Table Of Content3
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a Dynamics and Melting of Stripes, Crystals, and Bubbles with Quenched
J
4 Disorder
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C. J. Olson Reichhardt, C. Reichhardt, I. Martin, and A.R. Bishop a
]
ft aCenter for Nonlinear Science and Theoretical Division, Los Alamos National Laboratory, Los Alamos,
o
NM 87545.
s
.
at Two-dimensional systems in which there is a competition between long-range repulsion and short range at-
m traction exhibit a remarkable variety of patterns such as stripes, bubbles, and labyrinths. Such systems include
magnetic films, Langmuir monolayers, polymers, gels, water-oil mixtures, and two-dimensional electron systems.
-
d In many of these systems quenched disorder from the underlying substrate may be present. We examine the
n dynamics and stripe formation in the presence of both an applied dc drive and quenched disorder. When the
o disorder strength exceeds a critical value, an applied dc drive can induce a dynamical stripe ordering transition
c toa state that is more ordered than the originating undriven,unpinned pattern.
[
1
v
6
The growing interest in nonextensive thermo- that the former will exhibit mesoscopic statistics
3
dynamicsandstatisticalmechanicsrepresentedin driven by their mutual interactions, whereas the
2
1 these Proceedings is stimulated by observations latter should show mesoscopic statistics driven
0 (from real experiments or simulations of models) by the (colored, multiplicative) noise bath of the
3 of phenomena such as incomplete (e.g., fractal) non-collective degrees of freedom.
0
coverage of phase space, rare-event statistics, in- We have studied aspects of anomalous statis-
/
t termittency, stochastic resonance, and so forth. tics and dynamics in a number of physically mo-
a
m We can hope for, and certainly seek to find, un- tivated models, including: soliton diffusion [1];
derpinningmechanismsandclassificationsofsuch relaxation in two and three dimensional complex
-
d systems. Ginzburg-Landau equations [2]; multiscale struc-
n We suggest that among the relevant classi- ture and dynamics in elastic media [3]; filamen-
o fications are (a) systems exhibiting mesoscopic tary transportinsuperconductorflux lattices[4];
c
self-assembly (usually from nonlinearity or com- surface morphology and growth [5]; relaxation in
:
v peting scales at more microscopic scales), and magnets with planar spin anisotropy [6]; glassy
i
X (b) systems with long-rangeinteractions (and of- dynamics near a spinodal in a Van der Waals
ten coexisting short-range interactions and en- fluid [7]; charge, spin ordering and dynamics in
r
a tropic forces). Long-range interactions may be strongly correlated electronic materials [8]; and
in the microscopic variables of the system (e.g., large-scale materials deformation (fracture, fric-
Coulombfields),ortheymaybeinducedatmeso- tion) [9].
scopic scales because of local constraints or sym- Motivatedbythesimilaritiesinthephenomena
metries (e.g., vortices in fluids and magnets, pat- exhibited by these (and related) model systems,
terns in convection cells and vibrated granular wehavebegunstudiesofa“minimal”modelwith
materials,strain-straininteractionsinelasticme- coexistingshort-andlong-rangeinteractions. We
dia). It is important to distinguish topological introducethismodelhere,andbrieflyreportsome
(singularity) mesoscale excitations, which have of its properties.
long-range mutual interactions (such as vortices, Recent proposals for order-
dislocations, stress filaments) from mesoscopic ingintwo-dimensional(2D)electronsystemscon-
collective excitations with short-range interac- sider the idea ofanelectronliquidcrystalsystem
tions (such as solitons, breathers). We expect [10–12]. Such a system can form stripe phases
2
with varying degrees of disorder: An anisotropic impurities in the material, as well as an external
Wigner crystalis well-orderedin both directions; field, which would be an electric field in the case
a smectic phase has ordering along only one di- of holes.
rection; in a nematic, dislocations are introduced Theparticlesareassumedtoobeyoverdamped
into the smectic phase but there is still an over- Langevin dynamics given by
allpreferredorientationforthe system;finally,in
an isotropic liquid stripe phase, the stripe pat- Nv
f =− ∇U(ρ)+fT +fp+fd =ηv . (1)
tern is filled with dislocations and has no unique i X i
j=1
orientational direction.
Stripepatternsareverygeneralandareformed The potential between two particles is
in a wide rangeof systems[13]. Forexample, the
q
Swift-Hohenberg model produces glassy stripe U(ρ)= −Bexp(−ρ/ξ), (2)
ρ
configurations[14]. Otherstripe-formingsystems
include magnetic films [15], Langmuir monolay- where q = 1 is the charge on each particle, and
ers, polymers, gels, and water-oil mixtures [16]. ξ is the magnetic screening length. The long
One system in which stripes have attracted a range interactions are treated as in Ref. [17].
particularly large amount of interest is holes in Temperature is represented by random thermal
layered transition metal oxides. In a quasiclassi- kicks which have the property, < f(t)T >= 0,
caldescription,theinteractionbetweentwoholes < f(t)Tf(t′)T >= 2ηk Tδ δ(t−t′). The disor-
i j B ij
consists of both a long-rangerepulsiongenerated der is modeled as random, point-like pins, repre-
by the Coulomb interaction, as well as a short- sented by parabolic traps. The pinning force is
range attraction that arises due to the breaking given by
of the antiferromagnetic bonds surrounding the
holes. As a result, the holes can form linear ar- Np f
fp = p|r −r(p)|Θ ξ −|r −r(p) ˆr , (3)
rays termed stripes. There is experimental evi- i X ξ i k (cid:16) p i k (cid:17) i,k
p
dence for the presence of stripes in several of the k=1
lanthanum cuprate oxides, but it is not yet clear wheref =0.2isthepinningforce,ξ =0.125ξis
p p
whetherthehigh-Tcsuperconductingcompounds the pin radius, and Θ is the Heaviside step func-
such as YBCO contain stripes, nor what the role tion. The force from an applied field (voltage) is
of the stripes would be if they exist. Experimen- modeled as a uniform drive, f = f xˆ. We mea-
d d
tal detection of stripes is made more difficult by sure the velocitysignature<V > of the system,
x
the possible interaction of impurities in the ma- which would correspond to a current.
terialwiththestripes. Theeffectofdisorderona We initially consider a system with no pin-
stripephaseisnotwellunderstood. Thereforewe ning or external drive. We vary the strength of
have constructed a simple model containing the the short range attraction by changing the value
basicphysicsofthesysteminordertogainabet- of B, while holding the particle density fixed at
ter understanding of the ordering and dynamics n = 0.64/ξ2 As shown in Fig. 1(a), when B is
of a stripe phase. small,thelongrangerepulsiondominatesandthe
Our simple model consists of particles with system forms a Wigner crystal. At high values
competing long-range and short-range interac- of B, Fig. 1(c), the short range attraction dom-
tions. The particles experience a long-range re- inates and the particles form clumps which be-
pulsionofaCoulombform,andashort-rangeat- comelargerasthevalueofB isfurtherincreased.
tractionwhichweassumetobeasimpleexponen- The clumps organize into a superstructure of a
tial form. When these two interactions are com- triangularlattice of clumps. At intermediate val-
bined, a new length scaleemergeswhich depends ues of B, Fig. 1(b), the short and long range in-
on the density of the particles and the strength teractions are of comparable strength, and the
of the short-range attractive term. We further system forms a disordered stripe structure. We
add to our model quenched disorder to represent observe the same phases if we fix the value of B
3
and vary the density of the system. This is illus- phase that it is very difficult for them to adjust
trated in Fig. 2. in order to find the lowest energy pinning loca-
We have measured the melting temperature of tion. As a result many of the pins are ineffective
the three phases by computing the particle diffu- in the clump phase and the depinning force is
sion at each temperature. We define the melting low. This similarity between the melting and de-
temperature to be the temperature at which the pinning signaturesisparticularlyinterestingasit
particlesfirstdiffusemorethanalatticeconstant. implies that information about a thermal transi-
The melting temperature T as a function of B tion (melting) can be obtained from a dynamical
m
is plotted in Fig. 3. T decreases in the crystal experimentatconstanttemperature(depinning).
m
phase as B increases since the short range at- Athighapplieddrives,itispossibleto(dynam-
tractive term introduces distortions in the crys- ically) reorderthe phases, which are distortedby
tal, allowing it to melt more easily. The stripe the presence of the pinning at low drives. Dy-
phasehasthelowestmeltingtemperature,butT namical reordering of a triangular lattice of re-
m
increases dramatically upon entering the clump pulsively interacting particles has been observed
phase. This is because the clump phase contains previouslyinseveralsystems,includingsupercon-
a large amount of elastic energy. Within each ducting vortices [18,19]and Wigner crystals [20].
clump there is triangular ordering of the parti- At high drives, the force from the pinning begins
cles,andadditionallytheclumpsformanordered toresembleatemperaturewhichdecreasesasthe
superstructure. We find that the clump phase driveincreases,untilatinfinitedrivetheeffective
simultaneously melts and dissociates for the pa- temperatureiszero,leadingtotheconceptofdy-
rameters considered here. In contrast, in the namical freezing [18]. Dynamical freezing has al-
stripe phase melting first occurs along the length readybeenobservedintheWignercrystalphase.
of each stripe, as illustrated in Fig. 4. Particles We find that the clump phase also undergoes a
diffuse freely within the stripes but do not enter dynamical freezing transition, and that at high
the regions between the stripes until higher tem- drives the clump phase in the presence of disor-
peratures are applied. Thus we find evidence for der has the same form as the static clump phase
a two-stage melting of the stripe phase. in the absence of disorder, as shown in Fig. 6.
We next consider the effect of disorder on the When we consider the dynamical reorderingof
three phases. We add random point-like disor- thestripephase,illustratedinFig.7,wefindthat
der to the system,andthen apply a driving force thefinalorderedstripephaseismoreorderedthan
to the particles. At small drive strengths, the the corresponding stripe phase that appears in a
particles are trapped by the disorder and do not static system without pinning. The stripes are
move. We measure the threshold force required alignedduetothedrivingforce,ratherthanbeing
todepin the particlesineachphase. As shownin randomlyalignedaswasthecaseinthethermally
Fig.5,thedepinningforceincreaseswithincreas- annealed system. This indicates that, in a stripe
ingB inthe crystalphase,ishighestinthestripe formingsystem,itmaybepossibletouseacombi-
state, and then drops dramatically upon enter- nationofquencheddisorderandanexternaldrive
ing the clump phase. This is the same behavior, to produce well-ordered stripes. The reordering
but inverted, that was observed for the melting of the stripe phase occurs only if the disorder is
transition in Fig. 3. This similarity is due to the sufficiently strong to cause the stripe to undergo
fact that the softness of the stripe phase, which disordered, plastic flow at low drives. The orig-
allowsittomeltatlowertemperature,alsoallows inal ordering of the stripe must be destroyed by
itto be wellpinnedby therandomlylocatedpin- the disorder in order to allow for a new order-
ning sites. The stripe state is disorderedand can ing of the stripe to form aligned with the drive.
readily readjust to take advantage of the max- As a result of the reordering of the stripe phase,
imum number of pinning sites. In contrast, in only this shows hysteresis if the driving force is
the highly elastic and stiff clump phase, the par- increased and then decreased again. The other
ticles are so constrained by the ordering of the two phases, clump and Wigner, do not exhibit
4
any hysteresis. 1770;A. Sanchez,A.R.Bishop, andE.Moro,
In summary, the competition between short Phys. Rev. E 62 (2000) 3219.
range attraction and long range repulsion leads 6. S. Komineas et al., Europhys.Lett., in press.
to the formation of Wigner crystal, stripe, and 7. K.-K. Loh, K. Kawasaki, A.R. Bishop, T.
clump phases as a function of either the strength Lookman, A. Saxena, J. Schmalian, and Z.
of the short range term or the particle density. Nussinov, cond-mat/0206494.
The melting temperature is lowest in the stripe 8. B.P. Stojkovic, Z.G. Yu, A.L. Chernyshev,
phase, while the depinning force is highest in the A.R. Bishop, A.H. Castro Neto, and N.
stripe phase, and lower in the Wigner and clump Grønbech-Jensen, Phys. Rev. B 62 (2000)
phases. Quenched disorder distorts the ordered 4353; S. Teber, B.P. Stojkovic, S.A. Bra-
phases, but the structure of all three phases can zovskii,andA.R.Bishop,J.Phys.Cond.Mat.
beregainedthroughdynamicalreorderingathigh 13 (2001) 4015.
drives. Thedynamicallyreorderedstripephaseis 9. J.Ro¨der,J.E.Hammerberg,B.L.Holian,and
moreorderedthantheequilibriumstripephasein A.R.Bishop,Phys.Rev.B57(1998)2759;O.
a clean system, and as a result the stripe phases Braun,A.R.Bishop,andJ.Ro¨der,Phys.Rev.
exhibits hysteretic signatures which could be ob- Lett. 82 (1999) 3097.
served experimentally. 10. M.M. Fogler, A.A. Koulakov, and
Inthe contextof“nonextensivethermodynam- B.I. Shklovskii, Phys.Rev. B 54 (1996)1853.
ics,” it is nowclearlyofinterestto study the var- 11. S.A. Kivelson, E. Fradkin, and V.J. Emery,
ious mesoscopic phases of our model (and their Nature 393 (1998) 550.
dynamics), and to compare their statistics and 12. E. Fradkin and S.A. Kivelson, Phys. Rev. B
responses with predictions of Tsallis or other su- 59 (1999) 8065.
perstatistics. 13. M.SeulandD.Andelman,Science267(1995)
This work was supported by the US Depart- 476.
ment of Energy under contract W-7405-ENG-36. 14. D. Boyer and J. Vin˜als, cond-mat/0110254.
15. M.SeulandR.Wolfe,Phys.Rev.A46(1992)
REFERENCES 7519.
16. W.M. Gelbart and A. Ben Shaul, J. Phys.
1. See, e.g., K.Ø. Rasmussen, S. Aubry, A.R. Chem. 100 (1996) 13169.
Bishop, and T. Tsironis, Europ. J. Phys. B 17. N. Grønbech-Jensen, Int. J. Mod. Phys. C 7
15 (2000) 169. (1996) 873.
2. I.S. Aranson, A.R. Bishop, and L. Kramer, 18. A.E.KoshelevandV.M.Vinokur,Phys.Rev.
Phys. Rev. E 57 (1998) 5276; P. Kevrekidis, Lett. 73 (1994) 3580.
A.R. Bishop, and K.Ø. Rasmussen, Phys. 19. K. Moon, R.T. Scalettar, and G.T. Zim´anyi,
Rev. E 65 (2001) 16122. Phys. Rev. Lett. 77 (1996) 2778; C.J. Olson,
3. S.R. Shenoy, T. Lookman, A. Saxena, and C. Reichhardt, and F. Nori, Phys. Rev. Lett.
A.R.Bishop,Phys.Rev.B60(1999)R12537; 81 (1998) 3757; A.B. Kolton, D. Dom´ınguez,
K.Ø. Rasmussen, T. Lookman, A. Saxena, andN.Grønbech-Jensen,Phys.Rev.Lett.83
A.R. Bishop, R.C. Albers, and S.R. Shenoy, (1999) 3061.
Phys. Rev. Lett. 87 (2001) 5704; T. Look- 20. C. Reichhardt, C.J. Olson, N. Grønbech-
man, S.R. Shenoy, K.O. Rasmussen, A. Sax- Jensen, and F. Nori, Phys. Rev. Lett. 86
ena,andA.R.Bishop,Phys.Rev.B,inpress. (2001) 4354.
4. N. Grønbech-Jensen, A.R. Bishop, and D.
Dom´ınguez, Phys. Rev. Lett. 76 (1996)2985;
C.J.Olson,C.Reichhardt,andF.Nori,Phys.
Rev. Lett. 80 (1998) 2197.
5. I.S. Aranson, A.R. Bishop, I. Daruka, and
V.M. Vinokur, Phys. Rev. Lett. 80 (1998)
5
12
Stripe Clump
Crystal
10
a b c Melting temperature 468
Figure1. Staticphasesasafunctionofincreasing
2
strength of the short range attraction, B. (a)
Wigner crystal phase at low B. (b) Stripe phase
0
0 1 2 3 4 5 6
at intermediate B. (c) Clump phase at high B.
B
Figure 3. Melting temperature for the three
phases as a function of B.
Low T
High T
Figure 4. Snapshots at different temperatures il-
lustrating the melting of the stripe phase. Black
dots are the particles; lines indicate the trajecto-
ries of the particles over a fixed interval of time
Figure 2. Static phases for increasing particle
which is the same for all panels.
density with fixed B.
6
Crystal Stripe Clump
0.02
p
d
F
0.01
0
0 0.1 0.2 0.3 0.4 0.5
B
Figure5. DepinningforceF forthethreephases a b c
dp
as a function of B.
d e f
Figure7. Dynamicreorderingofthestripephase.
The external driving force increases from panel
(a) to panel (f).
a b c
d e f
Figure6. Dynamicreorderingoftheclumpphase.
The external driving force increases from panel
(a) to panel (f).
