Modeling and simulation on the magnetization in field-cooling and zero-field-cooling processes

Transcription

1 Ž. Physia C Modeling and simulation on the magnetization in field-ooling and zero-field-ooling proesses S.L. Li, H.H. Wen ), Z.X. Zhao National Laboratory for SuperondutiÕity, Institute of Physis and Center for Condensed Matter Physis, Chinese Aademy of Sienes, PO Box 63, Beijing 18, China Reeived 9 February 1999; aepted 23 Marh 1999 Abstrat By using Monte Carlo simulation and thermally ativated flux motion Ž TAFM. model, we alulate the magneti field profile and thus the magnetization in the field-ooling Ž FC. and zero-field-ooling Ž ZFC. proesses. Based on the simple Kim Anderson UŽ j. dependene, the major experimental observations an be niely reprodued. It is found that the alulated field profile BŽ x. is lose to the Bean ritial state model in the ZFC proess, in sharp ontrast to that in the FC proess whih shows a urved frozen flux pattern. The magnetization in both the ZFC and FC proesses is strongly dependent on the pinning potential and the ritial urrent density, but weakly dependent on the temperature-sweeping rate. q 1999 Elsevier Siene B.V. All rights reserved. Keywords: Magnetization; Flux pinning; Monte Carlo simulation 1. Introdution Magneti expulsion is one of the key properties of a superondutor; therefore, for both the fundamental researh and the high power appliation, it is neessary to measure and investigate the magnetization. Among many methods, the so-alled zero-field-ooling Ž ZFC. and field-ooling Ž FC. proesses Žfor a review, see Ref. wx. 1 have been widely used. In a ZFC proess, the sample is firstly ooled down from above T to a desired temperature in zero field, then an external field is applied and the data are olleted in the warming up proess. In a FC proess, the sample is ooled down from above T to a desired ) Corresponding author. Tel.: q ; Fax: q ; hhwen@aphy.iphy.a.n temperature with an external field, and the data are olleted in either the ooling down Ž FCC. or the warming up Ž FCW. proess. The ZFC and FC magneti moments have often been measured to determine the superonduting transition temperature T wx 2. In addition, they have been extensively measured to investigate the flux dynamis, suh as, the irreversibility point Ž T, B. irr irr below whih the flux motion is hindered by the pinning wx 3. For high power appliation, suh as the superonduting magneti levitation, it is believed that the levitation fore is diretly related to the ZFC magneti moment wx 4, whih stimulates enormous efforts on the magnetization relaxation. The idea about the so-alled superonduting permanent magnet is based on the remanent magnetization frozen in a superondutor in the FC proess wx 5. For all the purposes mentioned above r99r$ - see front matter q 1999 Elsevier Siene B.V. All rights reserved. Ž. PII: S

2 294 ( ) S.L. Li et al.rphysia C a detailed investigation and better understanding on the FC and ZFC magnetization is strongly needed. In this paper, by performing the Monte Carlo simulation, we alulate the magneti field profile and thus the magnetization in the FC and ZFC proesses based on the thermally ativated flux motion Ž TAFM. model. We will illustrate the major experimental observations, suh as the hysteresis loops, the temperature dependene of the magnetization and their relations with the pinning potential and ritial urrent density. 2. Modeling Before presenting our simulation results, we first show one typial example of the temperature dependene of the ZFC and FC magneti moments for a more 3D-like sample YBa 2Cu 3O 7. As shown in Fig. 1, in both the ZFC and FC proesses, the MT Ž. urve keeps flat at temperatures up to about 85 K, then the urve rises up rapidly and turns flat again at T. As shown below the flatness of the ZFC MT Ž. and the FC MT Ž. have very different physial origin. Shown in Fig. 2 are the shemati field profiles in the ZFC, FCC and FCW proesses, respetively. If we apply a small field Ž H -H. 1 to a zero-fieldooled sample, the Meissner shielding urrent will prevent the flux from entering, leading to a perfet diamagneti state Žorresponding to the status 1 in Fig. 2. a. As the temperature inreases, the H Ž T. 1 drops, eventually the flux starts to penetrate into the sample as soon as H Ž T. 1 sh. Due to the external magneti pressure, the flux will quikly move into the interior. The moving speed and the loal field gradient d BŽ x. rd x are determined by the flux motion properties whih an be simulated by using the Monte Carlo method instead of using a pre-assumed Bean ritial state model Ž status 2 to 5.. When the temperature is further inreased, the pinning beomes very weak delivering a simultaneously uniform distribution of the flux lines within the sample Ž status 6 and 7.. The threshold for this uniform distribution an be understood as the irreversibility point T irr. Between Tirr and T, the magnetization is reversible for all the ZFC, FCC and FCW proesses. In the FCC proess, the sample is ooled down with a field Žstatus 7 to 1 in Fig. 2Ž b... In the temperature range between Tirr and T, the flux is very easy to move out of the sample Ž status 7 to 6.. One the pinning is established, the flux in the deep interior annot esape easily. Again the moving speed and the loal field gradient d BŽ x. rd x is determined by the flux dynamis whih an be arried out by Monte Carlo simulation Ž status 5 to 1.. In the low temperature region, this flux pattern will be frozen showing a flat temperature dependene of the magnetization. In the FCW proess, this frozen pattern will be gradually melted when the temperature is high enough and eventually leading to the free motion in the irreversible region. The shemati temperature dependenes of the magnetization in the ZFC and FCW proesses are shown in Fig. 2 d with the orresponding field profiles as depited in Fig. 2 a to. 3. Simulation Ž. Ž. Fig. 1. The FC filled irles and ZFC open irles magneti moment of an Y-123 single rystal measured in an external field of 2 Oe. The sample onsidered here is a infinitely large slab with thikness 2a along the x-diretion and limited by infinite planes at xsya and xsa. At the surfae of the sample, it is assumed that the magneti indution B is equal to the externally applied field, BŽ xs"a. sm H Ž t.. However, beause of the Meissner shielding urrent within the penetration depth l, the indution B should sharply drop from m H to m wh yh x; therefore, for the 1

3 ( ) S.L. Li et al.rphysia C Fig. 2. The shemati field profiles in a ZFC, b FCC and FCW proesses; d is the shemati temperature dependene of the magnetization in the ZFC and FCW proesses. onveniene of our simulation and onsidering that l<a, we use BŽ xs"a, t. sm w H t yh Ž T.x 1, where H is the lower ritial field, H Ž T. sh Ž w Ž. 4 x y2 = 1y TrT Al, and H is the external mag- neti field along the z-axis. It is assumed that there are N periodi pinning sites along the x-axis in the sample and the distane between two neighbors is 2 x. For a given site, a vortex bundle an jump either downhill or uphill to its neighboring sites. The Monte Carlo method was hosen to simulate the movements of vorties. In our alulation, a one-dimensional array represents a superondutor with N pinning sites. Eah pinning site has a loal field indution BŽ x.. During eah step of the alulation, a vortex bundle with flux f is randomly seleted from eah site, and then its probability of jumping out of the site is ompared with a random number between and 1. The result will determine whether the vortex bundle jump out of its site or not: if the random number is less than the probability, the vortex will jump out; and the ontrary situation will make the vortex stay in its original site. Our simula-

4 296 ( ) S.L. Li et al.rphysia C Fig. 3. The magneti hysteresis loop obtained from our alulation. The values of parameters are hosen as follows: U rkbt 11 2 y6 s16.5, J s1 Arm, as.1 mm, x s2.5=1 m. tion method is the same as that used by Shnak et al. wx 6 and van der Beek et al. wx 7. The probability of a vortex jump is given by U Ž j Psexp y., Ž 1. k T ž B / where UŽ j. is the ativation energy and taken the form of the Kim Anderson model, UT, Ž J. s U Ž T.w1" JrJ Ž T.x, where q means the uphill jump and y the downhill jump. The loal urrent density J is determined by using the Maxwell s equation, JŽ x. syž 1rm. d Brd x. In all the alulation, we presumably use the relation U Ž T. s U Ž. = w1yž TrT. 2 x 3r2, J Ž T. sj Ž.w1yŽ TrT. 2 x 2 wx 8. The magnetization M is alulated by 1 a Ms H mhybž x. d x. Ž 2. 2 a ya To hek our simulation method, a hysteresis loop is obtained and shown in Fig. 3. As a result, it resembles the experimental data. The parameters 11 hosen here are U rkbts 16.5, J s 1 Arm 2. If not speially mentioned, in the oming part of the paper we will take other parameters as y6 follows: as.1 mm, x s2.5=1 m Ž with 4 pinning sites aross the sample., the mirosopi 7 attempting frequeny v s 1 Hz, m H 1 s.1 T, the external field mhs.5 T. Fig. 4. The temperature dependene of ZFC and FC suseptibili ties with U s1 K and J s1 Arm. The open symbols denote the ZFC proess and the symbols represent the FC proess J s1 Arm,Ts9 K. As we an see, these urves look quite similar to the experimental data as shown in Fig. 1. One may argue, however, that the U is too large ompared to the experimental 4. Results and disussion Fig. 4 shows the temperature dependene of the 5 ZFC and FC suseptibilities with U s1 K and Fig. 5. The magneti field profiles of a ZFC b FCC and FCW in our simulation. The parameters used here are the same as in Fig. 3.

5 ( ) S.L. Li et al.rphysia C Fig. 6. The suseptibilities with different U of a ZFC and b FC. The open symbol in the FC proess denotes the FCW proess and the solid one represents the FCC proess. A small differene has been found between FCC and FCW near the irreversibility temperature, whih is indued by the different field profiles near the edge in these two proesses. value U Ž 1 1 K.wx 9. We give answers to this argument as follows. The real distane between two neighboring pinning sites is about 1 nm for high temperature superondutors, that is, there are more than 1 pinning sites between our two virtual pinning sites. If the number of real pinning sites between two virtual sites is n, a vortex should pass n real sites to jump between two virtual sites; therefore, the nominal pinning energy in our formula Eq. 1 should be muh higher than that for a single jump between two real pinning sites. A simple estimation an be made by assuming that the vortex jumps downhill only and the probability of the vortex jump between our virtual pinning sites is n r u Ž j Ps exp y. kt Ł ž / i s 1 Ł ž ž / is1 / nuž T. J ž kt ž J // n U Ž T. J s exp y 1y kt J sexp y 1y. Ž 3. Compared with Eq. 1, the value of the nominal pinning energy U in the present alulation is n times the real value U f1 1 K. If taking the reverse hopping into aount in the above estimation, Eq. 3 will beome very ompliated but with an unhanged onlusion that the pinning energy in our alulation should be muh larger than the experimental real value U for a single real hopping. The alulated flux density profiles of ZFC, FCW and FCC orresponding to Fig. 2 are shown from Fig. 5 a to, respetively. The upper part of the field patterns, as expeted, represent the reversible region in whih the inner magneti fields are uniform and hange synhronously with the surfae field, thus the properties of flux dynamis in this area are all the same for every proesses. A striking differene between the ZFC and FC ases is the field profile in the low temperature region where, for the ZFC proess, BŽ x. shows a behavior lose to the Bean ritial model even in the non-fully-penetrated state, while that for the FC proess shows an extremely urved field pattern. From these results, we Fig. 7. The suseptibilities with different J of a ZFC and b FC. The open symbol in FC proess denotes the FCW proess and the solid one represents the FCC proess.

6 298 ( ) S.L. Li et al.rphysia C know that the flatness of the MT Ž. urves measured in the ZFC and FC proesses in the low-temperature region Ž as shown in Fig. 1. have very different origins: in the ZFC proess, it is due to the Meissner shielding while in the FC proess it is due to the freezing of the flux pattern. The profiles of FCW and FCC are very similar to eah other: the vorties are onfined in the sample at a low temperature, and eventually it enters the reversible region. A lose inspetion finds, however, a differene just before entering the reversible region: near the surfae of the sample, BŽ x. in the FCC proess shows a straight line but in the FCW proess shows a V-shape. Now, the influene of hanging parameters on ZFC and FC urves will be disussed. At first, the ZFC and FC suseptibilities with different U are shown in Fig. 6 a and Fig. 6 b, respetively. In Fig. 6 a the urves overlap eah other in the low-temperature region, while in Fig. 6 b the magnitude of the plateau hanges with the pinning potential dramatially. This differene is indued by the different origins of the plateau part in ZFC and FC proesses. In the ZFC proess, the plateau is due to the omplete Meissner shielding whih is ertainly not dependent on the pinning potential. In the FC proess, the plateau a result of the frozen flux pattern whih depends strongly on the pinning strength. In the ZFC proess, one an see that the MT Ž. urve shifts to higher temperature whih is due to more damped motion with a stronger U. For the same reason, the absolute values of suseptibility in Fig. 6 b beome smaller with larger U. A similar behavior an be observed for hanging the ritial urrent density. As shown in Fig. 7 a and b, with the inrease of J, the ZFC urve beomes sharper and FC urves move upward. There is another parameter that will affet the MT Ž. urves in the ZFC and FC proesses, that is, the rate of inreasing or dereasing the temperature. As shown in Fig. 8 a and b, if the rate is smaller, the magnetization will relax more due to flux motion. From our data, it is found that the temperature sweeping rate dtrdt does not alter the MŽ T. urves greatly, although it is hanged for two orders of magnitude. This may manifest that in ahieving a ertain magnetized state, the temperature-sweeping rate plays a minor role. It should be mentioned here that the temperature-sweeping rate in Fig. 8 is a Fig. 8. The suseptibilities with different temperature-sweeping rate of a ZFC b FC. The open symbol in the FC proess denotes the FCW proess and the solid one represents the FCC proess. relative value and it should be onsidered with the mirosopi attempting frequeny v. In summary, Monte Carlo simulations on the magnetization in the ZFC and FC proesses have been arried out based on TAFM model. The major experimental observations have been niely reprodued. It is found that the alulated field profile BŽ x. is lose to the Bean ritial state model in the ZFC proess, in sharp ontrast to that in the FC proess whih shows a urved frozen flux pattern. The magnetization in both the ZFC and FC proesses is strongly dependent on the pinning potential and the ritial urrent density, but weakly dependent on the temperature sweeping rate. Referenes wx 1 Y. Yeshurun, A.P. Malozemoff, A. Shaulov, Rev. Mod. Phys. 68 Ž wx 2 U. Welp, W.K. Kwok, G.W. Crabtree, K.G. van der Voort, J.Z. Liu, Phys. Rev. Lett. 62 Ž wx 3 I. Chong, Z. Hiroi, M. Izumi, J. Shimoyama, Y. Nakayama,

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