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8/3/2019 Jack A. Tuszynski- NL2664: Domain Walls
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NL2664 Domain Walls 1
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
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