Table Of ContentNumerical study of the depinning transition of a ferromagnetic magnetic domain wall
in films
Bin Xi1, Meng-Bo Luo2, Valerii M. Vinokur3 & Xiao Hu1
1International Center for Materials Nanoarchitectonics (WPI-MANA),
National Institute for Materials Science, Tsukuba 305-0044, Japan
2Department of Physics, Zhejiang University, Hangzhou 310027, China
3Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
5 (Dated: January 20, 2015)
1
0 We report first principle numerical study of domain wall (DW) depinning in two-dimensional
2 magneticfilm,whichismodeledby2Drandom-fieldIsingsystemwiththedipole-dipoleinteraction.
We observe nonconventional activation-type motion of DW and reveal its fractal structure of DW
n
near the depinning transition. We determine scaling functions describing critical dynamics near
a
thetransition and obtain universalexponentsestablishing connection between thermalsoftening of
J
pinning potential and critical dynamics. We observe that tuning the strength of the dipole-dipole
9
interaction switches DW dynamics between two different universality classes corresponding to two
1
distinct dynamic regimes, motion in therandom potential and that in the random force.
]
h
c Introduction – Motion of domain walls in magnetic this challenge we reveal the fractal structure of the do-
e nanowires and films is a key component of operation mainwallanduncoverthecriticaldynamicsatthedepin-
m
of any magnetic memory and logic device[1–3]. To ningtransition,andrelatetheobservedcriticalexponents
- a great extent DW dynamics is governed by pinning- to those of finite temperature creep dynamics. Further-
t
a depinning processes which control the operational speed more,weuncovertheroleofstrengthofthedipole-dipole
t and power consumption of a device and thus play cen- interactionindeterminingtheproperdynamicuniversal-
s
. tral role in device performance[4, 5]. There has been ity class.
t
a remarkable progress in description of pinned DW dy-
m namics based mostly on the elastic manifold model in Model and method –Wemodelthetwo-dimensional(2D)
- a random environment[6]. A key property of such a sys- magnetsubjectto quencheddisorderbythe 2Drandom-
d tem is that at zero temperature it experiences the dy- field Ising model with the dipole-dipole interaction, and
n namic phase transition (depinning transition): At small the dynamics is controlledby the externaldriving field:
o
external drives, F 6 F , where F is the critical pin-
c c c
[ ning force, DW is immobilized (pinned) by disorder, S S
1 whereas at F > Fc it acquires a finite velocity v. The H =−J SiSj +Vdd ri3j − (hi+H)Si, (1)
v thresholddepinning force Fc is a criticalpoint in a sense hXi,ji Xi<j ij Xi
that at F & F , the velocity exhibits critical behavior
6 c
3 v ∼(F−Fc)β,aswasproposedbyFisher[7]inthecontext with Si = ±1 at site i. The first term of the Hamilto-
nian is the ferromagnetic coupling between one spin and
4 of depinning of charge density waves. At finite temper-
4 atures the velocity is always finite, and at F ≪ F the its nearest neighbors. Hereafter we measure the energy
c
0 domain wall exhibits highly nonlinear glassy response, in the units of the coupling J. The second term is the
1. so-called creep dynamics, with v ∝ exp(−const/Fµ)[6]. magnetic dipole-dipole interaction with rij =|i−j| and
0 The depinning transition gets rounded and acquires a Vdd a parameter for interaction strength. The on-site
5 meaning of the intermediate region separating the low random field hi distributes uniformly within an interval
1 forcecreepdynamicsfromtheasymptoticlinearresponse [−∆,∆] which generates random pinning potentials. H
v: v ∝ F at F ≫ Fc. The critical depinning behavior has isauniformmagneticfieldwhichdrivesthedomainwall.
i to include temperature dependence and was conjectured Our simulations are performed on L×L square lat-
X
to be of the form: v ∼Ψ[(F/F −1)/Tη] [8, 9]. tice. A flat domain wall between spin +1 and spin −1
c
r is created along y axis at x = 1 as the initial condition.
a All the above results were obtained within the elas-
The magnetic field is applied to drive the domain wall
tic manifold approach, where the DW was modeled as
in the positive direction of x axis in accordance to the
an elastic membrane (in 3D) or an elastic string, if we
Metropolis algorithm with single-spin flip[10]. Periodic
discuss one-dimensional DW in a magnetic film. How-
boundarycondition(PBC)isadoptedatthedomain-wall
ever fundamental and successful, this description misses
(y) direction, whereas Anti-periodic boundary condition
important processes that may become essential for the
(APBC)[17] at the moving (x) direction. The number
DW dynamics at elevated temperatures. Domain wall is
of independent runs is at least 3000. The time unit is
multi-valued, so ahead boundaries can merge with the
defined by a sweep of Monte Carlo trials over the whole
‘main’ interface behind.
system, and the velocity is defined by v = dM/2Ldt in
This poses a challenge of developing first principle ap- steady states, with M the total magnetization. For a
proachstartingfromthemicroscopicmodelthatcaptures smallsystemunderlargedrivingfield,itreachesasteady
basic physics of the magnetic system. Taking up upon state quickly (t ∼ 10 for warm-up and ∼ 102 for statis-
2
1 so do the corresponding values of the depinning field.
1
Itisobservedthatforagivenfieldandsystemsize,the
domain wall may either be pinned inside the sample, or
P00.5
it may go through from edge to edge. We call the latter
P00.5 0-8 (H/Hc0-1)L1/ν 8 cdaespeinanidnegppinrnoibnagbielivteyntP0a.ndToevdaeluteartmeitnheeacotrrureespvaolnudeinogf
H ,onethushastoperformthefinite-sizescalinganalysis
c
Hc = 1.214 ofP0 whichwouldcontainHc as a parameter. As shown
(a) ν = 1.33 in Fig.1a, P0(L,H) increases sharply as function of the
0 magnetic field in the interval H =1.1∼1.3. The curves
1.1 1.15 1.2 1.25 1.3
corresponding to different system sizes cross at point of
0.6 L = 64 P0 =0.38±0.04 at H =1.214±0.006. This determines
L = 128
L = 256 thedepinning fieldwhichdoesnotdependonthesystem
L = 512 size and thus can be taken as a depinning field H of a
c
L = 1024
macroscopic system.
v
0.3 The problem of depinning at zero temperature is in-
timately related to percolation problem[13]. We thus
H = 1.214 assume that the depinning probability function has the
c
β = 0.36 form characteristic to the percolation problem[14]:
(b)
0
1.1 1.2 1H.3 1.4 1.5 P0(L,H)=Φ[(H/Hc−1)L1/ν], (2)
FIG. 1. (Color on line). (a): The probability P0 versus the where ν is a universal exponent. By choosing the vari-
drivingfieldwithdifferentsystemsizesandthecorresponding able (H/Hc−1)L1/ν andν asanadjustment parameter,
scaling plot (inset). V /J = 0.1 and ∆ = 1.5J are used we find that at ν = 1.33±0.05, all the data points of
dd
all through this work. (b): v − H characteristics at zero P0(L,H) collapse onto a single curve as shown in the
temperature with difference system sizes. The solid line is inset of Fig.1a. This procedure defines the exponent ν.
thefitting function (Eq. (3)) for L=1024. Now we are equipped to study the v-H characteris-
tics in the depinning regime. We start with the zero-
temperature behavior. The results for v-H for systems
tics), whereas typically 103 time steps for warm-up and of different sizes are displayed in Fig.1b. One sees that
statisticswithregardtolargesystemsundercriticaldriv- for L>128 the curves do not practically depend on the
ing field. The time scalein MC technique shouldbe pro- sizeofthesystem. Assumingthestandardv-H depinning
portionalto the realtime, but a straightforwardrelation relation[15]
is not easy. In order to derive the correspondence, one
need to compare the simulation results and experiments v =v0(H/Hc−1)β, (3)
at least once.
where β is a universal exponent, one finds v0 = 1.16±
The system adopted in the present work is a coarse- 0.01,H =1.214±0.006andβ =0.36±0.01forL=1024.
c
grained one. The on-site spin is represented in terms of Thisvalueisinafairagreementwiththe β =0.31result
a block spin which contains nz × nxy × nxy unit cells obtained in two-loops RG calculations[16] showing that
in a thin magnetic film, with nz and nxy the number the elastic manifold approximation works pretty well at
of unit cells along the out-of-plane and in-plane direc- zero temperature.
tion, respectively. Then the energy unit E =n n aA,
J z xy
with a the lattice constant of a real material and A Finite-temperaturedepinning –Nowweturntoourmain
the corresponding exchange stiffness. For Nd2Fe14B, task, the finite temperature motion. To reduce the com-
a = 0.88 nm and A = 7.7 pJ/m[11], whereas a = 0.25 putation time we choose L = 512 system. Figure2a
nm and A = 10.3 pJ/m for Co layer in Pt/Co/Pt thin showsthe expected increasein velocity at the givenfield
films[11, 12]. The dimensionless temperature T in the upon increasing temperature and an appreciable tail be-
present work can be related to the real temperature low the depinning field due to thermally activation pro-
through: T ·E /k , with k the Boltzmann constant. cesses.
J B B
Takingn =n =1,onecanhaveT =0.1approximates We use the standard scaling ansatz[17–19]:
z xy
to 49 K for Nd2Fe14B, whereas 19 K for Pt/Co/Pt thin
film.
v(T,H)=T1/δΨ (H/H −1)T−1/βδ , (4)
c
Zero-temperaturedepinning –Tocomeupwiththequan- h i
titativedescriptionofdepinning,wehavetoknowitskey with Ψ(x)∼xβ as x→∞. We achieve the best collapse
characteristic, the zero-temperature depinning field H . of the data to a single curve with δ = 2.76± 0.02 by
c
Finding its true value is a challenge since in finite sys- adopting the values of H and β determined above, see
c
tems realizations of the random potential fluctuate, and Fig.2b. Note,thattheseresultsceasetoholdforlargedd
3
0.6
0.6
T = 0.05
T = 0.08
T = 0.1
T = 0.125
v 0.3
0.3
(a) (c)
V /J = 0.1 V /J = 0
dd dd
0 0
0.8 1 1.2 1.4 0.7 1.05 1.4
H H
2 1
w ∝ exp[-a(-u)5/3] w ∝ exp[-a(-u)5/3]
δ H=1.214 H=1.289
-1/T1 β c= 0.36 1 β c= 0.33 1 0.5
v
βδ = 1 0.1 βδ = 5/3
0.1
0.01
0.01
(b) 0.001 (d)
-4 -2 0 -1.8 -0.9 0
0 0
-4 -2 0 2 4 -1.8 -0.9 0 0.9
-1/βδ -1/βδ
(H/H -1)T (H/H -1)T
c c
FIG. 2. (Color on line). (a): Finite temperature v−H characteristics for V /J = 0.1. (b): Scaling plot of v−H curves
dd
as vT−1/δ vs. (H/H −1)T−1/βδ. The inset shows the same scaling behavior replotted in the semi-log scale. (c): Finite
c
temperaturev−H characteristics for V =0. (d): Thecorresponding scaling plot for V =0, theinset shows thesame data
dd dd
in semi-log scale.
interaction, V /J > 0.5 where the ferromagnetic order cies that at the first glance are not that different from
dd
is broken. those of Fig.2a. The scaling treatment, however, yields
At x < 0 the scaling function exhibits the asymptotic v ∝exp −U (1−H/H )5/3/T , see Fig.2d, i.e. the Ar-
c c
behavior Ψ(x) = 1.05exp −0.70(−x)5/3 , see Fig.2b, rhenius activation behaviour with the barrier that scales
andonearrivesatthedynamicsofthedomainwallacross as U (1−(cid:2) H/H )5/3, where U (cid:3)≈0.67 is the bare energy
c c c
(cid:2) (cid:3)
the transition[17] given by: barrier, H = 1.289, β = 0.33, and βδ = 5/3. We now
c
recall that this kind of thermally activated behavior at
V = 0 is expected for the case of the motion of DW
5/3 dd
v =v1T1/δexp − Ec 1− H , (5) in the field of the random force[6], where an impurity
( (cid:20) T (cid:18) Hc(cid:19)(cid:21) ) “knows” to which magnetic domain it belongs in. This
impliesthatthepinningenergybarriercomprisestheen-
where Ec ≈ 0.81 is an energy barrier which governs the ergies of all pinning sites located in the area spanned by
domain-wallvelocityatfinitetemperatures,andthecon- the domain wall during an elemental activation jump.
ditionβδ =1istakenintoaccount. Notably,thedomain- In this case the contribution of the fluctuation thermal
wallmotionis not the conventionalArrhenius-type. The broadeningofthe domainwallpositiongives the negligi-
origin of this nontrivial temperature dependence is the ble contribution into the total pinning energy thus ther-
renormalization of the random potential landscape by mal fluctuations cannot appreciably reduce the depin-
thermal fluctuations. ning field. Hence the thermally activated motion retains
The exponent 5/3 is in an excellent agreement with its Arrhenius-like character. We conclude that varying
the prediction of Ref. [6] (see Ref. [20] for thermal V one tunes the system between the random force and
dd
depinning of vortex systems). This establishes an inti- random potential pinning behaviors. Hence our findings
mate connection between the critical depinning behav- provideanirreplaceabletoolforidentifyingthesedistinct
ior of the domain wall at finite temperatures and ther- pinning mechanisms in the experiment. The behaviors
mal softening of the pinning potential. Furthermore, summarized in Fig.2 constitute the main results of our
juxtaposing our v-H curves with those obtained ear- work.
lier for the case V = 0[17], one makes a remark-
dd
able observation. Figure2c shows the v(H) dependen- Another comment in order is that at finite temper-
4
4
10
(a) (e)
t = 100
103 t = 200
t = 400
t = 800
N t = 1000
2
10
t d=1.25
x f
y 101
0.01 0.1 1
1/l
80 1000 80
(b) (c) (d)
w~u0.82 w~u0.75
60 60
z 0.1
C(r,t)40 400 C(t)s40 α/-r,t)t
(
300 C
z = 1.43
20 200 20 α = 1.17
t=100 0.01
00 200 400 600 00 500 1000 0.01 0.1 1 10 100
r -1/z
t rt
FIG. 3. (Color on line). (a): Time evolution of domain wall near the depinning threshold at zero temperature. (b): Height-
differencecorrelation functionC(r,t)versusr fordifferentvaluesoftnearthedepinningthreshold. (c): SaturatedvalueC (t)
s
versust. (d): Scaling plot for datain (b) with thedashed liney∼x2−df. (e): Log-log plot of N versus1/l at differenttimes,
with l therulersize andN themeasured length in unitsof l. Parameters used in calculations are: L=1024, V /J =0.1 and
dd
H =1.22.
atures the definition of the depinning field H is not modified height function h(y,t) [22]
c
straightforward. Our approach offers a systematic way
foranalyzingdataatfinitetemperaturesyieldingH and L
c
the thermal activationenergy barrier Ec simultaneously. h(y,t)= θ[Sx,y(t)], (6)
Importantly,Ec dependsnotonlyonthestrengthofran- xX=1
domness ∆ but also on the competition of the exchange
withθ(x)theunitstepfunctionandS (t)thespinvalue
couplingJ andthedipole-dipoleinteractionstrengthV . x,y
dd
onthe site (x,y)attime t. The function h(y,t)describes
the totalnumberofflippedspinsalongtheline y attime
Domain-wall morphology – Next we investigate the DW t. It is obvious that h(y,t) describes the position of do-
morphologyduring the depinning process for the system main wall if there is no “ lakes” and “ overhangs”.
with V /J = 0.1. To this end we set a flat domain The height-difference correlation function C(r,t) de-
dd
wall along y axis at x = 1 at t = 0 with H = 1.22 scribes the domain-wall roughness characteristics and is
just above H =1.214,and drive it along x direction at define as[22, 23]:
C
zero temperature. As shown in Fig.3a, the domain wall
evolves rougher with time, and develops a fractal struc- 2
C(r,t)= [h(y+r,t)−h(y,t)] , (7)
ture. Moreover, there remain several small unflipped-
r
spin areas (black puddles) forming the “ lakes” inside D E
the domain. The multiconnected nature of the flipped with r the distance between two points in y direction.
domainoriginates fromspatialfluctuations of the depin- As shown in Fig.3b, for fixed t, C(r) increases from
ning fielddue to randomcharacterof the pinning poten- zero with r and saturates at large r, and the saturated
tial: therearelacoonswherethelocaldepinningfieldstill valueC (t)increaseswitht. Thesepropertiescanbeun-
s
exceedsthedrivingfield. The“overhangs”ofthefrontier derstoodfromthe time-evolutionprocessof domain-wall
of the domain wall are of the same origin. To quantify morphology as displayed in Fig.3a. The initial domain
this multi-valueddomain-wallmorphology,we define the wall is a straight blue line with no height difference. By
5
applying driving field, locally meandering segments ap- vestigated the depinning dynamics of magnetic domain
pear first (see the light blue regions in Fig.3a). Correla- wallwiththedipole-dipoleinteraction,andobtainedfive
tionofdomain-wallpositionsonlyexistsinasmalllength critical exponents. It is confirmed that the scaling rela-
scale. As time evolves, the meandering segments spread tion β/ν = z −α[24] is satisfied. The case without the
out along both the domain-wall (k) and moving (⊥) di- dipole-dipoleinteractionhasbeeninvestigatedpreviously
rections, leading to rougher structures. There are two by MC simulations[17] and Langevin dynamics[25, 26].
t-dependent correlation lengths: ξ (t) and ξ (t), grow- The critical exponents are different for the two cases,
k ⊥
ing with time as ξ (t) ∼ ξ (t)α ∼ tα/z with z the dy- indicating the two cases belong to different universality
⊥ k
namic exponent and α the roughness exponent, and the classes.
correlation function evolves as[22]:
The existence of two universal classes in elastic man-
ifolds in random potentials has been addressed by two
C(r,t)∼ξ (t)g[r/ξ (t)]∼tα/zg(r/t1/z), (8)
⊥ k of the present authors[19] for vortex dynamics in type-
II superconductors, where Bragg glass and amorphous
with g(x) saturates at constant as x>>1.
vortex glass (AVG) correspond to weak and strong ran-
As displayed in Fig.3c, we obtain α/z = 0.82±0.01
dom pinning potentials. In heavily disordered AVG the
in terms of C (t) ∼ tα/z in the large r limit of Eq.(8).
s depinningdynamicsisofArrhenius-type,whileintheor-
By choosing z = 1.43±0.01, all the data collapse into
deredBraggglassstate oneobservesnon-Arrhenius-type
a single curve as displayed in Fig.3d, which determines
behavior,similar to the present domain-wallsystem. In-
the dynamics exponent z. The roughness exponent α is
terestingly,scalingfunctionsoftwouniversalityclassesin
then estimated as α=1.17±0.02.
vortex dynamics share a simple exponential form, which
We then study the fractal geometry of the domain
differsfromourresultsforthedifferentdimensionalityof
wall. For a fractal structure, the measured length N
the space and the elastic manifold; this intriguing issue
in units of ruler size is related to the ruler size l by:
calls for further investigation.
d =logN/log(1/l), with d the fractalindex. Through
f f
log-log plot of N versus 1/l as shown in Fig.3e, we ob- Acknowledgements – We are delighted to thank Andreas
tain df = 1.25±0.01. We notice that C(r,t) ∼ rD−df Glatz for useful discussion and critical reading of the
only holds for r < ξ (t)[22]. As shown by the dashed manuscript. This work was supported by the WPI ini-
k
lineinFig.3d,the exponent2−d ≈0.75appearsinthe tiativeonMaterialsNanoarchitectonics,MEXTofJapan
f
small scaling variable r/t1/z limit of function g(r/t1/z). (BX, MBL, and XH) and the Elements Strategy Initia-
tiveCenterforMagneticMaterialsundertheoutsourcing
Importantly,the fractalmorphologyis asignatureofthe
depinningregion. AtlargedrivesH,theDWgetsflat[24]. projectofMEXT,andbytheU.S.DepartmentofEnergy,
OfficeofScience,MaterialsSciencesandEngineeringDi-
Discussions and conclusions – To conclude, we have in- vision (VV).
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