Depinning of an anisotropic interface in random media The tilt effect

a r X i v :c o n d -m a t /0008225v 1 [c o n d -m a t .s t a t -m e c h ] 16 A u g 2000

Depinning of an anisotropic interface in random media:The tilt e?ect

K.-I.Goh 1,H.Jeong 2,B.Kahng 3and D.Kim 1

1.Center for Theoretical Physics and Department of Physics,Seoul National University,Seoul 151-742,Korea

2.Department of Physics,University of Notre Dame,Notre Dame,IN 46556

3.Department of Physics and Center for Advanced Materials and Devices,Konkuk University,Seoul 143-701,Korea

(February 1,2008)

We study the tilt dependence of the pinning-depinning transition for an interface described by the anisotropic quenched Kardar-Parisi-Zhang equation in 2+1dimensions,where the two signs of the nonlinear terms are di?erent from each other.When the substrate is tilted by m along the positive sign direction,the critical force F c (m )depends on m as F c (m )?F c (0)～?|m |1.9(1).The interface velocity v near the critical force follows the scaling form v ～|f |θΨ±(m 2/|f |θ+φ)with θ=0.9(1)and φ=0.2(1),where f ≡F ?F c (0)and F is the driving force.PACS numbers:68.35.Fx,05.40.+j,64.60.Ht

The pinning-depinning (PD)transition from a pinned to a moving state is of interest due to its relevance to many physical systems.Typical examples include inter-face growth in porous (disordered)media under external pressure [1,2],dynamics of a domain wall under exter-nal ?eld [3–5],dynamics of a charge density wave under external ?eld [6,7],and vortex motion in superconduc-tors under external current [8,9].In the PD transition,there exists a critical value,F c ,of the external driving force F ,such that for F F c ,it moves forward with a constant velocity v ,leading to a transition across F c .The velocity v plays the role of the order parameter and typically behaves as

v ～(F ?F c )θ

(1)

with the velocity exponent θ.

The interface dynamics in disordered media may be described via the Langevin-type continuum equation for the interface position h (x,t );

?t h (x,t )=K [h ]+F +η(x,h ).

(2)The ?rst term on the right hand side of Eq.(2)describes the con?guration dependent force,the second is the ex-ternal driving force,and the last,the quenched random noise,independent of time,describes the ?uctuating force due to randomness or impurities in the medium.The ran-dom noise is assumed to have the properties, η(x,h ) =0and η(x,h )η(x ′,h ′) =2Dδd (x ?x ′)δ(h ?h ′),where the angular brackets represent the average over di?erent realizations and d is the substrate dimension.When K [h ]=ν?2h,the resulting linear equation is called the quenched Edwards-Wilkinson equation (QEW)[10].Recently,a couple of stochastic models mimicking the interface dynamics in disordered media have been intro-duced [11,12],displaying the PD transition di?erent from the QEW universality behavior [13].It was proposed that

the models are described by the Kardar-Parisi-Zhang equation with quenched noise (QKPZ)[14],where

K [h ]=ν?2h +

λ

insensitive to F.When the substrate-tilt m is smaller than s c,the PD transition is discontinuous,and F c is independent of m,while the transition is continuous and F c increases with m for m>s c.The discontinuous tran-sition is caused by the presence of a critical pinning force due to both the negative nonlinear term and random noise,which is localized.Once this pinning force is over-come by increasing external force F,the surface moves forward abruptly,yielding the velocity jump at F c.The amount of the velocity jump decreases with increasing m,and vanishes at the characteristic tilt m c=s c.For m>m c,the velocity increases continuously from zero, and the PD transition is continuous.Accordingly,the characteristic substrate-tilt m c is a multicritical point [16,17].

Since the sign of the nonlinear term is relevant in the quenched case,it would be interesting to consider the case of the anisotropic QKPZ equation in2+1dimen-sions,where the signs of the nonlinear terms are alterna-tive.Thus,we consider

?t h=νx?2x h+νy?2y h+λx

2

(?y h)2

+F+η(x,y,h),(6)

whereλx>0andλy<0.The anisotropic case is in particular interesting due to its application to the vortex motion in disordered system[9]and the adatom motion on step edge in epitaxial surface[19].It has been shown [19]that for the thermal case,the two nonlinear terms cancel each other e?ectively,thereby the anisotropic KPZ equation is reduced to the linear equation,the EW equation.For the quenched case,however,the interface dynamics in each direction are di?erent from each other and the surface morphology is anisotropic:the surface is gently sloping in the positive sign direction(x-direction), and is of the shape of a mountain range with steep slope in the negative sign direction(y-direction).In spite of the facet shape in the negative sign direction,the PD transition is continuous due to the critical behavior in the positive sign direction.Consequently,one may ex-pect that the critical force and the e?ective nonlinear coe?cient along the positive sign direction can be de-scribed by the scaling theory introduced forλ>0in 1+1dimensions.In this Brief Report,we show,from extensive numerical simulations,that this is indeed the case and determine the scaling exponentsφandν(1?α) independently.

Direct numerical integration has been carried out using standard discretization techniques[20,21],in which we choose the parameters,νx=νy=1,λx=?λy=1,and a temporal increment?t=0.01.The noise is discretized asη(x,y,[h]),where[···]means the integer part,andηis uniformly distributed in[?a/2,a/2]with a=(10)2/3. In order to consider the tilt-dependence,we tilted the substrate as h(x,y,0)=mx along the positive sign direction,and used the helicoidal boundary condition, h(L+x,y,t)=h(x,y,t)+Lm.We measured the growth velocity as a function of the external force F for several values of substrate-tilt m,which is shown in Fig.1.For m=0,we found thatθ=0.9(1)as in Ref.[16].But for non-zero m,the exponentθ(m)is generically1,di?erent from its m=0value[15].This picture can be seen in Fig.1,which shows straight lines for large m.

The critical force F c is estimated as the maximum value of F for which all samples of10are pinned until a large Monte Carlo steps,typically105?t.In this way, we?nd F c(0)≈0.51(1).This value is slightly larger than that obtained by extrapolating the velocity curve,≈0.50.Also,we estimated the critical force F c(m)as a function of the substrate-tilt m.The critical force F c(m) decreases with increasing substrate-tilt m with the expo-nent1/ν(1?α)≈1.9(1)as shown in Fig.2.

The PD transition is continuous due to the positive nonlinear term.To determine the exponentφindepen-dently,we assume the scaling form for the interface ve-locity v(F,m)as[13,15]

v～|f|θΨ± m2

the scaling relation,Eq.

(5),

within the errors.Alternatively,one may put the scaling form as

v ～m 2θ/(θ+φ)Φ

f

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0.10.20.30.40.5

0.6v

F

FIG.1.The interface velocity as a function of F for various m .The curves correspond to the cases of m =0.0,0.2,0.4,0.6,0.8,0.9,from right to left.

0.01

0.1

1

0.1

1

F c (0)-F c (m )

m

FIG.2.Log-log plot of F c (0)?F c (m )versus m .The dashed line has slope 1.9,drawn for the eye.

0.11

10

100

1000

0.01

0.1

1

101001000

v /|f |θ

m 2

/|f|

θ + φ

FIG.3.Data collapse for the interface velocity using Eq.(7)with the exponents θ=0.9(1)and φ=0.2(1).The dashed lines are drawn for the approximate form of the scaling functions,Eqs.(8)and (9).

0.1

1

10

100

0.1

1

10100

v /m 2 θ/(θ+φ)

f/m

2/(θ+φ)

+c 0

FIG.4.Data collapse for the interface velocity using Eq.(10)with θ=0.9(1)and φ=0.2(1).The dotted (dashed)line has slope 0.9(1.0),drawn for the eye.