Head-to-Head Domain Wall Structures in Wide Permalloy Strips

1.1 MB·English

by  Virginia Estevez

Head-to-Head Domain Wall Structures in Wide Permalloy Strips
Virginia Est´evez∗ and Lasse Laurson
COMP Centre of Excellence and Helsinki Institute of Physics,
Department of Applied Physics, Aalto University,
P.O.Box 11100, FI-00076 Aalto, Espoo, Finland.
WeanalyzetheequilibriummicromagneticdomainwallstructuresencounteredinPermalloystrips
of a wide range of thicknesses and widths, with strip widths up to several micrometers. By per-
forminganextensivesetofmicromagneticsimulations,weshowthattheequilibriumphasediagram
of the domain wall structures exhibits in addition to the previously found structures (symmetric
5 and asymmetric transverse walls, vortex wall) also double vortex and triple vortex domain walls
1 for large enough strip widths and thicknesses. Also several metastable domain wall structures are
0 found for wide and/or thick strips. We discuss the details of the relaxation process from random
2 magnetizationinitialstatestowardsthestabledomainwallstructure,andshowthatourresultsare
robust with respect to changes of e.g. the magnitude of the Gilbert damping constant and details
n
of the initial conditions.
a
J
PACSnumbers: 75.60.Ch,75.78.Cd
0
3
I. INTRODUCTION ergy may be found18,28–31. For H >HW or J >JW, the
]
DW structures exhibit dynamical evolution: for TWs,
l
l repeated nucleation and propagation of an antivortex
a During the last decade, a lot of effort has been de-
h across the strip width takes place19. Similarly, in VWs
voted to understand static and dynamic properties of
- the vortex core performs oscillatory back and forth per-
s magnetic domain walls (DWs) in ferromagnetic nanos-
e tructures such as nanowires and -strips. These studies pendicular motion19.
m
have been largely driven by promising technological ap- In Permalloy strips with even larger widths and/or
. plications based on domain walls and their dynamics, in thicknesses, one might expect also other, possibly more
t
a particular memory1,2 and logic devices3–5. In typical ex- complicatedequilibriumDWstructures. Forwiderstrips
m
periments DWs are driven by either applied magnetic shape anisotropy is less important, implying that energy
- fields6,7 or spin-polarized electric currents8–11. The re- minima with more complex spin structures closing the
d
sulting DW dynamics depends crucially on the micro- flux more efficiently than TWs, ATWs or VWs may ap-
n
magnetic DW structure, typically involving various in- pear. Indeed, e.g. double and triple vortex DWs have
o
c ternal degrees of freedom. These are essential e.g. for been observed in experiments on wide strips28, but they
[ the emergence of the Walker breakdown12, an instability have been attributed to current-induced vortex nucle-

occurring when the DW internal degrees of freedom get ation resulting in metastable DW structures. Conse-
1
v excited by a strong external drive (a magnetic field H quently, a pertinent and fundamental question is what
1 or a spin-polarized current J exceeding the Walker field are the possible intermediate equilibrium DW structures
3 H or current J , respectively), limiting the propaga- observable when the lateral Permalloy strip dimensions
W W
7 tion velocity of the DWs. increase from those corresponding to the typical nanos-
7 tripgeometry(withTW,ATWorVWasthestableDW
Twomainclassesofferromagneticmaterialshavebeen
0
structure) to strip widths of micrometers and beyond.
extensively studied within the strip geometry. Ma-
.
1 terials with a high perpendicular magnetic anisotropy In this paper we present an extensive numerical study
0
(PMA13–16) exhibit simple and narrow DWs of the of the equilibrium and metastable micromagnetic DW
5
Bloch and/or N´eel type. For H > H or J > J , structuresinPermalloystrips,withthestripwidthsupto
1 W W
: repeated transitions between these two structures are an order of magnitude larger than before19–24. Contrary
v observed16. The second class of systems includes soft to previous studies focusing on comparing the energies
i
X (lowanisotropy)magneticmaterials13,14 suchasPermal- of different a priori known DW structures19–24, we per-
loy, where in-plane domain magnetization along the long form micromagnetic simulations of relaxation dynamics
r
a axis of the strip is induced by shape anisotropy. By us- from random initial states towards the stable DW struc-
ing various experimental techniques17,18 and micromag- tures. In addition to the previously observed TW, ATW
netic simulations, it has been established that the equi- andVWDWs,wefindalsoDWswithequilibriumdouble
libriumDWstructuresseparatingthesein-planedomains andtriplevortexstructuresforwideand/orthickenough
are more complex, and depend crucially on the sample strips. Thelaststructureisencounteredonlyinthevery
geometry19–22. Transverse DWs (TWs) and asymmetric largest system sizes we were able to simulate. Moreover,
transverse DWs (ATWs) are observed for narrow and for wide strips we find a rich variety of metastable DWs
thin strips22–27, while in wider and thicker strips one with even more complex micromagnetic structures. We
encounters the vortex DW (VW)22,24–26,28,29. In addi- demonstrate that our results are robust with respect to
tion, various metastable DW structures with higher en- changes of the magnitude of the Gilbert damping con-
2
(a) thickness Δ 0 ns 0.2 ns
z
width w
length l 1.0 ns 2.0 ns
l
(b) r
3.0 ns 4.0 ns
FIG. 1. (color online) (a) Geometry of the Permalloy strip.
(b)Atopviewofthemagnetizationintheinitialstate. Mag- 6.0 ns 12.0 ns
netization points along the long axis of the strip within the
two domains (as indicated by the arrows) forming a head-
FIG. 2. (color online) An example of the temporal evolution
to-head configuration. In between them, a region of random
of the relaxation, with w=420 nm, ∆ =10 nm, α=3 and
magnetization (of length l ) has been included. z
r l = 2w. Relaxation towards the equilibrum DW structure
r
(here, a VW) takes place via coarsening dynamics of the de-
fectstructureinthemagnetictexture. Thecolorwheelinthe
stant or using different initial conditions for the relax- middle shows the mapping between magnetization directions
ation process. Our results underline the crucial role of and colors.
topological defects for physics of DWs in soft strips, and
that of micromagnetic simulations for finding the true
equilibrium DW structure. contributions due to exchange, Zeeman, and demagne-
tizing energies. The size of the discretization cell used
depends on the system size, but is always bounded by
II. MICROMAGNETIC SIMULATIONS the exchange length, Λ=(2A/µ0Ms2)1/2 ≈5 nm, in the
in-planedirections,andequals∆ inthetheout-of-plane
z
direction.
ThesystemstudiedisaPermalloystripofwidthwand
thickness ∆ , satisfying ∆ (cid:28) w, see Fig. 1 (a). In the
z z
micromagnetic simulations, magnetic charges are com-
III. RESULTS
pensatedontheleftandrightendsofthestrip,tomimic
an infinitely long strip; the actual simulated length sat-
isfies l ≥ 4w for all cases considered. The initial state We start by considering the effect of varying α and lr
from which the relaxation towards a stable DW struc- on the relaxation process. Fig. 2 shows an example of
ture starts is an in-plane head-to-head domain struc- the time evolution of m(r,t) for w = 420 nm, ∆z = 10
ture, with a region of random magnetization of length nm, α = 3 and lr = 2w. The initially random magne-
l in the middle of the sample, see Fig. 1 (b). If not tization evolves via coarsening of the defect structure of
r
specified otherwise, we consider l = 2w. Material pa- the magnetization texture towards the stable DW (here,
r
rameters of Permalloy are used, i.e. saturation magne- a VW). During the relaxation, the total energy E of the
tization M = 860 × 103 A/m and exchange constant system decreases in a manner that for a given geometry
s
Aex = 13 × 10−12 J/m. The typical Gilbert damping (w and ∆z) depends on both α and lr, see Fig. 3 (a) and
constant for Permalloy is α = 0.01, but here we analyze (b) where w =5120 nm and ∆z =20 nm, is considered.
alsotheinfluenceofαontherelaxationprocess,andthus For instance, E decreases faster for an intermediate α
consideralsoothervalues. Forsimplicity,wesetthetem- [Fig. 3 (a)]. We attribute this behavior to the balance
perature T to zero, and focus on the ideal case of strips between inertial effects related to precession favored by
free of any structural disorder or impurities. a small α, helping to overcome energy barriers, and the
The simulations are performed using the GPU- higher rate of energy dissipation due to a large α. Thus,
accelerated micromagnetic code MuMax332–34, offering the relaxation time to reach a (meta)stable DW struc-
a significant speedup as compared to CPU codes for the ture depends on α. Fig. 3 (b) illustrates that for a fixed
largesystemsizesweconsiderhere. Tocalculatethemag- α, systems with a larger lr relax more slowly. Fig. 3
netization dynamics of the system, the Landau-Lifshitz- (c) shows that on average, the early-time relaxation of
Gilbert equation35,36, E towards its final value Ef exhibits temporal power-law
decay, (cid:104)E−E(cid:105)∝t−β with β ≈1.3 for the α=0.3 case
f
∂m/∂t=γH ×m+αm×∂m/∂t, (1) shown, possibly related to collective effects due to inter-
eff
actions between several topological defects during early
is solved numerically. Here, m is the magnetization, γ stages of relaxation (Fig. 2).
the gyromagnetic ratio, and H the effective field, with In general, the final (meta)stable DW structure may
eff
3
9×10-13 (a) α = 3 (b) l = w 6×10-13
α = 0.3 lr = 2w 25
E [J]63××1100--1133 lr =α 2 =w 0.03 α l=rr =0 .33w 3×10-13E [J] 20 Vortex wall DTroipulbel ev ovrotretxe xw walalll
]
m
n15
0 0 Δt [z
Asymmetric transverse wall
0 t [s]2×10-9 0 t [s] 2×10-9 10
(0)> 100 (c) 5
E Transverse wall
E> / <f1100--21 lllrr === 23www 100 w [nm10]00
- r α = 0.3
Et() 10-3 t-1.3 FIG. 4. (color online) Phase diagram of the equilibrium
< 10-10 10-9 10-8 DWstructureinPermalloystripsofvariousthicknesses(from
t [s] ∆z =5to25nm)andwidthsrangingfromw=120nmupto
5120nm. Thesymbolscorrespondtoobservationsofthevari-
ousequilibriumDWstructures,withphaseboundariesshown
FIG. 3. (color online) The energy E(t) as a function of time
as solid lines. Examples of the DW structures corresponding
t for w=5120 nm and ∆ =20 nm. (a) For different values
z
to the 5 different phases are shown in Fig. 5.
of α and l = 2w. (b) For different values of l , and α = 0.3
r r
[resulting in the fastest relaxation in (a)]. (c) shows that on
average, the early time decay of E(t) towards its final value Transverse Wall
E obeys(cid:104)E(t)−E(cid:105)∝t−β. Fortheα=0.3caseshownhere,
f f
β ≈1.3. Empty(filled)symbolsin(c)correspondtow=420,
∆ =10 nm (w=860, ∆ =20 nm).
z z (a)
depend on the realization of the random initial state. Asymmetric Transverse Wall Vortex Wall
Thus, we consider 21 realizations of the initial random
magnetization for each w and ∆ , and compare the en-
z
ergies of the resulting relaxed configurations. The struc- (b) (c)
ture with the lowest energy is chosen as the equilibrium
structure,whileotherswithhigherenergyaremetastable
Double Vortex Wall Triple Vortex Wall
states. Although, as discussed above, the relaxation
times depend on α and l , the equilibrium DW struc-
r
ture is found to be independent of α and l in the range
r (d) (e)
considered, i.e. α ∈ [0.01,3] and l ∈ [w,3w]. Thus, in
r
what follows, we will use α=3 and l =2w.
r
The main results of this paper are summarized in FIG. 5. (color online) Examples of the different equilibrium
Figs. 4 and 5, showing the phase diagram of the equi- micromagnetic DW structures: (a) TW for w =120 nm and
librium DW structures for w ranging from 120 to 5120 ∆z =5 nm, (b) ATW for w =160 nm and ∆z =10 nm, (c)
VWforw=640nmand∆ =15nm,(d)DVWforw=2560
nm, and ∆ from 5 to 25 nm, and examples of these z
z nm and ∆ = 20 nm, and (e) TVW for w = 5120 nm and
structures, respectively. Forsmallw, werecoverthepre- z
∆ = 25 nm. The colorwheel (top left) shows the mapping
viousresults19–22,i.e. phasescorrespodingtoTW,ATW z
between magnetization directions and colors.
and VW, shown in Fig. 5 (a), (b) and (c), respectively.
For larger strip widths (w approaching or exceeding
1µm,dependingon∆z,seeFig.4),anewequilibriummi- is found for the very largest system sizes we have been
cromagneticDWstructure,adoublevortexwall(DVW), able to simulate. The middle vortex of the TVW has an
is observed. This structure consists of two vortices with oppositesenseofrotationtotheothertwo. Forw =5120
opposite sense of rotation of the magnetization around and∆ =25nm, DVWandTVWhavethesameenergy
z
the vortex core, see Fig. 5 (d). At the phase boundary (thecyansquaresymbolinthetoprightcornerofFig.4),
(blue triangle symbols pointing left in Fig. 4), VW and suggestingthepresenceofaphaseboundarybetweenthe
DVW have the same energy. The DVW phase spans a two structures. Indeed, by performing a set of 10 ad-
relatively large area within the (w,∆z) space, highlight- ditional simulations with w = 6144 and ∆z = 25 nm
ing the robustness of our results. (i.e. outside the phase diagram in Fig. 4), suggests that
In addition, a second new phase, with a triple-vortex TVWistheequilibriumDWstructureforverylargestrip
wall (TVW) as the equilibrium structure [see Fig. 5 (e)], widths. Thisstructurehasbeenobservedinexperiments
4
2V+AV 3V+AV vortices(withthesamesenseofrotation)andanantivor-
(a) (b) tex [2V+AV, see Fig. 6 (a)], there are two vortices and
only two edge defects. Thus, in order to compensate the
topological defects, also an antivortex appears. In gen-
eral,wehaveobservedthatinaDWwithN vorticeswith
the same sense of rotation, there must be N−1 antivor-
3V+AV 3V+2AV
tices to get a zero total winding number, see Fig. 6 (a)
(c) (d)
and (d) for examples with 2V+AV and 3V+2AV config-
urations, respectively. When some of the vortices have
oppposite sense of rotation, more complex scenarios are
encountered,withexamplesshowninFigs.6(b),(c),(e),
4V+AV 4V+2AV
(f),(g)and(h). NoticealsothattwoDWstructureswith
the same elements can look very different, see e.g. the
two 3V+AV DWs shown in Figs. 6 (b) and (c). All the
(e) (f)
DWstructuresfound,boththeequilibriumonesinFig.5
and the metastable states in Fig. 6 obey the principle of
4V+3AV 5V+2AV
compensationoftopologicaldefectstoyieldatotalwind-
ingnumberofzero. Therichnessoftheequilibriumphase
diagram and the large collection of metastable states in-
(g) (h)
dicatethatforwide/thickstripsinparticular,themicro-
magneticenergylandscapeisquitecomplex,withalarge
number of local minima. This is also in agreement with
FIG.6. (coloronline)ExamplesofmetastableDWstructures
our observations of power-law energy relaxation.
observed for a system with w=5120 nm and different thick-
nesses ∆ : (a) Two vortices and an antivortex (2V+AV),
z
∆ = 25 nm, (b) and (c) three vortices and an antivortex
z
(3V+AV),∆z =5nm,(d)threevorticesandtwoantivortices IV. SUMMARY AND CONCLUSIONS
(3V+2AV), ∆ =10 nm, (e) four vortices and an antivortex
z
(4V+AV), ∆ = 5 nm, (g) four vortices and three antivor-
z To summarize, we have performed an extensive set
tices (4V+3AV), ∆ = 5 nm, and (h) five vortices and two
z
of micromagnetic simulations to study the equilibrium
antivortices (5V+2AV), ∆ =5 nm.
z
and metastable DW structures in Permalloy strips of a
wide range of widths and thicknesses, as well as the re-
laxation dynamics starting from random magnetization
asametastablestateforsmallersystems28,29. Noticealso initial states. The general trend of our results is that
that the middle part of the TVW [Fig. 5 (e)], exhibiting both the equilibrium and metastable DW configurations
four line-like 90◦ DWs meeting at a vortex core in the become increasingly complex (i.e. they consist of an in-
middle of the TVW, resembles the typical Landau flux- creasingnumberoftopologicaldefects)asthelateralstrip
closure magnetization patterns observed for rectangular dimensions increase. We note that somewhat analogous
Permalloy thin films37–39. behaviour - i.e. existence of equilibrium magnetization
Followingtherelaxationfromarandommagnetization configurations with increasing complexity as the system
initial state, the system may in general end up into var- size increases - is observed also in some other systems
ious metastable states with higher energy than that of such as three-dimensional cylindrical elements with per-
the equilibrium DW. Sometimes these metastable states pendicular anisotropy41,42.
have even a higher probability than the equilibrium one, Several remarks are in order: first, for strips with
estimated here from the sample of 21 relaxed configura- even larger lateral dimensions one may in principle ex-
tions. Fig. 6 shows some of the metastable states found pect more complex DW patterns - possibly with four or
foralargestripwithw =5120nmanddifferentvaluesof more vortices with alternating sense of rotation. These,
∆ ; for strips with smaller lateral dimensions, different however, are currently beyond the reach of our available
z
metastable states tend to be less numerous and have a computing resources. Second, our phase diagram allows
simpler structure. Despite their apparent complexity, all one to check if experimental observations of the various
themetastableDWstructuresshowninFig.6respectthe DW structures in wider strips are equilibrium configura-
basic principles of topology of DWs. Each of the DWs tions or metastable states. According to our review of
are composed of topological defects, with an associated the experimental literature, most observations of DVWs
windingnumber: +1forvortices, -1forantivortices, and and TVWs appear to be metastable states28,29. Third,
±1/2 for edge defects40. In a DW all the topological whiletheequilibriumstructureswefindarecertainlysta-
defects have to be compensated, i.e. the total winding ble in the absence of external perturbations such as ap-
number is equal to zero. In the case of the DVW, the pliedmagneticfields,itremainstobeseenhowtheirfield
two topological vortex defects are compensated by four driven dynamics is like, and whether wide strips with a
edge defects [Fig. 5 (d)]. For the metastable state of two relatively weak shape ansitropy are able to support the
5
DWs as compact objects also when external perturba- ACKNOWLEDGMENTS
tions are being applied43.
We thank Mikko J. Alava for a critical reading of
the manuscript. This work has been supported by
the Academy of Finland through its Centres of Excel-
lence Programme (2012-2017) under project no. 251748,
and an Academy Research Fellowship (LL, project no.
268302). We acknowledge the computational resources
provided by the Aalto University School of Science
“Science-IT” project, as well as those provided by CSC
(Finland).
∗ virginia.esteveznuno@aalto.fi 33, 4167 (1997).
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