Memory and rejuvenation in a spin glass
aa r X i v : . [ c ond - m a t . d i s - nn ] M a y epl draft Memory and rejuvenation in a spin glass
R. Mathieu ∗ , M. Hudl, and P. Nordblad ∗ Electronic address: [email protected] of Engineering Sciences, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden
PACS – Spin glasses and other random magnets
PACS – Spin-glass and other random models
PACS – Dynamic properties
Abstract. - The temperature dependence of the magnetisation of a Cu(Mn) spin glass ( T g ≈
57 K)has been investigated using weak probing magnetic fields ( H = 0.5 or 0 Oe) and specific thermalprotocols. The behaviour of the zero-field cooled, thermoremanent and isothermal remanentmagnetisation on (re-)cooling the system from a temperature (40 K) where the system has beenaged is investigated. It is observed that the measured magnetisation is formed by two parts: (i) atemperature- and observation time-dependent thermally activated relaxational part governed bythe age- and temperature-dependent response function and the (latest) field change made at a lowertemperature, superposed on (ii) a weakly temperature-dependent frozen-in part. Interestingly weobserve that the spin configuration that is imprinted during an elongated halt in the cooling, if it isaccompanied by a field induced magnetisation, also includes a unidirectional excess magnetisationthat is recovered on returning to the ageing temperature. Introduction. - Spin glasses continue to fascinate sci-entists. The origin and nature of some basic experimentalobservations, such as the ageing, memory and rejuvena-tion phenomena, are still being discussed and investigated[1]. New experimental realisations of model systems suchas the three-dimensional (3 D ) XY and two-dimensional(2 D ) Ising model systems were recently discovered andcharacterised [2–4], and novel nano-structured mesoscopicspin glasses were investigated [5]. Spin glasses and glassyproperties were recently even associated with exchange-biased spintronic devices [6] and multiferroics [7].On the theoretical front, atomic and magnetic orderingsof spin glasses were recently studied using first-principlescalculations [8]. Also, a new supercomputer with the abil-ity to carry out Monte Carlo simulations up to experimen-tal time scales [9] was built, and may permit the theoreti-cal observation of chaos and rejuvenation effects [10], andgive insights on the validity of the different views on thenature of the spin-glass state [11].The non-equilibrium properties of spin glasses can beinvestigated experimentally by recording the temperature-dependent magnetisation M on reheating after spin con-figurations are imprinted while halting the cooling at con-stant temperatures T h below the spin-glass phase transi-tion temperature T g [12, 13]. These equilibrations, or age-ings, are kept in memory on further cooling and retrieved on reheating. Due to the chaotic nature of the spin-glassphase, this memory of the equilibration at T h is observedonly in a finite temperature range around T h , defining“memory dips” with a finite width. Outside this tem-perature range, the magnetisation recovers its referencelevel and the system appears to be rejuvenated. Theseso-called dc-memory experiments have been employed e.g.to investigate spin-glass model systems [1, 12, 13], super-spin glasses [14], geometrically frustrated systems [15] andexotic superconductors [16].It was shown recently that the dynamical propertiesof different glassy and superparamagnetic systems couldbe compared by employing specific in-field temperaturecycling-protocols [17, 18], in which the magnetisation ismeasured on reheating up to a given temperature afteran initial cooling to the lowest temperature. M is thenmeasured on repeated cooling down and reheating thesystem from different increasing temperatures.We have here combined the two above procedures,adding in-field temperature-cycling procedures to dc-memory experiments. Using a weak magnetic field (0.5Oe) in the linear response regime to magnetise thesample we are able to distinguish between a part of themagnetisation that is controlled by the dynamic responseof the spin glass and a part that is frozen in and fadesp-1. Mathieu et al. away when heating above the temperature where itwas attained. Our experiments surprisingly uncover aunidirectional excess magnetisation associated with theapplication of the magnetic field. Experimental. - We here investigate the temperature-dependent magnetisation of a Heisenberg-like Cu(Mn)spin glass, recorded after specific protocols on a noncom-mercial low-field superconducting quantum interferencedevice (SQUID) [19]. Small magnetic fields of H =0.5Oe were employed to probe the magnetisation, yieldinga linear response of the system [20]. The magnetic fieldis generated by a small superconductive magnet withtime constant ∼ T g ∼
57 K) employed in the experiments was prepared bydrop-synthesis method using an induction furnace.
Results and discussion. - The typical features ofthe ageing, memory and rejuvenation phenomena areexemplified in Fig. 1. The left panel shows dc-relaxationexperiments, in which the system is cooled down rapidlyto a temperature below T g , here 40 K. After a wait timeof 3 s or 3000 s, a dc-field of H =0.5 Oe is applied andthe magnetisation M is recorded as a function of time t in the case of zero-field cooled experiments. In thecase of thermoremanent (TRM) experiments, the field isswitched from 0.5 Oe to 0, while in the case of field-cooled(FC) ones the field is always 0.5 Oe. One can notice inthe left panel of Fig. 1 that at an observation time ofabout 30 s, corresponding to the effective observationtime of the magnetisation measurement on heating, theZFC and TRM curves recorded after a 3000 s stop liessignificantly below (resp. above) the curve recordednearly immediately ( t w = 3 s) after reaching 40 K. Thisreflects the ageing or equilibration that occurred whilethe spin glass was left at a constant temperature, and theassociated rearrangement of its spin configuration. TheFC magnetisation M FC shows in this context marginalrelaxation behaviour, however, the principle of superposi-tion is applicable and the fundamental relation: M ZFC ( t ) ∼ M FC ( t ) - M TRM ( t ) is obeyed [21]. The temperature(right) and observation time (left) dependence of thedifferent ZFC and TRM curves in Fig. 1 are governed bythe temperature- and age-dependent response functionof the spin glass. The initial magnetisation, zero in theZFC case and M F C in the TRM case, does not includeany component that has been attained by earlier fieldchanges at a temperature below T g ; the magnetisationchanges are dynamically limited and governed by theobservation time that corresponds to the heating rate inthe temperature dependent experiments. We also recallfrom these and earlier experiments, that the thermalhistory governs the global evolution of the spin state and the response function and that this occurs independentlyof any field changes within the linear response regime.Let us now add in-field temperature cyclings to dc-memory experiments on the ZFC magnetisation. Table 1lists and Fig. 2 illustrates the different protocols that wehave considered. R1 and R2 correspond to conventionalZFC experiments, without (R1) and with (R2) halt dur-ing the initial cooling, akin to the ones shown in the rightpanel of Fig. 1. We will use these curves as referencesfor the other procedures. In procedure A1, the system iscooled down below T g to 40 K (see Table 1). The mag-netic field H =0.5 Oe is switched on and the magnetisationis recorded in that field on resuming the cooling down to20 K, as well as on reheating to 70 K. Procedure A2 isidentical to A1, albeit the system is kept for 3000 s at 40K during the initial cooling (i.e. just before turning themagnetic field on) as in memory experiments. ProcedureC is performed for comparison. It is similar to A2, exceptthat the field is turned and kept on during the 3000 s longwait time at 40K.As seen in the left panel of Fig. 3, the equilibrationwhich occurred during the halt in the initial cooling (A2or R2) is kept in memory in both types of experiments, asobserved in conventional temperature cycling experiments(i.e. thermal cycling without magnetic field change) ontime-dependent dc- or ac-magnetisation experiments[1, 13]. The final reheating curves obtained in each caseeventually merge with their respective references. In thecase of procedure C, the result of the large relaxationof the magnetisation akin to the one shown in the leftpanel of Fig. 1 can be appreciated. It can be observedin the left panel of Fig. 3 that the magnetisation curvesrecorded following procedures A (and C) become flat atlow temperatures, with different magnetisation values.These weakly temperature-dependent magnetisationcurves are reminiscent of the field-blocked field cooled [22]and TRM [17] magnetisation, weakly affected by the spinreorganisation on short-length scales occurring on coolingdown the system. If instead we consider the differenceplots of the curves recorded with and without a 3000 shalt at 40 K for the different procedures (R2-R1, A2-A1),we obtain the dc-memory curves which are shown in theright panel of Fig. 3. The curve labelled R2-R1 depictsa conventional dc-memory experiment, as for the insetof Fig. 1. The other curves correspond to dc-memoryexperiments with in-field temperature cyclings. Thefrozen-in magnetisation implies that ∆ M/H remainsnearly constant as the temperature decrease below 40 K,outside the width of the memory dip exhibited by thereference R2-R1 curve [23].In procedure A1, M is recorded as soon as the tempera-ture reaches 40 K. Let us consider another procedure, likeprocedure B1 (and associated B2) in which the final re-heating curve is measured after first lowering the temper-ature to 20 K, as illustrated in the left panel of Fig. 4. Inp-2emory and rejuvenation in a spin glassprocedure B1, the system is cooled down to 20 K, and asin a conventional ZFC measurement (R1), the magneticfield H =0.5 Oe is switched on and the magnetisation isrecorded in that field on reheating. In this case howeverthe reheating stops at 40 K. While still recording the mag-netisation in H =0.5 Oe, the system is cooled again downto 20 K, and reheated to 70 K. Procedure B2 is identicalto B1, albeit the system is kept for 3000 s at 40 K duringthe initial cooling as in memory experiments.As for procedures A1 and A2, the final reheatingcurves obtained in each case eventually merge with theirrespective references. The magnetisation curves becomeflat at low temperatures, with different magnetisationvalues (with M C > M B1 > M A1 > M B2 > M A2 , see alsothe right panel of Fig. 4). As seen in the right panel ofFig. 4 which shows the difference plots of the reheatingcurves obtained for the five procedures, regardless of thehalt at 40 K during the initial cooling, the reheatingcurves measured on the system using protocols A1and A2 merge earlier with the reference than thosemeasured using protocols B1 and B2. In the latter case,a measurable ∆ M/H is observed way above 40 K. Thissubtle difference of the essentially ZFC magnetisationthat is recorded well above the temperature 40 K (belowwhich the thermal and field history differs in protocolsA and B) is mainly associated with the increase of themagnetisation that occurs around 40 K in protocol Bbefore the second cooling to 20 K.Illustrated in the left panel of Fig. 5, are ZFC, FC andTRM curves recorded using the same thermal protocols(R, A, B and C) as described above for the ZFC case.The TRM behaviour corresponds nicely with the ZFC be-haviour, whereas there is little effect of wait times andtemperature cyclings on the FC magnetisation. As seenin the inset of the left panel of Fig. 5, we can see that thedifference curves obey as well the linear relation mentionedearlier, with ∆ M ZF C ∼ − ∆ M T RM .In the right panel of Fig. 5 results of the IRM aftersimilar thermal protocols R, A, and B are shown. IRMrefers to isothermal remanent magnetisation [12] and ismeasured in H =0 after cooling the system also in H =0; ahalt is performed at constant temperature for a time t ∆ H during which a magnetic field ∆ H is applied (See Fig. 2).The IRM experiments are interesting, because in that casethe magnetic field is applied only to imprint the IRM dur-ing the halt (and turned off elsewhere). The behaviour ofthe IRM curves looks at a glance similar to those of theA2-A1 and B2-B1 curves of the right hand panel of Fig. 4,as illustrated in the inset of Fig. 5. This may be antici-pated; it is however worth to note that the thermal historydiffers in the IRM case, as there is always an additional3000 s stay at 40 K during which the field is applied. Themagnetisation that is attained during the field applicationis frozen in on cooling. This magnetisation has a fun-damentally different nature than that of the dynamicallycontrolled magnetisation that occurs with time at constant temperature after a field change or on heating after a fieldchange. A frozen-in magnetisation state is also attainedwhen the sample is cooled and reheated from 40 to 20 Kin protocol A and B. The weak temperature-dependenceof the FC, ZFC, TRM and IRM curves between 20 and 35K mimics frozen-in magnetisation states. The frozen-inmagnetisation rapidly fades away when the temperatureis increased above the temperature where it is attained.Looking again at the right panel of Fig. 5 one noticesa sharp upward temperature dependence on approaching40 K on all the reheating curves. IRM curves withsimilar features were observed for another canonicalAg(Mn) spin glass [12]. The spin configuration thatwas originally imprinted by the magnetic field at 40 Kbrings forth an excess magnetisation in the direction ofthe applied field. This excess magnetisation is slowlydrained by random spin fluctuations without preferreddirection as the temperature is shifted away from 40 K.However, when coming back towards 40 K the memoryof the spin configuration attained during the in-field stopat this temperature is recovered and this includes theunidirectional excess moment, of which only a minor parthas been lost by the random spin fluctuations during thetemperature cycling to lower temperatures. Above 40 Kthe excess moment rapidly fades away as the memory ofthe low temperature configuration is washed out. Conclusion. - The temperature dependence of themagnetisation of a Cu(Mn) spin glass on its thermal andmagnetic field history has been studied using various pro-tocols. The results show that rejuvenation does not af-fect the “frozen-in” magnetisation. On the other hand,the thermal history on cooling, the heating rate and thewait time at constant temperatures govern the responsefunction, that is reflected in the ordinary ZFC (or TRM)magnetisation response.A major finding is that the spin configuration thatis imprinted during elongated stop during cooling, ifit is accompanied by a field-induced magnetisation,also includes a unidirectional excess magnetisation thatis recovered on returning to the ageing temperature.This occurs in spite of the fact that substantial partsof that magnetisation appears to be lost through spinfluctuations at lower temperatures. Even when rapidrelaxation due to slow cooling occurs, this unidirectionalexcess magnetisation is almost perfectly recovered whenthe sample returns to the temperature where it was aged(see right panel of Fig. 5). ∗ ∗ ∗
We thank the Swedish Research Council (VR) and theG¨oran Gustafsson Foundation for financial support.p-3. Mathieu et al.
REFERENCES[1]
P. E. J¨onsson, R. Mathieu, P. Nordblad, H. Yoshino,H. Aruga Katori, and A. Ito , Phys. Rev. B , (2004)174402.[2] R. Mathieu, A. Asamitsu, Y. Kaneko, J. P. He, andY. Tokura , Phys. Rev. B , (2005) 014436.[3] R. Mathieu, J. P. He, Y. Kaneko, H. Yoshino, A.Asamitsu, and Y. Tokura , Phys. Rev. B , (2007)014436.[4] R. Mathieu and Y. Tokura , J. Phys. Soc. Jpn. , (124706) 2007.[5] K. Komatsu, H. Maki, and T. Sato , J. Phys. Soc. Jpn. , (2008) 124710.[6] M. Ali, P. Adie, C. H. Marrows, Denis Greig, B. J.Hickey, and R. L. Stamps , Nature Mater. , (2007) 70.[7] V.V. Shvartsman, S. Bedanta, P. Borisov, W. Klee-mann, and A. Tkach and P. M. Vilarinho , Phys. Rev.Lett. , (2008) 165704.[8] O. E. Peil, A. V. Ruban, B. Johansson , Phys. Rev. B , (2009) 024428.[9] F. Belletti et al. , Phys. Rev. lett. , (2008) 157201.[10] T. Komori, H. Yoshino, and H. Takayama , J. Phys.Soc. Jpn. Suppl. A , (2000) 228.[11] J.- P. Bouchaud, V. Dupuis, J. Hammann, and E.Vincent , Phys. Rev. B , (2001) 024439[12] R. Mathieu, P. J¨onsson, D. N. H. Nam, and P. Nord-blad , Phys. Rev. B , (2001) 092401.[13] R. Mathieu, P. E. J¨onsson, P. Nordblad, H. ArugaKatori, and A. Ito , Phys. Rev. B , (2002) 012411.[14] M. Sasaki, P. E. J¨onsson, H. Takayama, and H.Mamiya , Phys. Rev. B , (2005) 104405; D. Peddis,M. Hudl, C. Binns, D. Fiorani and P. Nordblad , J.Phys.: Conf. Ser. , (2010) 072074.[15] V. Dupuis, E. Vincent, J. Hammann, J. E. Greedan,and A. S. Wills , J. Appl. Phys. , (2002) 8384.[16] A. Gardchareon, R. Mathieu, P. E. J¨onsson, andP. Nordblad , Phys. Rev. B , (2003) 052505.[17] H. Mamiya, S. Nimori, M. Ohnuma, I. Nakatani, M.Demura, and T. Furubayashi , J. Magn. Magn. Mater. , (2007) e535.[18] M. Sawicki, D. Chiba, A. Korbecka, Y. Nishitani, J.A. Majewski, F. Matsukura, T. Dietl, and H. Ohno , Nature Phys. , (2009) 22.[19] J. Magnusson, C. Djurberg, P. Granberg, and P.Nordblad , Rev. Sci. Instrum. , (1997) 3761.[20] L. Lundgren, P. Nordblad, and L. Sandlund , Euro-phys. Lett. , (1986) 529.[21] P. Nordblad, L. Lundgren, and L. Sandlund , J.Magn. Magn. Mater. , (1986) 185.[22] P. E. J¨onsson and H. Takayama , J. Phys. Soc. Jpn. , (2005) 1131.[23] K. Jonason, E. Vincent, J. Hammann, J. P.Bouchaud, and P. Nordblad , Phys. Rev. Lett. , (1998) 3243 p-4emory and rejuvenation in a spin glass Table 1: Description of the different cooling and measurement protocols employed in the study. Procedures R2, A2 and B2 areidentical to R1, A1, and B1 respectively, adding a halt at 40 K for 3000 s during the initial cooling down to 20 K. In procedureC, the halt is performed with H = 0. Initial cool down Measure M ( T ) in H =0.5 OeProcedure in H =0 to T = Wait at 40 K with T varying as:R1 20 K 0 s 20 K →
70 KR2 20 K 3000 s ( H = 0) 20 K →
70 KA1 40 K 0 s 40 K →
20 K →
70 KA2 40 K 3000 s ( H = 0) 40 K →
20 K →
70 KC 40 K 3000 s ( H = 0.5 Oe) 40 K →
20 K →
70 KB1 20 K 0 s 20 K →
40 K →
20 K →
70 KB2 20 K 3000 s ( H = 0) 20 K →
40 K →
20 K →
70 K −1 −5−2.502.55 t (s) ∆ M / H ( a r b . un i t s ) TRMZFCFC3000s 3000s3st w =3s3s 3000s
20 30 40 50 60 70051015202530 T(K) M / H ( a r b . un i t s )
20 40 60−2−10 ∆ M / H ( a r b . un i t s ) T (K)FC ZFCTRM
Fig. 1: Left: Time t dependence of the ZFC, FC and TRM magnetisation M , recorded in H =0.5 Oe (ZFC, FC) or 0 Oe (TRM)after a quench from 70 K to 40 K, and a wait time t w , plotted as ∆ M = M − M ( t = 0 . s ). The vertical dotted line indicatesthe experimental time scale of temperature-dependent measurements. Right: Temperature T dependence of the ZFC, FC, andTRM magnetisation. The M ( T ) curves are measured on reheating after cooling the sample to the lowest temperature directly(open symbols, same symbols as in the left panel) or including a stop at 40 K for a time t s =3000 s (filled symbols). A magneticfield H =0.5 Oe is employed to record the magnetisation. The inset shows the corresponding difference curve between ZFCmeasurements with and without stop at 40 K. p-5. Mathieu et al. time T H w A1,A2 B1,B2
ZFC time
T H w TRM IRM t ∆ H A1,A2
Fig. 2: Schematic representation of the different procedures. The arrows indicate the time a which the magnetisation starts tobe recorded. Left: Variation of the temperature T and magnetic field H as a function of time during the procedures A1, A2(continuous line) and B1, B2 (dotted line) in the ZFC case. In the case of A1, B1, t w =0, while t w =3000 s for A2, B2. Right:Idem for the procedures A1, A2 in the IRM (continuous line) and TRM (dotted line) cases.
20 25 30 35 40 45 508101214161820 M / H ( a r b . un i t s ) T(K)CA1
H applied after t s =3000s A2R1,R2
References H applied after t s =0s
20 30 40 50 60−2−1.5−1−0.50 ∆ M / H ( a r b . un i t s ) T (K)A2−A1
A2−A1(300s) R2−R1(300s)
R2−R1
Fig. 3: Left: Temperature dependence of the ZFC magnetisation recorded under the A, C and R heating/cooling protocols (seetable 1 and Fig. 2). R1 and R2 are the ZFC reference curves shown in Fig. 1(right). Right: Difference plots of the A2-A1 curves(large symbols). The difference plots of the references R2-R1 (a conventional “dc-memory” plot) is added in thick continuousline. Corresponding data obtained from experiments employing a shorter wait time of 300 s are included (small symbols, thincontinuous line) for comparison. Only difference plots of reheating curves are depicted. p-6emory and rejuvenation in a spin glass
20 25 30 35 40 45 508101214161820 M / H ( a r b . un i t s ) T(K)
20 40 60−2−10 ∆ M / H ( a r b . un i t s ) T (K)
A2−A1B2−B1R2−R1 t s =0s B1 t s =3000s B2 References
R1,R2 20 30 40 50 60 70012345678 ∆ M / H ( a r b . un i t s ) T (K)
40 50 6000.51 ∆ M / H ( a r b . un i t s ) T (K)
C−R1A1−R1 t s =0s t s =3000s B2−R2A2−R2B1−R1
Fig. 4: Left: Temperature dependence of the ZFC magnetisation recorded under the B and R heating/cooling protocols (seetable 1 and Fig. 2). The inset shows the difference plots of the B2-B1 (open symbols), A2-A1 (filled symbols), as well as R2-R1(continuous line) curves. Right: Difference plots of the reheating A1, B1, C, A2, B2 curves with their respective reference R1and R2. The inset shows an enlarged view of the plot in the main frame.
20 30 40 50 60 700510152025 T(K) M / H ( a r b . un i t s )
30 40 50024 T (K) ∆ M / H ( a r b . un i t s ) TRMFCZFC ZFC, TRM
20 30 40 5000.511.522.533.54 M / H ( a r b . un i t s ) T(K) 20 30 40 50T(K)
20 35 50012
T(K) − ∆ M / H ( a r b . un i t s ) A2R1,R2 B1B2A1
A2−A1 (ZFC)
Fig. 5: Left: Temperature dependence of the ZFC, FC, and TRM magnetisation recorded under the same heating/coolingprotocols. For clarity no symbols are employed to plot the M FC data, and only reheating data for procedures A2 (open symbols),C (pluses), and B1 (filled symbols), and reference curves (continuous lines) are shown. The inset shows the correspondingdifference plots (A2-R1, C-R1, and B2-R2) for the ZFC and TRM (plotted as − M TRM ) data, which are virtually identical in allthree cases. Right: Temperature dependence of the IRM magnetisation recorded under the same heating/cooling protocols. Inthe reference IRM measurements, a magnetic field of H =0.5 Oe is applied for 3000 s at 40 K and removed, and M is recorded(in H =0) on reheating to 70 K after resuming the cooling to 20 K (also in H =0). The A1, A2, B1 and B2 procedures areperformed akin to the ZFC case, e.g. in the case of A1 by starting to measure M in H =0 after the field cycling at 40 K, oncooling to 20 K and subsequent reheating to 70 K.The inset shows the difference plot A2-A1 obtained for ZFC experiments forcomparison (from Fig. 4, plotted as − ∆ M ))