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Low (Reduced) dimensionality of magnetic materials can lead to interesting physical effects:
The spatial arrangement of atoms with a permanent magnetic moment requires consideration of the orientation of their magnetic moments with respect to one another. In bulk samples of magnetic materials energy considerations dictate that the material is organized into small domains in which the magnetic moments are aligned, but the axes of the domains are randomly oriented with respect to one another. In quasitwo dimensional films energy may be minimized by an orientation of the magnetic moments of the atoms perpendicular to the plane of the film.
Howson develops a model of the energy of the two types of magnetic structures for films. Although Howson's article is excellent there are some points at which its derivations are obscure. Below I have tried to make the derivation clearer and, in some cases, more rigorous.
The total energy density of a wall E_{w} is the sum of the exchange energy E_{ex} and uniaxial anisotropy energy E_{K}. Thus,
where δ is the domain wall thickness and A and K are parameters characteristic of the material. The wall energy density E_{w} is minimizes when
which Howson asserts is approximately equal to 4(AK)^{1/2}, although 2.5 would be a better choice than 4 for the approximation. While Howson made a small error in presenting the formula for δ his approximation for E_{w} is based upon the correct formula. Howson notes that for iron the optimal domain wall thickness is about 50 nm.
The selfenergy of a sample with magnetization M and an internal field H_{D} can be expressed as:
where the coefficient D depends upon the fraction of the sample's area where magnetization terminates.
For a single domain film with the magnetization axis perpendicular to the plane of the film D is essentially equal to 1.0, but if the film is partitioned into many subdomain the value of D becomes small. Small subdomains would reduce the selfenergy but lead to a large exchange energy from the energy of the walls of the domain. If t is the thickness of the film, and W and L are the dimensions of the sample surface then the volume of the sample is tWL. The selfenergy is then (1/2)DM^{2}tWL. The energy in the domain walls will depend upon the cross section of the domains. For purposes of this derivation it is assumed that the domains are square prisms of width d and height t. The areal energy density of the domain walls was found to be approximately 4(AK)^{1/2}. The area of a square prism is 4td, but to eliminate double counting we should multiply the areal density by 2td. The final factor is the number of domains, which is equal to the area of the sample, WL, divided by the cross section area of a prism, d^{2}. Thus the energy of the domain walls is:
The coefficient D in the selfenergy term is a function of d and t; specifically for square prisms it is,
Howson takes the case in which d is much less than t, in which case D is approximately (1/2)(d/t). The total energy is then:
When this is divided by the sample area WL the result is that, for d much smaller than t
The domain width d may be chosen to minimize E. The value of d which minimizes E is proportional to t^{1/2} and the minimum E is also proportional to t^{1/2}.
For the case where d is not necessarily much smaller than t the functional relation between E and d is:
The accompanying graph shows the relationship of E to d.
Hexagonal prisms would probably allow a lower energy than square prisms, the essential relationships are captured by the mathematically simpler square prisms. Generally the energy of the domain walls will be decreased if the perimeter enclosing a minimum area is minimized. A circle would provide the absolute minimum but an area cannot be divided into circular areas without leaving out some of the area. But a plane can be exactly partitioned by either triangles, rectangles or hexagons. Among the class of rectangles the perimeter is minimized for a given enclosed area if the rectangles are squares.
Howson argues that for very thin films there is a lower energy if the axis of magnetization is normal to the plane of the film. The source of the equation upon which he bases his argument is not clear.
Source:
Mark Howson, "Magnetism of thin films and multilayers,"Contemporary Physics (1994) Vol 35, no. 5, pp. 347359.
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