n2 n3 nf ni3 ns CIsland ni1 ni2 n1 n2 n3 nf ni3 Length of the system along ​x1​ direction Length of the system along ​x2​ direction Length of the system along ​x3​ direction 0 Thickness of the island along ​x3​ direction Thickness of the substrate along ​x3​ direction 1 Length of the island along ​x1​ direction Length of the island along ​x2​ direction Length of the system along ​x1​ direction 1 Length of the system along ​x3​ direction 0 Thickness of the island along ​x3​ direction Bulk, 1­D Thin film, 3­D Thin film, 2­D Island(s)­on­substrate, 3­D Island(s)­on­substrate, 2­D 8 Freestanding finite­size magnet, 3­D Freestanding finite­size magnet, 2­D Freestanding finite­size magnet, 1­D ns CIsland ni1 ni2 Thickness of the substrate along ​x3​ direction 1 Length of the island along ​x1​ direction 1 n1 n2 n3 nf ni3 ns CIsland ni1 ni2 n1 n2 n3 nf ni3 ns CIsland ni1 ni2 n1 n2 n3 nf ni3 ns CIsland ni1 ni2 Length of the system along ​x1​ direction Length of the system along ​x2​ direction Length of the system along ​x3​ direction 0 Thickness of the magnet along ​x3​ direction 0 1 Length of the magnet along ​x1​ direction Length of the magnet along ​x2​ direction Length of the system along ​x1​ direction Length of the system along ​x2​ direction 1 0 1 0 1 Length of the magnet along ​x1​ direction Length of the magnet along ​x2​ direction Length of the system along ​x1​ direction 1 1 0 1 0 1 Length of the magnet along ​x1​ direction 1 CIsland on input specifies whether the in­plane (i.e., in ​x1​­​x2 plane) shape of an island or a freestanding finite­size magnet be a rectangle (​CIsland​=1), an ellipse (​CIsland​=2), or any other arbitrary shape (​CIsland​=0), respectively. On setting ​CIsland​=2, ​ni1 and ni2 would specify the major or minor axes of the ellipse along ​x1 and ​x2 directions, respectively. On setting ​CIsland​=0, an arbitrary in­plane shape defined in an input file ​islandShape.in would be adopted (see Section 3 for details). Cases for ​CIsland​=2 and ​CIsland​=0 are omitted in Table 2.1. For thin films and island(s)­on­substrate systems, the thickness of the substrate should be at least 11 grids, i.e., ns ≥ 11 . At all actual surfaces of a magnet, a number of at least 4 stacking layers of vacuum is needed, if one or more of the following components is considered: (demagnetizing) stray field, magnetoelastic interaction, exchange interaction, or Dzyaloshinskii­Moriya interaction (DMI). For example, for simulating a cuboid island with a length of ​ni1 grids along ​x1 direction, ​n1​ should be chosen following (n1 − ni1)/2 ≥ 4 . 9 Spatial distribution of the local magnetization vector M (x) = M Sm(x) is used to describe magnetic domain structure, where ​x is the position vector, ​MS is the spontaneous magnetization, and ​m is the normalized magnetization. The SI units are adopted in ​μ­Pro​® Mag and in this documentation. A set of Euler angle arrays including ϕ(x), θ(x), and ψ(x) are introduced to treat a polycrystal. These angle arrays rotate the system coordinate axes to the local crystallographic coordinate axes. The transformation matrix ​a from the system coordinates to the local crystallographic coordinates is (1) For example, transformation of a vector ​v from the system coordinates to local crystallographic coordinates follows vi′ = aijvj (i, j = 1, 2, 3) . Here a prime (‘) in a subscript following an index 1, 2, or 3 indicates the local crystalline coordinate, e.g., ​m1’​(​x​) is the component of ​m​(​x​) along the local crystallographic coordinate axis x1′ at position ​x​. Note that Einstein summation convention is adopted throughout the documentation. 2.2 Magnetization dynamics Temporal evolution of ​m​ is governed by the Landau­Lifshitz­Gilbert (LLG) equation, i.e., (2) where ​t is the time, τ is the total torque, α is the damping constant, ​γ0 is the gyromagnetic ratio in m/(A ∙ s) . ​H​eff​ is the effective magnetic field given by . (3) Where μ0 = 4π × 10 N/A is the vacuum permeability, and ​F[​M​] is the Helmholtz free energy of the system, as a functional of the magnetization distribution. ​H​eff includes the following contributions: ● External field ​Hext​
​ ; ● Magnetic stray field or demagnetizing field ​H​d​; ● Magnetocrystalline anisotropy field ​Hanis​
​ ; ● Magnetoelastic field ​H​elas​; ● Exchange interaction field ​H​exch​; ● Dzyaloshinskii­Moriya interaction (DMI) field ​H​DMI​; ● Thermal fluctuation field ​H​therm​; −7
2
10 ● Effective field from spin­transfer torque or spin­orbit torque ​H​ST​, as below, . (4) μ­Pro​®​ Mag provides two numerical methods for solving the LLG equation: ● IGS, implicit Gauss­Seidel projection method[1] implemented with Fourier­Spectral approach[2][3]. Contribution of the short­range interaction H exch is implicitly considered, while other contributions to ​H​eff​ are explicitly evaluated. ● RK4, the Runge­Kutta method. All contributions to ​H​eff​ are explicitly evaluated. The time during magnetization evolution is discretized into time steps with a fixed duration Δt , i.e., t = ktΔt where kt is the step number. The recommended range for the value of Δt is 10−14s ≤ Δt ≤ 10−12s , for numerical stability and accuracy. 2.3 External field, stray field, and magnetostatic boundary condition The magnetic field ​H consists of external field and magnetostatic stray field, i.e., H = H ext + H stray . In ​μ­Pro​® Mag, the external field ​H​ext is considered spatially uniform in the simulation system, in the unit of A/m. The volume density of external field energy is given by . (5) μ­Pro​ Mag provides two options of specifying an option of choosing ​H​ext​ on input: ● As a combination of a DC and an AC component, i.e., H ext = H DC + H ACsin(2πf AC ∙ t) ● As an array of the external magnetic field sequence H ext(t) The magnetostatic stray field energy is given by ®​
(6) The energy density is written as . (7) H​stray​ is obtained at each evolution step by solving the magnetostatic equilibrium equation (8) Two types of boundary condition are used for ​H​stray​. In a period boundary condition, the simulation system is considered as a building block that appears repeatedly appear in 3­D space, ˉ , where ϕ is the magnetic scalar and the stray field is expressed as H stray =− ∇ϕ + N D + M
potential with a periodic boundary condition solved using the Fourier spectral method, as given 11 in Ref. [3], the 3×3 symmetric matrix ​N​D is the demagnetizing factor which depends only on the macroscopic shape of the actual sample (not the shape of the simulation system, e.g., see Table ˉ is the average magnetization of the simulation system. In a finite­size boundary 2.2), and M
condition, space outside the simulation system is considered to be filled by vacuum without magnetization, electric charge, or current etc., and the stray field is solved based on convolution theorem accelerated by FFT[4], without explicitly utilizing demagnetizing factors. Table 2.2 lists the recommended stray field boundary conditions and (when applicable) demagnetizing factors for typical types of systems. Table 2.2​ Recommended stray field boundary condition and demagnetizing factor System type Boundary condition Bulk (periodic) Periodic Thin film (periodic in­plane) Island(s)­on­substrate arrays (periodic in­plane) Single group of island(s)­on­substrate Freestanding finite­size magnet Periodic Periodic Diagonal components (N​D11​, N​D22​, N​D33,​ N​D23​, N​D13​, N​D12​) (0, 0, 0, 0, 0, 0) or (1/3, 1/3, 1/3, 0, 0, 0), or calculated based on the macroscopic shape of the sample (0, 0, 1, 0, 0, 0) (0, 0, 1, 0, 0, 0) Finite­size / Finite­size / 2.4 Magnetocrystalline anisotropy Two types of magnetocrystalline anisotropy are considered, the cubic anisotropy with magnetic easy axes along <100>, <110>, or <111> crystal axes, and the uniaxial anisotropy in ultra­thin films with a magnetic easy/hard axis perpendicular to the film plane. The volume density of magnetocrystalline anisotropy energy is given by Cubic anisotropy (9A) Uniaxial anisotropy (9B) where ​K​ is the magnetocrystalline anisotropy coefficient. The anisotropy effective field is calculated as (10) 12 2.5 Magneto­Elastic interaction The elastic energy density is given by (11) where ​c​ is the elastic stiffness tensor, ​ε​ is the strain, and ​ε0​​ is the stress­free strain calculated as (12) where ​λ100 and λ111 are saturation magnetostriction along <100> and <111> crystalline axes, respectively. The magnetoelastic effective field is written as (13) c(ε − ε0) . where ​σ​ is the stress field given by σ =
Assuming that the elastic equilibrium condition holds at each evolution step, the strain and stress are obtained at each evolution step through solving the mechanical equilibrium equation (14) using a Fourier spectral method[5][6] based on Khachaturyan’s elasticity theory[7]. Table 2.3 summarizes the boundary conditions implemented in typical types of systems. Examples showing the influence of elastic boundary condition on the magnetic domain structure can be found in Ref.[8]. Table 2.3​ Elastic boundary conditions System type Bulk (periodic) Thin film (periodic in­plane) Island(s)­on­substrate Freestanding finite­size magnet Boundary condition 3­d periodic, with specified applied strain​ ε​a or applied stress ​σa​ Thin film boundary condition (see Ref.​ ​[5] for details), with specified a
a
a
in­plane substrate strain​ ε​a (i.e., ε11
, ε22
, and ε12
) Periodic in­plane, stress­free island surfaces, with specified in­plane a
a
a
substrate strain​ ε​a (i.e., ε11
, ε22
, and ε12
) Stress­free An iterative Fourier­Spectral method is used (see details in Ref. [6]) to solve equation (14) in elastically inhomogeneous systems (that is, spatially variant ​c​), including: ● Island­on­substrate system ● freestanding finite­size systems ● Polycrystals ● Film­on­substrate system where the magnet and substrate have different elastic stiffness. 13 A convergence of the iterative approximation is claimed when the difference of total elastic energy between adjacent recursion loops is within a tolerance value Δe (arbitrary unit), i.e.,
(n)
(n)
|F elas
− F (n−1)
| ≤ Δe , where F elas
is the total elastic energy of the ​n­th iterative approximation. If elas
convergence is not reached after the allowed maximum number n​Recurs of recursion loops, the program claims to fail to solve the mechanical equilibrium equation and stops. An output file fort.72 will be generated and updated on solving the mechanical equilibrium equation with inhomogeneous elasticity, containing the total elastic energy (arbitrary unit) in every recursion loop. Both n​Recurs and
Δe are user adjustable (see Section 3) with recommended range of ​
­5​
­3​
100~2000 and 10​ ~10​ , respectively. 2.6 Magnetic exchange interaction The density of magnetic exchange energy is given by (15) where ​A is the exchange constant in the unit of J/m. The exchange field is expressed as (16) 2.7 Dzyaloshinskii­Moriya interaction (DMI) The DMI module in ​μ­Pro​® Mag belongs to the interface type[9], which can be invoked when simulating an ultrathin magnetic thin film or island with perpendicular magnetic anisotropy. Consider a homogenous effective DMI constant ​D, the interface DMI energy density is given by (17) The effective magnetic field due to DMI is therefore expressed as (18) where ​e​3​ is the unit vector along the ​x3​ direction. 14 2.8 Thermal fluctuation The thermal fluctuation field is given by H therm = η
√
2αkBT
μ0M Sγ0ΔV Δt (19) where kB = 1.38064853 × 10−23J ∙ K −1 is the Boltzmann constant, ​T is the Kelvin temperature, ΔV = Δl1 × Δl2 × Δl3 is the volume of a grid, and ​η is a random vector with three independent components ​η1​, ​η2​, and ​η3 all obeying standard normal distribution. Values of these three components are independent at each evolution step. 2.9 Spin torque The spin­orbit torque and Slonczewski spin­transfer torque are given by τ ST =
τ0
(20) 1+α2 (m × (mp × m) − α(mp × m)) Where mp is the normalized fixed­layer magnetization in the case of spin­transfer torque, while represents the direction of spin current generated through Spin Hall Effect; and the pre­factor τ 0 depends on the type of the spin torque. For spin­orbit torque[10], (21) Where μB = 9.27400968 × 10 A ∙ m2 is the Bohr magneton, ​J is the electric current density, θSH is the spin Hall angle, e = 1.6021766209 × 10−19C is the elementary charge. ​d is the thickness of the free magnetization layer, as defined with the type of the system (see Table 2.1). For Slonczewski spin­transfer torque[11][12], −24
(22) where is the spin polarization constant. ​
The effective field corresponding to spin­orbit torque or Slonczewski spin­transfer torque is given by (23) 15 The Zhang­Li spin­transfer torque is given by[13] 1
B
(24) τ ST = 1+α
2 2eM (1+ξ2) ((1 + ξα)m × (m × (J ∙ ∇)m) + (ξ − α)m × (J ∙ ∇)m) ​
μ
S
where ξ is the degree of non­adiabaticity. The effective field corresponding to a Zhang­Li spin­transfer torque is given by (25) 3. Input files Users need to prepare one to five files as input: parameter.in Declares the size of the system, the type of properties considered, properties of each phase, and external fields applied. This file can be written using the GUI provided in ​μ­Pro​®​ package. The format is as follows: Table 3.1​ Format of the input file ​parameter.in Data in the file l​2 n​2 n​f C​Island n​i1 n​i2 l​1 n​1 n​s l​3 n​3 n​i3 C​Grain φ​C θ​C ψ​C M​S γ α Fl​Anis C​Anis K​1 K​2 K​3 Explanation System real size along x​1​, x​2​, and x​3​ directions (nm) Total number of simulation grids along x​1​, x​2​, and x​3​ directions See Section 2.1 See Section 2.1 See Section 2.1 Choice of input type of crystalline grain structure: 0­Euler angles φ, θ, and ψ (°) array read from file ​eulerAng.in; 1­single crystal with specified Euler angles (For C​Grain​=1) Euler angles φ, θ, and ψ (°) of the single crystal orientation M​S​ – saturation magnetization (A/m); γ – electron gyromagnetic ratio (m/(A.s)); α – damping constant (unitless) Flag of whether to consider magnetocrystalline anisotropy (For Fl​Anis​=true) Choice of type of magnetocrystalline anisotropy: 1­cubic; 2­uniaxial (For Fl​Anis​=true and C​Anis​=1) K​1​, K​2​, K​3​ – cubic magnetocrystalline anisotropy coefficient ​K ​ (J/m​3​); 16 Fl​Stray Fl​Per N​D11 N​D23 N​D22 N​D13 N​D33 N​D12 C​Hext H​DC1 H​AC1 f​AC A Fl​Elas λ​100 c​1’1’ H​DC2 H​AC2 λ​111 c​1’2’ H​DC3 H​AC3 c​4’4’ c​s1’1’ c​s1’2’ c​s4’4’ Fl​Strain α ε11
α
( σ11 ) α ε22
α ) ( σ22
α ε12
α ) ( σ12
α ε33
α ) ( σ33
α ε13
α ) ( σ13
α ε23
α ) ( σ23
n​Recurs Δe F​ST C​ST θ​SH η​SP ξ​STT J​1 J​2 J​3 mP 1 mP 2 mP 3 Fl​DMI D Fl​Therm T (For Fl​Anis​=true and C​Anis​=2) K​1​, K​2​ – uniaxial magnetocrystalline anisotropy coefficient ​K ​ (J/m​3​) Flag of whether to consider stray field (For Fl​Stray​=true) Flag of the magnetostatic boundary condition: whether to use a periodic (or otherwise finite­size) boundary (For Fl​Stray​=true and Fl​Per​=true) demagnetizing factor ​N​D​ (unitless) Choice of input type of external magnetic field: 0­an array of field sequence read from file ​hExt.in with possible linear interpolations; 1­DC and AC components defined in following lines, i.e., H ext = H DC + H ACsin(2πf AC ∙ t) (For C​Hext​=1) DC component of external magnetic field ​HDC​
​ (A/m) (For C​Hext​=1) magnitude of AC component of external magnetic field ​HAC​
​ (A/m) (For C​Hext​=1) frequency of AC component of external magnetic field (Hz) Magnetic exchange energy coefficient (J/m) Flag of whether to consider magnetoelastic effect (For Fl​Elas​=true) Saturation magnetostriction ​λ​ (unitless) (For Fl​Elas​=true) Elastic stiffness ​c​ of the magnet in Voigt notation (Pa) (For film or island, and Fl​Elas​=true) Elastic stiffness ​cs​​ of the substrate in Voigt notation (Pa) (For bulk and Fl​Elas​=true) Flag of the mechanical boundary condition: whether to use an applied strain (or otherwise an applied stress) (For film or island, and F​Elas​=true) in­plane substrate strain ​εa​ ; (For bulk, Fl​Elas​=true, and Fl​Strain​=true) in­plane components of applied strains ​εa​ (unitless); (For bulk, Fl​Elas​=true, and Fl​Strain​=false) in­plane components of applied stress ​σa​ (Pa) (For bulk, Fl​Elas​=true, and Fl​Strain​=true) out­of­plane components of applied strains ​εa​ (unitless); (For bulk, Fl​Elas​=true, and Fl​Strain​=false) out­of­plane components of applied stress ​σa​ (Pa) (For Fl​Elas​=true, and with inhomogeneous elastic stiffness in the system) n​Recurs​ – maximum number of recursion loops for elastic solver; (For Fl​Elas​=true, and with inhomogeneous elastic stiffness in the system) Δe – maximum error for elastic solver Flag of whether to consider spin torque (For Fl​ST​=true) type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer­torque (For Fl​ST​=true, and C​ST​=1) spin Hall angle in a spin­orbit torque (For Fl​ST​=true, and C​ST​=2) spin polarization constant in a Slonczewski spin­transfer torque (For Fl​ST​=true, and C​ST​=3) degree of non­adiabaticity in a Zhang­Li spin­transfer torque (For Fl​ST​=true, and C​ST​=1 or 2) J​3​ – spin­polarized electric current density along x​3​ direction For Fl​ST​=true, and C​ST​=3) J​1​, J​2​, J​3​ – spin­polarized electric current density ​J (For Fl​ST​=true, and C​ST​=1 or 2) normalized magnetization ​mP​ in the fixed layer in a spin torque structure (unitless) Flag of whether to consider Dzyaloshinskii­Moriya interaction (DMI) (For Fl​DMI​=true) continuous effective DMI constant Flag of whether to consider thermal fluctuation field (For Fl​Therm​=true) temperature (K) 17 C​LLG Δt k​t0 k​tMax k​tTable k​tDist C​InitialM a​1 a​2 a​3 Fl​OutH Fl​OutE Fl​OutS Type of LLG numerical solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 Time per evolution step (s) k​t0​ – number of the starting evolution step; k​tMax​ – number of the finishing evolution step k​tTable​ – output step interval for data table; k​tDist​ – output step interval for spatial distribution Choice of input type of initial magnetization distribution: 0­an array of magnetization distribution read from file ​Magnt.in; 1­random orientation; 2­specified uniform orientation; 3­specified vortex domain (For C​InitialM​=2) axis of ​a​ uniform orientation of the initial magnetization; (For C​InitialM​=3) axis of ​a​ vortex of the initial magnetization Fl​OutH​ – flag of whether to output distribution of effective fields Fl​OutE​ – flag of whether to output distribution of energy densities (For Fl​Elas​=true) flag of whether to output distribution of eigenstrain, strain, and stress islandShape.in (optional) Contains an array Om2d(x1, x2) ​describing the 2­d in­plane shape of a magnetic island. This file is used only in an island­on­substrate or a finite­size magnet system with C​Island​=0, as defined in the file ​parameter.in. The format is as follows: Table 3.2​ Format of the input file ​islndShape.in 1 ⋮ 1 ⋮ n​1 Data in the file 1 o​m2d​(1, 1) ⋮ n​2 ⋮ n​2 ⋮ o​m2d​ (1, n​2​) ⋮ o​m2d​ (n​1​, n​2​) Explanation If o​m2d​(1, 1)=1, grid points (1, 1, k) where k is within the thickness of the island are considered within the magnetic island, i.e., o​m​(1, 1, k)=1. If o​m2d​(1, 1)=0, grid points (1, 1, k) where k is within the thickness of the island are considered vacuum, i.e., o​m​(1, 1, k)=0. eulerAng.in (optional) Contains an array of the distribution of the Euler angles ϕ(x), θ(x), and ψ(x) of grains in polycrystals, arranged in a row­major order. This file is used only with C​Grain​=0, as defined in the file ​parameter.in. The format is as follows: Table 3.3​ Format of the input file ​eulerAng.in n​1 n​2 n​3 1 1 1 Data in the file φ(1, 1, 1) θ(1, 1, 1) ψ(1, 1, 1) Explanation Total number of simulation grids in each direction φ, θ, ψ – Euler angles of the grain containing the grid point (1, 1, 1) (°) 18 ⋮ 1 1 n​3 φ(1, 1, n​3​) θ(1, 1, n​3​) ψ(1, 1, n​3​) θ(1, n​2​, n​3​) ψ(1, n​2​, n​3​) θ(n​1​, n​2​, n​3​) ψ(n​1​, n​2​, n​3​) ⋮ 1 n​2 n​3 φ(1, n​2​, n​3​) n​1 n​2 n​3 φ(n​1​, n​2​, n​3​) ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ hExt.in (optional) Contains an array of the external magnetic field sequence H ext(kt) . This file is used only with C​Hext​=0, as defined in the file ​parameter.in. The format is as follows: Table 3.4​ Format of the input file ​hExt.in k​t1 Data in the file H​ext1​(k​t1​) H​ext2​(k​t1​) H​ext3​(k​t1​) k​t2 ⋮ k​tn H​ext1​(k​t2​) ⋮ H​ext1​(k​tn​) H​ext2​(k​t2​) ⋮ H​ext2​(k​tn​) H​ext3​(k​t2​) ⋮ H​ext3​(k​tn​) Explanation H​ext1​(k​t1​), H​ext2​(k​t1​), and H​ext3​(k​t1​) – external magnetic field ​H​ext at the k​t1​­th evolution step (A/m) ⋮ ⋮ ⋮ In obtaining H ext(kt) at all evolution steps k​t​, the program considers linear interpolations of H ext(kt) between adjacent evolution steps k​t​ provided in this file. magnt.in (optional) Contains an array of the distribution of initial magnetization m(x, kt0) , arranged in a row­major order. This file is used only with C​InitialM​=0, as defined in the file ​parameter.in. The format is as follows: Table 3.5​ Format of the input file ​magnt.in Data in the file n​1 n​2 n​3 1 1 1 m​1​(1, 1, 1) 1 1 n​3 m​1​(1, 1, n​3​) m​2​(1, 1, 1) m​3​(1, 1, 1) m​2​(1, 1, n​3​) m​3​(1, 1, n​3​) m​2​(1, n​2​, n​3​) m​3​(1, n​2​, n​3​) m​2​(n​1​, n​2​, n​3​) m​3​(n​1​, n​2​, n​3​) ⋮ ⋮ 1 n​2 n​3 m​1​(1, n​2​, n​3​) ⋮ n​1 n​2 n​3 m​1​(n​1​, n​2​, n​3​) Explanation Total number of simulation grids in each direction m​1​, m​2​, m​3 – normalized initial magnetization ​m at grid point (1,1,1) (unitless) ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 4 Ouput files 19 4.1 Files generated before doing evolution oMag.00000000.dat Contains an array of Om(x) (unitless), arranged in a row­major order. The data follow a similar format with those in ​magnt.in. The file is generated before simulating evolution steps. cGlob.00000000.dat (optional) Contains an array of C 11(x), C 22(x), and C 44(x) (Pa) in Voigt notation, arranged in a row­major order. The data follow a similar format with those in ​magnt.in. The file is generated before simulating evolution steps and only under Fl​Elas​=true. hExt.dat Contains an array of the external magnetic field sequence H ext1(kt), H ext2(kt), and H ext3(kt) (A/m) at all evolution steps. The data follow a similar format with those in ​hExt.in. The file is generated before simulating evolution steps. 4.2 Files generated during evolution steps avMagntz.dat Contains an array of average values of the normalized magnetization m
ˉ 1(kt), m
ˉ 2(kt), and m
ˉ 3(kt)
(unitless) inside the magnet at every k​tTable evolution steps. The data follow a similar format with those in ​hExt.in. The file is updated every k​tTable​ steps. avHEff.dat Contains an array of average values of the effective fields H ext1(kt), H ext2(kt), and H ext3(kt) , ˉ eff3(kt), Hˉ stray1(kt), H
ˉ stray2(kt), H
ˉ stray3(kt), H
ˉ anis1(kt), Hˉ anis2(kt), Hˉ anis3(kt), Hˉ eff1(kt), Hˉ eff2(kt), H
ˉ ST3(kt), Hˉ therm1(kt), H
ˉ therm2(kt), Hˉ therm3(kt)
Hˉ elas1(kt), Hˉ elas2(kt), Hˉ elas3(kt), Hˉ ST1(kt), Hˉ ST2(kt), H
(A/m) inside the magnet at every k​tTable evolution steps. The data follow a similar format with those in ​hExt.in. The file is updated every k​tTable​ steps. avStrain.dat (optional) Contains an array of average values of the strain and eigenstrain εˉ11(kt), εˉ22(kt), εˉ33(kt),
0 (k ), ε0 (k ), ε0 (k ), ε0 (k ), ε0 (k ), ε0 (k ) (unitless) inside the εˉ23(kt), εˉ13(kt), εˉ12(kt), ε11
t
22 t
33 t
23 t
13 t
12 t
magnet at every k​tTable evolution steps. The data follow a similar format with those in ​hExt.in. The file is updated every k​tTable​ steps, and only under Fl​Elas​=true. avStress.dat (optional) Contains an array of average values of the stress σˉ11(kt), σˉ 22(kt), σˉ 33(kt), σˉ23(kt), σˉ 13(kt), and σˉ12(kt) (Pa) inside the magnet at every k​tTable evolution steps. The data follow a similar format with those in ​hExt.in. The file is updated every k​tTable​ steps, and only under Fl​Elas​=true. 20 magnt.**.dat Contains an array of m1(x), m2(x), and m3(x) (unitless) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist​ steps. hEff.**.dat (optional) Contains an array of H ′eff1(x), H ′eff2(x), and H ′eff3(x) (A/m) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutH​=true. hStra.**.dat (optional) Contains an array of H stray1(x), H stray2(x), and H stray3(x) (A/m) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutH​=true and Fl​Stray​=true. hAnis.**.dat (optional) Contains an array of H anis1(x), H anis2(x), and H anis3(x) (A/m) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutH​=true and Fl​Anis​=true. hElas.**.dat (optional) Contains an array of H elas1, H elas2, and H elas3 (A/m) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutH​=true and Fl​Elas​=true. hST.**.dat (optional) Contains an array of H ST1(x), H ST2(x), and H ST3(x) (A/m) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutH​=true and Fl​ST​=true. hTher.**.dat (optional) Contains an array of H therm1(x), H therm2(x), and H therm3(x) (A/m) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutH​=true and Fl​Therm​=true. eigStn.**.dat (optional) 21 0 (x), ε0 (x), ε0 (x), ε0 (x), ε0 (x), and ε0 (x) (unitless) at a certain Contains an array of ε11
22
33
23
13
12
evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutS​=true and Fl​Elas​=true. strain.**.dat (optional) Contains an array of ε11(x), ε22(x), ε33(x), ε23(x), ε13(x), and ε12(x) (unitless) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutS​=true and Fl​Elas​=true. stress.**.dat (optional) Contains an array of σ11(x), σ22(x), σ33(x), σ23(x), σ13(x), and σ12(x) (Pa) at a certain evolution step, arranged in a row­major order, where ** represents the 8­digit evolution step number. The data follow a similar format with those in ​magnt.in. A file is generated every k​tDist steps and only under Fl​OutS​=true and Fl​Elas​=true. References [1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
X.­P. Wang, C. J. Garcıá ­Cervera, and W. E, “A Gauss–Seidel Projection Method for Micromagnetics Simulations,” ​J. Comput. Phys., vol. 171, no. 1, pp. 357–372, Jul. 2001. L.­Q. Chen and J. Shen, “Applications of semi­implicit Fourier­spectral method to phase field equations,” ​Comput. Phys. Commun., vol. 108, no. 2–3, pp. 147–158, 1998. J. Zhang and L.­Q. Chen, “Phase­field microelasticity theory and micromagnetic simulations of domain structures in giant magnetostrictive materials,” ​Acta Mater., vol. 53, no. 9, pp. 2845–2855, May 2005. K. Fabian and A. Kirchner, “Three­dimensional micromagnetic calculations for magnetite using FFT,” ​Geophys. J. …, pp. 89–104, 1996. Y. Li, S. Y. Hu, Z.­K. Liu, and L.­Q. Chen, “Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films,” ​Acta Mater., vol. 50, pp. 395–411, 2002. S. Y. Hu and L.­Q. Chen, “A phase­field model for evolving microstructures with strong elastic inhomogeneity,” ​Acta Mater., vol. 49, no. 11, pp. 1879–1890, Jun. 2001. A. G. Khachaturyan, ​Theory of Structural Transformation in Solids. New York: Wiley, 1983. J.­M. Hu, T. N. Yang, L.­Q. Chen, and C.­W. Nan, “Engineering domain structures in nanoscale magnetic thin films via strain,” ​J. Appl. Phys., vol. 114, no. 16, p. 164303, 2013. S. Rohart and A. Thiaville, “Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii­Moriya interaction,” ​Phys. Rev. B ­ Condens. Matter Mater. Phys., vol. 88, no. 18, pp. 1–8, 2013. G. Finocchio, M. Carpentieri, E. Martinez, and B. Azzerboni, “Switching of a single 22 [11]
[12]
[13]
ferromagnetic layer driven by spin Hall effect,” ​Appl. Phys. Lett., vol. 102, no. 21, 2013. J. C. Slonczewski, “Current­driven excitation of magnetic multilayers,” ​J. Magn. Magn. Mater., vol. 159, no. 1–2, pp. L1–L7, Jun. 1996. L. Torres, L. Lopez­Diaz, E. Martinez, M. Carpentieri, and G. Finocchio, “Micromagnetic computations of spin polarized current­driven magnetization processes,” ​J. Magn. Magn. Mater., vol. 286, no. SPEC. ISS., pp. 381–385, 2005. S. Zhang and Z. Li, “Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets,” ​Phys. Rev. Lett., vol. 93, no. 12, pp. 1–4, 2004. 23 MuPro­Mag examples 24 μMag Standard Problem #1 μMag Standard Problem #1 considers the M­H hysteresis loop of a 1μm × 2μm × 20nm
permalloy rectangle with the following material parameters: M S = 8 × 105A/m A = 1.3 × 10−11 J/m K 1 = 5 × 102 J/m3 (Uniaxial anisotropy) on applying a magnetic field approximately parallel to the long and short axis of the magnet. Result The solution using μ­Pro® Mag is presented in the following figures. Since submitted solutions on the μMag website don’t agree with each other, the correctness of the solution couldn’t be tested. Figure M­H hysteresis loop on applying a magnetic field parallel to the long axis (H // x1) and the short axis (H // x3), respectively. 25 Figure Remnant magnetic domain configuration on applying a magnetic field parallel to the long axis (left) and the short axis (right), respectively. Input Parameter.in 1100. 20. 2100. !simulation size (nm) 110 2 210 !# of simulation grids 0 0 !# of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 26 1 !island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 100 2 200 !# of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 !grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. !Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 800.E3 2.211E5 0.5 !saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) true !whether to consider magnetocrystalline anisotropy 2 !type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 5.E2 0. 0. !magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true !whether to consider stray field false !periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 27 0. 0. 0. !demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0. 0. 0. !demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0 !external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. !DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. !peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. !frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 1.3E­11 !magnetic exchange constant (J/m) false !whether to consider magnetoelastic effect 0. 0. !saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. !elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. !elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true !strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 28 0. 0. 0. !in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 0. 0. 0. !out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. !recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false !whether to consider spin torque 1 !type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. !spin Hall angle (only for flagST=true and choiceST=1) 0. !spin polarization constant (only for flagST=true and choiceST=2) 0. !degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. !spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. !magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true and choiceST=1,2) false !whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. !continuous effective DMI constant (J/m^2) (only for flagDMI=true) 29 false !whether to consider thermal fluctuation 0. !temperature (K) (only for flagThermalFluc=true) 1 !type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 1.E­13 !time per evolution step (s) 0 1250000 !starting time step #; finishing time step # 500 50000 !output step interval for data table; output step interval for spatial distribution 1 !initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 0. 0. 0. !axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false !whether to output distribution of effective fields and energy densities false !whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) For magnetic field applied parallel to the long axis hExt.in 4 0 0.E3 0.E3 0.E3 30 250000 39.789E3 0.E3 0.E3 750000 ­39.789E3 0.E3 0.E3 1250000 39.789E3 0.E3 0.E3 For magnetic field applied parallel to the short axis hExt.in 4 0 0.E3 0.E3 0.E3 250000 0.E3 0.E3 39.789E3 750000 0.E3 0.E3 ­39.789E3 1250000 0.E3 0.E3 39.789E3 31 μMag Standard Problem #2 μMag Standard Problem #2 considers the scaling effect on remnant magnetization mR and coercive field H C of a magnet with the size of 5d × d × 0.1d , i.e., mR and H C as a function of d/lex , with exchange length lex = √A/K m , where K m is the magnetostatic energy density defined as K m = 12 μ0M S2 . The magnetic field is applied in the [1,1,1] direction. Result The solution using μ­Pro® Mag is presented in the following figures. The remnant magnetization agrees well with Streibl and Donahue’s solution submitted to the μMag group. 32 Figure Remnant magnetization mx−R ​as a function of d/lex . Figure Remnant magnetization my−R ​ as a function of d/lex . 33 Figure Coercive field H C ​as a function of d/lex . Input For d/lex = 15 parameter.in 825. 225. 15. !simulation size (nm) 110 30 1 !# of simulation grids 0 0 !# of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 34 1 !island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 100 20 1 !# of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 !grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. !Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 500.E3 2.211E5 0.5 !saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) false !whether to consider magnetocrystalline anisotropy 1 !type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 0. 0. 0. !magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true !whether to consider stray field false !periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 0. 0. 0. !demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 35 0. 0. 0. !demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0 !external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. !DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. !peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. !frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 15.708
E­12 !magnetic exchange constant (J/m) !!Exchange length = 10nm false !whether to consider magnetoelastic effect 0. 0. !saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. !elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. !elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true !strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. !in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 36 0. 0. 0. !out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. !recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false !whether to consider spin torque 1 !type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. !spin Hall angle (only for flagST=true and choiceST=1) 0. !spin polarization constant (only for flagST=true and choiceST=2) 0. !degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. !spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. !magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false !whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. !continuous effective DMI constant (J/m^2) (only for flagDMI=true) false !whether to consider thermal fluctuation 37 0. !temperature (K) (only for flagThermalFluc=true) 1 !type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 5.E­14 !time per evolution step (s) 0 1500000 !starting time step #; finishing time step # 2500 125000 !output step interval for data table; output step interval for spatial distribution 1 !initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 1. 0. 0. !axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false !whether to output distribution of effective fields and energy densities false !whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) hExt.in 5 0 50.E3 50.E3 50.E3 124999 50.E3 50.E3 50.E3 125000 0.E3 0.E3 0.E3 250000 0.E3 0.E3 0.E3 1500000 ­50.E3 ­50.E3 ­50.E3 38 μMag Standard Problem #3 μMag Standard Problem #3 models the transition of the stable magnetic domain structure from a flower state to a vortex state on increasing size of a magnet. A cubic magnet with edge length L expressed in units of lex = √A/K m , where $$K_m$$ is the magnetostatic energy density defined as K m = 12 μ0M S2 , is considered. The magnet has a uniaxial magnetocrystalline anisotropy with K u = 0.1 K m and easy axis parrellel to the principle axis x3 of the cube. Result Flower and vortex state domains simulated using μ­Pro® Mag are plotted. Transition size, corresponding average magnetization, and energy densities are listed and compared with solutions submitted to the μMag group[1]. Figure Stable flower state (left) and vortex state (right) magnetic domain structure in cubic magnet with L/lex = 8.40 ​and L/lex = 8.50 , respectively. 39 Table Transition size, corresponding average magnetization, and energy densities Input For L/lex = 8.46 , flower state parameter.in 105.75 105.75 105.75 !simulation size (nm) 50 50 50 !# of simulation grids 0 0 !# of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 1 !island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 40 40 40 !# of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 !grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 40 0. 0. 0. !Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 500.E3 2.211E5 0.5 !saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) true !whether to consider magnetocrystalline anisotropy 2 !type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 15.708E3 0. 0. !magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true !whether to consider stray field false !periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 0. 0. 0. !demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0. 0. 0. !demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 1 !external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. !DC component of external magnetic field (A/m) (only for choiceHExt=1) 41 0. 0. 0. !peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. !frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 15.708E­
12 !magnetic exchange constant (J/m) !!Exchange length = 10nm false !whether to consider magnetoelastic effect 0. 0. !saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. !elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. !elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true !strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. !in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 0. 0. 0. !out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. !recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false !whether to consider spin torque 42 1 !type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. !spin Hall angle (only for flagST=true and choiceST=1) 0. !spin polarization constant (only for flagST=true and choiceST=2) 0. !degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. !spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. !magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false !whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. !continuous effective DMI constant (J/m^2) (only for flagDMI=true) false !whether to consider thermal fluctuation 0. !temperature (K) (only for flagThermalFluc=true) 1 !type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 1.E­14 !time per evolution step (s) 43 0 500000 !starting time step #; finishing time step # 10000 50000 !output step interval for data table; output step interval for spatial distribution 2 !initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 0. 0. 1. !axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false !whether to output distribution of effective fields and energy densities false !whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) For L/lex = 8.46 , vortex state parameter.in 105.75 105.7
5 105.
75 !simulation size (nm) 50 50 50 !# of simulation grids 0 0 !# of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 1 !island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 40 40 40 !# of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is 44 considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 !grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. !Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 500.E3 2.211
E5 0.5 !saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) true !whether to consider magnetocrystalline anisotropy 2 !type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 15.708E3 0. 0. !magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true !whether to consider stray field false !periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 0. 0. 0. !demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0. 0. 0. !demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 1 !external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 45 0. 0. 0. !DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. !peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. !frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 15.708E­1
2 !magnetic exchange constant (J/m) !!Exchange length = 10nm false !whether to consider magnetoelastic effect 0. 0. !saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. !elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. !elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true !strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. !in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 0. 0. 0. !out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. !recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false !whether to consider spin torque 46 1 !type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. !spin Hall angle (only for flagST=true and choiceST=1) 0. !spin polarization constant (only for flagST=true and choiceST=2) 0. !degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. !spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. !magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false !whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. !continuous effective DMI constant (J/m^2) (only for flagDMI=true) false !whether to consider thermal fluctuation 0. !temperature (K) (only for flagThermalFluc=true) 1 !type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 1.E­14 !time per evolution step (s) 0 5000
00 !starting time step #; finishing time step # 47 10000 5000
0 !output step interval for data table; output step interval for spatial distribution 3 !initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain ­1. 0. 0. !axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false !whether to output distribution of effective fields and energy densities false !whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) References [1] ​http://www.ctcms.nist.gov/~rdm/results3.html 48 μMag Standard Problem #4 μMag Standard Problem #4 considers the magnetization dynamics in a of a 500nm × 125nm × 3nm permalloy magnet. The magnet is first relaxed to a equilibrium state by applying a saturating magnetic field along the [1, 1, 1] direction, which is then slowly reduced to zero. A magnetic field is further applied to reverse the magnetization, where the magnetization dynamics is assessed. The following material parameters are adopted: M S = 8×105 A/m A = 1.3×10−11 J/m K = 0 α = 0.02 γ0 = 2.211×105 m/(A.s) Two switching events with different magnetic fields are considered: Field (1): μ0H x = − 24.6 mT , μ0H y = 4.3 mT , μ0H z = 0.0 mT , which is approximately a 25mT field directly 170° counterclockwise from the positive ​x axis. Field (2): μ0H x = − 35.5 mT , μ0H y = − 6.3 mT , μ0H z = 0.0 mT , which is approximately a 36mT field directly 190° counterclockwise from the positive ​x axis. Result Time sequence of magnetization during switching is modeled using μ­Pro® Mag. Results from both the implicit Gauss­Seidel projection method and the Runge Kutta method (RK4) for solution of the LLG equation are shown, both of which agree well with solutions submitted to the μMag group. 49 50 Figure Time sequence of average magnetization on applying field (1). 51 52 Figure Time sequence of average magnetization on applying field (2). Input For solving the LLG equation using the implicit Gauss­Seidel projection method parameter.in 546.875 171.875 3. simulation size (nm) 140 44 1 # of simulation grids 0 0 # of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 53 1 island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 128 32 1 # of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 800.E3 2.211E5 0.02 saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) false whether to consider magnetocrystalline anisotropy 1 type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 0. 0. 0. magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true whether to consider stray field false periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 0. 0. 0. demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 54 0. 0. 0. demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0 external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 13.E­12 magnetic exchange constant (J/m) false whether to consider magnetoelastic effect 0. 0. saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 55 0. 0. 0. out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false whether to consider spin torque 1 type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. spin Hall angle (only for flagST=true and choiceST=1) 0. spin polarization constant (only for flagST=true and choiceST=2) 0. degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. continuous effective DMI constant (J/m^2) (only for flagDMI=true) false whether to consider thermal fluctuation 56 0. temperature (K) (only for flagThermalFluc=true) 1 type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 1.E­14 time per evolution step (s) 0 900000 starting time step #; finishing time step # 100 100000 output step interval for data table; output step interval for spatial distribution 2 initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 1. 1. 1. axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false whether to output distribution of effective fields and energy densities false whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) For switching under field (1) hExt.in 4 0 1.E5 1.E5 1.E5 400000 0.E3 0.E3 0.E3 800000 0.E3 0.E3 0.E3 57 800001 ­19.57606E3 3.42183E3 0.E3 For switching under field (2) hExt.in 4 0 1.E5 1.E5 1.E5 400000 0.E3 0.E3 0.E3 800000 0.E3 0.E3 0.E3 800001 ­28.2500E3 ­5.01338E3 0.E3 58 μMag Standard Problem #5 μMag Standard Problem #5 considers effect the Zhang­Li spin­transfer­torque. A spin­polarized electric current of j = 1012 A/m along x1 direction is applied to 100nm × 100nm × 10nm permalloy magnet with an initial vortex structure. The following parameters are adopted: M S = 8 × 105A/m A = 1.3 × 10−11J/m K = 0 α = 0.1 γ0 = 2.211 × 105m/(A ∙ s) Result The obtained magnetic vortex structure and time­dependent average magnetization is shown below. It agrees well with the OOMMF solution. Figure Initial vortex structure (left) and the vortex structure after applying the current for 5ns (right). 59 Figure Time­dependent average magnetization under the applied electric current. Input For initial relaxation of the magnetic vortex domain structure parameter.in 125. 125. 1
0. simulation size (nm) 50 50 2 # of simulation grids 60 0 0 # of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 1 island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 40 40 2 # of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 800.E3 2.211
E5 0.
1 saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) false whether to consider magnetocrystalline anisotropy 1 type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 0. 0. 0. magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true whether to consider stray field false periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 61 0. 0. 0. demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0. 0. 0. demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 1 external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 1.3E­11 magnetic exchange constant (J/m) false whether to consider magnetoelastic effect 0. 0. saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 62 0. 0. 0. out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false whether to consider spin torque 1 type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. spin Hall angle (only for flagST=true and choiceST=1) 0. spin polarization constant (only for flagST=true and choiceST=2) 0. degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. continuous effective DMI constant (J/m^2) (only for flagDMI=true) false whether to consider thermal fluctuation 0. temperature (K) (only for flagThermalFluc=true) 63 1 type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 1.E­13 time per evolution step (s) 0 1000
00 starting time step #; finishing time step # 100 1000
0 output step interval for data table; output step interval for spatial distribution 3 initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 0. 0. 1. axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false whether to output distribution of effective fields and energy densities false whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) For domain evolution under the applied electric current parameter.in 125. 125. 10. simulation size (nm) 50 50 2 # of simulation grids 0 0 # of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 1 island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 64 40 40 2 # of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 800.E3 2.211E5 0.1 saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) false whether to consider magnetocrystalline anisotropy 1 type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 0. 0. 0. magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true whether to consider stray field false periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 0. 0. 0. demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0. 0. 0. demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 1 external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 65 0. frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 1.3E­1
1 magnetic exchange constant (J/m) false whether to consider magnetoelastic effect 0. 0. saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 0. 0. 0. out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) true whether to consider spin torque 3 type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. spin Hall angle (only for flagST=true and choiceST=1) 0. spin polarization constant (only for flagST=true and choiceST=2) 0.05 degree of non­adiabaticity (only for flagST=true and choiceST=3) 66 1.E12 0. 0. spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. continuous effective DMI constant (J/m^2) (only for flagDMI=true) false whether to consider thermal fluctuation 0. temperature (K) (only for flagThermalFluc=true) 1 type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 1.E­13 time per evolution step (s) 0 50000 starting time step #; finishing time step # 100 10000 output step interval for data table; output step interval for spatial distribution 0 initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 0. 0. 0. axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false whether to output distribution of effective fields and energy densities false whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) 67 Example #6 This example considers the relaxed magnetic domain structure of a 100nm­thick nickel thin film. The nickel is considered isotropic, with the following parameters adopted: M S = 4.85×105A/m A = 8.2×10−12J/m K = 0 α = 0.3 γ0 = 2.42×105m/(A.s) λ100 = λ111 =− 32.9×10−6 c11 = 246.5 GP a c12 = 147.3GP a c44 = 49.6GP a The substrate is assumed to possess an elastic stiffness equal to the thin film magnet. The effect of biaxial substrate mismatch strains εS(= ε11 = ε22) are investigated. Result The obtained equilibrium magnetization distribution at εS = 0.003 and εS = 0.004 are shown below. The out of plane component of the magnetization is measured by calculating $$$$ averaged throughout the sample, which shows a trend to increase with increasing εS . 68 Figure Equilibrium magnetization distribution, which shows striped out­of­plane domains with Neel type domain walls at εS = 0.004 , and a similar structure with relatively smaller out­of­plane component at εS = 0.003 . 69 Figure Average out­of­plane magnetization component $$$$ of the equilibrium magnetization as a function of substrate strain εS . It shows that $$$$ increases with increasing εS and dominates at εS > 0.004 . Input For biaxial substrate strain εS > 0.004 parameter.in 3000. 3000. 250. simulation size (nm) 300 300 25 # of simulation grids 11 10 # of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 70 1 island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 0 0 0 # of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 4.85E5 2.42E
5 0.3 saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) false whether to consider magnetocrystalline anisotropy 1 type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 0. 0. 0. magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true whether to consider stray field true periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 0. 0. 0. demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0. 0. 0. demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 1 external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. DC component of external magnetic field (A/m) (only for choiceHExt=1) 71 0. 0. 0. peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 8.2E­12 magnetic exchange constant (J/m) true whether to consider magnetoelastic effect ­32.9E­6 ­32.9
E­6 saturation magnetostriction (unitless) (only for flagElastic=true) 246.5E9 147.3
E9 49.6
E9 elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 246.5E9 147.3
E9 49.6
E9 elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0.0040 0.004
0 0. in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 0. 0. 0. out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 1000 1.E­4 recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false whether to consider spin torque 1 type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. spin Hall angle (only for flagST=true and choiceST=1) 0. spin polarization constant (only for flagST=true and choiceST=2) 0. degree of non­adiabaticity (only for flagST=true and choiceST=3) 72 0. 0. 0. spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. continuous effective DMI constant (J/m^2) (only for flagDMI=true) false whether to consider thermal fluctuation 0. temperature (K) (only for flagThermalFluc=true) 1 type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 5.E­13 time per evolution step (s) 0 50000 starting time step #; finishing time step # 1000 10000 output step interval for data table; output step interval for spatial distribution 1 initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 1. 0. 0. axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false whether to output distribution of effective fields and energy densities false whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) 73 Example #7 This is an example following the calculations in Ref. [1]. It studies the transition of a Bloch wall to a Neel wall on increasing coefficient D of DMI, in a 300nm × 500nm × 0.6nm magnet with the following parameters adopted: M S = 1.1×106 A/m A = 1.6×10−11J/m K 1 = 1.27×106J/m3 (Uniaxial anisotropy) Result The obtained magnetic domain wall moments (the integrated magnetization m across the domain wall ϕi =
∫
x1∖across∖domain∖wall
midx1 as a function of D are shown below. It agrees well with the solutions given by q­Φ model [1] and Mumax [2]. 74 Figure Simulated magnetization distribution across the domain wall. 75 Figure Simulated domain wall moments as a function of D. Input We start from a two domain structure with a sharp wall, and let the domain wall configuration first relax at D = 0, and then re­stabilize at certain D. For structure relaxation at D = 0 parameter.in 310. 500. 0.
6 simulation size (nm) 310 1 1 # of simulation grids 76 0 0 # of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 1 island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 300 1 1 # of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 1.1E6 2.42
E5 0.
3 saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) true whether to consider magnetocrystalline anisotropy 2 type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 1.27E6 0. 0. magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true whether to consider stray field false periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 77 0. 0. 0. demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0. 0. 0. demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 1 external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 1.6E­11 magnetic exchange constant (J/m) false whether to consider magnetoelastic effect 0. 0. saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 78 0. 0. 0. out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false whether to consider spin torque 1 type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. spin Hall angle (only for flagST=true and choiceST=1) 0. spin polarization constant (only for flagST=true and choiceST=2) 0. degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. continuous effective DMI constant (J/m^2) (only for flagDMI=true) false whether to consider thermal fluctuation 0. temperature (K) (only for flagThermalFluc=true) 79 1 type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 5.E­14 time per evolution step (s) 0 5000
00 starting time step #; finishing time step # 5000 5000
0 output step interval for data table; output step interval for spatial distribution 3 initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 0.01 1. 0.
01 axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false whether to output distribution of effective fields and energy densities false whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) For subsequent evolution at D = 0.08 mJ/m2 parameter.in 310. 500. 0.
6 simulation size (nm) 310 1 1 # of simulation grids 0 0 # of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 80 1 island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 300 1 1 # of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 1 grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 1.1E6 2.42
E5 0.
3 saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) true whether to consider magnetocrystalline anisotropy 2 type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 1.27E6 0. 0. magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true whether to consider stray field false periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 0. 0. 0. demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 0. 0. 0. demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 81 1 external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 1.6E­11 magnetic exchange constant (J/m) false whether to consider magnetoelastic effect 0. 0. saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 0. 0. 0. out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) 82 false whether to consider spin torque 1 type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. spin Hall angle (only for flagST=true and choiceST=1) 0. spin polarization constant (only for flagST=true and choiceST=2) 0. degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) true whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0.08E­3 continuous effective DMI constant (J/m^2) (only for flagDMI=true) false whether to consider thermal fluctuation 0. temperature (K) (only for flagThermalFluc=true) 1 type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 5.E­14 time per evolution step (s) 83 0 5000
00 starting time step #; finishing time step # 5000 5000
0 output step interval for data table; output step interval for spatial distribution 0 initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 0. 0. 0. axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false whether to output distribution of effective fields and energy densities false whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) References [1] Thiaville, A., Rohart, S., Jué, É., Cros, V., & Fert, A. (2012). Dynamics of Dzyaloshinskii domain walls in ultrathin magnetic films. EPL (Europhysics Letters), 100(5), 57002. [2] Vansteenkiste, A., Leliaert, J., Dvornik, M., Helsen, M., Garcia­Sanchez, F., & Van Waeyenberge, B. (2014). The design and verification of MuMax3. AIP Advances, 4(10), 0–22. 84 Example #8 This example demonstrates the magnetic domain structure in a single crystal or polycrystalline 850nm (minor axis) × 2200nm (major axis) × 5nm (thickness) elliptical cylinder permalloy island with the following parameters adopted: M S = 8.0×105 A/m A = 1.3×10−11 J/m Cubic magnetocrystalline anisotropy with different K 1 are used and compared with each other. Result 85 Figure Simulated magnetization distribution in polycrystal and single crystal islands with different K 1 . Input parameter.in 900. 2250. 5. simulation size (nm) 180 450 1 # of simulation grids 0 0 # of grid layers of substrate and film (if nf=0, the system is considered either a bulk or an island) (ns only for nf/=0 or ni3/=0) 86 2 island in­plane shape: 0­2d array in 'islandShape.in' 1­rectangular or 2­elliptical shape defined in the next line (only for nf=0 and ni3/=0) 170 440 1 # of grids number of grids in the magnetic island (only for nf=0; if nf=0 and ni3=0, the system is considered bulk; ni1 and ni2 only for ni3≠0 and choiceIslandShape=1,2) 0 grain structure: 0­Euler angles phi, theta and psi (degree) array in 'eulerAng.in' 1­single crystal with specified Euler angles 0. 0. 0. Euler angles phi, theta and psi (degree) of the single crystal orientation (only for choiceGrainStruct=1) 800.E3 2.211
E5 0.02 saturation magnetization (A/m), electron gyromagnetic ratio (m/(A.s)), & damping constant (unitless) true whether to consider magnetocrystalline anisotropy 1 type of magnetocrystalline anisotropy: 1­cubic 2­uniaxial (only for flagAnisotropy=true) 1.E5 0.E5 0.E5 magnetocrystalline anisotropy coefficient (J/m^3) (only for flagAnisotropy=true; kc3 only for choiceAnisotropy=1) true whether to consider stray field false periodic (or finite­size) magnetostatic boundary? (only for flagStrayField=true) 0. 0. 0. demagnetizing factor N11, N22, N33 (unitless) (only for flagStrayField=true and flagPeriodic=true) 87 0. 0. 0. demagnetizing factor N23, N13, N12 (unitless) (only for flagStrayField=true and flagPeriodic=true) 1 external magnetic field: 0­array in 'hExt.in' with linear interpolations 1­DC & AC components defined in following lines 0. 0. 0. DC component of external magnetic field (A/m) (only for choiceHExt=1) 0. 0. 0. peak amplitude of AC component of external magnetic field (A/m) (only for choiceHExt=1) 0. frequency of AC component of external magnetic field (Hz) (only for choiceHExt=1) 13.E­12 magnetic exchange constant (J/m) !!Exchange length = 10nm false whether to consider magnetoelastic effect 0. 0. saturation magnetostriction (unitless) (only for flagElastic=true) 0. 0. 0. elastic stiffness (Voigt) (Pa) (only for flagElastic=true) 0. 0. 0. elastic stiffness of substrate (Voigt) (Pa) (only for flagElastic=true) true strain (or stress) elastic boundary condition? (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0. 0. 0. in­plane substrate/applied strains/stress (1 or Pa) (only for flagElastic=true) 88 0. 0. 0. out­of­plane applied strains/stress (1 or Pa) (only for bulk, i.e. nf=0 and ni3=0, and flagElastic=true) 0 0. recursion limit & tolerance of convergence for iterative elastic solver (only when the elastic stiffness of the system is inhomogeneous) false whether to consider spin torque 1 type of spin torque: 1­spin­orbit torque 2­Slonczewski spin­transfer torque 3­Zhang­Li spin­transfer torque (only for flagST=true) 0. spin Hall angle (only for flagST=true and choiceST=1) 0. spin polarization constant (only for flagST=true and choiceST=2) 0. degree of non­adiabaticity (only for flagST=true and choiceST=3) 0. 0. 0. spin­polarized electric current density (only for flagST=true) (jElect(1) and jElect(2) only for choiceST=3) 0. 0. 0. magnetization in the fixed layer in a spin torque structure (unitless) (only for flagST=true) false whether to consider Dzyaloshinskii­Moriya interaction (DMI) 0. continuous effective DMI constant (J/m^2) (only for flagDMI=true) false whether to consider thermal fluctuation 89 0. temperature (K) (only for flagThermalFluc=true) 1 type of LLG Solver: 1­implicit Gauss­Seidel using Fourier spectral 2­explicit RK4 5.E­13 time per evolution step (s) 0 1000
00 starting time step #; finishing time step # 1000 1000
0 output step interval for data table; output step interval for spatial distribution 2 initial m distribution: 0­'magnt.in' 1­random orientation 2­specified uniform orientation 3­specified vortex domain 0. 1. 0. axis of a uniform orientation, or a vortex domain, for the initial m distribution (only for choiceInitialM=2,3) false false whether to output distribution of effective fields and energy densities false whether to output distribution of eigenstrain, strain, and stress (only for flagElastic=true) We also used the input file eulerAng.in as attached, which is generated by a grain growth program. 90