Phase transitions in a two-dimensional vortex lattice with defects: Monte Carlo simulation

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1 Physica C Phase transitions in a two-dimensional vortex lattice with defects: Monte Carlo simulation Igor A. Rudnev ), Vladimir A. Kashurnikov, Mariya E. Gracheva, Oksana A. Nikitenko Deartment of SuerconductiÕity and Physics of Nanostructures, Moscow State Engineering Physics Institute Technical UniÕersity, 31 Kashirskoe Sh., Moscow, Russia Abstract Phase diagrams and hase transitions in a two-dimensional vortex lattice with defects have been investigated by Monte Carlo Ž MC. simulation techniques. It was found that in the resence of defects, the melting of the vortex lattice roceeds in two stages. First, the ideal triangular lattice transforms at low temerature into islands, which are inned to the defects and rotate around them Ž rotating lattice hase.. Then, at a higher temerature, the boundaries of the vortex lattice islands become smeared and the system transforms into a vortex liquid. The deendencies of hase transition temeratures on inning otential have been obtained. The visual ictures of the flux lattice structures have been calculated. The current voltage characteristics Ž IVC. of the vortex structures with defects in forms of oints, lines and squares were simulated at different temeratures and defects concentration. We have found that the oint defects are more efficient than the other tye of defects with equivalent secific concentration. It was shown that the deendencies of critical current on temerature and defects concentrations are directly associated with the hase state of the vortex system. q 2000 Elsevier Science B.V. All rights reserved. PACS: Bz; Bj; Dm Keywords: Vortex; Numerical simulation; Critical current; IV characteristics 1. Introduction In recent years, considerable attention is aid to hase transformations and their dynamics in vortex lattice of HTSC. Wide alication of numerical methods Žesecially the Monte Carlo Ž MC. method, see the review in Ref. wx. 1 has made it ossible to simulate the hase states and hase transitions in various vortex systems, and to demonstrate the eculiarities of the vortex lattice melting dynamics in the ) Corresonding author. Fax: q address: rudnev@suercon.mehi.ru I.A. Rudnev. resence of the inning centers w2,3 x. The MC method was used also for simulation of I V characteristics Ž IVC. of model suerconductors w4 8 x. IVC were calculated in the resence of a large number of defects with different otential energies Žrelative to the number of vortices. wx 5. The results on various hase modes of current flow, obtained recently from an analysis of IVC of HTSC, deserve secial attention Žsee Ref. wx 8 as well as Refs. w9 11 x.. The modes of fixed vortex glass, lastic flow of vortex liquid, and moving vortex glass were observed. These hase states of the Abrikosov vortex, as well as transitions between them, are close to the hase r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S X

2 384 I.A. RudneÕ et al.rphysica C transitions between the states of rotating lattice and vortex liquid w2,3 x. IVC of layered suerconductors have been calculated at different temerature and defects concentration wx 4. In this aer, we reort the icture of hase transformation of two-dimensional vortex lattice and results of calculations of IVC and critical current of layered suerconductors. We shall demonstrate the influences of temerature; tye of defects and defects concentration on IVC are associated with the hase states of vortex lattice. 2. Model and comutational method Let us consider a two-dimensional vortex lattice simulating a suerconducting HTSC layer on a eriodic rectangular grid under the assumtion of weak couling between filaments in a direction erendicular to the ab-lane and in the resence of inning centers. The effective Hamiltonian of such a system has the form w12x Žwithout the account of interaction between vortices and external field B.: N Ý Hs0.5 HŽ r,r. n n q U Ž r. n, Ž 1. i/j N Ý i j i j i i is1 < < 0 0Ž i j. 2 2 HŽ r,r. s f dr2l Ž T. m K < r yr < rlž T. i j i j su T K r yr rl T. 2 Here U Ž r. i is the energy of interaction between a vortex and a defect at the ith lattice site, ni the occuation numbers of vortices Ž 0 or 1. at the ith site of the satial grid with the total number of nodes N, f0 the magnetic flux quantum, K 0 Bessel s function of the imaginary argument, d the suerconducting layer thickness, lž T. s lž 0. rž1 y Ž TrT. n. c 0.5 the deth of magnetic field enetration Ž q we used ns3.3 from data of m SR measurement w x. y7 13, and m0 s 4 = 10 Hrm. The energy of interaction of a vortex with a inning center was chosen in the form U Ž T. syu Ž T. 0. We have used for simulation the arameters of HTSC Bi Sr CaCu O w13 x: ds0.27 nm; lž s180 nm; Tc s84 K and external magnetic field Bs0.1 T. We have analysed various tyes of defects, but main calculations have been made for oint defects occuying a single cell of the satial grid. The range of two-dimensional concentrations of defects was y2 from 10 to 3=10 m Žfrom 1 to 100 defects corresonded to 150 vortices in the system under investigation.. Dynamic rocesses have been investigated by an external transort current alied to the system. In this case, the Hamiltonian of the system was sulemented with the added term resonsible for the action of the Lorentz force on each vortex. In the case of elementary motion of a vortex, the term dusf0 JDx was subtracted from the total energy if the direction of vortex movement coincided with the direction of the Lorentz force exerted to it and added if the vortex moved against the Lorentz force. Critical current was defined from IVC as a current at fixed voltage Ž as in the exeriment.. The calculations were carried out on a square grid under eriodic boundary conditions with the hel of the classical MC by using the Metroolis algorithm. Details of numerical simulation of vortex lattice and IVC were described elsewhere w2 4 x. 3. Results and discussion 3.1. Phases states and hase transitions Fig. 1 dislays the vortex density distribution obtained by summing the instantaneous states of the vortex lattice Ž instantaneous vortex density. every 100 MC stes. Ordinarily, 10 4 MC stes were required to thermalize the system and 10 4 MC stes were made for calculation. A ractically ideal triangular lattice is reroduced at Ts1 K Ž Fig. 1a.. Not all defects are occuied by vortices. This fact testifies for the benefit of the stiffness of the vortex lattice at such temeratures. Defects would have been occuied by vortices if the corresondence between the arrangement of defect and the centers of the triangular lattice were ideal. At a temerature of 3 K, all defects are occuied by vortices, which rigidly hold the lattice around themselves. On account of the irregularity of their arrangement, the defects seemingly ull the lattice aart, breaking it u at locations that are Ž still. far away from defects. The boundaries between the coherent regions are melted and the lattice loses stiffness. As the temera-

3 I.A. RudneÕ et al.rphysica C Fig. 1. Dynamics of melting of the vortex lattice with five-oint defects ŽU Ž Ts2 K. sy3.52 mev. marked as circles: Ž a. Ts1 K; Ž b. Ts5 K; Ž. c Ts35 K. ture increases further u to 5 K, Fig. 1b, islands of the triangular lattice, which are ket around the defects and which move relative to the defects as an axis of rotation, seemingly rotate, smear the vortex density in concentric circles with maximum on coordination sheres exactly. A vortex-deleted region forms around the defects themselves at a distance of one coordination shere. A coordination shere is equal to the eriod of the ideal triangular lattice since a stationary inned vortex revents other vortices from aroaching closer. In this new hase Žthe transition temerature is estimated to be T f 3K,. m

4 386 I.A. RudneÕ et al.rphysica C The alternative method is to calculate the secific heat CT of the system as Ž k s1.: <² 2 : ² : 2 < 2 CŽ T. s E y E rt. Our CT calculations show two clear eaks in the function CT, which corresond to hase transitions: triangular lattice rotating lattice vortex liquid. The deendency of temerature rotating lattice vortex liquid on defect otential is resented in Fig Current Õoltage characteristics and critical current B Fig. 2. The deendency of melting temerature on otential of defects. Inset: CŽ T. deendencies for U s1 mev Ž circles. and U s100 mev Ž triangles.. which we shall call a rotating lattice, a long-range order still is resent within the coherent regions, which are much larger than the average distance between the vortices and are rigidly couled to the inning centers. As the temerature increases further, the vortices begin to detach from defects, coherent regions break u, and at Ts35 K, for examle, a comletely melted vortex liquid is observed ŽFig. 1c.. Therefore, the rocess of melting of the lattice in the resence of inning centers roceeds in the hases: triangular lattice rotating lattice vortex liquid. As the inning otential increases, the temeratures of both hase transitions shift: The temerature of the oint triangular lattice rotating lattice decreases slightly and the temerature of hase transition rotating lattice vortex liquid increases substantially. So strong inning increases the temerature range of the rotating hase. For a quantitative investigation of the hase oints on melting of vortex lattice, either a structure factor Sq, for estimating the degree of a long-range order, or a hexagonal arameter S w14,15 x 6, for analyzing the short-range correlations, is ordinarily calculated. We have calculated IVC for systems containing Nd s1%100 defects at temeratures 10, 20, 30, 40 K, etc. Ž u to ;83 K.. In the vicinity of the critical temerature, the characteristics were calculated with an interval of ;1 K u to the critical region. In Fig. 3, the tyical IVC are resented at various temerature ŽN s 100, U Ž Ts 2K. sy100 mev. d. The actual scale of electric field strength, E s5=10 y2 Vrm, has been determined by comarison of theoretical with exerimental IVC of Bi 2Sr2CaCu 2O8 film w16,17 x. Our results of calculation of the motion of a vortex system in the field of defects in the resence of current demonstrate different modes of IVC be- Fig. 3. IVC at various temeratures and Nd s100. Scaling factors are J s5=10 10 Arm 2 and E s5=10 y2 Vrm.

5 I.A. RudneÕ et al.rphysica C haviour, deending on the hase state of the system. Fig. 3 shows two grous of curves conditionally divided by the temerature boundary Ž T ( 70 K. m, the temerature of transition from a rotating lattice to a vortex liquid. A visual analysis of the density of vortex distribution shows that a rotating lattice is observed at T-Tm and a vortex glass at T)T m.in the vortex lattice hase, IVC change slowly uon heating, which can be exlained by a still strong interaction with inning centers. On the contrary, at T)T m2, we observe a strong influence of temera- ture Ža virtually equidistant increase in voltage uon an increase in temerature by only 28. u to the critical region. Thus, the observed difference in the temerature behaviour of IVC of a real HTSC can be attributed to different states of the vortex system. We have simulated also IVC for defects in the forms of line and square and comare IVC of vortex systems with different defects. For this urose, we introduce the defects to the system by three different ways: Ž. a 90-oint randomly distributed defects; Ž. b 10 clusters er nine-oint defect in the form of square; c. 10 clusters er nine-oint defect in the form of line. So, in all cases, there are the exactly equal secific concentration of oint defects. Then, we calculate IVC at different temeratures for all tyes of defects and defined the temerature deendencies of critical current. The results of these calculations are resented in Fig. 4. We can see that at low and high temeratures, there are small differ- Fig. 4. J Ž T. c deendencies for different tyes of defects. Squares 90-oint defects, oen circles 10-line defects, closed circles 10-square defects. Fig. 5. J Ž T. c deendencies for different numbers of oint defects and temerature deendency of value Ž J y J. r J. c100 c20 c20 ences between current and voltage characteristics. At intermediate temeratures, we have found that oint defects are more efficient than other tyes of defects with equivalent density. We have calculated the deendencies of critical current on concentration of oint defects at different temeratures and found that the increase of concentration of oint defects results in an increased critical current. These deendencies are resented in Fig. 5 for Nd s20, 40, 60, 100. The most interesting result is that the degree of increase deends on temerature. Fig. 5 shows both J Ž T. c deendencies for Nd s20, 40, 60, 100 and the deendence of value Ž J y J. rj on temerature Ž c100 c20 c20 here, Jc100 and J c20 critical currents for Nd s100 and 20, subse- quently.. It is seen that the Jc increase has a maxi- mum in vicinity 40 K. This result can be understood qualitatively according to the hase state of vortex system. At low temerature, the vortex lattice is cruel and increase of defects concentration does not result in additional inning. At high temerature, when the vortex system is liquid, additional defect cannot in all vortex. But, at intermediate temerature where there are the islands of rotating lattice, additional defects increase the number of islands that result in the stabilisation of the vortex system. The observed result indicates that the increase of critical current by means of irradiation should de-

6 388 I.A. RudneÕ et al.rphysica C end on temerature. It may be tested by irradiation exeriments on erfect Bi-tye single crystal. The resembling exerimental data have been observed, for examle, in Refs. w18,19 x. 4. Conclusions We have resented the results of numerical simulation hase states and dynamic of a two-dimensional vortex lattice with defects. It was found that there are three main vortex states: triangular lattice rotating lattice vortex liquid. The I V characteristics Ž IVC. have been calculated over a wide range of temerature for different tyes and concentrations of defects. It was shown that the hase states of vortex lattice determine the dynamic characteristics of system such as IVC and critical current. Acknowledgements We thank Yu. Lozovik for useful discussions. This work was suorted in art by the Russian State Program Actual roblems in condensed-state hysics, subrogram Suerconductivity under Grant No and Federal Program Integration. References wx 1 M.E. Gracheva, M.V. Katargin, V.A. Kashurnikov, I.A. Rudnev, Fiz. Nizk. Tem. 23 Ž wlow Tem. Phys. 23 Ž x. wx 2 M.E. Gracheva, V.A. Kashurnikov, I.A. Rudnev, JETP Lett. 66 Ž wx 3 M.E. Gracheva, V.A. Kashurnikov, I.A. Rudnev, Phys. Low-Dimens. Struct. 8r9 Ž wx 4 M.E. Gracheva, V.A. Kashurnikov, I.A. Rudnev, Fiz. Nizk. Tem. 25 Ž wlow Tem. Phys. 25 Ž x. wx 5 E. Bonabeau, P. Lederer, Phys. Rev. Lett. 77 Ž wx 6 R. Sugano, T. Onogi, Y. Murayama, Physica C 263 Ž wx 7 K. Moon, R.T. Scalettar, G.T. Zimanyi, Phys. Rev. Lett. 77 Ž wx 8 S. Ryu, M. Hellerqvist, S. Doniach, A. Kaitulnik, D. Stroud, Phys. Rev. Lett. 77 Ž wx 9 A.N. Lykov, C. Attanasio, L. Maritato, S.L. Prischea, Suercond. Sci. Technol. 10 Ž w10x B. Khaykovich, M. Konczykowski, E. Zeldov, R.A. Doyle, D. Majer, P.H. Kes, T.W. Li, Phys. Rev. B 56 Ž R517. w11x C. Gouil, A. Ruyter, V. Hardy, Ch. Simons, Physica C 278 Ž w12x V.M. Vinokur, M.V. Feigel man, V.B. Geshkenbein, A.I. Larkin, Phys. Rev. Lett. 65 Ž w13x S.L. Lee, P. Zimmermann, H. Keller, M. Warden, I.M. Savic, R. Schauwecker, D. Zech, R. Cubitt, E.M. Forgan, P.H. Kes, T.W. Li, A.A. Menovsky, Z. Tarnawski, Phys. Rev. Lett. 71 Ž w14x S. Ruy, S. Doniach, G. Deutscher, A. Kaitulnik, Phys. Rev. Lett. 68 Ž w15x E. Zeldov, D. Majer, M. Konczykowski et al., Nature 375 Ž w16x M.E. Gracheva, V.A. Kashurnikov, I.A Rudnev, Phys. Low- Dimens. Struct. 9r10 Ž w17x V.F. Elesin, I.A. Esin, I.A. Rudnev, B.L. Krivts, E.P. Limitovskii, Sverkhrovodimost: Fiz., Khim., Tekh. 6 Ž w18x M.J. Kramer, J.W. Farmer, S.R. Arrasmith, R.W. McCalum, IEEE Trans. Al. Suercond. 3 Ž w19x H.R. Hart Jr., F.E. Luborsky, R.H. Arendt, R.L. Fleischer, J.E. Tkaczyk, D.A. Orsini, IEEE Trans. Magn. 27 Ž