Ising-like dynamics and frozen states in systems of ultrafine magnetic particles

Transcription

1 Isng-lke dynamcs and frozen states n systems of ultrafne magnetc partcles Stefane Russ and Armn Bunde Insttut für Theoretsche Physk III, Justus-Lebg-Unverstät Gessen, D Gessen, Germany Receved 5 October 2006; publshed 29 May 2007 We use Monte Carlo smulatons to study agng phenomena and the occurence of spnglass phases n systems of sngle-doman ferromagnetc nanopartcles under the combned nfluence of dpolar nteracton and ansotropy energy for dfferent combnatons of postonal and orentatonal dsorder. We fnd that the magnetc moments orent themselves preferably parallel to ther ansotropy axes and changes of the total magnetzaton are solely acheved by 180 degree flps of the magnetc moments, as n Isng systems. Snce the dpolar nteracton favorzes the formaton of antparallel chan-lke structures, antparallel chan-lke patterns are frozen n at low temperatures, leadng to agng phenomena characterstc for spn-glasses. Contrary to the ntuton, these agng effects are more pronounced n ordered than n dsordered structures. DOI: /PhysRevB PACS number s : a, Mg, Lk, Tt I. ITRODUCTIO In the last decade, systems of ultrafne magnetc nanopartcles have receved consderable nterest, due both to ther mportant technologcal applcatons manly n magnetc storage and recordngs and ther rch and often unusual expermental behavor, whch s related to ther role as a complex mesoscopc system. 1,2 It has been dscussed controversally n the past, under whch crcumstances these systems are able to show spn-glass phases. Whle experments on dsordered magnetc materals present ndcatons of a spnglass phase 2 4 or of a glassy-lke random ansotropy system, 5 the stuaton s less clear on the theoretcal sde. Smulatons of the zero-feld coolng ZFC and feld-coolng susceptblty showed no ndcaton of a spn-glass phase. 6,7 In contrast, smulatons on agng 8 on a smplfed system, where the dpolar nteracton was only consdered up to a cut-off radus and magnetc relaxaton 9,10 favorze the spn-glass hypothess, but the structure of the frozen hstory-dependent states, as well as the actual mechansm leadng to them, has not yet been clarfed. In ths letter, n order to clarfy these questons, we use Monte Carlo smulatons 11 to study agng phenomena on a large varety of systems of ultrafne magnetc nanopartcles see Fg. 1. Our smulatons do not only pont to the exstence of frozen hstory-dependent states at low temperatures that are characterstc for spn glasses, but also yeld an nsght nto the structure of the frozen states and the underlyng dynamcs. We fnd that under the combned nfluence of dpolar and ansotropy energy, the magnetc moments have a tendency to algn n an Isng-lke manner ether parallel or antparallel to ther ansotropy axes and change ther drectons by 180 degree flps as n Isng systems. Ths way, chan-lke structures are formed where all magnetc moments pont nto the same drecton and neghborng chans have the tendency to orent themselves n an antparallel way. These topologcal chans that freeze n at low temperatures, form smple straght lnes when the partcles are arranged on the stes of a cubc lattce 10 and form complex wnded curves when the arrangement of the partcles s lqud-lke. As a consequence, f a small external magnetc feld s appled, the magnetc moments can follow the feld more easly n a dsordered system than n the ordered confguraton. Ths leads, contrary to the ntuton, to more pronounced agng effects characterstc for spn glasses n ordered than n dsordered structures. II. MODEL SYSTEM AD UMERICAL SIMULATIOS For the numercal calculatons, we focus on the same model as n earler papers, 6,9 whch assumes a coherent FIG. 1. Two-dmensonal sketches of the geometres consdered n ths paper: a cubc arrangement of the partcles and all ansotropy axes algned nto the z-drecton, b lqud-lke arrangement and all axes arranged, c cubc arrangement and all axes randomly orented, and d lqud-lke arrangement and all axes randomly orented. In the smulatons, the systems were three-dmensonal 64 partcles per cube /2007/75 17 / The Amercan Physcal Socety

2 STEFAIE RUSS AD ARMI BUDE magnetzaton rotaton wthn the ansotropc partcles, and takes nto account the magnetc dpolar nteracton between them. Every partcle of volume V s consdered to be a sngle magnetc doman,, wth all ts atomc magnetc moments rotatng coherently and the V are taken from a Gaussan dstrbuton of wdth V =0.4 and V =1 see also Refs. 6 and 9. Ths results n a constant absolute value =M s V of the total magnetc moment of each partcle, where M s s the saturaton magnetzaton. The energy of each partcle conssts of three contrbutons: ansotropy energy, dpolar nteracton, and magnetc energy of an external feld. We assume a temperature ndependent unaxal ansotropy energy, E A = KV n / 2, where K s the ansotropy constant and the unt vector n denotes the easy drectons. Eventually, the magnetc moments are coupled to an external feld H leadng to the addtonal feld energy E H = H. Fnally, the energy of the magnetc dpolar nteracton between,j two partcles and j separated by r j s gven by E D = j /r 3 j 3 r j j r j /r 5 j. Addng up the three energy contrbutons and summng over all partcles we obtan the total energy E = E A + E H E,j D. j In the Monte Carlo smulatons, we concentrate on samples of =L 3 partcles placed nsde a cube of sde length, L=4, and average over 1000 confguratons. Durng the smulatons, both the postons of the partcles and ther easy axes are kept fxed. The untless concentraton c s defned as the rato between the total volume V occuped by the partcles and the volume V s of the sample. Here, we focus on the concentraton c/c 0 0.3, where c 0 =2K/M 2 s s a dmensonless materal-dependent constant, c for ron ntrde and c for maghemte nanopartcles. 9 We also tested systems wth hgher concentratons c/c and the extremely hgh concentraton c/c and found that the results reman qualtatvely unchanged. The temperature s measured n unts of the reduced temperature T 1/ 2 KV, where 2KV s the heght of the ansotropy barrer and =1/ k B T. Smlarly, the magnetc feld s measured n unts of the ansotropy feld H a =2K/M s. The relaxaton of the ndvdual magnetc moments s smulated by the standard Metropols algorthm. 12 In contrast to Ref. 8, where dpole nteractons between the partcles were only consdered up to a cut-off radus, we calculate the nteracton energes by the Ewald sum method wth perodc boundary condtons n x-, y-, and z-drecton 6,13 and, thus, are able to account fully for the long-range character of the dpole forces. The magnetc moment s characterzed by the sphercal angles and relatve to a coordnate frame, where the z-axs s parallel to the external feld. 9,14,15 To study the magnetc relaxaton, we determne as a functon of tme t number of Monte Carlo steps for each partcle the angle between the magnetc moment and the z-axs, from whch we obtan the relevant quanttes, as, e.g., the normalzed magnetzaton, 1 m t = 1 =1 V V cos t. To obtan the orentaton of relatve to n, we ntroduce the orentatonal order parameter O n,.e., the average of the absolute values of the scalar product n over all partcles and all confguratons. O does not dstngush between the parallel and the antparallel algnment. It s equal to zero when all are perpendcular to ther axes n and equal to 1 f they are all parallel or antparallel to them. To study agng phenomena, we determne the magnetzaton n a ZFC smulaton. Frst, startng n a random confguraton of the magnetc moments, the system s cooled down n the absence of an external feld, from T= to a reduced temperature T wth a constant coolng rate of / t=0.1, correspondng to 400 Monte Carlo steps for T =1/40 and ten steps for T =1. Second, the coolng process s stopped at T and the system s allowed to relax for a certan watng tme t w. Fnally, n the thrd step, a small external feld h=0.1h a s appled n z drecton. The magnetzaton m s determned as a functon of t t w number of Monte Carlo steps after swtchng on the feld. Agng effects are represented by dfferences between the m -curves for dfferent t w and occur, when many dfferent relaxaton rates exst n the system, so that after a gven watng tme t w, the system has only partly relaxed towards equlbrum. Expermentally, agng effects have already been found n several spn-glasses as, e.g., n Permalloy/alumna granular flms, 16 rare-earth manganates, 17 CuMn spn-glasses, 18 multlayer systems, 19 and n Fe 3 nanopartcle systems. 20 III. UMERICAL RESULTS Fgure 2 shows m for the systems of Fg. 1 wthout watng tme, t w =0 flled symbols, and for t w =10 4 open symbols. The dfferent colors and symbols stand for three dfferent temperatures T =1/5, 1/10, and 1/40. Clearly, all curves show agng effects smlar to the expermental results of Refs Systems wth no or only small t w follow the external feld faster than the systems wth longer watng tmes, ndcatng that the longer relaxaton leads to more stable chans. The agng effects are most pronounced for those systems where all ansotropy axes are orented nto the drecton of the external feld Fgs. 2 a and 2 b and less pronounced but stll vsble for the systems wth dsordered ansotropy axes Fgs. 2 c and 2 d. In these systems wth orentatonal dsorder, the m curves concde for small and show agng effects only after a certan crossover tme close to 10 2 Monte Carlo steps. Ths ndcates that n these systems a certan fracton of dpoles do not belong to quasstable chan-lke structures and can follow the external feld nearly nstantaneously, ndependently of the watng tme, and thus domnate the short-tme behavor. The agng effects decrease wth ncreasng T, when the order s destroyed by the thermal fluctuatons. In order to understand the dynamcal behavor n a more mcroscopc way, we compare m wth the tme

3 ISIG-LIKE DYAMICS AD FROZE STATES I FIG. 2. Color onlne The magnetzaton m after watng tmes t w =0 flled symbols and t w = Monte Carlo steps open symbols s plotted versus number of Monte Carlo steps wth appled external feld for a cubc lattce and algned axes, b lqud-lke system and algned axes, c cubc system and random axes, and d lqud-lke systems and random axes for the reduced temperatures T =k B T/ 2KV =5 black symbols, crcles, T =1/10 red symbols, squares, and T =1/40 blue symbols, damonds. dependence of the correspondng orentatonal order parameters O. Fgure 3 shows O n the thrd step of the agng process for t w =0 and t w = flled and open symbols, respectvely and for the same geometres as before see Fg. 1. The fgure shows that, qute contrary to the expectaton, apart from a slght mnmum at ntermedate, O s constant n tme for the systems of t w = Wthout watng tme, the curves start at much smaller values of O, but ncrease rapdly untl they reach at a crossover tme c of about 10 3 Monte Carlo steps the common plateau value. In the plateau regme, the dpolar moments are ether orented parallel or antparallel to ther easy axes n and do, therefore, flp only between these two drectons. Accordngly, the value of O nether depends on the external feld nor on the functonal form of m. Snce O stays constant for large t w or c, whle m ncreases wth tme see Fg. 2, the have already reached ther parallel or antparallel poston and can only perform spn flps by 180 degrees, thereby ncreasng m and leavng O unchanged. To make ths pont stll clearer, we plot n Fg. 4 the percentage up of partcles pontng upwards,.e., wth /2, agan for the geometres of Fg. 1. The smlarty between Fg. 4 and Fg. 2 s FIG. 3. Color onlne The order parameter O after a watng tme t w =0 flled symbols and t w =10000 open symbols s plotted versus number of Monte Carlo steps for the same geometres, temperatures, system parameters, and symbols and colors as n Fg

4 STEFAIE RUSS AD ARMI BUDE FIG. 4. Color onlne The percentage up of partcles per system pontng upwards after watng tmes t w =0 flled symbols and t w =10000 open symbols s plotted versus number of Monte Carlo steps for the same geometres, temperatures, system parameters and symbols and colors as n Fg. 2. obvous, showng that the number of the magnetc moments orented upwards determne the shape of m. We, therefore, arrve at a remarkably smple Isng-lke dynamcs of these ultrafne magnetc partcles. The amount of agng s drectly related to the degree of order a system can acheve durng t w. In the fully ordered system of Fgs. 1, 2, and 3 a, after a long watng tme, t w, the prefer to be algned n stable chans 10 along the z-drecton and, thus, cannot follow an external feld easly. Sngle magnetc moments nsde a chan wll hardly flp to the other sde and flps of whole chans possess extremely large relaxaton tmes. Wthout watng tme, on the other hand, the are n unstable postons whch allows them to follow the external feld qute rapdly, leadng to large agng effects n ordered systems. As Fgs. 2 b 2 d show, the stuaton s dfferent n systems wth postonal and/or orentatonal dsorder. The relaxaton tmes for spn flps decrease wth the amount of dsorder, n partcular wth the amount of orentatonal dsorder. When the chans are wnded and algned nto dfferent drectons, they are less stable and possess a large varety of ntermedate postons to flp to the other sde. Accordngly, agng effects become weaker wth ncreasng dsorder. For llustraton, we vsualze the agng process n Fg. 5 for the system wth the hghest order and the strongest agng effects,.e., for the cubc system wth algned ansotropy axes. For ths vsualzaton, we follow the defnton of the transversal order parameter of Ref. 10: each of the L 2 stes n the xy plane can be ether a ste or a ste, f a chan has already been formed and all magnetc moments n the chan pont nto the postve or negatve z drecton, respectvely whte stes. If ths s not the case, the ste s a 0 ste grey stes. The fgure shows that chans are qute obvously formed n the second step of the agng process durng the watng tme t w, as can most easly be seen by comparng Fg. 5 a, where t w = wth 5 d where t w =0. In a, many chans are formed durng t w that appear to be qute stable n the followng the thrd step of the agng process Fgs. 5 b and 5 c, when an external feld s appled n the drecton. We can see that most of the chans persst n spte of the external feld. The stuaton s dfferent n Fgs. 5 d 5 f, where only a few chans exst at the end of the second step of the agng process Fg. 5 d. Here, after FIG. 5. Vsualzaton of the chans perpendcular to the xy-plane n the cubc system wth algned ansotropy axes at T =1/5 for one typcal system. The complete chans are ndcated by whte stes and by or sgns, dependng on the drecton of the chan. Stes, where chans have not yet been bult are ndcated by the gray shade. a c System wth watng tme t w =10000,.e., a after the coolng process and t w =10000 b,c after an external magnetc feld n the drecton has been appled for b 1000 and c Monte Carlo steps. d-f System wthout watng tme t w =0,.e., d after the coolng process and t w =0 e,f after an external magnetc feld n the drecton has been appled for e 1000 and f Monte Carlo steps

5 ISIG-LIKE DYAMICS AD FROZE STATES I FIG. 6. Color onlne The magnetzaton m after a watng tme t w =0 flled symbols and t w = open symbols for T =1/10 red lght gray symbols and T =1/40 blue dark gray symbols of systems wth algned and randomly orented ansotropy axes red crcles and damonds, respectvely, for T =1/10 and blue squares and trangles, respectvely, for T =1/40 are plotted versus number of Monte Carlo steps for systems wthout dpole-nteracton. swtchng on the external magnetc feld, new chans can be bult from the 0 stes and the system therefore follows the feld much easer than n Fgs. 5 a 5 c. Recently, t has been argued that also a broad dstrbuton of ansotropy energy barrers mght lead to agng effects n superparamagnetc systems. 20 To show that these knds of agng effects are n fact neglgble compared wth systems where both energy contrbutons are present, we have studed systems wthout dpole nteracton solely ansotropy energy at temperatures T =1/10 and 1/40. In ths case, the partcle postons play no role, so that the geometry of Fgs. 1 a and 1 b as well as 1 c and 1 d are physcally dentcal. The results of m for these two geometres are shown n Fg. 6 for the same agng procedure as before. The fgure shows that the dfferences between the curves for t w =0 and t w = are orders of magntude smaller than n the systems wth dpolar nteracton. It s nterestng to note that, also for systems wth only dpolar nteracton, some knd of agng can be seen, but orders of magntude smaller than for systems wth both energy contrbutons. In summary, analyzng the mcroscopc dynamcs of ultrafne magnetc partcles, we found that rrespectve of the strength of the dpolar nteracton, the dpoles orent themselves ether parallel or antparallel to ther ansotropy axes. We, therefore, arrve at a remarkably smple pcture of the dpole dynamcs, where smlar to the Isng model, the perform spn flps between these two orentatons. Agng effects occur when after a certan watng tme, the magnetc dpoles have arranged themselves n stable confguratons and flps of sngle magnetc moments are suppressed. These agng effects ncrease n a counter-ntutve way wth the order of the system and are thus most pronounced n completely orderded systems wth cubc arrangement of the partcles and axes algned nto the drecton of the magnetc feld. ACKOWLEDGMETS We gratefully acknowledge very valuable dscussons wth W. Kleemann and fnancal support from the Deutsche Forschungsgemenschaft. 1 X. Batlle and A. Labarta, J. Phys. D 35, R X Chen, S. Sahoo, W. Kleemann, S. Cardoso, and P. P. Fretas, Phys. Rev. B 70, T. Jonsson, J. Mattsson, C. Djurberg, F. A. Khan, P. ordblad, and P. Svedlndh, Phys. Rev. Lett. 75, R. W. Chantrell, M. El-Hlo, and K. O. Grady, IEEE Trans. Magn. 27, W. Luo, S. R. agel, T. F. Rosenbaum, and R. E. Rosensweg, Phys. Rev. Lett. 67, J. Garca-Otero, M. Porto, J. Rvas, and A. Bunde, Phys. Rev. Lett. 84, M. Porto, Eur. Phys. J. B 45, J.-O. Andersson, C. Djurberg, T. Jonsson, P. Svedlndh, and P. ordblad, Phys. Rev. B 56, M. Ulrch, J. Garca-Otero, J. Rvas, and A. Bunde, Phys. Rev. B 67, S. Russ and A. Bunde, Phys. Rev. B 74, U. owak, R. W. Chantrell, and E. C. Kennedy, Phys. Rev. Lett. 84, In every step, we select a partcle at random and generate an attempted orentaton of ts magnetzaton, chosen n a sphercal segment around the present orentaton wth an aperture angle d see also Ref. 6. By varyng d,.e., the maxmum jump angle, t s possble to modfy the rate of acceptance and to optmze the smulaton. As a compromse between smulatons at low and hgh temperatures, we chose d =0.1 for all smulatons, ndependent of temperature, whch refers to an accecptance rate between 0.5 and 0.8 for T between 1/40 and 1/5. We also tested larger values of d wth consderably lower acceptaton rates and found that they dd not change the fnal states sgnfcantly. 13 M. P. Allen and D. J. Tldesley, Computer Smulaton of Lquds Clarendon, Oxford, R. V. Chamberln, G. Mozurkewch, and R. Orbach, Phys. Rev. Lett. 52, K. L. ga and U. Strom, Phys. Rev. B 38, E. Vncent, Y. Yuan, J. Hamman, H. Hurdequnt, and F. Guevara, J. Magn. Magn. Mater. 161, A. K. Kundu, P. ordblad, and C.. R. Rao, Phys. Rev. B 72, L. Lundgren, P. Svendlndh, P. ordblad, and O. Beckmann, Phys. Rev. Lett. 51, S. Bedanta, O. Petracc, E. Kentznger, W. Kleemann, U. Rücker, A. Paul, Th. Brückel, S. Cardoso, and P. P. Fretas, Phys. Rev. B 72, M. Sasak, P. E. Jönsson, H. Takayama, and H. Mamya, Phys. Rev. B 71,