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NL2664 DOMAIN WALLS

In ferromagnetic materials, small regions of correlated magnetic moments formed below the critical

temperature are called domains. Domain walls are two-dimensional structures that separate distinct domains of order

and form spontaneously when a discrete symmetry (such as time-reversal symmetry in magnets) is broken at a phase

transition. With each subdivision of a substance into distinct domains there is a decrease in the bulk energy since the

order parameter value inside each domain minimizes its free energy. However, there is a simultaneous increase inthe energy of interaction between differently aligned domains giving rise to an extra surface energy at the boundary

between two neighboring domains. Consequently, this competition leads to an average domain size that gives the

lowest overall free energy in a material sample. This is quantified below. The energy of a ferromagnetic domain

wall is calculated as arising from the exchange interactions between spins augmented by the anisotropy energy.

While the exchange energy for N spins comprising a domain wall varies as

N

JSEexch

= , (1)

the anisotropy energy is :

KNaEanis = (2)

where a is the lattice constant and K the anisotropy constant. Minimizing the sum with respect to N yields:

Ka

JSN

= (3)

giving the domain width as =N .aA Bloch domain wall is a region separating two (magnetic) domains within which magnetization changes

gradually by rotating in the plane perpendicular to the line along the direction from one domain to the next. This

way the magnetization direction experiences a reversal by 180 degrees without changing its magnitude. The energy

associated with a domain wall decreases with the width of the wall. However, domain wall thickness is found as a

minimization problem involving the anisotropy energy. The Neel domain wall involves magnetization reversal in the

plane perpendicular to the boundary between two domains.Domains undergo processes of reorganization under the effects of applied fields and can move in space

which occurs especially in the initial phase of remagnetization favoring those domains that are aligned with it and

thus setting their boundary in motion to occupy more space. This is followed by reorientation of the magnetization

within each domain that is not aligned with the field.

In other condensed matter systems, domain walls exist in crystals, ferroelectrics, metals, alloys, liquid

crystals, etc. In annealing metals, for example, domain walls appear as the grain boundaries between two sharply

different compositions. In each case, the underlying physical quantity is called the order parameterand is specific to

the given substance. For the annealing metal it is a real field, while in superfluid helium it is a complex-valued field.

Over most of the sample the order parameter has a constant magnitude. However, the sign (when is real) or the

phase (when is complex) is not fixed and can change from place to place. A real order parameter field may be

positive in one region of space and negative in its neighborhood, the continuity of the field implies that it must cross

the zero value on a surface between them. This transition region is a domain wall.

In all types of critical systems, domain walls arise due to the competition between the bulk part of the free

energy which in the Landau theory of phase transitions is a quartic polynomial in the order parameter and thesurface energy term that is due to inhomogeneities and varies as the square of the order parameter gradient following

Ginzburg. Minimizing this type of free energy functional

dxdx

dDAAF

++=

2

44

22

(4)

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leads to a stationary non-linear Klein-Gordon (NLKG) type equation for the order parameter as a function of the

spatial variable.

(5)3

'' BAD +=

One of its stable solutions is proportional to 0 tanh(x/) where AD/8= which describes a smooth function

that interpolates between the two homogeneous phases BA/0 = . For magnetization as an order parameter,this solution represents a magnetic domain wall (in 1D), for ferroelectrics where the order parameter is a

polarization vector, this represents a ferroelectric domain wall. For crystals undergoing structural phase transitions,

there can also be a kinetic energy term in the free energy functional leading to a standard form of the NLKG

equation

(6)3

'' BADm +=+&&

This solution is a moving domain wall (Krumhansl and Schrieffer, 1975)

=0

tanh [(x-vt)/] (7)as shown in Fig.1

Figure 1 A typical form of a domain wall.

Nonlinear traveling solitary waves have also been investigated in ferroelectrics where kinks representing domain

walls were shown do carry a dipole flip (Benedek et al, 1987).

Domain walls in ferroelectrics are typically several unit cells wide while in ferromagnets their thickness is

several hundred or even thousands of unit cells. This difference is due to the exchange interactions between spins

which are much stronger than the dipole-dipole interactions in ferroelectric crystals. It is worth noting that there also

exist cylindrical domains in magnets.

Modern particle physics predicts that phase transitions occurred in the early Universe following the Big

Bang. Of particular interest to cosmology is the production of topological defects, which are sheet-like, line-like or

point-like concentrations of energy.

Table 1 Geometry of space and the corresponding topological defects.

Geometry Names

sheet-like domain walls, membranes

line-like vortices, strings

point-like monopoles, hedgehogs

As can be seen from Table 1, domain walls are examples of topological defects and as such they are very common

in all broken-symmetry phenomena that take place slowly enough to allow for the generation of defects.

JACK A. TUSZYNSKI

See also order parameters, ferromagnetism and ferroelectricity, critical phenomena, nonlinear field equations

Further Reading

Anderson, P.W. 1984. Basic Notions of Condensed Matter Physics. Menlo Park, California: Benjamin/Cummings

Benedek, G., Bussmann-Holder, A. and Bilz, H. 1987. Phys. Rev. B36: 630

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Kittel, C. 1956. Introduction to Solid State Physics. New York: Wiley

Krumhansl, J.A. and Schrieffer, J.R. 1975. Phys. Rev., B11: 3535

White, R.H. and Geballe, T. 1979. Long Range Order in Solids. New York: Academic Press