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Magnetism of Thin Films and Multilayers
Source: Mark Howson,
"Magnetism of thin films and multilayers,"Contemporary Physics
(1994) Vol 35, no 5, pp. 347-359.
Low (Reduced) dimensionality of magnetic materials can lead to interesting
physical effects:
- Thin films can lead to surface anisotropy and an axis of magnetizatio
which is perpendicular to the plane of the film.
- The surface of a magnetic film may have enhanced magnetic moments.
- Unnatural but metastable lattice structures for thin films can be grown using molecular
epitaxy.
- Multilayer systems of magnetic and nonmagnetic materials can be produced
and the coupling of the magnetic layers can lead to interesting effects
such as giant magnetoresistance.
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 quasi-two dimensional films energy may be minimized by an
orientation of the magnetic moments of the atoms perpendicular to the
plane of the film.
Energy and Magnetic Structures
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.
Energy of Wall of Domains
The total energy density of a wall Ew is the sum of the
exchange energy Eex and
uniaxial anisotropy energy EK. Thus,
Ew = Eex + EK
= A
/δ + Kδ/2
where δ is the domain wall thickness and A and K are
parameters characterisitic of the material. The wall energy density Ew is
minimizes when
δ = (2)1/2(A/K)1/2
and thus
miminum Ew = (2)1/2(AK)1/2
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 Ew is based
upon the correct formula. Howson notes that for iron the optimal domain
wall thickness is about 50 nm.
The self-energy of a sample with magnetization M and an internal field
HD can be expressed as:
Eself = -(1/2)MHD = -(1/2)M(DM)
= - (1/2)DM2
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 self energy 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 self energy is then (1/2)DM2tWL.
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, d2. Thus the energy of the
domain walls is: 4(AK)1/2(2td)(WL/d2)
= 8(AK)1/2(WLt)/d.
The coefficient D in the self energy term is a function of d and t;
specifically for square prisms it is,
D = (2d2)/(2d2+4td)
= d2/(d2+2td)
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:
ETotal =
(1/2)D0(d/t)M2tWL + 8(AK)1/2(WLt)/d
When this is divided by the sample area WL the result is that, for
d much smaller than t
E = (1/2)D0M2d + 8(AK)1/2(t/d).
The domain width d may be chosen to minimize E. The value of d which
minimizes E is proportional to t1/2 and the mimimum E is
also proportional to t1/2.
For the case where d is not necessarily much smaller than t the
functional relation between E and d is:
E = (1/2)[d2/(d2+2td)]M2t + 8(AK)1/2(t/d).
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.
The Orientation of the Axis of Magnetization With Respect to the
Plane of the Film
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.
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