μMAG Standard Problem #2 studies the switching behaviour of a Permalloy thin-film rectangular element (length $5d$, width $d$, thickness $0.1d$) as a function of the particle size $d$ measured in exchange lengths $l_{\mathrm{ex}} = \sqrt{2A / \mu_0 M_s^2}$. A saturating field is applied along the diagonal of the film (1,1,0.1) direction, then reversed to obtain the hysteresis loop.

The key results are:

• $H_c / M_s$ — coercive field (normalised by $M_s$)
• $M_{rx} / M_s$ — remanent magnetisation, $x$-component
• $M_{ry} / M_s$ — remanent magnetisation, $y$-component

In the small-particle limit ($d/l_{\mathrm{ex}} \to 0$) the magnetisation is nearly uniform and a 3D Stoner-Wohlfarth analysis gives $H_c / M_s = 0.05707$.

μMAG Standard Problem #3 determines the single domain limit of a cubic magnetic particle — the critical edge length $L$ (in units of exchange length $l_{\mathrm{ex}} = \sqrt{A / K_m}$, where $K_m = \frac{1}{2} \mu_0 M_s^2$) at which the “flower state” and “vortex state” have equal total energy.

The uniaxial anisotropy constant is $K_u = 0.1,K_m$ with the easy axis along $z$. All energies are normalised by $K_m L^3$.

The expected result is $L \approx 8.47,l_{\mathrm{ex}}$.

μMAG Standard Problem #4 studies the time-dependent magnetisation reversal of a thin-film Permalloy rectangle (500 nm × 125 nm × 3 nm) with material parameters $M_s$ = 8 × 10⁵ A/m, $A$ = 1.3 × 10⁻¹¹ J/m, $\alpha$ = 0.02.

The system is initialised in an S-state (obtained by relaxing from saturation along the [1, 1, 1] direction) and then subjected to a reversing field:

• Field 1: $\mu_0 H$ = −25 mT at 170° from the $+x$ axis
• Field 2: $\mu_0 H$ = −36 mT at 190° from the $+x$ axis

The reported quantities are the spatially averaged magnetisation components $\langle M_x \rangle / M_s$, $\langle M_y \rangle / M_s$, $\langle M_z \rangle / M_s$ as a function of time.

The Heistracher et al. standard problem tests domain wall pinning at the interface of a two-phase magnetic rod. It is sensitive to discontinuities in the exchange constant $A$, uniaxial anisotropy $K$, and spontaneous polarisation $J_s$ between phases, making it a targeted test for correct interface handling.

P. Heistracher, C. Abert, F. Bruckner, T. Schrefl, D. Suess, J. Magn. Magn. Mater. 548, 168875 (2022).

Problem setup#

A cuboid rod (80 nm × 1 nm × 1 nm) is divided into two equal halves: