Phase transition in site-diluted Josephson junction arrays: A numerical study
aa r X i v : . [ c ond - m a t . s up r- c on ] M a r Phase transition in site-diluted Josephson-junction arrays: A numerical study
Jian-Ping Lv , Huan Liu , and Qing-Hu Chen , , † Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China Center for Statistical and Theoretical Condensed Matter Physics,Zhejiang Normal University, Jinhua 321004, P. R. China
Intriguing effects produced by random percolative disorder in two-dimensional Josephson-junctionarrays are studied by means of large-scale numerical simulations. Using dynamic scaling analysis, weevaluate critical temperatures and critical exponents in high accuracy. With the introduction of site-diluted disorder, the Kosterlitz-Thouless phase transition is eliminated and evolves into continuousphase transition with a power-law divergent correlation length. Moreover, genuine depinning transi-tion and the related creep motion are studied, distinct types of creep motion for different disorderedsystems are observed. Our results not only are in good agreement with the recent experimentalfindings, but also shed some light on the relevant phase transitions.
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I. INTRODUCTION
Understanding the critical behavior of Josephson-junction arrays (JJA’s) with various disorders is alwaysa challenging question and has been intensely studiedin recent years [1]–[10]. However, the properties of dif-ferent phases and various phase transitions are not wellunderstood. Josephson-junction arrays gives an excellentrealization to both two-dimensional (2D) XY model andgranular High- T c superconductors [11]. As is well knownthat the pure JJA’s undergoes celebrated Kosterlitz-Thouless (KT) phase transition [12] from the supercon-ducting state to the normal state, this transition is drivenby the unbinding of thermally created topological defects.When the disorder is introduced, the interplays betweenthe periodic pinning potential caused by the discretenessof the arrays, the repulsive vortex-vortex interaction andthe effects produced by the disorder provide a rich phys-ical picture.In diluted JJA’s, islands are randomly removed fromthe square lattice. Since it is a representative model forrealizing the irregular JJA’s systems, how the percola-tion influences the physical properties of JJA’s has at-tracted considerable attention [1, 2, 3, 4, 9]. Harris etal introduced random percolative disorder into Nb-Au-Nb proximity-coupled junctions, the current-voltage ( I - V ) characteristics were measured and the results demon-strated that the only difference of the phase transitioncompared with that in ideal JJA’s system is the decreaseof critical temperature, while the phase transition stillbelong to the KT-type with the disorder strength span-ning from p = 0 . p = 1 . − p is the fraction ofdiluted sites) [1]. However, a recent experimental studyby Yun et al showed that the KT-type phase transition inunfrustrated JJA’s was eliminated due to the introduc-tion of site-diluted disorder [9]. Therefore, the existenceof the KT-type phase transition in site-diluted JJA’s re-mains a topic of controversy.On the other hand, much attention has been paidto investigate the zero-temperature depinning transitionand the related low-temperature creep motion both the- oretically [13, 14, 15] and numerically [16, 17, 18] in alarge variety of physical problems, such as charge densitywaves [13], random-field Ising model [16], and flux linesin type-II superconductors [17, 18]. Since the non-lineardynamic response is a striking aspect, there is increasinginterest in these systems, especially in the flux lines oftype-II superconductors [17, 18]. In a recent numericalstudy on the three-dimensional glass states of flux lines,Arrhenius creep motion and non-Arrhenius creep motionwere observed with strong collective pinning and weakcollective pinning, respectively [17].In this work, we numerically investigate the finite-temperature phase transition in site-diluted JJA’s at dif-ferent percolative disorders, the zero-temperature depin-ning transition and the low-temperature creep motion arealso considered. The outline of this paper is as follows.Section II describes the model and the numerical methodbriefly. In section III, we present the simulation results,analyzing them by means of scaling analysis. Section IVgives a short summary of the main conclusions. II. MODEL AND SIMULATION METHOD
JJA’s can be described by the 2D XY model on a sim-ple square lattice, the Hamiltonian of which is [19, 20] H = − X J ij cos( φ i − φ j − A ij ) , (1)where the sum is over all nearest neighboring pairs on a2D square lattice, J ij denotes the strength of Josephsoncoupling between site i and site j, φ i specifies the phaseof the superconducting order parameter on site i, and A ij = (2 π/ Φ ) R A · d l is the integral of magnetic vectorpotential from site i to site j with Φ the flux quantum.The direct sum of A ij around an elementary plaquette is2 πf , with f the magnetic flux penetrating each plaquetteproduced by the uniformly applied field, measured in unitof Φ . In this paper, f = 0 and f = 2 / ×
128 for f = 0 and100 ×
100 for f = 2 /
5, the finite size effects in these sizesare negligible. Diluted sites are randomly selected, thenthe nearest four bonds of which are removed from thelattice. The same random-number seed is used to choosethe diluted sites, the percolative threshold concentrationis about 0 . σ ~ e X j ( ˙ φ i − ˙ φ j ) = − ∂H∂φ i + J ex ,i − X j η ij , (2)where σ is the normal conductivity, J ex,i refers to theexternal current, η ij denotes the thermal noise currentwith < η ij ( t ) > = 0 and < η ij ( t ) η ij ( t ′ ) > = 2 σk B T δ ( t − t ′ ).The fluctuating twist boundary condition is applied inthe xy plane to maintain the current, thus the new phaseangle θ i = φ i + r i · ∆ (∆ = ( △ x , △ y ) is the twist variable)is periodic in each direction. In this way, supercurrentbetween site i and site j is given by J si → j = J ij sin ( θ i − θ j − A ij − r ij · ∆) , and the dynamics of ∆ α can be writtenas ˙∆ α = 1 L X α [ J i → j + η ij ] − I α , (3)where α denotes the x or y direction, the voltage dropin α direction is V = − L ˙∆ α . For convenience, units aretaken as 2 e = ~ = J = σ = k B = 1 in the following.Above equations can be solved efficiently by a pseudo-spectral algorithm due to the periodicity of phase in alldirections. The time stepping is done using a second-order Runge-Kutta scheme with ∆ t = 0 .
05. Our runs aretypically (4 − × time steps and the latter half timesteps are for the measurements. The detailed procedurein the simulations was described in Ref. [20, 22]. In thiswork, a uniform external current I along x direction isfed into the system.Since RSJ simulations with direct numerical integra-tions of stochastic equations of motion are very time-consuming, it is practically difficult to perform any se-rious disorder averaging in the present rather large sys-tems. Our results are based on one realization of disor-der. For these very large samples, it is expected to exista good self-averaging effect, which is confirmed by twoadditional simulations with different realizations of dis-order. This point is also supported by a recent studyof JJA’s by Um et al [8], they confirmed that a well-converged disorder averaging for the measurement is notnecessary, and well-converged data for large systems ata single disorder realization leads to a convincing result.In addition, simulations with different initial states areperformed and the results are independent on the initialstate we used. Actually, the hysteric phenomenon is usu-ally negligible in previous RSJ dynamical simulations on JJA’s [7, 8]. For these reasons, the results from simula-tions with a unique initial state (random phases in thiswork) are accurate and then convincing. III. RESULTS AND DISCUSSIONA. Finite temperature phase transition
The I - V characteristics are measured at various disor-der strengths and temperatures. At each temperature,we try to probe the system at a current as low as possi-ble. To check the method used in this work, we investi-gate the I - V characteristics for f = 0 , p = 1 .
0. As shownin Fig. 1(a), the slope of the I - V curve in log-log plotat the transition temperature T c ≈ .
894 is equal to 3,demonstrating that the I - V index jumps from 3 to 1, inconsistent with the well-known fact that the pure JJA’sexperiences a KT-type phase transition at T c ≈ . I - V traces at different per-colative disorders in unfrustrated JJA’s, while Fig. 1(d)for f = 2 / , p = 0 .
65. It is clear that, at lower temper-atures, R = V /I tends to zero as the current decreases,which follows that there is a true superconducting phasewith zero linear resistivity.It is crucial to use a powerful scaling method to an-alyze the I - V characteristics. In this paper, we adoptthe Fisher-Fisher-Huse (FFH) dynamic scaling method,which provides an excellent approach to analyze thesuperconducting phase transition [23]. If the properlyscaled I - V curves collapse onto two scaling curves aboveand bellow the transition temperature, a continuoussuperconducting phase transition is ensured. Such amethod is widely used recently [6, 24], the scaling formof which in 2D is V = Iξ − z ψ ± ( Iξ ) , (4)where ψ +( − ) ( x ) is the scaling function above (below) T c , z is the dynamic exponent, ξ is the correlation length,and V ∼ I z +1 at T = T c .Assuming that the transition is continuous and char-acterized by the divergence of the characteristic length ξ ∼ | T − T c | − ν and time scale t ∼ ξ z , FFH dynamicscaling takes the following form( V /I ) | T − T c | − zν = ψ ± ( I | T − T c | − ν ) . (5)On the other hand, a new scaling form is successfullyadopted to certify a KT-type phase transition in JJA’sby [25] ( I/T )( I/V ) /z = P ± ( Iξ/T ) , (6)note that the Eq. (6) can be obtained directly from theFFH dynamic scaling form after some simple algebra.The correlation length of KT-type phase transition above T c is well defined as ξ ∼ e ( c/ | T − T c | ) / and Eq. (6) reads( I/T )( I/V ) /z = P + ( Ie ( c/ | T − T c | ) / /T ) . (7) -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -2 -1 -4 -3 -2 -1 (b) f=0, p=0.86T C =0.58- - - slope = 3.0 T=0.8 T=0.75 T=0.7 T=0.65 T=0.6 T=0.55 T=0.5 T=0.45 T=0.4 (d) f=2/5,p=0.65T C =0.14- - - slope=2.25 I T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1 T=0.075 T=0.05 T=0.025 II (c) I f=0,p=0.65T C =0.24 - - - slope=2.2 T=0.5 T=0.4 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1
T=1.1 T=1.0 T=0.95 T=0.9 T=0.894 T=0.85 T=0.8 VV f=0,p=1.0T C =0.894- - - slope = 3.0 V V (a) FIG. 1: I - V characteristics for different frustrations and per-colative disorders. The dash lines are drawn to show wherethe phase transition occurs, the slopes of which are equal to z + 1, z is the dynamic exponent. The transition temperatureand dynamic exponent for (a) are well consistent with thewell-known result, i. e. , T c = 0 . z = 2 .
0, for (b),(c),(d)are well consistent with those determined by FFH dynamicscaling analysis. Solid lines are just guide to eyes.
As shown in Fig. 2, using T c = 0 . ± . z =1 . ± .
02 and ν = 1 . ± .
02, we get an excellent col-lapse for f = 0 , p = 0 .
65 according to equation (5). Inaddition, all the low-temperature I - V curves can be fit-ted to V ∼ I exp( − ( α/I ) µ ) with µ = 0 . ∼ .
1. Theseresults certify a continuous superconducting phase withlong-rang phase coherence. The critical temperature forsuch a strongly disordered system is very close to that in2D gauge glass model ( T = 0 .
22) [26].For f = 0 , p = 0 .
86, firstly, we still adopt the scalingform in equation (5) to investigate the I - V character-istics. As displayed in Fig. 3, we get a good collapsefor T < T c with T c = 0 . ± . z = 2 . ± .
01 and ν = 1 . ± .
02, demonstrating a superconducting phasewith power-law divergent correlation for
T < T c . Notethat the collapse is bad for T > T c , indicating that thephase transition is not a completely non-KT-type one.Next, we use the scaling form in equation (7) to ana-lyze the I - V data above the critical temperature. In-terestingly, using T c = 0 .
58 and z = 2 . T > T c is achieved, which isshown in Fig. 4. That is to say, the I - V characteristicsat T < T c are like those of a continuous phase transi- -1 -2 -1 V | T - T c | - z υ / I I |T-T c | - υ f=0,p=0.65 υ =1.0,z=1.2,T c =0.24 T=0.5 T=0.4 T=0.35 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1
FIG. 2: Dynamic scaling of I - V data at various temperaturesaccording to equation (5) for f = 0 , p = 0 . tion with power-law divergent correlation length whileat T > T c are like those of KT-type phase transition,which are well consistent with the recent experimentalobservations [9]. Therefore, by the present model, werecover the phenomena in experiments and give someinsight into the phase transition. More information onthe low-temperature phase calls for further equilibriumMonte Carlo simulations as in Ref. [27].To make a comprehensive comparison with the exper-imental findings as in Ref. [9], we also investigated thefinite-temperature phase transition in frustrated JJA’s( f = 2 /
5) at a strong site-diluted disorder ( p = 0 . B. Depinning transition and creep motion
Next, we pay attention to the zero-temperature depin-ning transition and the related low-temperature creepmotion for the typical site-diluted JJA’s systems men- -1 f=0,p=0.86 υ =1.4,z=2.0,T c =0.58 T=0.8 T=0.75 T=0.7 T=0.65 T=0.6 T=0.55 T=0.5 T=0.45 T=0.4 V | T - T c | - z υ / I I |T-T c | - υ FIG. 3: Dynamic scaling of I - V data at various temperaturesaccording to equation (5) for f = 0 , p = 0 . , T < T c . Solidlines are just guide to eyes. -1 f=0,p=0.86c=0.8,z=2.0,T c =0.58 T=0.8 T=0.75 T=0.7 T=0.65 T=0.55 T=0.5 T=0.45 T=0.4 I (I / V ) / z / T I exp{ ( c/|T-T c | ) }/T FIG. 4: Dynamic scaling of I - V data at various temperaturesaccording to equation (7) for f = 0 , p = 0 . , T > T c . Solidlines are just guide to eyes. tioned above. Depinning can be described as a criticalphenomenon with scaling law V ∼ ( I − I c ) β , demonstrat-ing a transition from a pinned state below critical drivingforce I c to a sliding state above I c . The ( I − I c ) .vs.V traces at T = 0 for f = 0 , p = 0 . f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65 are displayed in Fig. 6, linear-fittingsof
Log ( I − I c ) .vs.LogV curves are also shown as solidlines. As for f = 0 , p = 0 .
86, the depinning exponent β is determined to be 2 . ± . I c is 0 . ± . f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65, the depinning exponents are evaluatedto be 2 . ± . . ± .
05 with the critical currents I c = 0 . ± .
001 and I c = 0 . ± . I - V traces are rounded near the zero-temperaturecritical current due to thermal fluctuations. Fisher firstsuggested to map such a phenomenon for the ferromag-net in magnetic field where the second-order phase tran-sition occurs [29]. This mapping was then extended tothe random-field Ising model [16] and the flux lines in -1 -1 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1 T=0.075 T=0.05 f=2/5,p=0.65=1.1,z=1.25,T c =0.14 I |T-T c | V | T - T c | - z / I FIG. 5: Dynamic scaling of I - V data at various temperaturesaccording to equation (5) for f = 2 / , p = 0 . -1 -3 -2 -1 -2 -1 -3 -2 -1 -1 -1 -4 -3 -2 Fitting V a Fitting c I-I c Fitting b V I-I c V I-I c FIG. 6: (a) IV characteristics for f = 0 , p = 0 .
86 with I c =0 . ± . β = 2 . ± .
1. (b) IV characteristics for f =0 , p = 0 .
65 with I c = 0 . ± . β = 2 . ± .
1. (c) IV characteristics for f = 2 / , p = 0 .
65 with I c = 0 . ± . β = 2 . ± . -1 -1 -4 -3 -2 -1 -4 -3 -1 -1 -4 -3 I=0.03 I c =0.034 I=0.04 I=0.05 c T V T V I=0.3 I c =0.302 I=0.31 I=0.32 V T a I=0.0375 I c =0.03875 I=0.04 I=0.0425 b FIG. 7: (a)LogV-LogT curves for f = 0 , p = 0 .
86 in thevicinity of I c with I c = 0 . ± . /δ = 1 . ± . f = 0 , p = 0 .
65 in the vicinity of I c with I c = 0 . ± . /δ = 2 . ± .
02. (c)LogV-LogT curves for f = 2 / , p = 0 .
65 in the vicinity of I c with I c = 0 . ± . /δ = 2 . ± . type-II superconductors [17]. For the flux lines in type-IIsuperconductors, if the voltage is identified as the orderparameter, the current and the temperature are taken asthe inverse temperature and the field respectively, anal-ogous to the second-order phase transition in the ferro-magnet, the voltage, current and the temperature willsatisfy the following scaling ansatz [17, 26] V ( T, I ) = T /δ S [(1 − I c /I ) T − /βδ ] . (8)The relation V ( T, I = I c ) = S (0) T /δ can be easily de-rived at I = I c , by which the critical current I c and thecritical exponent δ can be determined through the linearfitting of the LogT − LogV curve at I c . The LogT − LogV curves are plotted in Fig. 7(a) for f = 0 , p = 0 . T - V curves fromthe power law is calculated as the square deviations SD = P [ V ( T ) − y ( T )] between the temperature rangewe calculated, here the functions y ( T ) = C T − C are ob-tained by linear fitting of the LogT − LogV curves. Thecurrent at which the SD is minimum is defined as thecritical current. The critical current is then determinedto be 0 . ± . /δ = 1 . ± .
001 from the slope of
LogT − logV curve at I c = 0 . f = 0 , p = 0 .
65 and f = 2 / , p = 0 . I c and critical exponent 1 /δ for f = 0 , p = 0 .
65 are deter-mined to be 0 . ± . . ± .
02 respectively,for f = 2 / , p = 0 .
65, the result is I c = 0 . ± . /δ = 2 . ± . β , we get the bestcollapses of data to a single scaling curve with β =2 . ± .
02 and 2 . ± .
02 for f = 0 , p = 0 .
86 and f = 0 , p = 0 .
65 in the regime I ≤ I c , respectively, whichare shown in Figs. 8(a) and (b). For f = 0 , p = 0 .
86, thiscurve can be fitted by S ( x ) = 0 . exp (1 . x ), combinedwith the relation βδ = 1 .
55, suggesting a non-Arrheniuscreep motion. However, for the strongly site-diluted sys-tem with f = 0 , p = 0 .
65, the scaling curve can be fittedby S ( x ) = 0 . exp (0 . x ), combined with the relation βδ ≈ .
0, indicative of an Arrhenius creep motion. Inter-estingly, as displayed in Fig. 8(c) for f = 2 / , p = 0 . β is fitted to be 2 . ± .
02, which yields βδ ≈ .
0. The scaling curve in the regime I ≤ I c can befitted by S ( x ) = 0 . exp (0 . x ). These two combinedfacts suggest an Arrhenius creep motion in this case.It is worthwhile to note that both the finite-temperature phase transition and the creep motion forstrongly disordered JJA’s ( p = 0 .
65) with and with-out frustration are very similar. The I - V curves inlow temperature for all three cases can be described by V ∝ T /δ exp [ A (1 − I c /I ) /T βδ ], which is just one of themain characteristics of glass phases [17, 26]. While the I - V traces for KT-type phases can be fitted to V ∝ I a .Therefore, we have provided another evidence for the ex-istence of non-KT-type phases in the low-temperatureregime for these three cases ( f = 0 , p = 0 . f = 0 , p =0 . f = 2 / , p = 0 . IV. SUMMARY
To explore the properties of various phase transitionsin site-diluted JJA’s, we have performed large scale simu-lations at two typical percolative strengths p = 0 .
86 and p = 0 .
65 as in a recent experimental work [9]. The RSJdynamics was incorporated in our work, from which wemeasured the I - V characteristics at different tempera-tures. The critical temperature of the finite-temperaturephase transition was found to decrease as the dilutedsites increase. For f = 0 , p = 0 .
86, the phase transitionis the combination of a KT-type transition and a con-tinuous transition with power-law divergent correlation -1.5 0.0 -3 0 -10 -5 0 y=0.0994exp(1.9x)
T=0.01 T=0.02 T=0.04 T=0.06 T=0.08 T=0.1 T=0.2 T=0.3 a V T (1-I c /I)T - (1-I c /I)T - T=0.05 T=0.1 T=0.15 T=0.2 T=0.3 T=0.4 y=0.037exp(0.5x) V T (1-I c /I)T - b y=0.105exp(0.25x) c V T T=0.075 T=0.1 T=0.125 T=0.15 T=0.175 T=0.2 T=0.25 T=0.3
FIG. 8: (a) Scaling plot for f = 0 , p = 0 .
86 with I c = 0 . /δ = 1 .
688 and βδ = 1 .
55. (b) Scaling plot for f = 0 , p =0 .
65 with I c = 0 . /δ = 2 .
24 and βδ ≈ .
0. (c)Scalingplot for f = 2 / , p = 0 .
65 with I c = 0 . /δ = 2 .
29 and βδ ≈ . length. At strong percolative disorder ( p = 0 . z = a −
1, with a the I - V indexat the critical temperature, and all the static exponentsfall in the range of ν = (1 . , .
0) usually observed atvortex-glass transitions experimentally. Following table summarizes the critical temperatures at different frustra-tions and disorder strengths.
TABLE I: Summary of T c .f=0 f=2/5p=0.95 0.85(2) 0.16(2)p=0.86 0.58(1) 0.13(1)p=0.7 0.27(2) 0.12(1)p=0.65 0.24(1) 0.14(1) In a recent experiment, Yun et al [9] suggested a non-KT-type phase transition in unfrustrated JJA’s with site-diluted disorder for the first time, however the nature ofthese phase transitions and various phases is still in anintensive debate. Our results not only recover the re-cent experimental findings [9], but also shed some lighton the various phases. Non-KT-type finite-temperaturephase transition in site-diluted JJA’s was confirmed bythe scaling analysis. The different divergent correlationsat various disorder strengths were suggested, the crit-ical exponents were evaluated in high accuracy, whichare crucial for understanding such a critical phenomenon.Furthermore, the results in this paper are not only usefulfor understanding the site-diluted systems, but also use-ful for understanding the whole class of disordered JJA’s.For instance, the combination of two different phase tran-sitions may exist in other disordered JJA’s systems.In addition, the zero-temperature depinning transitionand the low-temperature creep motion are also touched.It is demonstrated by the scaling analysis that the creeplaw for f = 0 , p = 0 .
86 is non-Arrhenius type while thosefor f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65 belong tothe Arrhenius type. The evidence of non-KT-type phasetransition can also be provided by this scaling analysis. Itis interesting to note that the non-Arrhenius type creeplaw for weak disorder ( f = 0 , p = 0 .
86) is similar tothat in three-dimensional flux lines with a weak collectivepinning [17]. The product of the two exponents 1 .
55 isalso very close to 3 / f =0 , p = 0 .
65 and f = 2 / , p = 0 .
65, the observed Arrheniustype creep law is also similar to that in the glass statesof flux lines with a strong collective pinning as in Ref.[17]. Future experimental work is needed to clarify thisobservation.
V. ACKNOWLEDGEMENTS
This work was supported by National Natural Sci-ence Foundation of China under Grant Nos. 10774128,PCSIRT (Grant No. IRT0754) in University in China,National Basic Research Program of China (GrantNos. 2006CB601003 and 2009CB929104), and ZhejiangProvincial Natural Science Foundation under Grant No.Z7080203. † Corresponding author. Email:[email protected] [1] D. C. Harris, S. T. Herbert, D. Stroud and J. C. Garland,Phys. Rev. Lett. , 3606 (1991).[2] E. Granato and D. Dom´ınguez, Phys. Rev. B , 14671(1997).[3] M. Benakli and E. Granato, S. R. Shenoy and M. Gabay,Phys. Rev. B , 10314 (1998).[4] E . Granato and D. Dominguez, Phys. Rev. B , 094507(2001).[5] Y. J. Yun, I. C. Baek and M. Y. Choi, Phys. Rev. Lett. , 037004 (2002).[6] E. Granto and D. Dominguez, Phys. Rev. B , 094521(2005).[7] J. S. Lim, M. Y. Choi, B. J. Kim and J. Choi, Phys. Rev.B , 100505R (2005).[8] J. Um, B. J. Kim, P. Minnhagen, M. Y. Choi and S. I.Lee, Phys. Rev. B , 094516 (2006).[9] Y. J. Yun, I. C. Baek and M. Y. Choi, Phys. Rev. Lett. , 215701 (2006).[10] Y. J. Yun, I. C. Baek and M. Y. Choi, Europhys. Lett. , 271 (2006).[11] C. J. Lobb, D. W. Abraham and M. Tinkham, Phys.Rev. B , 150 (1983); M. Prester, Phys. Rev. B , 606(1996).[12] J. M. Kosterlitz and D. J. Thouless, J.Phys.C , 1181(1973); J. M. Kosterlitz, J. Phys. C , 1046 (1974); V.L. Berezinskii, Sov. Phys. - JETP, ,610 (1972); V. L.Berezinskii, Zh. Eksp. Teor. Fiz. ,1144 (1973).[13] T. Nattermann, Phys. Rev. Lett. , 2454 (1990).[14] P. Chauve, T. Giamarchi and P. L. Doussal, Phys. Rev.B. , 6241 (2000).[15] M. M¨uller, D. A. Gorokhov and G. Blatter, Phys. Rev.B. , 184305 (2001). [16] L. Rosters, A. Hucht, S. L¨ubeck, U. Nowak and K. D.Usadel, Phys. Rev. E. , 5202 (1999).[17] M. B. Luo and X. Hu, Phys. Rev. Lett. , 267002 (2007).[18] P. Olsson, Phys. Rev. Lett. , 097001 (2007); Q. H.Chen, Phys. Rev. B , 104501 (2008).[19] P. Olsson and S. Teitel, Phys. Rev. Lett. , 137001(2001).[20] Q. H. Chen and X. Hu, Phys. Rev. Lett. , 117005(2003); Q. H. Chen and X. Hu, Phys. Rev. B , 064504(2007).[21] T. Gebele, J. Phys. A: Math. Gen. , L51(1984); Y.Laroyer and E. Pommiers, Phys.Rev.B , 2795 (1994).[22] Q. H. Chen and L. H. Tang, Phys. Rev. Lett. , 067001(2001); L. H. Tang and Q. H. Chen, Phys. Rev. B ,024508 (2003).[23] D. S. Fisher, M. P. A. Fisher and D. A. Huse, Phys. Rev.B , 130 (1991).[24] H. Yang, Y. Jia, L. Shan, Y. Z. Zhang, H. H. Wen, C.G. Zhuang, Z. K. Liu, Q. Li, Y. Cui and X. X. Xi, Phys.Rev. B , 134513 (2007).[25] J. Holzer, R. S. Newrock, C. J. Lobb, T. Aouaroun andS. T. Herbert, Phys. Rev. B , 184508 (2001).[26] Q. H. Chen, J. P. Lv, H. Liu, Phys. Rev. B , 054519(2008).[27] H. G. Katzgraber, Phys. Rev. B , 180402R (2003); H.G. Katzgraber and A. P. Young, Phys. Rev. B
439 (2002).[29] D. S. Fisher, Phys. Rev. Lett. , 1486 (1983); Phys.Rev. B , 1396 (1985). r X i v : . [ c ond - m a t . s up r- c on ] M a r Phase transition in site-diluted Josephson-junction arrays: A numerical study
Jian-Ping Lv , Huan Liu , and Qing-Hu Chen , , † Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China Center for Statistical and Theoretical Condensed Matter Physics,Zhejiang Normal University, Jinhua 321004, P. R. China (Dated: October 28, 2018)We numerically investigate the intriguing effects produced by random percolative disorder intwo-dimensional Josephson-junction arrays. By dynamic scaling analysis, we evaluate critical tem-peratures and critical exponents with high accuracy. It is observed that, with the introductionof site-diluted disorder, the Kosterlitz-Thouless phase transition is eliminated and evolves into acontinuous transition with power-law divergent correlation length. Moreover, genuine depinningtransition and creep motion are studied, evidence for distinct creep motion types is provided. Ourresults not only are in good agreement with the recent experimental findings, but also shed somelight on the relevant phase transitions.
PACS numbers: 74.81.Fa,68.35.Rh,47.32.Cc
I. INTRODUCTION
Understanding the critical behavior of Josephson-junction arrays (JJA’s) with various disorders is alwaysa challenging issue and has been intensely studied in re-cent years [1]–[10]. However, the properties of differentphases and various phase transitions are not well under-stood. Josephson-junction arrays gives an excellent re-alization to both two-dimensional (2D) XY model andgranular High- T c superconductors [11]. As we know, thepure JJA’s undergoes the celebrated Kosterlitz-Thouless(KT) phase transition from the superconducting state tothe normal one, this transition is driven by the unbindingof thermally activated topological defects [12]. When thedisorder is introduced, the interplays among the repulsivevortex-vortex interaction, the periodic pinning potentialcaused by the discreteness of the arrays, and the defectsproduced by the disorder provide a rich physical picture.In site-diluted JJA’s, the crosses around the randomlyselected sites are removed from the square lattice. Sinceit is a representative model for realizing the irregularJJA’s systems, how the percolation influences the physi-cal properties of JJA’s has attracted considerable atten-tion [1, 2, 3, 4, 9]. Harris et al introduced random per-colative disorder into Nb-Au-Nb proximity-coupled junc-tions, the current-voltage ( I - V ) characteristics were mea-sured and the results demonstrated that the only dif-ference of the phase transition compared with that inideal JJA’s system is the decrease of critical tempera-ture, while the transition type still belongs to the KTone with the disorder strength spanning from p = 0 . p = 1 . − p is the fraction of diluted sites)[1]. However, in a recent experiment, Yun et al showedthat the phase transition changes into a non-KT-type onewhen the disorder strength increases to a moderate value( p = 0 .
86) [9]. Therefore, the existence of the KT-typephase transition in site-diluted JJA’s remains a topic ofcontroversy, the nature of these phase transitions and thevarious phases is not clear.On the other hand, much effort has been devoted to the zero-temperature depinning transition (ZTDT) andthe related low-temperature creep motion (LTCM) boththeoretically [13, 14, 15] and numerically [16, 17, 18] in alarge variety of physical problems, such as charge densitywaves [13], random-field Ising model [16], and flux lines intype-II superconductors [17, 18]. Since the non-linear dy-namic response is a striking problem, there is increasinginterest in its properties and characteristics, especiallyin the flux lines of type-II superconductors [17, 18]. Ina recent numerical study on the three-dimensional glassstates of flux lines, Arrhenius creep motion was observedat a strong collective pinning, while the non-Arrheniuscreep motion was demonstrated at a weak collective pin-ning [17].In this work, we numerically study the finite-temperature phase transition (FTPT) in site-dilutedJJA’s at different percolative disorder strengths, theZTDT and the LTCM are also investigated. The outlineof this paper is as follows. Sec. II describes the modeland the numerical method briefly. In Sec. III, we presentthe main results, where some discussions are also made.Sec. IV gives a short summary of the main conclusions.
II. MODEL AND SIMULATION METHOD
JJA’s can be described by the 2D XY model on asimple square lattice, the Hamiltonian reads [19, 20] H = − X J ij cos( φ i − φ j − A ij ) , (1)where the sum is over all nearest neighboring pairs ona 2D square lattice, J ij denotes the strength of Joseph-son coupling between site i and site j, φ i specifies thephase of the superconducting order parameter on site i,and A ij = (2 π/ Φ ) R A · d l is the integral of magneticvector potential from site i to site j, Φ denotes the fluxquantum. The direct sum of A ij around an elementaryplaquette is 2 πf , with f the magnetic flux penetratingeach plaquette produced by the uniformly applied field,which is measured in unit of Φ . f = 0 and f = 2 / ×
128 for f = 0 and 100 ×
100 for f = 2 /
5, wherethe finite size effects are negligible. We introduce thesite-diluted disorder similar to the previous experiments[1, 9]. We first select the diluted sites randomly with theprobability 1 − p , then remove the nearest four bondsaround the selected sites from the lattice. The distribu-tions of the diluted sites are the same for all the samplesconsidered. The percolative threshold concentration p c is about 0 .
592 [21].The resistivity-shunted-junction (RSJ) dynamics is in-corporated in the simulations, which can be described as[20, 22] σ ~ e X j ( ˙ φ i − ˙ φ j ) = − ∂H∂φ i + J ex ,i − X j η ij , (2)where σ is the normal conductivity, J ex,i refers to theexternal current, η ij denotes the thermal noise currentwith < η ij ( t ) > = 0 and < η ij ( t ) η ij ( t ′ ) > = 2 σk B T δ ( t − t ′ ).The fluctuating twist boundary condition is applied inthe xy plane to maintain the current, thus the new phaseangle θ i = φ i + r i · ∆ (∆ = ( △ x , △ y ) is the twist variable)is periodic in each direction. In this way, supercurrentbetween site i and site j is given by J si → j = J ij sin ( θ i − θ j − A ij − r ij · ∆) , and the dynamics of ∆ α can be writtenas ˙∆ α = 1 L X α [ J i → j + η ij ] − I α , (3)where α denotes the x or y direction, the voltage dropin α direction is V = − L ˙∆ α . For convenience, units aretaken as 2 e = ~ = J = σ = k B = 1 in the following.Above equations can be solved efficiently by a pseudo-spectral algorithm due to the periodicity of phase in alldirections. The time stepping is done using a second-order Runge-Kutta scheme with ∆ t = 0 .
05. Our runs aretypically (4 − × time steps and the latter half timesteps are for the measurements. The detailed procedurein the simulations was described in Ref. [20, 22]. In thiswork, a uniform external current I along x direction isfed into the system.Since RSJ simulations with direct numerical integra-tions of stochastic equations of motion are very time-consuming, it is practically difficult to perform any se-rious disorder averaging in the present rather large sys-tems. Our results are based on one realization of disor-der. For these very large samples, it is expected to exista good self-averaging effect, which is confirmed by twoadditional simulations with different realizations of dis-order. This point is also supported by a recent studyof JJA’s by Um et al [8]. In addition, simulations withdifferent initial states are performed and the results arenearly the same. Actually, the hysteric phenomenon is usually negligible in previous RSJ dynamical simulationson JJA’s [7, 8]. For these reasons, the results from simu-lations with a unique initial state (random phases in thiswork) are accurate and convincing. III. RESULTS AND DISCUSSIONSA. Finite temperature phase transition
The I - V characteristics are measured at different dis-order strengths and temperatures. At each temperature,we try to probe the system at a current as low as possi-ble. To check the method used in this work, we investi-gate the I - V characteristics for f = 0 , p = 1 .
0. As shownin Fig. 1(a), the slope of the I - V curve in log-log plotat the transition temperature T c ( ≈ . I - V index jumps from 3 to 1,consistent with the well-known fact that the pure JJA’sexperiences a KT-type phase transition at T c ≈ . I - V traces at different dis-order strengths in unfrustrated JJA’s, while Fig. 1(d) isfor f = 2 / , p = 0 .
65. It is clear that, at lower temper-atures, R = V /I tends to zero as the current decreases,which follows that there is a true superconducting phasewith zero linear resistivity.It is crucial to use a powerful scaling method to an-alyze the I - V characteristics. In this paper, we adoptthe Fisher-Fisher-Huse (FFH) dynamic scaling method,which provides an excellent approach to analyze thesuperconducting phase transition [23]. If the prop-erly scaled I - V curves collapse onto two scaling curvesabove and below the transition temperature, a continu-ous superconducting phase transition is ensured. Such amethod is widely used [6, 24], the scaling form of whichin 2D is V = Iξ − z ψ ± ( Iξ ) , (4)where ψ +( − ) ( x ) is the scaling function above (below) T c , z is the dynamic exponent, ξ is the correlation length,and V ∼ I z +1 at T = T c .Assuming that the transition is continuous and char-acterized by the divergence of the characteristic length ξ ∼ | T − T c | − ν and time scale t ∼ ξ z , FFH dynamicscaling takes the following form( V /I ) | T − T c | − zν = ψ ± ( I | T − T c | − ν ) . (5)On the other hand, to certify a KT-type phase tran-sition in JJA’s, a new scaling form [25] is proposed asfollows ( I/T )( I/V ) /z = P ± ( Iξ/T ) , (6)which can be derived directly from Eq. (4) after somesimple algebra. The correlation length of KT-type phasetransition above T c is well defined as ξ ∼ e ( c/ | T − T c | ) / and Eq. (6) is rewritten as( I/T )( I/V ) /z = P + ( Ie ( c/ | T − T c | ) / /T ) . (7) -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -2 -1 -4 -3 -2 -1 (b) f=0, p=0.86T C =0.58- - - slope = 3.0 T=0.8 T=0.75 T=0.7 T=0.65 T=0.6 T=0.55 T=0.5 T=0.45 T=0.4 (d) f=2/5,p=0.65T C =0.14- - - slope=2.25 I T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1 T=0.075 T=0.05 T=0.025 II (c) I f=0,p=0.65T C =0.24 - - - slope=2.2 T=0.5 T=0.4 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1
T=1.1 T=1.0 T=0.95 T=0.9 T=0.894 T=0.85 T=0.8 VV f=0,p=1.0T C =0.894- - - slope = 3.0 V V (a) FIG. 1: I - V characteristics for different frustrations and dis-order strengths. The dash lines are drawn to show wherethe phase transition occurs, the slopes of which are equal to z + 1, z is the dynamic exponent. The transition temperatureand dynamic exponent for (a) are well consistent with thewell-known result, i. e. , T c = 0 . z = 2 .
0, for (b),(c),(d)are well consistent with those determined by FFH dynamicscaling analysis. Solid lines are just guide to eyes.
We perform the dynamic scaling analysis at a strongdisorder ( p = 0 .
65) in unfrustrated system ( f = 0). Us-ing T c = 0 . ± . z = 1 . ± .
02 and ν = 1 . ± .
02, anexcellent collapse is achieved according to Eq. (5), whichis shown in Fig. 2. In addition, all the low-temperature I - V curves can be fitted to V ∼ I exp( − ( α/I ) µ ) with µ = 0 . ∼ .
1. These results certify a continuous super-conducting phase with long-rang phase coherence. Thecritical temperature for such a strongly disordered systemis very close to that in 2D gauge glass model ( T c = 0 . f = 0 , p = 0 .
86, we first still adopt the scalingform in Eq. (5) to investigate the I - V characteristics.As displayed in Fig. 3, we get a good collapse for T < T c with T c = 0 . ± . z = 2 . ± .
01 and ν = 1 . ± . T < T c . Note that the collapseis poor for T > T c , implying that the phase transitionis not a completely non-KT-type one. Next, we use thescaling form in Eq. (7) to analyze the I - V data above T c . Interestingly, using T c = 0 .
58 and z = 2 . T > T c is achieved, which isshown in Fig. 4. That is to say, the I - V characteristics -1 -2 -1 V | T - T c | - z υ / I I |T-T c | - υ f=0,p=0.65 υ =1.0,z=1.2,T c =0.24 T=0.5 T=0.4 T=0.35 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1
FIG. 2: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 0 , p = 0 . at T < T c are like those of a continuous phase transitionwith power-law divergent correlation length while at T >T c are like those of KT-type phase transition, which arewell consistent with the recent experimental observations[9].To make a comprehensive comparison with the experi-mental findings in Ref. [9], we also investigate the FTPTin frustrated JJA’s ( f = 2 /
5) at a strong site-diluted dis-order ( p = 0 . TABLE I: Summary of T c .f=0 f=2/5p=0.95 0.85(2) 0.16(2)p=0.86 0.58(1) 0.13(1)p=0.7 0.27(2) 0.12(1)p=0.65 0.24(1) 0.14(1) The systems considered in our work are site-dilutedJJA’s, which are not the same as bond-diluted JJA’s in -1 f=0,p=0.86 υ =1.4,z=2.0,T c =0.58 T=0.8 T=0.75 T=0.7 T=0.65 T=0.6 T=0.55 T=0.5 T=0.45 T=0.4 V | T - T c | - z υ / I I |T-T c | - υ FIG. 3: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 0 , p = 0 . , T < T c . Solid linesare just guide to eyes. -1 f=0,p=0.86c=0.8,z=2.0,T c =0.58 T=0.8 T=0.75 T=0.7 T=0.65 T=0.55 T=0.5 T=0.45 T=0.4 I (I / V ) / z / T I exp{ ( c/|T-T c | ) }/T FIG. 4: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (7) for f = 0 , p = 0 . , T > T c . Solid linesare just guide to eyes. Ref. [3, 4]. In bond-diluted systems the diluted bondsare randomly removed, while in the site-diluted systems,the diluted sites are randomly selected, then the nearestfour bonds around the selected sites are removed. Al-though the JJA’s in Ref. [3, 4] and the present work arediluted in different ways, it is interesting to note that theobtained exponents in FTPT are very close, possibly dueto the similar disorder effect produced.
B. Depinning transition and creep motion
Next, we turn to the ZTDT and the LTCM for thetypical site-diluted JJA’s systems mentioned above. De-pinning can be described as a critical phenomenon withscaling law V ∼ ( I − I c ) β , demonstrating a transitionfrom a pinned state below critical driving force I c toa sliding state above I c . The ( I − I c ) .vs.V traces at T = 0 for f = 0 , p = 0 . f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65 are displayed in Fig. 6, linear-fittingsof Log( I − I c ) .vs. Log V curves are also shown as solid -1 -1 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1 T=0.075 T=0.05 f=2/5,p=0.65=1.1,z=1.25,T c =0.14 I |T-T c | V | T - T c | - z / I FIG. 5: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 2 / , p = 0 . -1 -3 -2 -1 -2 -1 -3 -2 -1 -1 -1 -4 -3 -2 Fitting V a Fitting c I-I c Fitting b V I-I c V I-I c FIG. 6: (a) IV characteristics for f = 0 , p = 0 .
86 with I c =0 . ± . β = 2 . ± .
1. (b) IV characteristics for f =0 , p = 0 .
65 with I c = 0 . ± . β = 2 . ± .
1. (c) IV characteristics for f = 2 / , p = 0 .
65 with I c = 0 . ± . β = 2 . ± . -1 -1 -4 -3 -2 -1 -4 -3 -1 -1 -4 -3 I=0.03 I c =0.034 I=0.04 I=0.05 c T V T V I=0.3 I c =0.302 I=0.31 I=0.32 V T a I=0.0375 I c =0.03875 I=0.04 I=0.0425 b FIG. 7: (a)Log T -Log V curves for f = 0 , p = 0 .
86 around I c with I c = 0 . ± . /δ = 1 . ± . T -Log V curves for f = 0 , p = 0 .
65 around I c with I c = 0 . ± . /δ = 2 . ± .
02. (c)Log T -Log V curves for f =2 / , p = 0 .
65 around I c with I c = 0 . ± . /δ = 2 . ± . lines. As for f = 0 , p = 0 .
86, the depinning exponent β is determined to be 2 . ± . I c is 0 . ± . f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65, the depinning exponents are evaluatedto be 2 . ± . . ± .
05 with the critical currents I c = 0 . ± .
001 and I c = 0 . ± . I - V traces are rounded near the zero-temperaturecritical current due to thermal fluctuations. Fisher firstsuggested to map such a phenomenon for the ferromag-net in magnetic field where the second-order phase tran-sition occurs [29]. This mapping was then extended tothe random-field Ising model [16] and the flux lines intype-II superconductors [17]. For the flux lines in type-IIsuperconductors, if the voltage is identified as the orderparameter, the current and the temperature are taken asthe inverse temperature and the field respectively, anal-ogous to the second-order phase transition in the ferro-magnet, the voltage, current and the temperature will satisfy the following scaling ansatz [17, 26] V ( T, I ) = T /δ S [(1 − I c /I ) T − /βδ ] . (8)Where S ( x ) is a scaling function. The relation V ( T, I = I c ) = S (0) T /δ can be easily derived at I = I c , by whichthe critical current I c and the critical exponent δ can bedetermined through the linear fitting of the Log T -Log V curve at I c .The Log T -Log V curves are plotted in Fig. 7(a) for f = 0 , p = 0 .
86. We can observe that the critical currentis between 0.3 and 0.32. In order to locate the criti-cal current precisely, we calculate other values of voltageat current within (0.3,0.32) with a current step 0.01 byquadratic interpolation [26]. Deviation of the T - V curvesfrom the power law is calculated as the square deviations SD = P [ V ( T ) − y ( T )] between the temperature rangewe calculated, here the functions y ( T ) = C T − C areobtained by linear fitting of the Log T -Log V curves. Thecurrent at which the SD is minimum is defined as thecritical current. The critical current is then determinedto be 0 . ± . /δ = 1 . ± .
001 from the slope of Log T -Log V curve at I c = 0 . f = 0 , p = 0 .
65 and f = 2 / , p = 0 . I c and critical exponent 1 /δ for f = 0 , p = 0 .
65 are deter-mined to be 0 . ± . . ± .
02 respectively,for f = 2 / , p = 0 .
65, the result is I c = 0 . ± . /δ = 2 . ± . β ), we get the best col-lapses of data in the regime I ≤ I c with β = 2 . ± . . ± .
02 for f = 0 , p = 0 .
86 and f = 0 , p = 0 . f = 0 , p = 0 .
86, this curve can be fitted by S ( x ) =0 . . x ), combined with the relation βδ = 1 . f = 0 , p = 0 . S ( x ) = 0 . . x ),combined with the relation βδ ≈ .
0, indicative of anArrhenius creep motion. Interestingly, as displayed inFig. 8(c) for f = 2 / , p = 0 .
65, the exponent β isfound to be 2 . ± .
02, which yields βδ ≈ .
0. Thescaling curve in the regime I ≤ I c can be fitted by S ( x ) = 0 . . x ). These two combined facts sug-gest an Arrhenius creep motion in this case.It is worthwhile to note that both the FTPT and theLTCM for strongly disordered JJA’s ( p = 0 .
65) with andwithout frustration are very similar. The I - V curvesin low temperature for all three cases can be describedby V ∝ T /δ exp[ A (1 − I c /I ) /T βδ ], this is one of themain characteristics of glass phases [17, 26], while the I - V traces for KT-type phases can be fitted to V ∝ I a .Hence, by the scaling ansatze in Eq. (8), we have pro-vided another evidence for the existence of non-KT-typephases in the low-temperature regime for these threecases ( f = 0 , p = 0 . f = 0 , p = 0 . f = 2 / , p =0 . -1.5 0.0 -3 0 -10 -5 0 y=0.0994exp(1.9x) T=0.01 T=0.02 T=0.04 T=0.06 T=0.08 T=0.1 T=0.2 T=0.3 a V T (1-I c /I)T - (1-I c /I)T - T=0.05 T=0.1 T=0.15 T=0.2 T=0.3 T=0.4 y=0.037exp(0.5x) V T (1-I c /I)T - b y=0.105exp(0.25x) c V T T=0.075 T=0.1 T=0.125 T=0.15 T=0.175 T=0.2 T=0.25 T=0.3
FIG. 8: (a) Scaling plot for f = 0 , p = 0 .
86 with I c = 0 . /δ = 1 .
688 and βδ = 1 .
55. (b) Scaling plot for f = 0 , p =0 .
65 with I c = 0 . /δ = 2 .
24 and βδ ≈ .
0. (c)Scalingplot for f = 2 / , p = 0 .
65 with I c = 0 . /δ = 2 .
29 and βδ ≈ . IV. SUMMARY
To explore the properties of various critical phenom-ena in site-diluted JJA’s, we performed large-scale simu- lations for two typical percolative strengths p = 0 .
86 and p = 0 .
65 as in a recent experimental work [9]. We in-vestigated the FTPT, the ZTDT and the LTCM in thesesystems. The RSJ dynamics was applied in our work,from which we measured the I - V characteristics at dif-ferent temperatures.The results obtained in this work about FTPT arewell consistent with the recent experimental findings inRef. [9] and are inconsistent with the earlier experimen-tal study in Ref. [1], possibly due to the large noise in themeasurement of voltage in Ref. [1] (larger than 0.2nv),which was considerably reduced in the experiments byYun et al [9]. The evidence for non-KT-type phase tran-sition was revealed by two different scaling ansatzes (Eq.(5) and Eq. (8)). Our results also shed some light on thevarious phases and the phase transitions where the differ-ent divergent correlations at various disorder strengthswere suggested, and the critical exponents were evalu-ated. Furthermore, the results in this paper are usefulfor understanding not only the site-diluted systems, butalso the whole class of disordered JJA’s, for instance, thecombination of two different phase transitions may existin other disordered JJA’s systems.In addition, the ZTDT and the LTCM were alsotouched. It was demonstrated by the scaling analysisthat the creep law for f = 0 , p = 0 .
86 is non-Arrheniustype while those for f = 0 , p = 0 .
65 and f = 2 / , p = 0 . f = 0 , p = 0 . .
55 is also very close to 3 / f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65, the ob-served Arrhenius type creep law is also similar to that inthe glass states of flux lines with a strong collective pin-ning as in Ref. [17]. Future experimental work is neededto clarify this observation.
V. ACKNOWLEDGEMENTS
This work was supported by National Natural ScienceFoundation of China under Grant Nos. 10774128, PC-SIRT (Grant No. IRT0754) in University in China, Na-tional Basic Research Program of China (Grant Nos.2006CB601003 and 2009CB929104),Zhejiang ProvincialNatural Science Foundation under Grant No. Z7080203,and Program for Innovative Research Team in ZhejiangNormal University. † Corresponding author. Email:[email protected] [1] D. C. Harris, S. T. Herbert, D. Stroud and J. C. Garland,Phys. Rev. Lett. , 3606 (1991). [2] E. Granato and D. Dom´ınguez, Phys. Rev. B , 14671 (1997).[3] M. Benakli and E. Granato, S. R. Shenoy and M. Gabay,Phys. Rev. B , 10314 (1998).[4] E . Granato and D. Dominguez, Phys. Rev. B , 094507(2001).[5] Y. J. Yun, I. C. Baek and M. Y. Choi, Phys. Rev. Lett. , 037004 (2002).[6] E. Granto and D. Dominguez, Phys. Rev. B , 094521(2005).[7] J. S. Lim, M. Y. Choi, B. J. Kim and J. Choi, Phys. Rev.B , 100505R (2005).[8] J. Um, B. J. Kim, P. Minnhagen, M. Y. Choi and S. I.Lee, Phys. Rev. B , 094516 (2006).[9] Y. J. Yun, I. C. Baek and M. Y. Choi, Phys. Rev. Lett. , 215701 (2006).[10] Y. J. Yun, I. C. Baek and M. Y. Choi, Europhys. Lett. , 271 (2006).[11] C. J. Lobb, D. W. Abraham and M. Tinkham, Phys.Rev. B , 150 (1983); M. Prester, Phys. Rev. B , 606(1996).[12] J. M. Kosterlitz and D. J. Thouless, J.Phys.C , 1181(1973); J. M. Kosterlitz, J. Phys. C , 1046 (1974); V.L. Berezinskii, Sov. Phys. - JETP, ,610 (1972); V. L.Berezinskii, Zh. Eksp. Teor. Fiz. ,1144 (1973).[13] T. Nattermann, Phys. Rev. Lett. , 2454 (1990).[14] P. Chauve, T. Giamarchi and P. L. Doussal, Phys. Rev.B. , 6241 (2000).[15] M. M¨uller, D. A. Gorokhov and G. Blatter, Phys. Rev.B. , 184305 (2001).[16] L. Rosters, A. Hucht, S. L¨ubeck, U. Nowak and K. D.Usadel, Phys. Rev. E. , 5202 (1999). [17] M. B. Luo and X. Hu, Phys. Rev. Lett. , 267002 (2007).[18] P. Olsson, Phys. Rev. Lett. , 097001 (2007); Q. H.Chen, Phys. Rev. B , 104501 (2008).[19] P. Olsson and S. Teitel, Phys. Rev. Lett. , 137001(2001).[20] Q. H. Chen and X. Hu, Phys. Rev. Lett. , 117005(2003); Q. H. Chen and X. Hu, Phys. Rev. B , 064504(2007).[21] T. Gebele, J. Phys. A: Math. Gen. , L51(1984); Y.Laroyer and E. Pommiers, Phys.Rev.B , 2795 (1994).[22] Q. H. Chen and L. H. Tang, Phys. Rev. Lett. , 067001(2001); L. H. Tang and Q. H. Chen, Phys. Rev. B ,024508 (2003).[23] D. S. Fisher, M. P. A. Fisher and D. A. Huse, Phys. Rev.B , 130 (1991).[24] H. Yang, Y. Jia, L. Shan, Y. Z. Zhang, H. H. Wen, C.G. Zhuang, Z. K. Liu, Q. Li, Y. Cui and X. X. Xi, Phys.Rev. B , 134513 (2007).[25] J. Holzer, R. S. Newrock, C. J. Lobb, T. Aouaroun andS. T. Herbert, Phys. Rev. B , 184508 (2001).[26] Q. H. Chen, J. P. Lv, H. Liu, Phys. Rev. B , 054519(2008).[27] H. G. Katzgraber, Phys. Rev. B , 180402R (2003);H. G. Katzgraber and A. P. Young, Phys. Rev. B ,224507 (2002); Y. Deng, T. M. Garoni, W. A. Guo, H.W. J. Blote and A. D. Sokal, Phys. Rev. Lett. , 120601(2007).[28] P. Holme and P. Olsson, Europhys. Lett.
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439 (2002).[29] D. S. Fisher, Phys. Rev. Lett. , 1486 (1983); Phys.Rev. B31 , 1396 (1985). r X i v : . [ c ond - m a t . s up r- c on ] M a r Phase transition in site-diluted Josephson-junction arrays: A numerical study
Jian-Ping Lv , Huan Liu , and Qing-Hu Chen , , † Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China Center for Statistical and Theoretical Condensed Matter Physics,Zhejiang Normal University, Jinhua 321004, P. R. China
We numerically investigate the intriguing effects produced by random percolative disorder intwo-dimensional Josephson-junction arrays. By dynamic scaling analysis, we evaluate critical tem-peratures and critical exponents with high accuracy. It is observed that, with the introductionof site-diluted disorder, the Kosterlitz-Thouless phase transition is eliminated and evolves into acontinuous transition with power-law divergent correlation length. Moreover, genuine depinningtransition and creep motion are studied, evidence for distinct creep motion types is provided. Ourresults not only are in good agreement with the recent experimental findings, but also shed somelight on the relevant phase transitions.
PACS numbers: 74.81.Fa,68.35.Rh,47.32.Cc
I. INTRODUCTION
Understanding the critical behavior of Josephson-junction arrays (JJA’s) with various disorders is alwaysa challenging issue and has been intensely studied in re-cent years [1]–[10]. However, the properties of differentphases and various phase transitions are not well under-stood. Josephson-junction arrays gives an excellent re-alization to both two-dimensional (2D) XY model andgranular High- T c superconductors [11]. As we know, thepure JJA’s undergoes the celebrated Kosterlitz-Thouless(KT) phase transition from the superconducting state tothe normal one, this transition is driven by the unbindingof thermally activated topological defects [12]. When thedisorder is introduced, the interplays among the repulsivevortex-vortex interaction, the periodic pinning potentialcaused by the discreteness of the arrays, and the defectsproduced by the disorder provide a rich physical picture.In site-diluted JJA’s, the crosses around the randomlyselected sites are removed from the square lattice. Sinceit is a representative model for realizing the irregularJJA’s systems, how the percolation influences the physi-cal properties of JJA’s has attracted considerable atten-tion [1, 2, 3, 4, 9]. Harris et al introduced random per-colative disorder into Nb-Au-Nb proximity-coupled junc-tions, the current-voltage ( I - V ) characteristics were mea-sured and the results demonstrated that the only dif-ference of the phase transition compared with that inideal JJA’s system is the decrease of critical tempera-ture, while the transition type still belongs to the KTone with the disorder strength spanning from p = 0 . p = 1 . − p is the fraction of diluted sites)[1]. However, in a recent experiment, Yun et al showedthat the phase transition changes into a non-KT-type onewhen the disorder strength increases to a moderate value( p = 0 .
86) [9]. Therefore, the existence of the KT-typephase transition in site-diluted JJA’s remains a topic ofcontroversy, the nature of these phase transitions and thevarious phases is not clear.On the other hand, much effort has been devoted tothe zero-temperature depinning transition (ZTDT) and the related low-temperature creep motion (LTCM) boththeoretically [13, 14, 15] and numerically [16, 17, 18] in alarge variety of physical problems, such as charge densitywaves [13], random-field Ising model [16], and flux lines intype-II superconductors [17, 18]. Since the non-linear dy-namic response is a striking problem, there is increasinginterest in its properties and characteristics, especiallyin the flux lines of type-II superconductors [17, 18]. Ina recent numerical study on the three-dimensional glassstates of flux lines, Arrhenius creep motion was observedat a strong collective pinning, while the non-Arrheniuscreep motion was demonstrated at a weak collective pin-ning [17].In this work, we numerically study the finite-temperature phase transition (FTPT) in site-dilutedJJA’s at different percolative disorder strengths, theZTDT and the LTCM are also investigated. The outlineof this paper is as follows. Sec. II describes the modeland the numerical method briefly. In Sec. III, we presentthe main results, where some discussions are also made.Sec. IV gives a short summary of the main conclusions.
II. MODEL AND SIMULATION METHOD
JJA’s can be described by the 2D XY model on asimple square lattice, the Hamiltonian reads [19, 20] H = − X J ij cos( φ i − φ j − A ij ) , (1)where the sum is over all nearest neighboring pairs ona 2D square lattice, J ij denotes the strength of Joseph-son coupling between site i and site j, φ i specifies thephase of the superconducting order parameter on site i,and A ij = (2 π/ Φ ) R A · d l is the integral of magneticvector potential from site i to site j, Φ denotes the fluxquantum. The direct sum of A ij around an elementaryplaquette is 2 πf , with f the magnetic flux penetratingeach plaquette produced by the uniformly applied field,which is measured in unit of Φ . f = 0 and f = 2 / ×
128 for f = 0 and 100 ×
100 for f = 2 /
5, wherethe finite size effects are negligible. We introduce thesite-diluted disorder similar to the previous experiments[1, 9]. We first select the diluted sites randomly with theprobability 1 − p , then remove the nearest four bondsaround the selected sites from the lattice. The distribu-tions of the diluted sites are the same for all the samplesconsidered. The percolative threshold concentration p c is about 0 .
592 [21].The resistivity-shunted-junction (RSJ) dynamics is in-corporated in the simulations, which can be described as[20, 22] σ ~ e X j ( ˙ φ i − ˙ φ j ) = − ∂H∂φ i + J ex ,i − X j η ij , (2)where σ is the normal conductivity, J ex,i refers to theexternal current, η ij denotes the thermal noise currentwith < η ij ( t ) > = 0 and < η ij ( t ) η ij ( t ′ ) > = 2 σk B T δ ( t − t ′ ).The fluctuating twist boundary condition is applied inthe xy plane to maintain the current, thus the new phaseangle θ i = φ i + r i · ∆ (∆ = ( △ x , △ y ) is the twist variable)is periodic in each direction. In this way, supercurrentbetween site i and site j is given by J si → j = J ij sin ( θ i − θ j − A ij − r ij · ∆) , and the dynamics of ∆ α can be writtenas ˙∆ α = 1 L X α [ J i → j + η ij ] − I α , (3)where α denotes the x or y direction, the voltage dropin α direction is V = − L ˙∆ α . For convenience, units aretaken as 2 e = ~ = J = σ = k B = 1 in the following.Above equations can be solved efficiently by a pseudo-spectral algorithm due to the periodicity of phase in alldirections. The time stepping is done using a second-order Runge-Kutta scheme with ∆ t = 0 .
05. Our runs aretypically (4 − × time steps and the latter half timesteps are for the measurements. The detailed procedurein the simulations was described in Ref. [20, 22]. In thiswork, a uniform external current I along x direction isfed into the system.Since RSJ simulations with direct numerical integra-tions of stochastic equations of motion are very time-consuming, it is practically difficult to perform any se-rious disorder averaging in the present rather large sys-tems. Our results are based on one realization of disor-der. For these very large samples, it is expected to exista good self-averaging effect, which is confirmed by twoadditional simulations with different realizations of dis-order. This point is also supported by a recent studyof JJA’s by Um et al [8]. In addition, simulations withdifferent initial states are performed and the results arenearly the same. Actually, the hysteric phenomenon isusually negligible in previous RSJ dynamical simulations on JJA’s [7, 8]. For these reasons, the results from simu-lations with a unique initial state (random phases in thiswork) are accurate and convincing. III. RESULTS AND DISCUSSIONSA. Finite temperature phase transition
The I - V characteristics are measured at different dis-order strengths and temperatures. At each temperature,we try to probe the system at a current as low as possi-ble. To check the method used in this work, we investi-gate the I - V characteristics for f = 0 , p = 1 .
0. As shownin Fig. 1(a), the slope of the I - V curve in log-log plotat the transition temperature T c ( ≈ . I - V index jumps from 3 to 1,consistent with the well-known fact that the pure JJA’sexperiences a KT-type phase transition at T c ≈ . I - V traces at different dis-order strengths in unfrustrated JJA’s, while Fig. 1(d) isfor f = 2 / , p = 0 .
65. It is clear that, at lower temper-atures, R = V /I tends to zero as the current decreases,which follows that there is a true superconducting phasewith zero linear resistivity.It is crucial to use a powerful scaling method to an-alyze the I - V characteristics. In this paper, we adoptthe Fisher-Fisher-Huse (FFH) dynamic scaling method,which provides an excellent approach to analyze thesuperconducting phase transition [23]. If the prop-erly scaled I - V curves collapse onto two scaling curvesabove and below the transition temperature, a continu-ous superconducting phase transition is ensured. Such amethod is widely used [6, 24], the scaling form of whichin 2D is V = Iξ − z ψ ± ( Iξ ) , (4)where ψ +( − ) ( x ) is the scaling function above (below) T c , z is the dynamic exponent, ξ is the correlation length,and V ∼ I z +1 at T = T c .Assuming that the transition is continuous and char-acterized by the divergence of the characteristic length ξ ∼ | T − T c | − ν and time scale t ∼ ξ z , FFH dynamicscaling takes the following form( V /I ) | T − T c | − zν = ψ ± ( I | T − T c | − ν ) . (5)On the other hand, to certify a KT-type phase tran-sition in JJA’s, a new scaling form [25] is proposed asfollows ( I/T )( I/V ) /z = P ± ( Iξ/T ) , (6)which can be derived directly from Eq. (4) after somesimple algebra. The correlation length of KT-type phasetransition above T c is well defined as ξ ∼ e ( c/ | T − T c | ) / and Eq. (6) is rewritten as( I/T )( I/V ) /z = P + ( Ie ( c/ | T − T c | ) / /T ) . (7) -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -2 -1 -4 -3 -2 -1 (b) f=0, p=0.86T C =0.58- - - slope = 3.0 T=0.8 T=0.75 T=0.7 T=0.65 T=0.6 T=0.55 T=0.5 T=0.45 T=0.4 (d) f=2/5,p=0.65T C =0.14- - - slope=2.25 I T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1 T=0.075 T=0.05 T=0.025 II (c) I f=0,p=0.65T C =0.24 - - - slope=2.2 T=0.5 T=0.4 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1
T=1.1 T=1.0 T=0.95 T=0.9 T=0.894 T=0.85 T=0.8 VV f=0,p=1.0T C =0.894- - - slope = 3.0 V V (a) FIG. 1: I - V characteristics for different frustrations and dis-order strengths. The dash lines are drawn to show wherethe phase transition occurs, the slopes of which are equal to z + 1, z is the dynamic exponent. The transition temperatureand dynamic exponent for (a) are well consistent with thewell-known result, i. e. , T c = 0 . z = 2 .
0, for (b),(c),(d)are well consistent with those determined by FFH dynamicscaling analysis. Solid lines are just guide to eyes.
We perform the dynamic scaling analysis at a strongdisorder ( p = 0 .
65) in unfrustrated system ( f = 0). Us-ing T c = 0 . ± . z = 1 . ± .
02 and ν = 1 . ± .
02, anexcellent collapse is achieved according to Eq. (5), whichis shown in Fig. 2. In addition, all the low-temperature I - V curves can be fitted to V ∼ I exp( − ( α/I ) µ ) with µ = 0 . ∼ .
1. These results certify a continuous super-conducting phase with long-rang phase coherence. Thecritical temperature for such a strongly disordered systemis very close to that in 2D gauge glass model ( T c = 0 . f = 0 , p = 0 .
86, we first still adopt the scalingform in Eq. (5) to investigate the I - V characteristics.As displayed in Fig. 3, we get a good collapse for T < T c with T c = 0 . ± . z = 2 . ± .
01 and ν = 1 . ± . T < T c . Note that the collapseis poor for T > T c , implying that the phase transitionis not a completely non-KT-type one. Next, we use thescaling form in Eq. (7) to analyze the I - V data above T c . Interestingly, using T c = 0 .
58 and z = 2 . T > T c is achieved, which isshown in Fig. 4. That is to say, the I - V characteristics -1 -2 -1 V | T - T c | - z υ / I I |T-T c | - υ f=0,p=0.65 υ =1.0,z=1.2,T c =0.24 T=0.5 T=0.4 T=0.35 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1
FIG. 2: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 0 , p = 0 . at T < T c are like those of a continuous phase transitionwith power-law divergent correlation length while at T >T c are like those of KT-type phase transition, which arewell consistent with the recent experimental observations[9].To make a comprehensive comparison with the experi-mental findings in Ref. [9], we also investigate the FTPTin frustrated JJA’s ( f = 2 /
5) at a strong site-diluted dis-order ( p = 0 . TABLE I: Summary of T c .f=0 f=2/5p=0.95 0.85(2) 0.16(2)p=0.86 0.58(1) 0.13(1)p=0.7 0.27(2) 0.12(1)p=0.65 0.24(1) 0.14(1) The systems considered in our work are site-dilutedJJA’s, which are not the same as bond-diluted JJA’s in -1 f=0,p=0.86 υ =1.4,z=2.0,T c =0.58 T=0.8 T=0.75 T=0.7 T=0.65 T=0.6 T=0.55 T=0.5 T=0.45 T=0.4 V | T - T c | - z υ / I I |T-T c | - υ FIG. 3: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 0 , p = 0 . , T < T c . Solid linesare just guide to eyes. -1 f=0,p=0.86c=0.8,z=2.0,T c =0.58 T=0.8 T=0.75 T=0.7 T=0.65 T=0.55 T=0.5 T=0.45 T=0.4 I (I / V ) / z / T I exp{ ( c/|T-T c | ) }/T FIG. 4: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (7) for f = 0 , p = 0 . , T > T c . Solid linesare just guide to eyes. Ref. [3, 4]. In bond-diluted systems the diluted bondsare randomly removed, while in the site-diluted systems,the diluted sites are randomly selected, then the nearestfour bonds around the selected sites are removed. Al-though the JJA’s in Ref. [3, 4] and the present work arediluted in different ways, it is interesting to note that theobtained exponents in FTPT are very close, possibly dueto the similar disorder effect produced.
B. Depinning transition and creep motion
Next, we turn to the ZTDT and the LTCM for thetypical site-diluted JJA’s systems mentioned above. De-pinning can be described as a critical phenomenon withscaling law V ∼ ( I − I c ) β , demonstrating a transitionfrom a pinned state below critical driving force I c toa sliding state above I c . The ( I − I c ) .vs.V traces at T = 0 for f = 0 , p = 0 . f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65 are displayed in Fig. 6, linear-fittingsof Log( I − I c ) .vs. Log V curves are also shown as solid -1 -1 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1 T=0.075 T=0.05 f=2/5,p=0.65=1.1,z=1.25,T c =0.14 I |T-T c | V | T - T c | - z / I FIG. 5: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 2 / , p = 0 . -1 -3 -2 -1 -2 -1 -3 -2 -1 -1 -1 -4 -3 -2 Fitting V a Fitting c I-I c Fitting b V I-I c V I-I c FIG. 6: (a) IV characteristics for f = 0 , p = 0 .
86 with I c =0 . ± . β = 2 . ± .
1. (b) IV characteristics for f =0 , p = 0 .
65 with I c = 0 . ± . β = 2 . ± .
1. (c) IV characteristics for f = 2 / , p = 0 .
65 with I c = 0 . ± . β = 2 . ± . -1 -1 -4 -3 -2 -1 -4 -3 -1 -1 -4 -3 I=0.03 I c =0.034 I=0.04 I=0.05 c T V T V I=0.3 I c =0.302 I=0.31 I=0.32 V T a I=0.0375 I c =0.03875 I=0.04 I=0.0425 b FIG. 7: (a)Log T -Log V curves for f = 0 , p = 0 .
86 around I c with I c = 0 . ± . /δ = 1 . ± . T -Log V curves for f = 0 , p = 0 .
65 around I c with I c = 0 . ± . /δ = 2 . ± .
02. (c)Log T -Log V curves for f =2 / , p = 0 .
65 around I c with I c = 0 . ± . /δ = 2 . ± . lines. As for f = 0 , p = 0 .
86, the depinning exponent β is determined to be 2 . ± . I c is 0 . ± . f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65, the depinning exponents are evaluatedto be 2 . ± . . ± .
05 with the critical currents I c = 0 . ± .
001 and I c = 0 . ± . I - V traces are rounded near the zero-temperaturecritical current due to thermal fluctuations. Fisher firstsuggested to map such a phenomenon for the ferromag-net in magnetic field where the second-order phase tran-sition occurs [29]. This mapping was then extended tothe random-field Ising model [16] and the flux lines intype-II superconductors [17]. For the flux lines in type-IIsuperconductors, if the voltage is identified as the orderparameter, the current and the temperature are taken asthe inverse temperature and the field respectively, anal-ogous to the second-order phase transition in the ferro-magnet, the voltage, current and the temperature will satisfy the following scaling ansatz [17, 26] V ( T, I ) = T /δ S [(1 − I c /I ) T − /βδ ] . (8)Where S ( x ) is a scaling function. The relation V ( T, I = I c ) = S (0) T /δ can be easily derived at I = I c , by whichthe critical current I c and the critical exponent δ can bedetermined through the linear fitting of the Log T -Log V curve at I c .The Log T -Log V curves are plotted in Fig. 7(a) for f = 0 , p = 0 .
86. We can observe that the critical currentis between 0.3 and 0.32. In order to locate the criti-cal current precisely, we calculate other values of voltageat current within (0.3,0.32) with a current step 0.01 byquadratic interpolation [26]. Deviation of the T - V curvesfrom the power law is calculated as the square deviations SD = P [ V ( T ) − y ( T )] between the temperature rangewe calculated, here the functions y ( T ) = C T − C areobtained by linear fitting of the Log T -Log V curves. Thecurrent at which the SD is minimum is defined as thecritical current. The critical current is then determinedto be 0 . ± . /δ = 1 . ± .
001 from the slope of Log T -Log V curve at I c = 0 . f = 0 , p = 0 .
65 and f = 2 / , p = 0 . I c and critical exponent 1 /δ for f = 0 , p = 0 .
65 are deter-mined to be 0 . ± . . ± .
02 respectively,for f = 2 / , p = 0 .
65, the result is I c = 0 . ± . /δ = 2 . ± . β ), we get the best col-lapses of data in the regime I ≤ I c with β = 2 . ± . . ± .
02 for f = 0 , p = 0 .
86 and f = 0 , p = 0 . f = 0 , p = 0 .
86, this curve can be fitted by S ( x ) =0 . . x ), combined with the relation βδ = 1 . f = 0 , p = 0 . S ( x ) = 0 . . x ),combined with the relation βδ ≈ .
0, indicative of anArrhenius creep motion. Interestingly, as displayed inFig. 8(c) for f = 2 / , p = 0 .
65, the exponent β isfound to be 2 . ± .
02, which yields βδ ≈ .
0. Thescaling curve in the regime I ≤ I c can be fitted by S ( x ) = 0 . . x ). These two combined facts sug-gest an Arrhenius creep motion in this case.It is worthwhile to note that both the FTPT and theLTCM for strongly disordered JJA’s ( p = 0 .
65) with andwithout frustration are very similar. The I - V curvesin low temperature for all three cases can be describedby V ∝ T /δ exp[ A (1 − I c /I ) /T βδ ], this is one of themain characteristics of glass phases [17, 26], while the I - V traces for KT-type phases can be fitted to V ∝ I a .Hence, by the scaling ansatze in Eq. (8), we have pro-vided another evidence for the existence of non-KT-typephases in the low-temperature regime for these threecases ( f = 0 , p = 0 . f = 0 , p = 0 . f = 2 / , p =0 . -1.5 0.0 -3 0 -10 -5 0 y=0.0994exp(1.9x) T=0.01 T=0.02 T=0.04 T=0.06 T=0.08 T=0.1 T=0.2 T=0.3 a V T (1-I c /I)T - (1-I c /I)T - T=0.05 T=0.1 T=0.15 T=0.2 T=0.3 T=0.4 y=0.037exp(0.5x) V T (1-I c /I)T - b y=0.105exp(0.25x) c V T T=0.075 T=0.1 T=0.125 T=0.15 T=0.175 T=0.2 T=0.25 T=0.3
FIG. 8: (a) Scaling plot for f = 0 , p = 0 .
86 with I c = 0 . /δ = 1 .
688 and βδ = 1 .
55. (b) Scaling plot for f = 0 , p =0 .
65 with I c = 0 . /δ = 2 .
24 and βδ ≈ .
0. (c)Scalingplot for f = 2 / , p = 0 .
65 with I c = 0 . /δ = 2 .
29 and βδ ≈ . IV. SUMMARY
To explore the properties of various critical phenom-ena in site-diluted JJA’s, we performed large-scale simu- lations for two typical percolative strengths p = 0 .
86 and p = 0 .
65 as in a recent experimental work [9]. We in-vestigated the FTPT, the ZTDT and the LTCM in thesesystems. The RSJ dynamics was applied in our work,from which we measured the I - V characteristics at dif-ferent temperatures.The results obtained in this work about FTPT arewell consistent with the recent experimental findings inRef. [9] and are inconsistent with the earlier experimen-tal study in Ref. [1], possibly due to the large noise in themeasurement of voltage in Ref. [1] (larger than 0.2nv),which was considerably reduced in the experiments byYun et al [9]. The evidence for non-KT-type phase tran-sition was revealed by two different scaling ansatzes (Eq.(5) and Eq. (8)). Our results also shed some light on thevarious phases and the phase transitions where the differ-ent divergent correlations at various disorder strengthswere suggested, and the critical exponents were evalu-ated. Furthermore, the results in this paper are usefulfor understanding not only the site-diluted systems, butalso the whole class of disordered JJA’s, for instance, thecombination of two different phase transitions may existin other disordered JJA’s systems.In addition, the ZTDT and the LTCM were alsotouched. It was demonstrated by the scaling analysisthat the creep law for f = 0 , p = 0 .
86 is non-Arrheniustype while those for f = 0 , p = 0 .
65 and f = 2 / , p = 0 . f = 0 , p = 0 . .
55 is also very close to 3 / f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65, the ob-served Arrhenius type creep law is also similar to that inthe glass states of flux lines with a strong collective pin-ning as in Ref. [17]. Future experimental work is neededto clarify this observation.
V. ACKNOWLEDGEMENTS
This work was supported by National Natural ScienceFoundation of China under Grant Nos. 10774128, PC-SIRT (Grant No. IRT0754) in University in China, Na-tional Basic Research Program of China (Grant Nos.2006CB601003 and 2009CB929104),Zhejiang ProvincialNatural Science Foundation under Grant No. Z7080203,and Program for Innovative Research Team in ZhejiangNormal University. † Corresponding author. Email:[email protected] [1] D. C. Harris, S. T. Herbert, D. Stroud and J. C. Garland,Phys. Rev. Lett. , 3606 (1991). [2] E. Granato and D. Dom´ınguez, Phys. Rev. B , 14671 (1997).[3] M. Benakli and E. Granato, S. R. Shenoy and M. Gabay,Phys. Rev. B , 10314 (1998).[4] E . Granato and D. Dominguez, Phys. Rev. B , 094507(2001).[5] Y. J. Yun, I. C. Baek and M. Y. Choi, Phys. Rev. Lett. , 037004 (2002).[6] E. Granto and D. Dominguez, Phys. Rev. B , 094521(2005).[7] J. S. Lim, M. Y. Choi, B. J. Kim and J. Choi, Phys. Rev.B , 100505R (2005).[8] J. Um, B. J. Kim, P. Minnhagen, M. Y. Choi and S. I.Lee, Phys. Rev. B , 094516 (2006).[9] Y. J. Yun, I. C. Baek and M. Y. Choi, Phys. Rev. Lett. , 215701 (2006).[10] Y. J. Yun, I. C. Baek and M. Y. Choi, Europhys. Lett. , 271 (2006).[11] C. J. Lobb, D. W. Abraham and M. Tinkham, Phys.Rev. B , 150 (1983); M. Prester, Phys. Rev. B , 606(1996).[12] J. M. Kosterlitz and D. J. Thouless, J.Phys.C , 1181(1973); J. M. Kosterlitz, J. Phys. C , 1046 (1974); V.L. Berezinskii, Sov. Phys. - JETP, ,610 (1972); V. L.Berezinskii, Zh. Eksp. Teor. Fiz. ,1144 (1973).[13] T. Nattermann, Phys. Rev. Lett. , 2454 (1990).[14] P. Chauve, T. Giamarchi and P. L. Doussal, Phys. Rev.B. , 6241 (2000).[15] M. M¨uller, D. A. Gorokhov and G. Blatter, Phys. Rev.B. , 184305 (2001).[16] L. Rosters, A. Hucht, S. L¨ubeck, U. Nowak and K. D.Usadel, Phys. Rev. E. , 5202 (1999). [17] M. B. Luo and X. Hu, Phys. Rev. Lett. , 267002 (2007).[18] P. Olsson, Phys. Rev. Lett. , 097001 (2007); Q. H.Chen, Phys. Rev. B , 104501 (2008).[19] P. Olsson and S. Teitel, Phys. Rev. Lett. , 137001(2001).[20] Q. H. Chen and X. Hu, Phys. Rev. Lett. , 117005(2003); Q. H. Chen and X. Hu, Phys. Rev. B , 064504(2007).[21] T. Gebele, J. Phys. A: Math. Gen. , L51(1984); Y.Laroyer and E. Pommiers, Phys.Rev.B , 2795 (1994).[22] Q. H. Chen and L. H. Tang, Phys. Rev. Lett. , 067001(2001); L. H. Tang and Q. H. Chen, Phys. Rev. B ,024508 (2003).[23] D. S. Fisher, M. P. A. Fisher and D. A. Huse, Phys. Rev.B , 130 (1991).[24] H. Yang, Y. Jia, L. Shan, Y. Z. Zhang, H. H. Wen, C.G. Zhuang, Z. K. Liu, Q. Li, Y. Cui and X. X. Xi, Phys.Rev. B , 134513 (2007).[25] J. Holzer, R. S. Newrock, C. J. Lobb, T. Aouaroun andS. T. Herbert, Phys. Rev. B , 184508 (2001).[26] Q. H. Chen, J. P. Lv, H. Liu, Phys. Rev. B , 054519(2008).[27] H. G. Katzgraber, Phys. Rev. B , 180402R (2003);H. G. Katzgraber and A. P. Young, Phys. Rev. B ,224507 (2002); Y. Deng, T. M. Garoni, W. A. Guo, H.W. J. Blote and A. D. Sokal, Phys. Rev. Lett. , 120601(2007).[28] P. Holme and P. Olsson, Europhys. Lett.
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439 (2002).[29] D. S. Fisher, Phys. Rev. Lett. , 1486 (1983); Phys.Rev. B31 , 1396 (1985). r X i v : . [ c ond - m a t . s up r- c on ] M a r Phase transition in site-diluted Josephson-junction arrays: A numerical study
Jian-Ping Lv , Huan Liu , and Qing-Hu Chen , , ∗ Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China Center for Statistical and Theoretical Condensed Matter Physics,Zhejiang Normal University, Jinhua 321004, P. R. China (Dated: October 28, 2018)We numerically investigate the intriguing effects produced by random percolative disorder intwo-dimensional Josephson-junction arrays. By dynamic scaling analysis, we evaluate critical tem-peratures and critical exponents with high accuracy. It is observed that, with the introductionof site-diluted disorder, the Kosterlitz-Thouless phase transition is eliminated and evolves into acontinuous transition with power-law divergent correlation length. Moreover, genuine depinningtransition and creep motion are studied, evidence for distinct creep motion types is provided. Ourresults not only are in good agreement with the recent experimental findings, but also shed somelight on the relevant phase transitions.
PACS numbers: 74.81.Fa,68.35.Rh,47.32.Cc
I. INTRODUCTION
Understanding the critical behavior of Josephson-junction arrays (JJA’s) with various disorders is alwaysa challenging issue and has been intensely studied in re-cent years [1]–[10]. However, the properties of differentphases and various phase transitions are not well under-stood. Josephson-junction arrays gives an excellent re-alization to both two-dimensional (2D) XY model andgranular High- T c superconductors [11]. As we know, thepure JJA’s undergoes the celebrated Kosterlitz-Thouless(KT) phase transition from the superconducting state tothe normal one, this transition is driven by the unbindingof thermally activated topological defects [12]. When thedisorder is introduced, the interplays among the repulsivevortex-vortex interaction, the periodic pinning potentialcaused by the discreteness of the arrays, and the defectsproduced by the disorder provide a rich physical picture.In site-diluted JJA’s, the crosses around the randomlyselected sites are removed from the square lattice. Sinceit is a representative model for realizing the irregularJJA’s systems, how the percolation influences the physi-cal properties of JJA’s has attracted considerable atten-tion [1, 2, 3, 4, 9]. Harris et al introduced random per-colative disorder into Nb-Au-Nb proximity-coupled junc-tions, the current-voltage ( I - V ) characteristics were mea-sured and the results demonstrated that the only dif-ference of the phase transition compared with that inideal JJA’s system is the decrease of critical tempera-ture, while the transition type still belongs to the KTone with the disorder strength spanning from p = 0 . p = 1 . − p is the fraction of diluted sites)[1]. However, in a recent experiment, Yun et al showedthat the phase transition changes into a non-KT-type onewhen the disorder strength increases to a moderate value( p = 0 .
86) [9]. Therefore, the existence of the KT-typephase transition in site-diluted JJA’s remains a topic ofcontroversy, the nature of these phase transitions and thevarious phases is not clear.On the other hand, much effort has been devoted to the zero-temperature depinning transition (ZTDT) andthe related low-temperature creep motion (LTCM) boththeoretically [13, 14, 15] and numerically [16, 17, 18] in alarge variety of physical problems, such as charge densitywaves [13], random-field Ising model [16], and flux lines intype-II superconductors [17, 18]. Since the non-linear dy-namic response is a striking problem, there is increasinginterest in its properties and characteristics, especiallyin the flux lines of type-II superconductors [17, 18]. Ina recent numerical study on the three-dimensional glassstates of flux lines, Arrhenius creep motion was observedat a strong collective pinning, while the non-Arrheniuscreep motion was demonstrated at a weak collective pin-ning [17].In this work, we numerically study the finite-temperature phase transition (FTPT) in site-dilutedJJA’s at different percolative disorder strengths, theZTDT and the LTCM are also investigated. The outlineof this paper is as follows. Sec. II describes the modeland the numerical method briefly. In Sec. III, we presentthe main results, where some discussions are also made.Sec. IV gives a short summary of the main conclusions.
II. MODEL AND SIMULATION METHOD
JJA’s can be described by the 2D XY model on asimple square lattice, the Hamiltonian reads [19, 20] H = − X J ij cos( φ i − φ j − A ij ) , (1)where the sum is over all nearest neighboring pairs ona 2D square lattice, J ij denotes the strength of Joseph-son coupling between site i and site j, φ i specifies thephase of the superconducting order parameter on site i,and A ij = (2 π/ Φ ) R A · d l is the integral of magneticvector potential from site i to site j, Φ denotes the fluxquantum. The direct sum of A ij around an elementaryplaquette is 2 πf , with f the magnetic flux penetratingeach plaquette produced by the uniformly applied field,which is measured in unit of Φ . f = 0 and f = 2 / ×
128 for f = 0 and 100 ×
100 for f = 2 /
5, wherethe finite size effects are negligible. We introduce thesite-diluted disorder similar to the previous experiments[1, 9]. We first select the diluted sites randomly with theprobability 1 − p , then remove the nearest four bondsaround the selected sites from the lattice. The distribu-tions of the diluted sites are the same for all the samplesconsidered. The percolative threshold concentration p c is about 0 .
592 [21].The resistivity-shunted-junction (RSJ) dynamics is in-corporated in the simulations, which can be described as[20, 22] σ ~ e X j ( ˙ φ i − ˙ φ j ) = − ∂H∂φ i + J ex ,i − X j η ij , (2)where σ is the normal conductivity, J ex,i refers to theexternal current, η ij denotes the thermal noise currentwith < η ij ( t ) > = 0 and < η ij ( t ) η ij ( t ′ ) > = 2 σk B T δ ( t − t ′ ).The fluctuating twist boundary condition is applied inthe xy plane to maintain the current, thus the new phaseangle θ i = φ i + r i · ∆ (∆ = ( △ x , △ y ) is the twist variable)is periodic in each direction. In this way, supercurrentbetween site i and site j is given by J si → j = J ij sin ( θ i − θ j − A ij − r ij · ∆) , and the dynamics of ∆ α can be writtenas ˙∆ α = 1 L X α [ J i → j + η ij ] − I α , (3)where α denotes the x or y direction, the voltage dropin α direction is V = − L ˙∆ α . For convenience, units aretaken as 2 e = ~ = J = σ = k B = 1 in the following.Above equations can be solved efficiently by a pseudo-spectral algorithm due to the periodicity of phase in alldirections. The time stepping is done using a second-order Runge-Kutta scheme with ∆ t = 0 .
05. Our runs aretypically (4 − × time steps and the latter half timesteps are for the measurements. The detailed procedurein the simulations was described in Ref. [20, 22]. In thiswork, a uniform external current I along x direction isfed into the system.Since RSJ simulations with direct numerical integra-tions of stochastic equations of motion are very time-consuming, it is practically difficult to perform any se-rious disorder averaging in the present rather large sys-tems. Our results are based on one realization of disor-der. For these very large samples, it is expected to exista good self-averaging effect, which is confirmed by twoadditional simulations with different realizations of dis-order. This point is also supported by a recent studyof JJA’s by Um et al [8]. In addition, simulations withdifferent initial states are performed and the results arenearly the same. Actually, the hysteretic phenomenon is usually negligible in previous RSJ dynamical simulationson JJA’s [7, 8]. For these reasons, the results from simu-lations with a unique initial state (random phases in thiswork) are accurate and convincing. III. RESULTS AND DISCUSSIONSA. Finite temperature phase transition
The I - V characteristics are measured at different dis-order strengths and temperatures. At each temperature,we try to probe the system at a current as low as possi-ble. To check the method used in this work, we investi-gate the I - V characteristics for f = 0 , p = 1 .
0. As shownin Fig. 1(a), the slope of the I - V curve in log-log plotat the transition temperature T c ( ≈ . I - V index jumps from 3 to 1,consistent with the well-known fact that the pure JJA’sexperiences a KT-type phase transition at T c ≈ . I - V traces at different dis-order strengths in unfrustrated JJA’s, while Fig. 1(d) isfor f = 2 / , p = 0 .
65. It is clear that, at lower temper-atures, R = V /I tends to zero as the current decreases,which follows that there is a true superconducting phasewith zero linear resistivity.It is crucial to use a powerful scaling method to an-alyze the I - V characteristics. In this paper, we adoptthe Fisher-Fisher-Huse (FFH) dynamic scaling method,which provides an excellent approach to analyze thesuperconducting phase transition [23]. If the prop-erly scaled I - V curves collapse onto two scaling curvesabove and below the transition temperature, a continu-ous superconducting phase transition is ensured. Such amethod is widely used [6, 24], the scaling form of whichin 2D is V = Iξ − z ψ ± ( Iξ ) , (4)where ψ +( − ) ( x ) is the scaling function above (below) T c , z is the dynamic exponent, ξ is the correlation length,and V ∼ I z +1 at T = T c .Assuming that the transition is continuous and char-acterized by the divergence of the characteristic length ξ ∼ | T − T c | − ν and time scale t ∼ ξ z , FFH dynamicscaling takes the following form( V /I ) | T − T c | − zν = ψ ± ( I | T − T c | − ν ) . (5)On the other hand, to certify a KT-type phase tran-sition in JJA’s, a new scaling form [25] is proposed asfollows ( I/T )( I/V ) /z = P ± ( Iξ/T ) , (6)which can be derived directly from Eq. (4) after somesimple algebra. The correlation length of KT-type phasetransition above T c is well defined as ξ ∼ e ( c/ | T − T c | ) / and Eq. (6) is rewritten as( I/T )( I/V ) /z = P + ( Ie ( c/ | T − T c | ) / /T ) . (7) -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -1 -2 -1 -4 -3 -2 -2 -1 -4 -3 -2 -1 (b) f=0, p=0.86T C =0.58- - - slope = 3.0 T=0.8 T=0.75 T=0.7 T=0.65 T=0.6 T=0.55 T=0.5 T=0.45 T=0.4 (d) f=2/5,p=0.65T C =0.14- - - slope=2.25 I T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1 T=0.075 T=0.05 T=0.025 II (c) I f=0,p=0.65T C =0.24 - - - slope=2.2 T=0.5 T=0.4 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1
T=1.1 T=1.0 T=0.95 T=0.9 T=0.894 T=0.85 T=0.8 VV f=0,p=1.0T C =0.894- - - slope = 3.0 V V (a) FIG. 1: I - V characteristics for different frustrations and dis-order strengths. The dash lines are drawn to show wherethe phase transition occurs, the slopes of which are equal to z + 1, z is the dynamic exponent. The transition temperatureand dynamic exponent for (a) are well consistent with thewell-known result, i. e. , T c = 0 . z = 2 .
0, for (b),(c),(d)are well consistent with those determined by FFH dynamicscaling analysis. Solid lines are just guide to eyes.
We perform the dynamic scaling analysis at a strongdisorder ( p = 0 .
65) in unfrustrated system ( f = 0). Us-ing T c = 0 . ± . z = 1 . ± .
02 and ν = 1 . ± .
02, anexcellent collapse is achieved according to Eq. (5), whichis shown in Fig. 2. In addition, all the low-temperature I - V curves can be fitted to V ∼ I exp( − ( α/I ) µ ) with µ = 0 . ∼ .
1. These results certify a continuous super-conducting phase with long-range phase coherence. Thecritical temperature for such a strongly disordered systemis very close to that in 2D gauge glass model ( T c = 0 . f = 0 , p = 0 .
86, we first still adopt the scalingform in Eq. (5) to investigate the I - V characteristics.As displayed in Fig. 3, we get a good collapse for T < T c with T c = 0 . ± . z = 2 . ± .
01 and ν = 1 . ± . T < T c . Note that the collapseis poor for T > T c , implying that the phase transitionis not a completely non-KT-type one. Next, we use thescaling form in Eq. (7) to analyze the I - V data above T c . Interestingly, using T c = 0 .
58 and z = 2 . T > T c is achieved, which isshown in Fig. 4. That is to say, the I - V characteristics -1 -2 -1 V | T - T c | - z υ / I I |T-T c | - υ f=0,p=0.65 υ =1.0,z=1.2,T c =0.24 T=0.5 T=0.4 T=0.35 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1
FIG. 2: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 0 , p = 0 . at T < T c are like those of a continuous phase transitionwith power-law divergent correlation length while at T >T c are like those of KT-type phase transition, which arewell consistent with the recent experimental observations[9].To make a comprehensive comparison with the experi-mental findings in Ref. [9], we also investigate the FTPTin frustrated JJA’s ( f = 2 /
5) at a strong site-diluted dis-order ( p = 0 . TABLE I: Summary of T c .f=0 f=2/5p=0.95 0.85(2) 0.16(2)p=0.86 0.58(1) 0.13(1)p=0.7 0.27(2) 0.12(1)p=0.65 0.24(1) 0.14(1) The systems considered in our work are site-dilutedJJA’s, which are not the same as bond-diluted JJA’s in -1 f=0,p=0.86 υ =1.4,z=2.0,T c =0.58 T=0.8 T=0.75 T=0.7 T=0.65 T=0.6 T=0.55 T=0.5 T=0.45 T=0.4 V | T - T c | - z υ / I I |T-T c | - υ FIG. 3: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 0 , p = 0 . , T < T c . Solid linesare just guide to eyes. -1 f=0,p=0.86c=0.8,z=2.0,T c =0.58 T=0.8 T=0.75 T=0.7 T=0.65 T=0.55 T=0.5 T=0.45 T=0.4 I (I / V ) / z / T I exp{ ( c/|T-T c | ) }/T FIG. 4: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (7) for f = 0 , p = 0 . , T > T c . Solid linesare just guide to eyes. Ref. [3, 4]. In bond-diluted systems the diluted bondsare randomly removed, while in the site-diluted systems,the diluted sites are randomly selected, then the nearestfour bonds around the selected sites are removed. Al-though the JJA’s in Ref. [3, 4] and the present work arediluted in different ways, it is interesting to note that theobtained exponents in FTPT are very close, possibly dueto the similar disorder effect produced.
B. Depinning transition and creep motion
Next, we turn to the ZTDT and the LTCM for thetypical site-diluted JJA’s systems mentioned above. De-pinning can be described as a critical phenomenon withscaling law V ∼ ( I − I c ) β , demonstrating a transitionfrom a pinned state below critical driving force I c toa sliding state above I c . The ( I − I c ) .vs.V traces at T = 0 for f = 0 , p = 0 . f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65 are displayed in Fig. 6, linear-fittingsof Log( I − I c ) .vs. Log V curves are also shown as solid -1 -1 T=0.3 T=0.25 T=0.2 T=0.175 T=0.15 T=0.125 T=0.1 T=0.075 T=0.05 f=2/5,p=0.65=1.1,z=1.25,T c =0.14 I |T-T c | V | T - T c | - z / I FIG. 5: Dynamic scaling of I - V data at various temperaturesaccording to Eq. (5) for f = 2 / , p = 0 . -1 -3 -2 -1 -2 -1 -3 -2 -1 -1 -1 -4 -3 -2 Fitting V a Fitting c I-I c Fitting b V I-I c V I-I c FIG. 6: (a) IV characteristics for f = 0 , p = 0 .
86 with I c =0 . ± . β = 2 . ± .
1. (b) IV characteristics for f =0 , p = 0 .
65 with I c = 0 . ± . β = 2 . ± .
1. (c) IV characteristics for f = 2 / , p = 0 .
65 with I c = 0 . ± . β = 2 . ± . -1 -1 -4 -3 -2 -1 -4 -3 -1 -1 -4 -3 I=0.03 I c =0.034 I=0.04 I=0.05 c T V T V I=0.3 I c =0.302 I=0.31 I=0.32 V T a I=0.0375 I c =0.03875 I=0.04 I=0.0425 b FIG. 7: (a)Log T -Log V curves for f = 0 , p = 0 .
86 around I c with I c = 0 . ± . /δ = 1 . ± . T -Log V curves for f = 0 , p = 0 .
65 around I c with I c = 0 . ± . /δ = 2 . ± .
02. (c)Log T -Log V curves for f =2 / , p = 0 .
65 around I c with I c = 0 . ± . /δ = 2 . ± . lines. As for f = 0 , p = 0 .
86, the depinning exponent β is determined to be 2 . ± . I c is 0 . ± . f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65, the depinning exponents are evaluatedto be 2 . ± . . ± .
05 with the critical currents I c = 0 . ± .
001 and I c = 0 . ± . I - V traces are rounded near the zero-temperaturecritical current due to thermal fluctuations. Fisher firstsuggested to map such a phenomenon for the ferromag-net in magnetic field where the second-order phase tran-sition occurs [29]. This mapping was then extended tothe random-field Ising model [16] and the flux lines intype-II superconductors [17]. For the flux lines in type-IIsuperconductors, if the voltage is identified as the orderparameter, the current and the temperature are taken asthe inverse temperature and the field respectively, anal-ogous to the second-order phase transition in the ferro-magnet, the voltage, current and the temperature will satisfy the following scaling ansatz [17, 26] V ( T, I ) = T /δ S [(1 − I c /I ) T − /βδ ] . (8)Where S ( x ) is a scaling function. The relation V ( T, I = I c ) = S (0) T /δ can be easily derived at I = I c , by whichthe critical current I c and the critical exponent δ can bedetermined through the linear fitting of the Log T -Log V curve at I c .The Log T -Log V curves are plotted in Fig. 7(a) for f = 0 , p = 0 .
86. We can observe that the critical currentis between 0.3 and 0.32. In order to locate the criti-cal current precisely, we calculate other values of voltageat current within (0.3,0.32) with a current step 0.01 byquadratic interpolation [26]. Deviation of the T - V curvesfrom the power law is calculated as the square deviations SD = P [ V ( T ) − y ( T )] between the temperature rangewe calculated, here the functions y ( T ) = C T − C areobtained by linear fitting of the Log T -Log V curves. Thecurrent at which the SD is minimum is defined as thecritical current. The critical current is then determinedto be 0 . ± . /δ = 1 . ± .
001 from the slope of Log T -Log V curve at I c = 0 . f = 0 , p = 0 .
65 and f = 2 / , p = 0 . I c and critical exponent 1 /δ for f = 0 , p = 0 .
65 are deter-mined to be 0 . ± . . ± .
02 respectively,for f = 2 / , p = 0 .
65, the result is I c = 0 . ± . /δ = 2 . ± . β ), we get the best col-lapses of data in the regime I ≤ I c with β = 2 . ± . . ± .
02 for f = 0 , p = 0 .
86 and f = 0 , p = 0 . f = 0 , p = 0 .
86, this curve can be fitted by S ( x ) =0 . . x ), combined with the relation βδ = 1 . f = 0 , p = 0 . S ( x ) = 0 . . x ),combined with the relation βδ ≈ .
0, indicative of anArrhenius creep motion. Interestingly, as displayed inFig. 8(c) for f = 2 / , p = 0 .
65, the exponent β isfound to be 2 . ± .
02, which yields βδ ≈ .
0. Thescaling curve in the regime I ≤ I c can be fitted by S ( x ) = 0 . . x ). These two combined facts sug-gest an Arrhenius creep motion in this case.It is worthwhile to note that both the FTPT and theLTCM for strongly disordered JJA’s ( p = 0 .
65) with andwithout frustration are very similar. The I - V curvesin low temperature for all three cases can be describedby V ∝ T /δ exp[ A (1 − I c /I ) /T βδ ], this is one of themain characteristics of glass phases [17, 26], while the I - V traces for KT-type phases can be fitted to V ∝ I a .Hence, by the scaling ansatze in Eq. (8), we have pro-vided another evidence for the existence of non-KT-typephases in the low-temperature regime for these threecases ( f = 0 , p = 0 . f = 0 , p = 0 . f = 2 / , p =0 . -1.5 0.0 -3 0 -10 -5 0 y=0.0994exp(1.9x) T=0.01 T=0.02 T=0.04 T=0.06 T=0.08 T=0.1 T=0.2 T=0.3 a V T (1-I c /I)T - (1-I c /I)T - T=0.05 T=0.1 T=0.15 T=0.2 T=0.3 T=0.4 y=0.037exp(0.5x) V T (1-I c /I)T - b y=0.105exp(0.25x) c V T T=0.075 T=0.1 T=0.125 T=0.15 T=0.175 T=0.2 T=0.25 T=0.3
FIG. 8: (a) Scaling plot for f = 0 , p = 0 .
86 with I c = 0 . /δ = 1 .
688 and βδ = 1 .
55. (b) Scaling plot for f = 0 , p =0 .
65 with I c = 0 . /δ = 2 .
24 and βδ ≈ .
0. (c)Scalingplot for f = 2 / , p = 0 .
65 with I c = 0 . /δ = 2 .
29 and βδ ≈ . IV. SUMMARY
To explore the properties of various critical phenom-ena in site-diluted JJA’s, we performed large-scale simu- lations for two typical percolative strengths p = 0 .
86 and p = 0 .
65 as in a recent experimental work [9]. We in-vestigated the FTPT, the ZTDT and the LTCM in thesesystems. The RSJ dynamics was applied in our work,from which we measured the I - V characteristics at dif-ferent temperatures.The results obtained in this work about FTPT arewell consistent with the recent experimental findings inRef. [9] and are inconsistent with the earlier experimen-tal study in Ref. [1], possibly due to the large noise in themeasurement of voltage in Ref. [1] (larger than 0.2nv),which was considerably reduced in the experiments byYun et al [9]. The evidence for non-KT-type phase tran-sition was revealed by two different scaling ansatzes (Eq.(5) and Eq. (8)). Our results also shed some light on thevarious phases and the phase transitions where the differ-ent divergent correlations at various disorder strengthswere suggested, and the critical exponents were evalu-ated. Furthermore, the results in this paper are usefulfor understanding not only the site-diluted systems, butalso the whole class of disordered JJA’s, for instance, thecombination of two different phase transitions may existin other disordered JJA’s systems.In addition, the ZTDT and the LTCM were alsotouched. It was demonstrated by the scaling analysisthat the creep law for f = 0 , p = 0 .
86 is non-Arrheniustype while those for f = 0 , p = 0 .
65 and f = 2 / , p = 0 . f = 0 , p = 0 . .
55 is also very close to 3 / f = 0 , p = 0 .
65 and f = 2 / , p = 0 .
65, the ob-served Arrhenius type creep law is also similar to that inthe glass states of flux lines with a strong collective pin-ning as in Ref. [17]. Future experimental work is neededto clarify this observation.
V. ACKNOWLEDGEMENTS
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