In electromagnetism, the Stoner–Wohlfarth model is a widely used model for the magnetization of ferromagnets with a single domain. It is a simple example of magnetic hysteresis and is useful for modeling small magnetic particles in magnetic storage, biomagnetism, rock magnetism and paleomagnetism.
History The Stoner–Wohlfarth model was developed by Edmund Clifton Stoner and Erich Peter Wohlfarth and published in 1948. It included a numerical calculation of the integrated response of randomly oriented magnets. Since this was done before computers were widely available, they resorted to trigonometric tables and hand calculations.
Description
In the Stoner–Wohlfarth model, the magnetization does not vary within the ferromagnet and it is represented by a vector M. This vector rotates as the magnetic field H changes. The magnetic field is only varied along a single axis; its scalar value h is positive in one direction and negative in the opposite direction. The ferromagnet is assumed to have a uniaxial magnetic anisotropy with anisotropy parameter Ku. As the magnetic field varies, the magnetization is restricted to the plane containing the magnetic field direction and the easy axis. It can therefore be represented by a single angle φ, the angle between the magnetization and the field (Figure 1). Also specified is the angle θ between the field and the easy axis.
Equations The energy of the system is
where V is the volume of the magnet, Ms is the saturation magnetization, and μ0 is the vacuum permeability. The first term is the magnetic anisotropy and the second the energy of coupling with the applied field (often called the Zeeman energy). Stoner and Wohlfarth normalized this equation:
where h = μ0MsH/2Ku. A given magnetization direction is in mechanical equilibrium if the forces on it are zero. This occurs when the first derivative of the energy with respect to the magnetization direction is zero:
This direction is stable against perturbations when it is at an energy minimum, having a positive second derivative:
In zero field the magnetic anisotropy term is minimized when the magnetization is aligned with the easy axis. In a large field, the magnetization is pointed towards the field.
Hysteresis loops
For each angle θ between easy axis and field, equation (3) has a solution that consists of two solution curves. It is trivial to solve for these curves by varying φ and solving for h. There is one curve for φ between 0 and π and another for φ between π and 2π; the solutions at φ = 0 and π correspond to h = ±∞. Since the magnetization in the direction of the field is Ms cos φ, these curves are usually plotted in the normalized form mh vs. h, where mh = cos φ is the component of magnetization in the direction of the field. An example is shown in Figure 2. The solid red and blue curves connect stable magnetization directions. For fields −1/2 ≤ h ≤ 1/2, the two curves overlap and there are two stable directions. This is the region where hysteresis occurs. Three energy profiles are included (insets). The red and blue stars are the stable magnetization directions, corresponding to energy minima. Where the vertical dashed lines intersect the red and blue dashed lines, the magnetization directions are energy maxima and determine the energy barriers between states. In an ordinary magnetic hysteresis measurement, h starts at a large positive value and is decreased to a large negative value. The magnetization direction starts on the blue curve. At h = 0.5 the red curve appears, but for h > 0 the blue state has a lower energy because it is closer to the direction of the magnetic field. When the field becomes negative, the red state has the lower energy, but the magnetization cannot immediately jump to this new direction because there is an energy barrier in between (see the insets). At h = −0.5, however, the energy barrier disappears, and in more negative fields the blue state no longer exists. It must therefore jump to the red state. After this jump, the magnetization remains on the red curve until the field increases past h = 0.5, where it jumps to the blue curve. Usually only the hysteresis loop is plotted; the energy maxima are only of interest if the effect of thermal fluctuations is calculated. The energy barriers are of importance in determining the behavior of superparamagnetic systems, for example, in ensuring magnetic recording media have sufficient stability. An explicit solution was found in 2009 for the magnetizations and the energy barriers for a given applied field. The Stoner–Wohlfarth model is a classic example of magnetic hysteresis. The loop is symmetric (by a 180° rotation) about the origin and jumps occur at h = ± hs, where hs is known as the switching field. All the hysteresis occurs at ± hs.
Dependence on field direction
The shape of the hysteresis loop has a strong dependence on the angle between the magnetic field and the easy axis (Figure 3). If the two are parallel (θ = 0), the hysteresis loop is at its biggest (with mh = hs = 1 in normalized units). The magnetization starts parallel to the field and does not rotate until it becomes unstable and jumps to the opposite direction. In general, the larger the angle, the more reversible rotation occurs. At the other extreme of θ = 90°, with the field perpendicular to the easy axis, no jump occurs. The magnetization rotates continuously from one direction to the other (it has two choices of rotation direction, though). For a given angle θ, the switching field is the point where the solution switches from an energy minimum (∂2η/∂ φ2 > 0) to an energy maximum (∂2η/∂ φ2 < 0). Thus, it can be calculated directly by solving equation (3) along with ∂2η/∂ φ2 = 0. The solution is
where
In normalized units, 0.5 ≤ hs ≤ 1. An alternative way of representing the switching field solution is to divide the vector field h into a component h|| = h cos θ that is parallel to the easy axis, and a component h⊥ = h sin θ that is perpendicular. Then
If the components are plotted against each other, the result is a Stoner–Wohlfarth astroid. A magnetic hysteresis loop can be calculated by applying a geometric construction to this astroid.
Predictions for homogeneous, isotropic systems
Hysteresis
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