Approximate analytic solutions for transient and steady-state 180 deg domain wall motion in bulk magnetic material are obtained from the dynamic torque equations with a Gilbert damping term. The results for the Walker region in which the transient solution approaches the familiar Walker steady-state solution are presented in a slightly new form for completeness. An analytic solution corresponding to larger drive fields predicts an oscillatory motion with an average value which decreases with drive field for reasonable values of the damping parameter. These results agree with those obtained by a computer solution of the torque equation and those obtained with the assumption of a very large anisotropy field

Bartran, D. S.

Bourne, H. C., Jr.

English

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Transient and Steady-State Velocity of Domain Walls
for a Complete Range of Drive Fields
Henry C. Bourne, Jr., and David S. Bartran
S Abstract - Approximate analytic solutions for transient and steady-state
CN
r- Ln 180" domain wall motion in bulk magnetic material areobtained from the
= dynamic torque equations with a Gilbert damping term. The results for the
> Walker region in which the transient solution approaches the familiar Walker
r steady-state solutionare presented in a slightly new form for completeness.
An analytic solution corresponding to larger drive fields predicts an oscil-
o o latory motion with an average value which decreases with drive field for
( 0 _4 reasonable values of the damping parameter. These results agree with those
u obtained by a computer solution of the torque equation and those obtained with
the assumption of a very large anisotropy field.
r W I1TRODUCTIONHPRo
Z 0H (1)S>4 Qn In a previous paper , transient solutions for the motion of domain
on walls in bulk magnetic materials were obtained from the vector equation of0 1
r >1 = motion,
= -M x ( (1)
I E- H-
in which M is the saturation magnetization, y is the gyromagnetic ratio,
I PH 4.-
4 < ao H is the effective field including applied, stray, anisotropy, and exchange
1 44 U -I
Ln ' components, and a is the damping parameter. With the coordinate system
and wall configuration as represented in Fig. 1, the solutions obtained for
e, 8  and the corresponding wall velocity, v, are as follows
This work was supported by NSF Grant GH 34584 and NASA Grant NGR-44-006-
001. H. Bourne is with the Department of Electrical Engineering, Rice Univ-
ersity, Houston, Texas. D. Bartran is with Honeywell, Oklahoma City, Oklahoma.
https://ntrs.nasa.gov/search.jsp?R=19750004622 2020-03-23T00:43:00+00:00Z
t
'tn(tan 0/2) = C2(t)[Y- v(z)dz] (2)
0
C2(t )  2A (sin 2 +hk 1/2 (3)
and
v 2yCA ahz + sin cp cos c (4)
o (1+a2)(sin2+h k)l/2
in which q is obtained from
2h
(1 + a2 Id ' a ( - sin 2 9 ) (5)dt 2 o 1
In these equations A is the exchange constant, hk is the normalized anisotropy
-7field o Hk/M , h is the normalized applied field poHz/M, and .o= 4 X 10 7mks.
The only approximation involved concerns a consistency condition which requires
that
dC2  t
dt2 y - v(,r)dT << c2 (t) v(t)0
in order for 9 0 f(y), a basic assumption in the trial solution. The solution
under this assumption indicates that this approximation is an excellent one
for the wall widths encountered in magnetic materials. 'This condition is
equivalent to neglecting the incremental velocity along the wall compared to
the velocity of the wall center. An alternate assumption which gives the same
result concerns the partition of Eq. 1 such that the same terms equate to
determine the structure constant, C2, and the velocity at the wall center, v,
during the transient as in the steady-state solution, which involves no ap-
proximation at least in the region 2h z/ < 1
For a stationary wall hZ, z - 0 and the equations reduce to the familiar
n(tan /2) = y K/A . For 2h /a < 1, the Walker solution ( 2 ) corresponding
to 2h /t = sin 29 predicts a contracted wall moving at constant velocity.
The transient solution corresponding to this constant velocity solution is
the one examined theoretically and experimentally in previous papers.
For completeness and for comparison, this solution is repeated in a slightly
different form together with the solution for 2hz/a > 1. All solutions
assume a step function drive field, h
z
Walker Region: 2hz/01 < 1
Eq. 5, which predicts that cp increases monotonically in time until it
reaches a steady-state value corresponding to 2h /aI sin 2%9, is integrated
to yield
2h /oi
ztan 9 =
1 +Vl-(2hz/0)2 coth 5t (6)
.with
2o = 1- (2hz/ )2 /(1 + 2)
If the appropriate trigonometric functions of 9 obtained from Eq. 6 are
substituted in Eq. 4 and the hyperbolic function is expressed in terms of
exponentials, the "exact" solution of reference (1) is obtained. Note
that a factor in the effective damping 0 involves the drive field. With
2h / < I, the steady-state value of c9 is less than r/4. Although c9 itself
is well-behaved, Eq. 4 is sufficiently complex that the transient response
contains an overshoot for 2hk/4(1 + hk ) / (1 + hk)1/h < < ,
as previously shown.
Limiting Case: 2h /=  1
42h
In the limiting case, 1, either from Eq. 6 or directly from
integration of Eq. 5,
(yM/Po)at /2
tan y =
(yM/pLo)at/2 + (1 + a 2)
The angle cp increases monotonically to TT/4 and the solution is similar to
that of the previous case, with an overshoot in the transient response.
Oscillatory Region: 2h z/ > 1
In this particular case Eq. 5 predicts that (p continues to increase for
all time with a periodically changing rate. From Eq, 4 the velocity behaves
as sin 2p displaced by the relatively small term h and modified by a posi-
z
tive term of oscillating magnitude in the denominator so that the wall velocity
is alternately positive (0 < < rr/2) and negative (T/2 < cp Z rr) and repeats.
Again Eq. 5 may be integrated directly to obtain
2h /a
z
tan CP.=
1 + (2h/a) 2 - 1 cot wt
with
W o V(2h,/a)2 -1 /(l + 2)
The periodic nature of the solution is apparent. The system is in a
steady-state oscillatory condition from the beginning with no transient in-
volved. The frequency depends on the drive field, h . In the limit of
h z/ = 1 the previous solution is obtained.
The maximum velocity magnitude in time is the same as the maximum steady-
state velocity in the Walker region and corresponds to
tan = hkl/ 4/(l + hk) 1/ 4
sin ~ cos P =+ h 1 (1 + hk)1/4(1 + hk - hk /2
sin2  = hkl/2[(1 + hk) 1/2 - hkl/ 2]
and
vmax = [C1( + h ) 1/2 - hkl/2
independent of the drive field, hZ . However, the time at which the maximum
occurs depends on hk and h and is given by
k, z
- I +(2hz/)(1 + hk) /4/h k / 4
cot Wt =
(2hz/c) - 1
Figs 2, 3, and 4 give the velocity as a function of time for several values
of drive field and three values of hk corresponding to permalloys and
bubble-type materials.
2h,
For a (-~-) << , the velocity is approximately equal to zero for
S= 0,r which corresponds to wt = 0, W , The velocity is also approximately
equal to zero for 0 = rr/2 which corresponds to
cot tj = -
j(2hz / o) - 1
In general more time is spent in the positive velocity region and the velocity
has a nonzero average value which is shown in Fig. 5.
2h
In the limit of very large drives, 2h / >> I , but again 2 ( ) << 1,
the positive and negative half-cycles of velocty are an odd function about
wt = 7/2, p wt = y H t- and the average velocity is zero. If in addition
z
hk >> I the velocity behaves as sin 2wt at very high frequencies.
6Conclusion
With suitable approximations a complete solution of the vector equation
.of motion with a viscous damping parameter may be obtained which describes
the motion of a magnetic domain wall over a complc0' range of drive fields.
The solution pertains to a bulk magnetic material 'scribed by an exchange
constant, an anisotropy constant, a saturation magnetization, and a damping
parameter. The oscillatory response predicted in the high drive region
probably may best be explored experimentally in materials with relatively
small saturation magnetizations and large anisotropy constants.
These results seem to agree with those reported by Slonczewski (5) and
by Walker and Schryer (6) in a computer solution of the torque equation.
Acknowledgment
The authors are grateful to Dr. William Wilson for calculating several
of the curves and for helpful discussions.
7References
I. Henry C. Bourne, Jr. and David S. Bartran, "A Transient Solution for
Domain Wall Motion", IEEE Trans. on Magnetics, Vol. Mag-8, No. 4,
December 1972, pp. 741-743.
2. L. R. Walker, quoted by J. F. Dillon, Magnetism, Vol. III, edited by
G. Rado and H. Suhl, Academic Press, pp. 450-453, 1963.
3. David S. Bartran and Henry C. Bourne, Jr., "Wall Contraction in Bloch
Wall Films", IEEE Trans. on Magnetics, Vol. MAG-8, No. 4, December 1972,
pp. 743-746.
4. David S. Bartran and Henry C. Bourne, Jr., "Domain Wall Velocity and
Interrupted Pulse Experiments", IEEE Trans. on Magnetics, Vol. MAG-9,
No. 4, December 1973, pp. 609-613.
5. J. C. Slonczewski, "Dynamics of Magnetic Domain.Walls", AIP Conference
Proceedings No. 5, Magnetism and Magnetic Materials, pp. 170-174,
AIP, New York, 1972.
6. N. L. Schryer and L. R. Walker, "Motion of 180* Domain Walls in Uniform
Magnetic Fields", AIP Conference Proceedings No. 10, Pt II, Nov. 1972,
p. 1026.
Figure Captions
Fig. 1. Coordinate System and Wall Configuration
Fig. 2. Normalized Velocity as a Function of Time for hk oHk/M = 0.0005
and Various Drive Fields 2h / = 2poHz/ M
Fig. 3. Normalized Velocity as a Function of Time for hk  o Hk/M = 1.0
and Various Drive Fields 2hz / = 2P H z7aM
Fig. 4. Normalized Velocity as a Function of Time for hk = oHk/M = 5.0
and Various Drive Fields 2h z/ = 2 H z/IM
Fig. 5. Average Velocity as a Function of Drive Field for Various hk
in the Walker Region and in the Oscillatory Region
2
E (easy)
H _ X- Z wall plane
M
I
Mm
"-
Figure 1
Bourne, Bartran
1.0
2h
z 5.0 2.0 1.25 .1 1.01
0.4
h = 0.0005
> 0.2 -
-0.4 -
-0.6 -
vmaxz ( )2 (lt hk)2 'hk2
-0.8 -
-1.0 7r 7 37r
4 2 4
wt
Figure 2
Bourne, Barctran
0.8-
0.6
0.4
2hzh 5.0 2.0 1.25
0.2
-0.2 -
-0.4
-0.6 _
Ax )2 [(,hk)2 -hk2]
Lo
-0.8
-1.0 I V
77 "7
4 2 4
Figure 3
Bourne, Bartran
1.0
0.8-
0.6
0.4 2hz0.4 5.0 2.0 1.25a
0.2 i h k =5.0
m
- O
-0.4
-0.6 4 - I
-0.8
-1.0 y + 1
4 2 4r
Figure 4
Bourne, Bartran
1.0 T
0.9 -
0.8
0.7
WALKER REGION
- 0.5 OSCILLATORY REGION---
A 0.4
V
0 .3 -
0.2 k = 0.0005
h0.1 k 1.0
I hk 5.0
1.0 2.0 3.0 4.0 5.0
2h z
Figure 5
Boturne, Bartran


Transient and steady-state velocity of domain walls for a complete range of drive fields

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