A numerical study of planar arrays of correlated spin islands
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A numerical study of planar arrays of correlated spin islands
I. Maccari , A. Maiorano , E. Marinari , and J. J. Ruiz-Lorenzo Dipartimento di Fisica, Sapienza Universit`a di Roma, P. A. Moro 2, 00185 Roma, Italy Instituto de Biocomputaci´on y F´ısica de Sistemas Complejos (BIFI) 50018 Zaragoza, Spain Dipartimento di Fisica, IPCF-CNR and INFN, Sapienza Universit`a di Roma, P. A. Moro 2, 00185 Roma, Italy Departamento de F´ısica and Instituto de Computaci´on Cient´ıfica Avanzada (ICCAEx), Universidad de Extremadura, 06071Badajoz, SpainJuly 9, 2018
Abstract.
We analyze a system of interacting islands of XY spins on a triangular lattice. This model hasbeen introduced a few years ago by Eley et al. to account for the phenomenology in experiments on tun-able arrays of proximity coupled long superconductor-normal metal-superconductor junctions. The mainfeatures of the model are the separation of a local and a global interaction energy scale and the mesoscopiccharacter of the spin islands. Upon lowering the temperature the model undergoes two crossovers corre-sponding to an increasing phase coherence on a single island and to the onset of global coherence acrossthe array; the latter is a thermodynamical phase transition in the Ising universality class. The dependenceof the second transition on the island edge-to-edge spacing is related to the proximity-effect of the couplingconstant.
Recently Eley et al. [1] have introduced a model of cou-pled islands of XY spins. Their goal was explaining theresults of measurements of resistance in arrays of longsuperconducting-normal-superconducting junctions. Theexperimental devices are based on planar arrays of iden-tical islands made of superconducting (Nb) grains, dis-posed in a triangular matrix over a metal (Au) film. Theauthors studied the dependence of the system resistance R ( T ; h, ℓ ) on temperature T , on island (vertical) thick-ness h and on inter-island spacing ℓ . They found i) theresistance dropping to zero, by lowering T , in two steps,and they determined two transition temperatures T and T with T > T ; ii) an interesting dependence of both T and T on the island spacing, possibly (both) extrap-olating to T = 0 at large ℓ ; iii) a strong dependence ofthe behavior of the system on the island thickness. In afollowing paper [2], they discussed a more detailed com-parison between the experimental data and the predictionsabout the dependence of T on ℓ given by the conventionaltheory of Lobb, Abraham, and Tinkham (LAT) [3]. Theyargued that for large inter-island spacing the supercon-ducting transition is more likely to be driven by diffusiveeffects [4,5] in the normal metal substrate, and that itdoes not depend on the details of the superconducting is-lands, with the puzzling dependence on island height as anotable exception.The superconducting transition in proximity-coupled macroscopic grains embedded in normal metal films hasbeen the object of intensive work in the past years, [6,7, 8]. Tunable realizations of 2 D superconductivity were alsoobject of previous experiments [9]. The classical modelpresented in Ref. [1] to account for a novel phenomenol-ogy is at difference with previous theoretical and experi-mental work, as it takes into account the intrinsic fluctua-tions of the superconducting state inside the single meso-scopic islands (see also refs. [10,11] for recent theoreticaland experimental work on mesoscopic Sn islands laid ongraphene). It is clear that, because of many reasons wewill discuss in the following, this model does not try to re-produce faithfully the experimental situation (for examplethe use of an anisotropic coupling is not connected to thephysical form of the Josephson interaction but is a toolneeded to obtain a phase transition). The idea of [1], andour point of view here, is to analyze a very simple modelthat offers a behavior quite similar to the one detectedin the experiments, and to try to learn from this behav-ior. Here we will present a detailed analysis of the model,that corroborates and supplements the hints coming fromthe first analysis of [1]. It is also worth mentioning thattunable two-dimensional superconductors are also of inter-est in a revived search for a non-conventional 2 D , T = 0metallic phase. [12,13,14,15,16,17,18,19] The Hamiltonian of the model is based on O (2) vectorsliving on the individual grains (labeled with i, j ). Groups I. Maccari et al.: A numerical study of planar arrays of correlated spin islands of grains form islands (labeled by p ): H = − J X p X h ij i∈ p S i · S j − X h p,p ′ i M p · J ′ M p ′ , (1) M p ≡ X i ∈ p S i , (2)where by a dot we denote the scalar product in the internalspace and where J ′ is a 2 × D -dimensional hyper-cubic array of grains oflinear size I (and volume V I = I D , with either D = 1 or D = 2 in our computations). Islands are arranged on a(two dimensional) planar regular lattice of linear size L .Islands are mesoscopic: their linear size I is not larger thana few grains. Because of that they may have large globalphase fluctuations. The size of the underlying planar arrayis macroscopic, L ≫ I . The case of one-dimensional islandis an exercise useful to understand better the role of islanddimensionality, and does not try to be a description of theexperimental situation. On the contrary the case of twodimensional islands is probably closer to the experimentalsituation, where islands have many layers, but only one orfew conjure to build the inter-island interaction.The first term of the Hamiltonian is a sum of nearest-neighbor interactions between grains contained in the sameisland. The second term couples neighboring islands in thearray. Each spin in a given island interacts directly with itsneighboring spins in the same island and with the averagespin field of surrounding islands. In the model proposedin Ref. [1] the inter-island coupling matrix (in the internal O (2) space) J ′ is anisotropic: J ′ = (cid:18) J ′
00 0 (cid:19) . (3)This particular choice polarizes the islands in one spe-cific direction in internal vector space, changing the na-ture of the inter-island phase transition. This is a techni-cally useful choice (since it carries a phase transition inthe game), but it does not aim at reproducing the de-tails of the physical Josephson interaction. Finally, noticethat in the isotropic case and if the energy scales J and J ′ are far apart, i.e. J ′ ≪ J , so that at low tempera-tures all (mesoscopic) islands are magnetized, we recovera Kosterlitz-Thouless [20] phase transition.The island-island couplings depend on the temperatureand on the inter-island edge-to-edge spacing, according tothe theory of diffusion of electron pairs in SC-Normal-SCjunction. As in the work of [1] we take a “quasi- proximity-effect ” [1,3] form for both couplings; in a proximity in-teraction J ′ would depend on the inverse square of islandspacing when the latter is small, but following [1] andfor the same sake of simplicity we omit this part of theinteraction, that is not expected to change the nature ofthe phase transitions here. We assume the proximity-effectform also for the grain-grain coupling J and we take the grain-grain distance as the length (lattice) unit (and de-note the inter-island spacing as ℓ ). J = J exp (cid:16) −√ T (cid:17) , (4) J ′ = J ′ exp (cid:16) − ℓ √ T (cid:17) . (5)The choice of an interaction of a proximity-like form im-plies that physically grains of the islands are also im-mersed into a metallic matrix. The authors of Ref. [1] in-troduced the model defined in (1) to explain the presenceof two transitions (intra-island, T , and inter-island co-herence, T ) and the depression of T for increasing islandspacing ℓ . Such a dependence of T on ℓ has been observedand reported for the first time in [1] (for example in previ-ous experiments on lead disks on a thin substrate [9] whereislands were not mesoscopic, the effect was not observed).The energy scales J and J ′ must be well-separated: weadjust them in order to clearly split the high- T and thelow- T transition. We fix J = 1 and vary J ′ ; in order toeasily compare data for different island sizes, we also take J ′ = j ′ /V I and adjust the parameter j ′ .In Ref. [1] the authors also give some predictions byanalyzing a D = 1 islands model, where it turns out that: – T ≤ T provided J ≫ J ′ and islands are small; – T → J ′ → T ∼
0, and, aswe will see in the following, for large I the intra-islandcoherence is driven by inter-island ordering (also see thediscussion in the Conclusions section).Another striking aspect of the phenomenology of thesystem[1,2] is the dependence of its behavior from theheight of columnar grains. Realistic islands extend in morethan one dimension. We have analyzed by numerical sim-ulations the behavior of one and two-dimensional islands.The dependence on thickness may suggest that it wouldbe interesting to go to D = 3, too (in case of a very largevalue of I this should turn the T transition to a truesecond-order one). If energy scales are adequately sepa-rated (i.e. J ≫ J ′ ), this should not change the propertiesof the T (KT) transition, when phases of grains in thesame island are mutually locked. Mesoscopic islands canthen have a crossover at T from a disordered to an or-dered phase; for D ≥ I this crossover is relatedto a true thermodynamic transition. We have studied the model defined in (1). By following avery slow annealing protocol, with constant ratios between . Maccari et al.: A numerical study of planar arrays of correlated spin islands 3 adjacent temperature (a logarithmic annealing scale), wehave cooled down the system in order to get a signal for thetwo transitions. At each temperature we collected mea-surements during the evolution of the Monte Carlo dy-namics. Our Monte Carlo step consists of n m sweeps of thewhole lattice by single-spin moves Metropolis dynamics,followed by n o sweeps by over-relaxation [22]. The choiceof n m = 10 and n o = 12 have shown to be appropriatefor most island and array sizes considered (and an overkillfor the smaller sizes), and allowed an estimate of inte-grated auto-correlation times not larger than ten MonteCarlo steps at most temperatures. Although averages al-ways stabilize quickly after any temperature change, wedrop the first half of the collected measurements at all T values. The simulated annealing protocol, together withover-relaxation, is appropriate to the needs of this prob-lem. All observables of interest converge very fast to aplateau at all temperatures. Although averages always sta-bilize quickly after any temperature change, we drop thefirst half of the collected measurements at all T values.Since the devices in the experimental setup [1,2] aretriangular arrays, we consider a triangular lattice. A sim- χ I χ R I =144 T ℓ =2 ℓ =4 ℓ =8 ℓ =120.11100.1110100 0.1 1 χ I χ R I =36 T ℓ =2 ℓ =4 ℓ =8 ℓ =12 Fig. 1. D = 1 results for the largest simulated size L = 32. Weplot the connected susceptibilities χ I and χ R for two differentvalues of I . j ′ = 0 . ℓ . Notice that the peaksof χ I (which mark the T transition) depend strongly on ℓ . ple implementation choice in simulation is to consider atriangular array with regular hexagonal shape with heli-cal boundary condition (in this way we preserve the sym-metries of the triangular array and avoid involved bulkproperties extrapolations); each side of the hexagon has awidth of L islands, and the number of islands is V S =3 L ( L −
1) + 1. We simulated systems with L = 8 , D = 1 systems we have islands of sizes I =16 , , ,
100 and 144, while for D = 2 we have I =6 , ,
10 and 12. The inter-island edge-to-edge spacing ℓ has been varied in the set { , , , } for D = 1 and { , , , , , } for D = 2 (the SNS arrays in Ref. [1]had edge-to-edge spacings up to approximately 10 timesthe grain size in their experiments, and ℓ up to 20 inRef. [2]). We have considered both free (FBC) and pe-riodic (PBC) boundary conditions on the single islands.Although we found no qualitative differences, FBC is amore realistic choice when dealing with mesoscopic ob-jects, for which we expect finite-size effects to play a role.We measured the following quantities. – The single island magnetization magnitude (averagedover islands): M I = 1 V S X p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V I X i ∈ p S i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (6)This should be, in the limit of infinitely extended is-lands, a good order parameter for island internal order-ing (in any direction in internal spin space, and glob-ally over the array: it has a non-zero value wheneverany island starts to order internally and it is maximumwhen all islands are locally ordered, independently ofthe relative orientation between different islands). – The total magnetization: M = 1 V S V I X p X i ∈ p S i . (7) – A renormalized magnetization: µ p = P i ∈ p S i (cid:12)(cid:12)(cid:12)P i ∈ p S i (cid:12)(cid:12)(cid:12) , (8)which is a unit vector on the single island, and itsaverage over the array. – M R , which characterizes the globally-ordered phase,even if islands are not yet internally fully ordered: M R = 1 V S X p µ p . (9)We also consider the fluctuations of the magnetizations χ ≡ V I V S h(cid:10) M (cid:11) − h| M |i i , (10) χ I ≡ V I h(cid:10) M I (cid:11) − h M I i i , (11) χ R ≡ V S h(cid:10) M R (cid:11) − h M R i i , (12) I. Maccari et al.: A numerical study of planar arrays of correlated spin islands where (cid:10) M I (cid:11) and h M I i are averaged over all islands. χ isthe total susceptibility of the system. At very low temper-atures, when M I ∼
1, we have χ R ∼ χ/V I . At T , thatwe define as the location of the peak of the inter-islandsusceptibility χ R , χ R and χ have very similar sharp peaks(both in shape and location). We take the location of the(very smooth) maximum of χ I as a rough estimate of thetemperature T at which islands order internally (in thisway we give an operative definition of T in our model:since islands are of finite extent the T defined in this wayis indeed a crossover temperature). In Fig. 1 we report the results for arrays of one-dimensionalchains. The T temperature value goes to zero very fast asthe island size grows, as expected for linear spin chains.Upon lowering T , coherence between island builds up andalso drives the internal ordering; the two transitions canbe resolved only for very small island sizes and by loweringconsiderably the value of coupling constant j ′ o . The effectis also strongly dependent on island size.The situation is far clearer for two-dimensional islands(see Fig. 2), where we still have a finite temperature ther-modynamic transition for isolated islands in the limit oflarge sizes. For mesoscopic islands, the crossover betweenunordered and ordered island depends more weakly on is-land size than in the linear chains case. Our numerical sim-ulations show that the temperature T does not depend onthe island spacing, or the dependence is very weak. Thiseffect has been also reported in experimental results onnon-mesoscopic island samples [9]. Also the dependenceof T on island size is very weak.We try a more quantitative approach studying the de-pression of T by increasing the inter-island spacing. Wetake as an estimate for T the midpoint of the temperatureinterval bracketing the peak at its half-height. The depen-dence of T on ℓ and I for the largest simulated array size L = 32 is shown in Fig. 3.Following Ref. [1], we notice that T ( ℓ ) compares wellto a proximity-effect prediction T = ∆ exp( − Cℓ p T ) , (13)corresponding to the solid straight line in Fig. 3, suggest-ing a diverging ℓ ( T = 0); we report in Table 1 our bestfit estimates of the parameters ∆ and C .We have run more accurate numerical simulations inthe temperature region close to the T transitions, with afour times smaller cooling rate and ten times more mea-surements. We measured the Binder cumulant G = 12 − D(cid:0) M (cid:1) E h M i . (14)The value of G at the T transition point is universal [22];we report data for I = 6, ℓ = 4 and various array sizes L in Fig. 4. Note that the dips in the curves of G for the I =12 T ℓ =2 ℓ =4 ℓ =8 ℓ =12 ℓ =16 ℓ =240.11100.1 0.1 1 I =10 T ℓ =2 ℓ =4 ℓ =8 ℓ =12 ℓ =16 ℓ =240.11100.1 0.1 1 I =6 T ℓ =2 ℓ =4 ℓ =8 ℓ =12 ℓ =16 ℓ =24 Fig. 2. D = 2 results for the largest simulated size L = 32.We plot the connected susceptibilities χ I (filled symbols) and χ R (empty symbols) for three different pairs of the parame-ters I and j ′ . On each panel we plot the curves for six valuesof the island spacing l . The peaks of χ I (which mark the T transition) depend only weakly on ℓ . In all panels j ′ = 0 . largest system sizes in Fig. 4 are due to the breakdown ofthe O (2) internal symmetry introduced by the anisotropicform of J ′ in Eq. (3). This is the behavior one would ex-pect, since at high temperature the G value depends onthe symmetry of the system. The fluctuations of the mag-netization in the infinite volume limit are Gaussian dis- . Maccari et al.: A numerical study of planar arrays of correlated spin islands 5 T ∆ exp(- C ℓ T ) I =6 I =8 I =10 I =12 Fig. 3.
A scaling plot of the T transition temperature forarrays of D=2 island based on the behavior ∆ exp( − CℓT / )as suggested by Eq. 5 ( L = 32 data). tributed and the Binder parameter for the two-componentmagnetization of a XY system should approach the value G ( T → ∞ ) = 0 .
5, whereas for Ising spins the correspond-ing high-temperature value is G ( T → ∞ ) = 0. When longrange order in the system builds up at low temperatures,the value of the Binder parameter must approach unity: G ( T →
0) = 1. The data in Fig. 4 show that G is notmonotonically increasing when the temperature decreases:it starts at a value around 0 . T , the value of G drops to low values,as expected for an Ising system. The minimum of the dipdecreases as the system size L increases.Moreover, the critical value of the Binder cumulant(which is universal) for the two-dimensional Ising model isknown to great accuracy [23]. The values G = 0 . T ∼ .
335 (for comparison,from the position and width at half-height of the peak ofthe susceptibility for the same simulated system we ob-tain T ≃ . ± . I ∆ C
Table 1.
Best fit estimates of ∆ , C parameters in Eq. 5 from T data for various island size I ( D = 2, L = 32, ℓ = 8 , , , ℓ = 2 , . . .
11 and 0 .
32. Uncertainties are gnuplot estimates cor-rected as in [21]. numerical evidence for a second order phase transition inthe two-dimensional Ising universality class. The asymp-totic value of the crossing points of the Binder cumulantscurves (see inset of Fig. 5), T c ( L, L ) which asymptoti-cally tends to T , is clearly different from zero. We finallyremark that the value of the Binder cumulant below thecritical temperature is almost unity as expected asymp-totically (as the size of the system goes to infinity). ℓ =4, I =6 G T L =2 L =4 L =8 L =16 L =32 Fig. 4.
Binder cumulant G versus T for D = 2-islands with I = 6 and ℓ = 4 and for the five simulated sizes ( L ). In the insetwe show in detail the region near T ∼ . It has been very difficult to resolve the crossover at T (internal island ordering) and the T transition (inter-island ordering) in the case of one-dimensional islands.In D = 1 and in the limit of large islands sizes we ex-pect T → T . The island inter-nal magnetization remains small and no clear maximumof the susceptibilities signals a crossover down to T . At T the inter-island interaction couples the fluctuations ofthe magnetizations of neighboring islands, making themcoherent: at this point spins inside each islands starts toalign to the average field of neighboring islands. The meso-scopic character of the islands is crucial: in the limit oflarge islands we expect the fluctuations of local magneti-zation to be too small (we did not try an experiment inthat direction). Then, although J ′ ≪ J , it is the T tran-sition that drives both inter-island and internal ordering.As the inter-island spacing grows the depression of T im-plies the depression of T , too. This is compatible with thecounterintuitive requisite that T → J ′ → I. Maccari et al.: A numerical study of planar arrays of correlated spin islands G ( T c ( L , L ) ) L -7/4 T ( L ,2 L ) L -11/4 Fig. 5.
Binder cumulant G ( T c ( L, L )), for D = 2-islandswith I = 6 and ℓ = 4, computed at the crossing pointof the Binder curves of lattice sizes L and 2 L , denoted as T c ( L, L ), as a function of L − / . We have marked, with agreen horizontal line, the G value for the two dimensionalIsing model: G = 0 . T c ( L, L ) ofthe Binder curves as a function of L − / . We have assumedthe scaling of the two dimensional Ising model: T c ( L, L ) − T ∼ L − ∆ − /ν and G ( T c ( L, L )) − G ( T ) ∼ L − ∆ . We haveused the exact value ν = 1 and the conjectured one for thecorrection-to-scaling exponent ∆ = 7 /
4. For more details, seeRef. [23]. grains, the 1 D system is not really connected to the exper-imental setup we analyze here. The 2 D island system is,on the contrary, closer to the physical system we want tounderstand, and resolving the two transitions is easier inthe case of the D = 2 system. The T transition for D = 2macroscopic islands and for J ′ ≪ J is expected to be of theKosterlitz-Thouless type. Since we consider mesoscopic,finite islands, we observe, as expected, a “long-range” or-dering of the islands. We expect a spin ordering crossoverat a temperature value T which does not go to zero withthe island size as fast as in the D = 1 case. The modelwith planar islands is capable of describing the depressionof the global superconductivity transition down to verylow temperature at large lattice spacing. The transitiontemperature we measure by locating the peaks of responsefunctions remains finite for moderate-to-large inter-islandedge-to-edge spacing. T approaches a zero value only forvery large inter-island spacing. We cannot detect any de-pendence of the T transition temperature on ℓ . At small ℓ values, the value of T is influenced by nearby islands onlybecause of global coherence driving internal ordering. Wedid not try a full finite-size scaling analysis of the T tran-sition, which for well-separated coupling scales j and j ′ is in the Ising universality class. On the simulated lengthscales there are no relevant effects of the mesoscopic char-acter of the islands on the behavior of T , which behavesat large inter-island spacings as expected by the choice ofthe dependence of the coupling constants on ℓ and T . Appropriate variations of the basic model we have dis-cussed here could lead to interesting developments in thestudy of the superconducting transition in arrays of SNSjunctions. We think it is an interesting starting point tounderstand many striking experimental evidences, as forinstance the dependence of the transition temperatures onisland thickness, or the strong depression of the T and T transitions. We thank Kay Kirkpatrick for introducing us to the modelstudied in this work and her and Jack Weinstein for interestingdiscussions. This work was partially supported by EuropeanUnion through Grant No. PIRSES-GA-2011-295302, and ERCGrant No. 247328, by the Ministerio de Ciencia y Tecnolog´ıa(Spain) through Grant No. FIS2013-42840-P, and by the Juntade Extremadura (Spain) through Grant No. GRU10158 (par-tially founded by FEDER).
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