Material parameters

All material parameters may be specified per cell, i.e. as an array with one entry per grid cell, which is what makes it possible to model composite materials, grain structures and phase boundaries. In Matlab a scalar value is expanded to a full array by struct(problem); in Python a scalar passed to the constructor is expanded on assignment.

Matlab	Python	Unit	Description
`Ms`	`Ms`	A/m	Saturation magnetization. Enters the demagnetization field directly and scales the exchange and anisotropy fields.
`A0`	`A0`	J/m	Exchange stiffness constant. Where two cells with different values share a face, the coupling between them is the harmonic mean of the two, which can be overridden as seen in Exchange across a material interface.
`K0`	`K0`	J/m³	Uniaxial anisotropy constant, used together with the easy axis `u_ea`.
`K1`, `K2`	`K1`, `K2`	J/m³	Cubic anisotropy constants, used together with `CrysAxis`.
`K0_arr`	`K0_arr`	J/m³	General anisotropy expansion, an `(ntot,6,3)` array.
`CrysAxis`	`CrysAxis`	-	Local crystal coordinate system, an `(ntot,3,3)` array.
`u_ea`	`u_ea`	-	Uniaxial easy-axis direction of every cell, an `(ntot,3)` array.
`alpha`	`alpha`	m/(A s)	Damping constant of the Landau-Lifshitz equation.
`gamma`	`gamma`	m/(A s)	Precession constant. Set to zero to remove precession.
`temperature`	`T`	K	Per-cell temperature, see Thermal fluctuations.

Internally the solver forms the scaled coefficients \(J = A_0/(\mu_0 M_s)\), \(K = K_0/(\mu_0 M_s)\) and \(M = M_s\), so that all field terms come out in A/m.

Note

For the tetrahedron and unstructuredPrisms grids the exchange prefactor is normalised by the largest value of \(A_0\) in the mesh, and the spatial variation of \(A_0\) is instead carried inside the exchange operator itself. On a uniform grid the local \(A_0\) enters the prefactor directly.

The exchange length that sets the required cell size follows from \(A_0\) and \(M_s\),

\[l_\mathrm{ex} = \sqrt{\frac{A_0}{\tfrac{1}{2}\mu_0 M_s^2}} .\]

Magnetocrystalline anisotropy

MagTense offers three ways of specifying the anisotropy. The cubic constants K1/K2 and the general expansion K0_arr are expressed in the same rotated crystal frame and are summed, so they can be given together. The uniaxial constant K0 is its own path with its own easy axis; a uniaxial term that is to be combined with the cubic or the general expansion is written into K0_arr(:,1,:) instead, as shown under General anisotropy, and the solver points this out if K0 is set together with the others.

Uniaxial anisotropy

Specify a local anisotropy constant K0(i) at every grid point together with an easy axis u_ea(i,:) = [easyX(i), easyY(i), easyZ(i)]. The anisotropy field is then

\[\mathbf{H}_\mathrm{ani} = \frac{2 K_0}{\mu_0 M_s} \left( \mathbf{u}_\mathrm{ea}\cdot\mathbf{m} \right) \mathbf{u}_\mathrm{ea},\]

i.e. a positive \(K_0\) makes \(\mathbf{u}_\mathrm{ea}\) an easy axis. The easy axis is given directly in the global coordinate system, per cell.

Cubic anisotropy

Specify K1(i) and K2(i) at every grid point together with a local coordinate system

\[\mathrm{CrysAxis}(i,:,:) = \begin{pmatrix} x_1 & x_2 & x_3 \ y_1 & y_2 & y_3 \ z_1 & z_2 & z_3 \end{pmatrix},\]

whose rows are the crystal axes of cell \(i\) expressed in the global coordinate system. The default is the three Cartesian axes. The energy density follows the standard convention

\[E_\mathrm{cubic} = K_1 \left( m_x^2m_y^2 + m_y^2m_z^2 + m_z^2m_x^2 \right) + K_2\, m_x^2m_y^2m_z^2 ,\]

with \(m\) expressed in the local crystal frame, so that \(K_1 > 0\) gives easy axes along \(\langle 100 \rangle\).

General anisotropy

If the anisotropy is neither uniaxial nor cubic, a general matrix formulation is available. The anisotropy energy density is written as a polynomial expansion [following Eq. (2) in https://doi.org/10.1088/1361-665X/aafff8, generalized to independent coordinates]

\[E_\mathrm{an}(\mathbf{m}) = - \big[ \alpha_{1,x}m_x^2 + \alpha_{1,y}m_y^2 + \alpha_{1,z}m_z^2 + \alpha_{11,x}m_x^4 + \alpha_{11,y}m_y^4 + \alpha_{11,z}m_z^4 \ + \alpha_{12,x}m_y^2m_z^2 + \alpha_{12,y}m_z^2m_x^2 + \alpha_{12,z}m_x^2m_y^2 + \alpha_{111,x}m_x^6 + \alpha_{111,y}m_y^6 + \alpha_{111,z}m_z^6 \ + \alpha_{112,x}m_x^4(m_y^2 + m_z^2) + \alpha_{112,y}m_y^4(m_z^2 + m_x^2) + \alpha_{112,z}m_z^4(m_x^2 + m_y^2) \ + \alpha_{123}(m_x^2m_y^2m_z^2) \big],\]

again with \(\mathbf{m}\) in the local crystal frame given by CrysAxis. The anisotropy field is \(\mathbf{H}_\mathrm{ani} = -\frac{1}{\mu_0 M_s}\,\partial E_\mathrm{an}/\partial \mathbf{m}\). The overall minus sign in front of the bracket is what makes a positive coefficient an easy direction, consistent with the uniaxial convention above.

The coefficients are specified per grid point as a 6x3 matrix

\[K_{0,arr}(i,:,:) = \begin{pmatrix} \alpha_{1,x} & \alpha_{1,y} & \alpha_{1,z} \ \alpha_{11,x} & \alpha_{11,y} & \alpha_{11,z} \ \alpha_{12,x} & \alpha_{12,y} & \alpha_{12,z} \ \alpha_{111,x} & \alpha_{111,y} & \alpha_{111,z} \ \alpha_{112,x} & \alpha_{112,y} & \alpha_{112,z} \ \alpha_{123} & 0 & 0 \end{pmatrix}.\]

In this notation a uniaxial anisotropy along the local z-direction is

\[K_{0,arr} = \begin{pmatrix} 0 & 0 & K_0 \ 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix},\]

and a cubic anisotropy with the standard sign convention is

\[K_{0,arr} = \begin{pmatrix} 0 & 0 & 0 \ 0 & 0 & 0 \ -K_1 & -K_1 & -K_1 \ 0 & 0 & 0 \ 0 & 0 & 0 \ -K_2 & 0 & 0 \end{pmatrix},\]

which is exactly what the solver inserts when K1 and K2 are given directly. Because the two are added, an arbitrary higher-order term may be placed in K0_arr while the cubic part is still given through K1 and K2.

Time-dependent damping

The damping constant may be made a function of time. This is used to speed up relaxation towards an equilibrium state: a large damping early on kills the precession quickly, and it is then lowered towards the physical value.

If alpha is set to zero, the solver instead interpolates the tabulated values in alphat. In Matlab this table is filled by

problem = problem.setAlpha( @(t) alpha_of_t(t), t_alpha );

and in Python through the t_alpha and alpha_fct constructor arguments. A non-zero alpha is used as a constant damping.