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Generic Dynamic Scaling in Kinetic Roughening José J. Ramasco1,2,a, Juan M. Lo´pez3,b and Miguel A. Rodr´ıguez1 1 Instituto de F´ısica de Cantabria, CSIC-UC, E-39005 Santander, Spain 2 Departamento de F´ısica Moderna, Universidad de Cantabria, E-39005 Santander, Spain 0 3 Dipartamento di Fisica and Unità INFM, Università di Roma ”La Sapienza”, I-00185 Roma, Italy 0 Westudythedynamicscalinghypothesisininvariantsurfacegrowth. Weshowthattheexistence 0 of power-law scaling of the correlation functions (scale invariance) does not determine a unique 2 dynamic scaling form of the correlation functions, which leads to the different anomalous forms of n scaling recentlyobservedin growth models. Wederivealltheexistingforms ofanomalous dynamic a scaling from a newgeneric scaling ansatz. The different scaling forms are subclasses of this generic J scaling ansatz associated with bounds on the roughness exponent values. The existence of a new 0 class of anomalous dynamicscaling is predicted and compared with simulations. 1 ] Thetheoryofkineticrougheningdealswiththefateof A most intringuing feature of some growth models is h surfaces growing in nonequilibrium conditions [1,2]. In a that the above standard scaling of the global width dif- c typicalsituationaninitiallyflatsurfacegrowsandrough- fers substancially fromthe scaling behaviourof the local e m ens continuously as it is driven by some external noise. interfacefluctuations(measuredeitherbythelocalwidth Thenoisetermcanbeofthermalorigin(likeforinstance ortheheight-heightcorrelation). Moreprecisely,insome - t fluctuations in the flux of particles in a deposition pro- growthmodelsthelocalwidth(andtheheight-heightcor- a t cess),oraquencheddisorder(likeinthemotionofdriven relation) scales as in Eq.(1), i.e. w(l,t) = tβfA(l/ξ(t)), s interfaces through porous media). A rough surface may but with the anomalous scaling function . t a becharacterizedbythefluctuationsoftheheightaround uαloc if u≪1 m itsmeanvalue. So,abasicquantitytolookatistheglobal f (u)∼ , (3) A (cid:26)const if u≫1 - interface width, W(L,t) = h[h(x,t)−h]2i1/2, where the d overbar denotes average over all x in a system of size L where the new independent exponent α is called the n loc o and brackets denote average over different realizations. local roughness exponent. This is what has been called c Rough surfaces then correspond to situations in which anomalous roughening in the literature, and has been [ thestationarywidthW(L,t→∞)growswiththesystem found to occur in many growth models [4–10] as well 1 size. Alternatively,onemaycalculateotherquantitiesre- as experiments [11–15]. Moreover ,it has recently been v latedtocorrelationsoveradistancelastheheight-height shown [16,17] that anomalous roughening can take two 1 correlationfunction,G(l,t)=h[h(x+l,t)−h(x,t)]2i,or different forms. On the one hand, there are super-rough 1 the local width, w(l,t) = hh[h(x,t)−hhi ]2i i1/2, where processes, i.e. α>1, for which always α =1. On the 1 l l loc h···i denotes an average over x in windows of size l. other hand, there are intrinsically anomalous roughened 1 l 0 In absence of any characteristic length in the prob- surfaces, for which αloc < 1 and α can actually be any 0 lem growth processes are expected to show power-law α>αloc. 0 behaviour of the correlation functions in space and time Anomalous scaling implies that one more independent t/ and the Family-Vicsek dynamic scaling ansataz [3,1,2] exponent, αloc, may be needed in order to asses the uni- a versality class of the particular system under study. In m W(L,t)=tα/zf(L/ξ(t)), (1) other words, some growth models may have exactly the - ought to hold. The scaling function f(u) behaves as same α and z values seemingly indicating that they be- d n long to the same universality class. However, they may uα if u≪1 o f(u)∼ , (2) have different values of α showing that they actually c (cid:26)const. if u≫1 belong to distict classes olofcgrowth. As for the experi- : v where α is the roughnessexponentand characterizesthe ments, only the local roughness exponent is measurable i stationary regime, in which the horizontal correlation by direct methods, since the system size remains nor- X length ξ(t) ∼ t1/z (z is the so called dynamic exponent) mallyfixed. Fractureexperiments[14]insystemsofvary- r a has reached a value larger than the system size L. The ing sizes have succeded in measuring both the local and ratio β = α/z is called growth exponent and character- global roughness exponents in good agreement with the izes the short time behavior of the surface. As occurs in scaling picture described above. equilibrium critical phenomena, the corresponding crit- InthisLetterweintroduceanewanomalousdynamics ical exponents do not depend on microscopic details of in kinetic roughening. We show that, by adopting more the system under investigation. This has made possible generalforms of the scaling functions involved,a generic to divide growth processes into universality classes ac- theory of dynamic scaling can be constructed. Our the- cording to the values of these characteristic exponents oryincorporatesallthedifferentformsthatdynamicscal- [1,2]. ing can take, namely Family-Vicsek, super-rough and 1 intrinsic, as subclasses and predicts the existence of a assumptions (like for instance surface self-affinity or im- newclassofgrowthmodelswithnovelanomalousscaling plicitboundsfortheexponentsvalues)areimposed,then properties. Simulations of the Sneppen model (rule A) we propose that the structure factor is given by [18] of self-organized depinning (and other related models) are presented as examples of the new dynamics. S(k,t)=k−(2α+1)s(kt1/z), (6) Firstly, let us consider the Fourier transform of the where the scaling function has the general form heightofthe surfaceinasystemofsizeL,whichisgiven by h(k,t)=L−1/2 x[h(x,t)−h(t)]exp(ikx), where the u2(α−αs) if u≫1 spabtial average of tPhe height has been substracted. The s(u)∼(cid:26) u2α+1 if u≪1 , (7) scaling behaviour of the surface can now be investigated by calculating the structure factor or power spectrum and the exponent α is what we will call the spectral s roughness exponent. This scaling ansatz is a natural S(k,t)=hh(k,t)h(−k,t)i, (4) generalization of the scaling proposed for the structure factor in Ref. [16,17]for anomalous scaling. b b whichis relatedto the height-heightcorrelationfunction In the case of the global width, one can make use of G(l,t) defined above by 1 dk W2(L,t)= S(k,t)= S(k,t), (8) 4 L Z 2π G(l,t)= [1−cos(kl)]S(k,t) Xk L 2π/LX≤k≤π/a to prove easily that the global width scales as in Eqs.(1) π/a dk and (2), independently of the value of the exponents α ∝ [1−cos(kl)]S(k,t), (5) Z 2π and α . 2π/L s However, the scaling of the local width is much more where a is the lattice spacing and L is the system size. involved. The existence of a generic scaling behaviour In order to explore the most general form that kinetic like(7)forthestructurefactoralwaysleadstoadynamic roughening can take, we study the scaling behaviour of scaling behaviour, surfaces satisfying what we will call a generic dynamic scaling form of the correlation functions. We will con- w(l,t)∼ G(l,t)=tβg(l/ξ) (9) sider that a growing surface satisfies a generic dynamic p oftheheight-heightcorrelation(andlocalwidth),butthe scaling when there exists a correlation lenght ξ(t), i.e. correspondingscalingfunctiong(u)is notunique. When the distance over which correlationshave propagatedup substitutingEqs.(7)and(6)into(5),onecanseethatthe to time t, and ξ(t) ∼ t1/z, being z the dynamic ex- variouslimitsinvolved(a→0,ξ(t)/L→∞andL→∞) ponent. If no characteristic scale exists but ξ and the donotconmute[16,17]. Thisresultsinadifferentscaling system size L, then power-law behaviour in space and behaviourofg(u)dependingonthevalueoftheexponent time is expected and the growth saturates when ξ ∼ L α . and the correlations (and from Eq.(5) also the structure s Let us now summarize how all scaling behaviours re- factor) become time-independent. The global roughness ported in the literature are obainedfrom the generic dy- exponent α can now be calculated in this regime from namic scaling ansatz (7). We shall also show how a new G(l = L,t ≫ Lz) ∼ L2α (or W(L,t ≫ Lz) ∼ Lα). In roughening dynamics naturally appears in this scaling general, as we will see below, the scaling function that theory. Two major cases can be distinguised, namely enters the dynamic scaling of the local width (or the α < 1 and α > 1. On the one hand, for α < 1 the height-height correlation) takes different forms depend- s s s integralinEq.(5)hasalreadybeencomputed[16,17]and ing on further restrictions and/or bounds for the rough- one gets ness exponentvalues. These kind ofrestrictionsare very oftenassumedandnotvalidforeverygrowthmodel. For uαs if u≪1 instance, only ifthe surface wereself-affine saturationof gαs<1(u)∼(cid:26)const if u≫1 . (10) thecorrelationfunctionG(l,t)wouldalsooccurforinter- mediate scales l at times t ∼ lz and with the very same So,the correspondingscalingfunctionisg ∼f and αs<1 A roughness exponent. However, the latter does not hold α = α , i.e. the intrinsic anomalous scaling function s loc when anomalous roughening takes place as can be seen inEq.(3). Moreover,inthis casethe interface wouldsat- from the scaling of the local width in Eq.(3). isfy a Family-Vicsek scaling (for the local as well as the Our aim here is to investigate all the possible forms global width) only if α = α were satisfied for the par- s thatthescalingfunctionscanexhibitwhensolelytheex- ticular growth model under study. Thus, the standard istence of generic scaling is assumed. So, if the roughen- Family-Vicsekscalingturns outto be one ofthe possible ing process under consideration shows generic dynamic scaling forms compatible with generic scaling invariant scaling (in the sense above explained), and no further growth, but not the only one. 2 On the other hand, a new anomalous dynamics shows theconditions|h(i ,t)−h(i ±1,t)|<2aresatisfied. Pe- 0 0 up for growth models in which α > 1. In this case, riodic boundary conditions are assumed. We have stud- s one finds that, in the thermodynamic limit L → ∞, the ied the behaviour of the model in systems of different integral Eq.(5) has a divergence coming from the lower sizes from L = 26 up to L = 213. From calculations of integration limit. To avoid the divergence one has to thesaturatedglobalwidthW(L)forvarioussystemsizes compute the integral keeping L fixed. We then obtain we find a global roughness exponent α = 1.000±0.005 the scaling function in agreement with previous simulations [18]. We have checked that the scaling of the global width is given by u if u≪1 Eq.(1) with a scaling function like (2). Also in agree- g (u)∼ . (11) αs>1 (cid:26)const if u≫1 ment with previous work [18] we find that the time exponent α/z =0.95±0.05. The local width w(l,t) scales So that in this case one always gets αloc = 1 for any asw(l,t)=tα/zg(l/ξ)wherethescalingfunctionisgiven αs > 1. Thus, for growth models in which α = αs one by Eq.(11), and also αloc =1. recovers the super-rough scaling behaviour [16,17]. From these simulation results one could conclude that However, it is worth noting that neither the spectral the behaviour of the Sneppen growth model is rather exponent αs or the global exponent α are fixed by the trivialand that the exponentsα=αloc =z =1 describe scaling in Eqs(7) and (11) and, in principle, they could its scaling properties. Quite the opposite, this model ex- be different. Therefore, growth models in which αs > 1 hibits no trivial features that can be noticed when the but α 6= αs could also be possible and represent a new structure factor is calculated. In Figure 1 we show our type ofdynamicswithanomalousscaling. The mainfea- numericalresultsforthestructurefactorS(k,t)inasys- ture of this new type of anomalous roughening is that it tem of size L = 2048. Note that in Figure 1 the curves can be detected only by determining the scaling of the S(k,t) for different times are shifted downwards reflect- structure factor. Whenever such a scaling takes place ing that α < α . This contrasts with the case of intrin- s in the problem under investigation the new exponent αs sic anomalous roughening [16,17] where αs = αloc and will only show up when analyzing the scaling behaviour α ≤ 1. The slope of the continuous line is −3.7 and loc of S(k,t) and will not be detectable in either W(L,t), indicates that a new exponent α =1.35 enters the scal- s w(l,t) or G(l,t). In fact, as we have shown, the station- ing. This can be better appreciatedby the data collapse ary regime of a surface exhibiting this kind of anoma- shown in Figure 2, where one can observe that, instead lous scaling will be characterized by W(L) ∼ Lα and of being constant, the scaling function s(u) has a nega- w(l,L) ∼ G(l,L) ∼ lLα−1, however, the structure tive exponent u−0.7 for u ≫ 1. The exponents used for factor scaleps as S(k,L) ∼ k−(2αs+1)L2(α−αs) where the the data collapse are α = 1, z = 1 and the scaling func- spectral roughness exponent αs is a new and indepen- tion obtained is in excellent agreement with Eq.(7) and dent exponent. We can summarize our analitycal results a spectral exponent α =1.35±0.03. s as follows The interface in the Sneppen model A is formed by facets with constant slope ±1 [18]. The value of the α =α⇒Family−Vicsek if α <1⇒α =α s exponents α = α = 1 and α = 1.35 is related to  s loc s (cid:26)αs 6=α⇒Intrinsic the faceted form loofc the interfacse at saturation. It is  if α >1⇒α =1 αs =α⇒Super−rough easy to understand how the anomalous spectral rough- s loc (cid:26)αs 6=α⇒New class ness exponent appears due to the faceted form of the  (12) interface. For the simpler (and trivial) case of a faceted interfaceformedbyafinitenumberofidenticalsegments, In the following we present simulations of a one- N,ofconstantslope,±m,onecanshowanaliticallythat dimensional growth model that is a nice example of the the global width W(L) ∼ m2L2/N2, and the height- new dynamics. We have performed numerical simula- height correlation function G(l) ∼ l2m2 − Nm2l3/L, tions of the Sneppen model of self-organized depinning which leads to α = α = 1, while the spectrum loc (model A) [18]. We have found that this model exhibits S(k,L) ∼ k−4L−1 as k → 0. A simple comparison with anomalous roughening the type described by Eq.(7) for the anomalous scaling form for the stationary spectrum α > 1 and α 6= α. In this model the height of the S(k,L)∼k−(2αs+1)L2(α−αs) leadstoα =1.5. Actually, s s s interface h(i,t) is taken to be an integer defined on a the facetsoccuringinthe Sneppen modelarenotformed one-dimensional discrete substrate i = 1,···,L. A ran- by identicalsegments,but rather followa randomdistri- dom pinning force η(i,h) is associated with each lattice bution [20], which leads to a spectral exponent different site [i,h(i)]. The quenched disorder η(i,h) is uniformly from the trivial case. distributed in [0,1] and uncorrelated. The growth algo- In summary, we have presented a generic theory of rithm is then as follows. At every time step t, the site i scaling for invariant surface growth. We have shown 0 with the smallest pinning force is chosen and its height that the existence of power-law scaling of the correla- h(i ,t)isupdatedh(i ,t+1)=h(i ,t)+1providedthat tion functions (scale invariance) does not determine a 0 0 0 3 unique formofthe scalingfunctions involved. This leads (1998). to the different dynamic scaling forms recently observed [14] S. Morel et. al.,Phys.Rev. E 58, 6999 (1998). ingrowthmodels[4–10]andexperiments[11–15]exhibit- [15] A. Bru et. al. Phys. Rev.Lett. , 1998 ing anomalous roughening. In particular interface scale [16] J.M. López, M.A. Rodr´ıguez andR.Cuerno, Phys. Rev. E 56, 3993 (1997); invariance does not necessarily imply Family-Vicsek dy- [17] J.M. López, M.A. Rodr´ıguez and R. Cuerno, Physica A namic scaling. We have derived all the types of scal- 246, 329 (1997). ing (Family-Vicsek, super-rough and intrinsic anoma- [18] K. Sneppen,Phys.Rev.Lett. 69, 3539 (1992). lous) from a unique scaling ansatz, which is formulated [19] J.M. López, Phys.Rev. Lett. 83, 4594 (1999). in the Fourier space. The different types of scaling are [20] J.J. Ramasco, J.M. López and M.A. Rodr´ıguez, unpub- subclasses of our generic scaling ansatz associated with lished. bounds on the values that the new spectral roughness exponent α may take. This generalization has allowed s us to predict the existence of a new kind of anomalous 10.0 scaling with interesting features. Simulations of a model t = 7.8125 for self-organized interface depinning have been shown t = 3 x 7.8125 to be in excellent agreement with the new anomalous t = 8 x 7.8125 ) 5.0 t = 20 x 7.8125 dynamics. It has recently been shown [19] that anoma- ,t t = 100 x 7.8125 k lousrougheningstemsfromanontrivialdynamicsofthe ( S mean local slopes h(∇h)2i. In contrast, the new anoma- 0 1 lous dynamics can be pinned down to growth models in g 0.0 o which the stationary state consits of faceted interfaces. l The authorswouldliketo thankR.Cuernoforanear- lier collaboration that led to Ref. [16,17]. This work has −5.0 beensupportedbytheDGESoftheSpanishGovernment −3.0 −2.0 −1.0 0.0 1.0 (project No. PB96-0378-C02-02). JJR is supported by log k 10 the Ministerio de Educación y Cultura (Spain). JML is supportedbyaTMRNetworkoftheEuropeanCommis- sion (contract number FMRXCT980183). FIG.1. StructurefactoroftheSneppenmodelforinterface depinning at different times. The continuous straight line is a guide to the eye and has a slope -3.7. Note the anomalous downwards shift of thecurvesfor increasing times. 2.0 a Electronic address: [email protected] b Electronic address: [email protected] 1)] 0.0 [1] A.-L. Barabási and H. E. Stanley, Fractal Concepts + α in Surface Growth, Cambridge University Press, Cam- (2k bridge, (1995). ) −2.0 [2] J. Krug, Adv.Phys.46, 139 (1997). ,t k [3] F. Family and T. Vicsek, J. Phys.A 18, L75 (1985). S( t = 7.8125 [[45]] JM..KSrcuhgr,oPedheyrs,.eRt.eval..,LEetutr.o7p2h,y2s.90L7et(t1.92944,).563 (1993). [10 −4.0 ttt === 382 0xx x77 ..788.1182215525 g [6] J.M.LópezandM.A.Rodr´ıguez,Phys.Rev.E54,R2189 o (1996). l −6.0 [7] S.Das Sarma, et. al., Phys.Rev.E 53, 359 (1996). −2.0 −1.0 0.0 1.0 2.0 3.0 1/z log (kt ) [8] C.Dasgupta,S.DasSarmaandJ.M.Kim,Phys.Rev.E 10 54, R4552 (1996). [9] J.M. López and M.A. Rodr´ıguez, J. Phys. I 7, 1191 (1997). FIG. 2. Data collapse of the graphs in Fig. 1. The expo- [10] M. Castro, et. al. Phys. Rev.E 57, R2491 (1998). nents used for the collapse are α = 1.0 and z = 1.0. The [11] H.-Yang,G.-C.WangandT.-M.Lu,Phys.Rev.Lett.73, straight lines have slopes −0.7 (solid) and 3.0 (dashed) and 2348 (1994). are a guide to the eye. The scaling function is given by Eq. [12] J.H.Jeffries,J,-K.ZuoandM.M.Craig,Phys.Rev.Lett. (7)withaspectralroughnessexponentα =1.35±0.03. The s 76, 4931 (1996). deviations from the scaling for large values of the argument [13] J.M. López and J. Schmittbuhl, Phys. Rev E 57, 6405 kt1/z are duetothe finitelattice spacing. 4