History-Induced Critical Behavior in Disordered Systems

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by  John H. Carpenter

History-Induced Critical Behavior in Disordered Systems
John H. Carpenter, Karin A. Dahmen, Andrea C. Mills, and Michael B. Weissman
University of Illinois at Urbana-Champaign, Department of Physics, 1110 West Green Street, Urbana, IL 61801.
Andreas Berger and Olav Hellwig
Hitachi Global Storage Technologies, San Jose Research Center, 650 Harry Road, E3 San Jose, CA 95120.
4 (Dated: February 2, 2008)
0
0 Barkhausen noise as found in magnets is studied both with and without the presence of long-
2 range (LR) demagnetizing fields using the non-equilibrium, zero-temperature random-field Ising
model. Two distinct subloop behaviors arise and are shown to be in qualitative agreement with
n
experimentsonthinfilmmagnetsandsoft ferromagnets. With LRfieldspresentsubloopsresemble
a
aself-organized criticalsystem,whiletheirabsenceresultsinsubloopsthatreflectthecriticalpoint
J
seen in the saturation loop as the system disorder is changed. In the former case, power law
3
distributions of noise are found in subloops, while in thelatter case history-induced critical scaling

isstudiedinavalanchesizedistributions,spin-flipcorrelationfunctions,andfinite-sizescalingofthe
n] second momentsof thesize distributions. Results are presented for simulations of over108 spins.
n
- PACSnumbers: 75.60.Ej,75.70.Ak,64.60.Ht,75.60.Ch
s
i
d
. Hysteresis occurs when a system far from equilibrium pictureforsubloops,consistentwiththesaturationloop,
t
a is driven by an external force. The system state then isintroduced. Forbothcaseswereportnewexperimental
m
depends on the history of the system. In many sys- results for subloops showing qualitative agreement with
- tems such as ferromagnets [1], superconductors [2, 3], the two behaviors of the RFIM.
d
and martensites [4], the response to the driving force is In the non-equilibrium, zero-temperature RFIM spins
n
o not continuous but occurs in discrete jumps which is of- si = ±1 are placed on a hyper-cubic lattice. At each
c ten referred to as “crackling noise”, or specifically for site i a quenched random local field h is chosen from
i
1 [ mofapgonwetesr,-lBawarskchaaliunsgenofnnooisisee. hIanvembaegennetast,trbirbouatdedratongeeis- afieGldauhssaiacntsdaisstraibsuotuirocne,oρf(hdii)so=rde√r21fπoRretxhpe(−sy2shRt2ie2m).,Tanhde
i
v ther a disorder induced critical point, such as found in
the standard deviation R of the random distribution is
3 the non-equilibriumzero temperature random-fieldIsing
termed the ’disorder’. The energy of a system with N
2 model (RFIM) [5], or to self-organized criticality (SOC)
0 spins is given by [15]
[6], as found in soft magnets.
1
0 JLR
4 Thedisorderinducedcriticalpointfoundinthesatura- H=−JXsisj −X(hi+H)si+ N Xsisj, (1)
0 tionmagnetizationcurve,tracedoutasanexternalmag- ij i ij
h i
/ netic field drives the system from one saturated state to
t
a theoppositelymagnetizedstateandback,hasbeenstud- where H is a tunable external magnetic field. The first
m ied in detail for the RFIM [5, 7, 8, 9] and its existence termofEq.1couplesnearest-neighborspinsferromagnet-
- has been experimentally confirmed [10]. However, tak- ically (hiji implies summing over nearest-neighbor pairs
d
ing magnetsto saturationoftenis impracticaldue to the of spins) while the last term provides an infinite range
n
o largemagneticfieldsrequired,sothebehaviorofsubloops anti-ferromagnetic (AF) coupling, where one sums over
c isofgreatinteresttoexperimentsandapplicationsalike. all pairs of spins regardless of their relative distance.
v: TheRFIMmaybeusedtomodelsubloops,andmagneti- This AF coupling models the dipolar interactions rele-
i zationcurves havebeen computed exactly in one dimen- vant in soft ferromagnets, which Zapperi, et. al. [6, 16]
X
sion [11] and on a Bethe lattice [12] and have also been have shown in three dimensions to have the same effect
ar collapsed near the demagnetized state using Rayleigh’s onlong lengthscalesasinfinite rangemeanfieldinterac-
Law [13]. Also, the idea of history acting as a source of tions. Note that by choosing JLR = 0 one recovers the
effectivedisorderwasrecentlyintroduced[14]. Inthislet- nearest-neighbor RFIM. In order to model hysteresis we
terwereporthowthepresenceorlackoflongrange(LR) study the model at zero temperature, far from thermal
demagnetizing fields produces two distinct behaviors in equilibrium [5, 8], and for convenience set J =1.
subloops of the RFIM. In particular, in the presence of Simulations of the above model were run by starting
sufficiently large LR fields, subloops are found to resem- with the external field at H = −∞ with all spins down,
ble a SOC system. However,in the absence of LR fields, and then adiabatically slowly moving the external field
wefindacriticalsubloopinsidethesaturationloopwhere throughaparticularhistory. AsthefieldH ischanged,a
the system history acts as a tuning parameter instead of given spin will flip (either upwards or downwards) when
thesystemdisorder. Correspondingly,amodifiedscaling itseffectivelocalfield,heiff =H+hi+JP ij sj−JLRM,
h i
2
1 a very different picture in the absence of LR fields. In
(a) 80 (c)
Fig. 1(b), obtained for a system with J = 0, R = 2.7
LR
40
u) and subloop spacing ∆M = 0.1, as one moves inwards,
m
M 0 e 0 subloops begin to resemble the saturation loop at an
µ
M ( -40 effectively higher disorder, i.e. R > 2.7. Indeed, pre-
flipped spins not participating in a given subloop may
-80
-1 actasanadded,possiblycorrelated,“effectivedisorder.”
-3 -1.5 0 1.5 3 -3 -1.5 0 1.5 3 Tuning the history in this way, we were unable to di-
H H (kOe) rectly observe a transition from loops with a jump in
1 80 magnetization to smooth inner subloops. This is due to
(b) (d)
the inability to break up the system spanning avalanche
40
u) present below the critical disorder even for the largest
m
M 0 e 0 simulated system sizes (4803 spins) [13, 14, 19].
µ
M ( Experimental magnetization curves for thin films also
-40
exhibit these two types of behavior depending on the
-1 -80 presence of LR fields. Fig. 1(c) displays magnetiza-
-3 -1.5 0 1.5 3 -5 -2.5 0 2.5 5 tion curves for a Co/Pt-multilayerwith the field applied
H H (kOe)
along the surface normal. Due to the strong interface
anisotropy of such multilayer samples, the easy axis of
FIG. 1: Hysteresis loops with concentric inner subloops for
(a) RFIM with 2403 spins, JLR = 0.25, and R = 1.8, (b) magnetization is perpendicular to the film plane even
RFIM with 2403 spins, JLR = 0, and R = 2.7, (c) a Co/Pt though the LR demagnetizing effect is largest in this di-
multilayer thin film, and (d) a CoPtCrB alloy thin film. rection. Despite the fairlyrectangularmajorloopshape,
indicating easy axis orientation, the loop exhibits an ex-
tended linear segment on which all minor loops merge
changessign. Here M = 1 s is the magnetizationof due to the presence of LR dipolar effects. On the other
N Pi i
a system with N spins. When a spin flips, it may induce hand,filmswithin-planemagnetizationbehavequitedif-
its neighboring (or for J 6= 0 even distant) spins to ferently,becauseLRdemagnetizingeffectsareextremely
LR
flip, creating an avalanche of flipping spins, which is the small in this geometry. Fig. 1(d) shows magnetization
analogofaBarkhausenpulse. Thesimulationsarebased curves for a CoPtCrB-alloy film with the field applied
on the code available on the web [17] which has been in the film plane. This film is polycrystalline with grain
modified to allow for subloops in the history. The code sizes narrowly distributed around 10 nm diameter and
uses the sorted list algorithm, which stores the random exhibits strong exchange coupling between grains. Due
fields, and is described in depth in Ref. [18]. to the lack of LR coupling, subsequent minor loops ap-
In the absence of LR fields, as one tunes the system pear increasingly sheared with decreasing coercive field,
disorder (R), the RFIM exhibits a non-equilibrium sec- very similar to the curves shown in Fig. 1(b) for the
ond order phase transition at a critical disorder R =R J =0 RFIM. All experimentaldata were measuredat
c LR
[5]. Below R the coupling of nearest-neighbors domi- room temperature using an Alternating Gradient Mag-
c
nate the dynamics, and there is a finite jump in magne- netometer.
tization of the saturation loop. However, above R the Much information on these two behaviors can be ob-
c
random fields dominate, resulting in smooth hysteresis tainedbyexaminingtheBarkhausennoisepresentinthe
curvesandmostly smallavalanches. Manyquantitiesas- system. With LR fields present, the subloops display a
sociated with this critical point display scaling behavior power-law avalanche size distribution, all with the same
for R→R . A detailed discussion is given in Ref. [8]. exponent and cutoff size, indicating that the system is
c
The two distinct behaviors of subloops in the RFIM SOC [6, 15, 16]. Figure 2 displays the avalanche size
can be seen in Figure 1. In Fig. 1(a) subloops spaced by distribution for subloops of a 21 cm x 1 cm x 30 µm
∆M = 0.1 are shown for a system with J = 0.25 and ribbon of a Fe Co B amorphous alloy. In the ex-
LR 21 64 15
a disorder of R = 1.8, chosen smaller than R = 2.16 as periment, a solenoid provides a triangular driving field
c
manyexperimentalsamplessuchassoftferromagnetsare along the long axis of the sample at 0.03 Oe/s. The first
believed to be below the critical disorder. The subloops cycle drives the sample to saturation, while subsequent
showa linearM(H)behaviorasthe fieldis increased,as cycles drive the field to successively smaller amplitudes
one would expect for a system undergoing domain wall (subloops). The driving rate is slow enough to ensure
propagation while being exposed to a demagnetizing ef- that avalanches do not overlap. The Barkhausen noise
fect. Only the first few subloops show some effects of was measured with a pick-up coil of 150 turns of cop-
the initialcondition(allspinsdown)before thesweeping per wire woundaroundthe middle 0.3cm ofthe sample.
fieldhascreatedalargeenoughdomaintoallowforsingle By integrating the pick-up voltage one may obtain the
domain wall propagation. On the other hand one finds magnetization. The resulting hysteresis loops are shown
3
10-3 1.0 )x100
a
m
H
utionution A/m) 00..05 100 intG(x,f10-2
e Size Distribe Size Distrib 1100--96 M (M --10..05 -40-2H0 (A0/m2)0 40 intintGG(x,H)(x,H)maxmaxff 10-5 βνd+/x ll 10-410-2x (Hmaxc-H1m00ax)νf
hh
cc
nn
aa
valval 10-10 HHmax == 11..4421297647
AA max
H = 1.41317
10-12 max
100 102 104 1 10 100
Avalanche Size (pWb) Distance (x)
FIG. 2: Experimental avalanche size distributions for FIG. 3: Integrated spin-flip correlation functions for 4803
subloops of a soft ferromagnet. The three subloops analyzed systems at R = 2.198 and averaged over 20 random seeds.
are shown in the inset, with the largest corresponding to the Curves are given for subloops starting at values of Hmax =
saturationloop. Thedistributionswereextractedfromawin- 1.42274,1.41967,and1.41317. Theinsetcontainsacollapseof
dow of width ∆H ≈20A/m which started near H = 0A/m. thethreerespectivedistributions,yieldingd+β/ν =3.0±0.2
A power law of −1.75 is show by theoffset, dash-dot line. and 1/νf =1.28±0.40 with Hmaxc =1.427. l l
in the inset of Fig. 2. The avalanche distributions for
magnetization curve [20]. According to scaling theory
the subloops are all identical, affirming the SOC behav-
one expects the exponents d+β /ν, ν, and ν as well
l l l f
ior. The measured power law exponent (1.75) is larger
as the scaling function Gint to be universal. Here a sub-
than has been predicted and measured for the satura-
scriptl denotes anexponentassociatedwith the random
tion loop (1.25−1.5) [15, 16]. This is due to M(H) not
disorder scaling of the critical subloop and subscript f
being strictly linear in the window analyzed. Choosing
one associated with the history-induced “disorder”. For
a smaller window close to the linear regime results in a
the collapses, the system is run at the effective critical
powerlawexponentof1.3,consistentwithpreviousmea-
disorderofthe saturationloop,R (L),for the linearsys-
surements. The RFIMwith LRfields presentsanalmost c
temsizeLwithR (L)→R asL→∞. Thissystemsize
identical picture of subloops. For a 2403 system with c c
dependent criticaldisorderR (L)is definedas the disor-
c
R = 1.8 and J = 0.25, subloops displayed identical
LR der at which the maximum number of system spanning
power laws in their avalanche size distributions with a
avalanches in the saturation loop is observed [8].
loopintegratedexponentof 1.7±0.2,in goodagreement
The integrated spin-flip correlation function,
with the experimental results.
Gint(x,H ), was measured for subloops spaced
In the absence of LR fields the case is much different; f max
by ∆M = 0.025 in their maximum magnetizations
the system history acts as a tuning parameter affecting max
(see Fig. 3). A similar behavior to the saturation
a subloop dependent cutoff. Although a transition like
loop is found. As the history-induced disorder is
the one found in the saturation loop cannot be directly
increased, by moving to inner subloops, the curves
observed,its existence may be ascertained by examining
display a decreasing cutoff. As the system is started
the scaling behavioron one side of (“above”)the critical
at the effective critical disorder, Eq. 2 reduces to
point [5, 8]. Thus we present a scaling collapse of loop
integratedspin-flipcorrelationfunctions anda finite-size Gifnt(x,Hmax) ∼ x−(d+βl/νl)Gfint(xfνf). This scaling
ansatzresultsinthecollapseshownintheinsetofFig.3.
scaling collapse of the second moments of the integrated
Only subloops void of spanning avalanches were used in
avalanche size distribution for subloops with J =0.
LR the collapses to remove the effects of the finite system
For the spin-flip correlation function, in analogy with
size, equal to 4803 spins.
the saturation loop [8] one may assume a scaling form
The integrated avalanche size distribution,
Gint(x,R,Hmax)∼x−(d+βl/νl)Gint(x(R−Rcd)νl,xfνf), with a similar scaling ansatz Dfint(S,Hmax) ∼
(2) S−(τ+σβδ)lDfint(Sσff), was previously analyzed in
where H is the maximum field along the subloop Ref. [14]. The conclusions therein remain valid, and the
max
and f = H − H is the history-induced analog exponents(seeTableI)havebeenupdatedtocorrespond
maxc max
of the reduced saturation loop disorder [21]. Here Rd is with the use of the scaling variable f.
c
the criticaldisorderforthe demagnetizationcurvewhich Table I lists the exponents from the above two col-
is numerically within the error bars of R . Similarly, lapsesalongwiththe saturationloopvalues. While both
c
H = Hd is the critical field associated with the de- power law behaviors, given by τ +σβδ and d+β/ν, are
maxc c
4
firmed by Barkhausen noise measurements on soft mag-
TABLEI: Universalcriticalexponentsfromscalingcollapses
nets. However, in the absence of such forces the disor-
in three dimensions for the history-induced disorder present
deredcriticalpointwasreflectedinsubloops. Scalingcol-
in subloops and the random disorder of thesaturation loop.
lapseswereperformedforintegratedavalanchesizedistri-
a
Exponent History-Induced Saturation butions, integratedcorrelationfunctions, andthe second
τ +σβδ 2.01±0.10 2.03±0.03 moments of the avalanche size distributions. Finally, it
1/σ 2.3±0.5 4.2±0.3 was shownthat only one of the new critical exponents is
d+β/ν 3.0±0.2 3.07±0.30 independent of those found in the critical subloop.
1/ν 1.28±0.40 0.71±0.09 WethankJimSethnaandGaryFriedmanforveryuse-
ρ 2.6±0.40 2.90±0.16 fuldiscussionsandFerencPazmandiforsuggestingtouse
H asthe scalingvariableinsteadofM . The work
aReferences[7,8] atmUaxIUC was supported by NSF grant DmMaRxs 99-76550
(MCC), 00-72783, and 02-40644, an A. P. Sloan fellow-
ship (to K. D.), and an equipment awardfrom IBM.
identical within error, the exponents governing the cut-
off behavior, σ and ν , are found to differ from their
f f
saturation loop counterparts with only a slight overlap
in their error estimates. One would expect identical val-
ues if the pre-flippedspins at the startofa subloopwere [1] P. J. Cote and L. V. Meisel, Phys. Rev. Lett. 67, 1334
randomly distributed about the lattice, thus preserving (1991).
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der. However, this is not the case, as avalanches have (1995).
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(1995).
up to the system’s correlation length. So the difference
[4] E. Vives, J. Ort´ın, L. Man˜osa, I. R`afols, R. P´erez-
inexponentsis notsurprisingasthe history-induceddis-
Magran´e, and A. Planes, Phys. Rev. Lett. 72, 1694
order acts more like a correlated disorder as opposed to
(1994).
the uncorrelated, random system disorder R. [5] J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl,
The second moments of the integrated avalanche size B. W. Roberts, and J. D. Shore, Phys. Rev. Lett. 70,
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loop and Eq. 2, hS2iifnt ∼ L−ρlSf2(fL1/νf) where ρl = [7] ORe.vP.eLrketotv.i´c7,9K, 4.6D69ah(m19e9n7,).and J. P. Sethna, Phys. Rev.
−(τ +σ β δ −3)/σν. Table I contains the exponents
l l l l l l Lett. 75, 4528 (1995).
from a collapse for systems with sizes from L = 80 to
[8] O.Perkovi´c,K.A.Dahmen,andJ.P.Sethna,Phys.Rev.
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required in the collapse. The exponent ν was chosen to (1996).
f
be consistent across all collapses of the correlation func- [10] A. Berger, A. Inomata, J. S. Jiang, J. E. Pearson, and
S. D. Bader, Phys. Rev.Lett. 85, 4176 (2000).
tionandsecondmoments. Itsvaluediffersfromthevalue
[11] P. Shukla,Phys. Rev.E 62, 4725 (2000).
of ν for the saturation loop, which reaffirms the differ-
[12] P. Shukla,Phys. Rev.E 63, 027102 (2001).
ences between the history-inducedand random disorder.
[13] L. Dante, G. Durin, A. Magni, and S. Zapperi, Phys.
Two new exponents, σf and νf, were introduced to Rev. B 65, 144441 (2002).
describe the history-induced scaling. However, only one [14] J. H. Carpenter, K. A. Dahmen, J. P. Sethna, G. Fried-
is an independent exponent as they obey the exponent man, S. Loverde, and A. Vanderveld,J. Appl. Phys. 89,
relation σ ν = σ ν, which may be derived from the 6799 (2001).
f f l l [15] M. C. Kuntz and J. P. Sethna, Phys. Rev. B 62, 11699
relations in Ref. [20]. Additionally, the critical subloop
(2000).
exponents may be close to those of the saturation loop.
[16] S.Zapperi,P.Cizeau,G.Durin,andH.E.Stanley,Phys.
Indeed, within error bars, numerical results have found Rev. B 58, 6353 (1998).
equalpower-lawexponents. However,currentlynoexpo- [17] M. C. Kuntz and J. P. Sethna, URL
nent relations are known that require equality between http://www.lassp.cornell.edu/sethna/hysteresis/code.
the saturationloop and subloop(subscript l) exponents. [18] M.C.Kuntz,O.Perkovi´c,K.A.Dahmen,B.W.Roberts,
The RFIM has been used to investigate subloops of and J. P. Sethna,Comput. Sci. Eng. 1, 73 (1999).
[19] M. C. Kuntzand J. P. Sethna, unpublished.
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surementofsubloopsonthinfilmmagnetsbothwithand
020412(R) (2003).
withoutLRforcesshowedqualitativeagreementwiththe [21] Ref.[14]usedǫ=(Mmaxc−Mmax)/Mmaxforthehistory
RFIM. In the presence of LR forces, subloops may be ’disorder’.However,f providesasounderscalingpicture.
explained by simple domain wall propagation, as con-