Time relaxation of interacting single-molecule magnets
Julio F. Fernández1,*and Juan J. Alonso2,†
1ICMA, CSIC and Universidad de Zaragoza, 50009-Zaragoza, Spain 2Física Aplicada I, Universidad de Málaga, 29071-Málaga, Spain
共Received 14 June 2005; revised manuscript received 10 August 2005; published 26 September 2005兲
We study the relaxation of interacting single-molecule magnets in both spatially ordered and disordered systems. The tunneling window is assumed to be, as in Fe8, much narrower than the dipolar field spread. We show that relaxation in disordered systems differs qualitatively from relaxation in fully occupied cubic and Fe8 lattices. We also study how line shapes that develop in “hole-digging” experiments evolve with timetin these fully occupied lattices. We show共1兲that the dipolar fieldhscales astpin these hole line shapes and共2兲how
pvaries with lattice structure. Line shapes are not, in general, Lorentzian. More specifically, in the lower portion of the hole, they behave as共兩h兩/tp兲共1/p兲−1ifhis outside the tunnel window. This is in agreement with experiment and with our own Monte Carlo results.
DOI:10.1103/PhysRevB.72.094431 PACS number共s兲: 75.45.⫹j, 75.50.Xx
I. INTRODUCTION
Magnetic relaxation in crystals of single-molecule mag-nets 共SMM’s兲, such as Fe8, has become a subject of great interest.1–4 At low temperature T, each SMM behaves ap-proximately as a single spinS. Magnetic relaxation at kBT
ⱗ0.1U/S, where U is a magnetocrystalline anisotropy bar-rier, is temperature independent, and is duly attributed to quantum tunneling under the barrier. Hyperfine interactions with nuclear spins as well as dipole-dipole interactions among all SMM electronic spins give rise to a variety of phenomena that are not yet fully understood. Hyperfine in-teractions enable spins to tunnel even when the ensuing Zee-man energy change 2h is much larger than the tunnel split-ting energy⌬, provided 兩h兩 is smaller than some w.5 For Fe8, for instance, ⌬⬃10−4mK, w⬇10 mK, and the rms value of the Zeeman energy␦h is approximately 400 mK. We shall restrict ourselves to systems withwⰆ␦h. In these systems, the relaxation of the magnetization goes on long after time⌫−1, where ⌫ is the spin tunneling rate for spins within the tunneling energy window共that is, spins for which 兩h兩⬍w兲. Tunneling spins give rise to changing dipolar fields, which in turn bring new spins into the tunneling en-ergy window, thus keeping a magnetic relaxation process from extinction. In this paper, we focus our attention on ef-fects that stem from this process.
We consider here experiments of the sort that were re-ported by Wernsdorferet al.in Ref. 6. In them, a crystalline sample of SMM’s is first quenched fromkBT⬃U/S to kBT
ⱗ0.1U/Sin either共a兲a weak applied magnetic field of a few millitesla, and later observed after the field is switched off at timet= 0 or共b兲a zero field, and later observed after a weak field is applied at t= 0. We shall refer to the former as a field-cooled共FC兲experiment and to the latter as a zero-field-cooled共ZFC兲experiment. Both types of experiments can be lumped into one by definingm˜⬅1 −m/m0, wherem0 is the initial magnetization in a FC experiment, and m˜⬅m/ms, where ms is the final steady-state magnetization in a ZFC experiment. Wernsdorfer et al. observed that m˜⬀t1/2 over roughly two time decades. The time evolution of “holes”
that, under suitable conditions, develop in the magnetization density function, have also been reported.6,7 The existing theory at the time5,8,9 predicted a universal
冑
t short time relaxation from a fully polarized system, but said little about relaxation from or into weakly polarized states.10 Other theories11take hyperfine interactions into account but disre-gard dipole-dipole interactions. They therefore apply if ␦hⱗwwhich is not within the scope of this paper.
We have developed a theory12,14that gives the time evo-lution both of m˜ and of the holes’ line shapes in weakly polarized systems of interacting SMMs, such as Fe8, in whichwⰆ␦h. There are three clearly discernible time re-gimes. For⌫tⱗ1,m˜⬀⌫t. In the second time stage, when 1
ⱗ⌫t, up to some time before m˜⬃1, m˜⬀tp, at least for all fully occupied cubic lattices. Moreover, the theory gives a simple relation that specifies howpvaries with lattice struc-ture. In FC experiments,m˜⬃1 in the third time stage, that is,
m⬇0. The third time stage is more interesting in ZFC ex-periments. Thenm共t兲settles down temporarily to a quasista-tionary value ms, which the theory predicts, if either kBT
Ⰷw or if the heat exchange rate with the lattice is much smaller than⌫; on the other hand, ifkBTⰇwis not satisfied and heat exchange rate with the lattice is not much smaller than⌫, then the relaxation of the magnetization shifts into a thermally driven approach to equilibrium, skipping the qua-sistationary state. A quite different treatment of relaxation from weakly polarized states that gives 1 −m/m0⬀
冑
t, inde-pendently of the spins’ spatial distribution, is given by Tupi-tsyn, Stamp, and Prokof’ev 共TSP兲 in Ref. 15, criticized in Ref. 13, and defended in Ref. 16. TSP’s treatment of relax-ation fromweaklypolarized states is rather unrelated to the earlier theory5 of Prokofe’v and Stamp for a冑
t relaxation fromfullypolarized states, about which we have nothing to say here. According to TSP,冑
trelaxation in weakly polarized systems holds as long as 1ⱗ⌫t andm˜Ⰶ1, that is, roughly, over the second time stage. This is also the time domain where our theory givesm˜⯝共w/␦h兲共⌫t兲pfor fully occupied lattices. This time stage, sometimes referred to as “short times,” can in fact be arbitrarily large, for arbitrarily smallw, sincem˜Ⰶ1 only implies⌫tⰆ共␦h/w兲1/p. Note also that
冑
trelaxation fromfullypolarized systems ends sooner, whent⬃␦h/w.5This may explain why Wernsdorfer6 et al.were able to observem˜⬃
冑
t in Fe8 共p⬇0.5 for Fe8兲under weak applied magnetic fields for up to some 103s while Ohmetal.17observed
冑
trelaxation from fully polarized Fe8for only up to 102s.
The two approaches, ours and TSP’s, are quite different. The underlying assumptions are as follows. In Ref. 15, the main result follows from a claim that is made on the function
f共h,t兲=p↓共h,t兲−p↑共h,t兲, where p↑共h,t兲 and p↓共h,t兲 are the number densities of up and down spins, respectively, with a dipolar fieldh acting on them at timet. In its latest form,16 the claim is that the scaling formf共h,t兲=f共h/t2兲“was found to be valid in our MC关Monte Carlo兴simulations for different lattice types…” when 1ⱗ⌫tandm˜Ⰶ1. Since, the magneti-zationmand f共h,t兲are clearly related by m共t兲= −兰dhf共h,t兲, a
冑
t relaxation follows immediately. As we have shown recently,13the above scaling form holds approximately for sc and Fe8lattices共as defined in Ref. 18兲, but not in general. It fails, for example, for fcc, bcc, and diamond lattices.19 Our theory also gives the time evolution of f共h,t兲, but follows from a more fundamental assumption: that the dipolar field on any one given site changes by somerandomamount⌬h, whenever a spin flips somewhere else for the first time, and that ⌬h follows a Lorentzian distribution 共for more details see Secs. II A and III A, and Ref. 14兲. The integro-differential equations for the evolution of f共h,t兲 and of the magnetization that obtain in our theory follow from this as-sumption.Unfortunately, as far as we know, only experiments on a crystalline Fe8structure have thus far been performed. How-ever, MC simulations have been performed for various fully occupied cubic lattices, which have given values of p that agree with our predictions.13,14 This paper’s first aim is to extract from our theory how the magnetization is supposed to relax in Fe8, compare this with experiment over the time span where published experimental data exist,6 and make predictions for later times. It is also our purpose to predict, and check with MC simulations, howm relaxes with t for other spatial distributions, namely, under full spatial disorder, thus providing another test for our theory.
The holes observed in the experiments described above6 are also of interest. They correspond to “wells” that develop in the function f共h,t兲=p↓共h,t兲−p↑共h,t兲 共defined above兲. From the relationm共t兲= −兰dh f共h,t兲, the time evolution ofm
follows, butf共h,t兲provides additional information about the magnetic evolution of the system that m共t兲 does not.20 For short times, that is, for⌫tⱗ1, the hole’s width is equal to
w.21 This was first surmised by Wernsdorferet al.6 to pro-pose a number, approximately 10 mK, for the tunneling en-ergy windoww. However, we know of no published data for the hole line shape evolution well into the intermediate time range, that is, for 1Ⰶ⌫t. Our second aim is to fill this gap. To this end, we work out from our theory the time evolution of the hole line shapes in this time stage in fully occupied cubic systems and Fe8 crystals, and check the results obtained against our MC results. We also obtain results of a more general nature for the hole line shape. Before we state our results, we specify the model.
A. The model
All spins are on a lattice, they point along the easy aniso-tropy axis, and interact as magnetic dipoles. We consider both fully and partially occupied lattices. Let the magnetic field at sitei produced by spin Sj at site j be given, in the usual notation, by hij=hd共a/rij兲3关1 − 3共zij/rij兲2兴, where rij is the distance between thei and j sites, a is the distance be-tween nearest neighbor sites, hd=共0/ 4兲gBS/a3, and Si = ±S for alli. Furthermore, let the magnetic fieldhiat sitei be given byhi=兺jhij, where 兺j is over all occupied lattice sites. The tunnel window size and tunneling rate⌫ are de-fined next. At very low temperature, that is, if kBT
ⱗ0.1U/S, a spin can flip only if the field h acting on it satisfies 兩h兩⬍hw.22 The flipping rate is ⌫ if upon tunneling the energy decreases, but if the energy increases by⌬E, then the rate is ⌫exp共−⌬E/kBT兲, following detailed balance. 关Even though 兩⌬E兩⬍w, and usually kBTⰇw, exp共−⌬E/kBT兲 is not quite equal to 1, and this makes a dif-ference after a sufficiently long time, as is shown below.兴We also simulate relaxation processes in which the energyE is assumed to remain constant共such as if no spin-lattice relax-ation takes place兲. Then, we assume a value ofTis such that
Eremains approximately constant. No tunneling window re-striction applies for spin flips ifkBTⲏU/S.
We let p共h,t兲 be the probability density function 共PDF兲 that any one given spin have fieldhat timet, letp0共h兲be the same distribution for a completely random spin configura-tion, and let
=关
冑
2p0共0兲兴−1. 共1兲For a Gaussian field distribution, is equal to the dipole field rms value␦hfor a random spin configuration共see Table I兲, but this is not so in general. Values ofthat follow from MC simulations for cubic and Fe8 lattices with randomly oriented spins are given in Table I. Finally, let h0
=共82/ 35/2兲hd˜n, where˜n is the number of dipoles per unit cubic cell.25Randomly oriented spins on a cubic lattice give a Lorentzian field distribution ofh0half width at half maxi-TABLE I. Quantitiesh0,, and
冑具
h2典0are given for randomly oriented spins on the lattices specified. h0=˜8n 2/ 35/2, ⬅关
冑
2p共0兲兴−1, where p共0兲 stands for the probability distribution function ath= 0, and冑具
h2典0is the rms spatial average of the dipolar field. For all lattices except for Fe8,h0,, and
冑具
h2典0are given in terms ofhd. All lattices are fully occupied, except for the “dilutesc,” which stands for a sc lattice with˜nⰆ1 sites occupied. Finally,
␣= 82/ 35/2.
Lattice h0
冑具
h2典0 sc ␣ 3.655 3.83共2兲
bcc 2␣ 3.864 4.03共2兲
fcc 4␣ 8.303 8.44共2兲
Fe8a 47共1兲mT 46共1兲mT 31共1兲mT
Dilute sc ␣˜n
冑
/ 2h0aWe assume an easy anisotropy axis as given in Refs. 23, 24, and
mum if˜nⰆ1.26Values we will be using forh
0are given in Table I. From here on, unless otherwise stated, all magnetic fields and energies are given in termshdandgBhdS, respec-tively.
B. Plan and main results
Section II is devoted to the relaxation of the magnetiza-tion. The equations from our theory14 which we use to cal-culate the time evolution of the magnetization are restated in Sec. II A. The results that follow from them for Fe8crystals are shown to agree rather well with experiment and with our own MC results in Sec. II B. The evolution we predict for
m共t兲as well as our MC results cover a time span that is two orders of magnitude longer than the experimental one E共E⬇20 min兲. Letwbe the end time for the regime where
m⬀tp. We showed in in Ref. 14 thatw⬇⌫−1共/h
w兲1/p, from which we obtainw⬃10E. For wⱗt, the evolution ofm共t兲
is shown to depend sensitively onTif good thermal contact with a heat reservoir is assumed. If on the other hand we assume constant energy processes共i.e., no spin-lattice relax-ation兲, thenmlevels off, if only temporarily, to a stationary valuemswhenwⱗt. The value ofmswe obtain from theory is unrelated to the equilibrium value of m, which only ob-tains much later. This stage, whenm→ms, sets in after most spins in the system have tunneled at least once after the magnetic field is applied. Simulations bear this out. We also obtain, from theory as well as from MC simulations,m共t兲for spatially disordered systems. More specifically, we make a random selection of a fraction˜n ofL⫻L⫻L sc lattice sites and place spins on them. For˜nⱗ0.1, we assume full disor-der. Then, theory predicts a magnetic relaxation that bears no resemblance to a
冑
t rule, not even to the tp rule that we obtain for fully occupied sc lattices. Instead, for˜nⱗ0.1,m ˜ ⯝80hw
e−共
t/m兲q, 共2兲
approximately, where m⯝106⌫−1 and q⯝−0.105. Results from our MC simulations are in fair agreement with this. In Sec. III we report results for the hole line shapes in fully occupied crystal lattices. The main results for the line shapes, which are derived in Sec. III A, follow. When 1Ⰶ⌫t
Ⰶ共/hw兲1/pandhwⰆ兩h+H兩Ⰶ,
f共h,t兲/f共h,0兲⬀兩兩1/p−1 共3兲 if兩兩Ⰶ1, where
⬅ h+H
hw共⌫t兲p
, 共4兲
His an applied field, andp is given by
p−1sinp=
冑
2/h0. 共5兲 In Sec. III A we also derive relation for hole line shapes that hold over longer time spans: 1ⱗ⌫tⱗ共/w兲1/p. In Sec. III B we apply these results to published6 experimental data for Fe8and to MC data for Fe8 as well as to fully occupied fcc lattices. Finally, concluding remarks appear in Sec. IV.II. RELAXATION OF THE MAGNETIZATION
A. Theory
We first describe a stochastic model which helps to under-stand the physics of the problem as well as the statistical assumption we have made in order to solve it. Consider two tracks, both filled with particles. Let there be one particle on the “up track” for each up spin on a lattice, and one particle in the “down track” for each down spin. Let all particles on each track be ordered according to the value of the magnetic field H+h acting on each spin. To mimic tunneling, ran-domly select a particle within the tunnel window, that is, a particle satisfying −hw⬍H+h⬍hw, whether on the up or down track, and move it to the “point”H+hon the opposite track. In order to mimic the effect such a spin flip has on the dipolar fields acting on other spins, draw a random value of
⌬h for each particle from a Lorentzian distribution of half-width h0/N, where N is the total number of spins, and let
h→h+⌬h if the particle that just shifted track has done so for the first time. This latter proviso is related to the fact that no effect on the dipolar field follows when the same spin flips twice 共for a more detailed explanation, see Ref. 14兲. Repeat this whole process at every tick of a clock. Clearly, the whole process stops when all particles have jumped track at least once.
We can get a feeling for the relevance of lattice structure or spatial spin distribution from the following simple consid-eration. Np共0兲2hw is the number of particles in the tunnel window, which is the number of times the clock ticks in time
⌫−1, which, in turn, would be the relaxation time for the equilibration of the number of up and down particles in the tunneling window if there were no “field shifting.” Now, the median field shift, or diffusion length, in time ⌫−1 is 2p共0兲h0hw, that is, a fractionp共0兲h0 of the tunnel window’s width. Thus,p共0兲h0is a measure of the amount by which the unbalance between up and down particles is restored in the tunnel window in time ⌫−1. Therefore, the relaxation rate clearly depends onp共0兲h0, which in turn depends on lattice structure. This shows why the latter is relevant to the relax-ation of the magnetizrelax-ation.
The simple statistical assumption above has enabled us to derive14the equations we need for the calculation of the time evolution of the magnetization. These equations, which we reproduce in a compact form immediately below, give the magnetizationm共t兲at timet, andn共t兲, the fractional number of spins that flip at least once in time t. We first recall an important ingredient of the theory for ZFC experiments:14 the energy per spin at the time when the system is quenched, which we refer to as the “annealing energy,” is −a. Letx1 =mgBS具h2典0/共ahwH兲 and x1= 2共/hw兲共1 −m/m0兲, for ZFC and FC experiments 共defined in the Introduction兲, re-spectively, andx2=n/hwfor both FC and ZFC experiments. The desired equation follows,
dxj
dt ⯝aj
冑
2 −bj
冕
0 t
ddxj共兲 d
1
共t−兲+ 1, 共6兲
共t
⬘
兲 ⯝min冉
h02x2共t
⬘
兲,冑
2hwx2共t
⬘
兲冊
, 共7兲a1= 4, b1= 2. In order to obtain x1, Eq. 共6兲 must first be solved for j= 2, lettinga2= 1 andb2= 1, in order to then use
x2共t兲in Eq.共7兲, and thus enable substitution ofinto Eq.共6兲 for j= 1.
The theory applies ifhwⰆ and the energy is constant, that is, if no energy transfer to the phonon bath takes place. This is also approximately so ifkTⰇw. If the constant en-ergy condition is not met in a ZFC experiment, a linear in time magnetization relaxation that is thermally driven takes over beforem共t兲→ms.14The theory also gives
ms⯝2Ha/共gSB具h2典0兲. 共8兲 Note that the definition ofx1共t兲, together with Eqs.共6兲–共8兲 imply that the time variation ofm共t兲/msin a ZFC experiment is the same as 1 −m/m0in a FC experiment.
A few remarks about Eqs.共6兲and共7兲are in order. Clearly,
x1共t兲only depends on two parameters: h0/ and/hw. The latter only comes into play at the later portion of the time evolution, whenn共t兲⬎22/共h
0
2兲, which is whenm共t兲starts leveling off. Now, 0.4ⱗ/h0艋
冑
/ 2 for all cubic lattices, whether fully occupied or not. It follows that leveling off ofm共t兲 is triggered some time共see below兲while n共t兲艋1. This has to do with the fact that a spin flip contributes to hole widening in f共h,t兲the first time it takes place after the field is switched on. When a spin flips a second time, it only returns to its initial state, thus canceling the effect of the first flip. Sincemis proportional to the width of the hole inf共h,t兲
when ⌫tⲏ1 it follows that m共t兲 becomes constant when n
⬃1. The time whenm共t兲levels off is illustrated in Fig. 1, for fully occupied fcc lattices. For instance, forhw= 0.01, theory
predicts m共t兲 to start leveling off when n共t兲⯝0.11 since 22/共h
0
2兲⯝0.11 then. Thus, a significant portion of the evo-lution, up ton⬃1, only depends onh0/. Information about the lattice or degree of spatial disorder only comes into the equations through this number. Other numbers, such ashw,
a, andHonly come into the proportionality factor between
x1and eithermor 1 −m/m0. The temperature does not enter anywhere into the equations.
We have shown analytically in Ref. 14 thatm⬀tp andn
⬀tp satisfy Eq. 共6兲 for j= 1 and 2 in the 1Ⰶ⌫tⰆ共/h w兲1/p time range if p is given by Eq. 共5兲. Numerical solutions of Eq. 共6兲 show that m⬀tp ensues in the wider 1ⱗ⌫t
ⱗ共/hw兲1/ptime span for all fully occupied cubic lattices.14 We show below that this is also so for a fully occupied Fe8 lattice.
Spatially disordered systems behave differently. Consider a very small fraction 共n˜Ⰶ1兲 of sites of a sc lattice to be occuppied by randomly oriented spins. Then, a Lorentzian dipolar magnetic field distribution26of half-widthh0ensues. It follows then, from the definition of and of h0, that /h0=
冑
/ 2. The number p= 0 follows then from Eq. 共5兲. Recall, however, that theory implies that this ensues only when ⌫tⰇ1. For earlier times, more specifically, for all ⌫tⲏ1, we find that the numerical solution from Eq.共6兲is well fitted by Eq.共2兲ifn˜ⱗ0.1. Numerical solutions of Eq.共6兲are plotted in Fig. 2共a兲forn˜= 1 and 0.6, both forhw= 0.02 and in Fig. 2共b兲 for 共1兲 n˜= 0.1 and hw= 0.02, and 共2兲 ˜n= 0.03 and
hw= 0.006. Note that the same solution obtains for the latter two cases. We come back to these figures in Sec. II B.
B. Comparison with experiments and simulations
We first make use of Eqs.共6兲and共7兲to obtain m共t兲for Fe8. Some numbers must first be fed into Eqs. 共6兲and 共7兲. For hw, we use27 0.8 mT, as given in Ref. 6. We use ⌫ = 0.04 s−1共see Refs. 14 and 21兲. With the numbers given in Table I for andh0, we obtainx1共t兲 andx2共t兲 numerically from Eqs.共6兲and共7兲. Finally, the value of −a, the annealing energy,14is needed in order to obtainmfromx
1. Not know-ing a, we treat it as a fitting parameter. We find a ⯝36 mK fits best the experimental data points from Ref. 6, which are shown in Fig. 3 for a few applied fields. The energy −36 mK may be compared to the approximate value −500 mK of the ground-state energy.28
The MC data points shown in Fig. 3 follow from simula-tions in which the system first evolves at some high tempera-ture共a few kelvin兲for a short time共less than 1 MC sweep兲 until the energy equals −36 mK. At such temperatures, Fe8 cluster spins are not forced to tunnel through the ground-state doublet. Accordingly, all spins are allowed to flip, re-gardless of the dipolar field acting on them. We explore dif-ferent scenarios after quenching. In our theory, we assume no energy exchange takes place between the spin system and a heat reservoir. We have also performed MC simulations un-der this assumption. This is approximately realized for the time range exhibited in Fig. 3 by flipping only spins within the tunnel window, and then with equal probabilities for up-ward or downup-ward flips. If, on the other hand, heat exchange does take place readily, as one gathers from Ref. 29, where FIG. 1. 共Color online兲m˜ andn共the fractional number of spins
that have flipped in timetat least once兲versus time. Full lines are for theory and symbols stand for MC results for 65 536 spins that are initially up or down at random, with probabilities 0.6 and 0.4, respectively, on a fcc lattice.⽧共䊊兲stand form˜ forhw= 0.1共0.01兲;
䊐stand fornforhw= 0.01. The dash-dotted lines stand for slopes
p= 0.73, andp= 0.5, which follow for fcc lattices from Eq.共5兲and from Ref.关15兴, respectively. Data points taken from Ref.关16兴are shown as〫. Our data points for ⌫t⬍1024 come from averages over 1400 MC runs, and for 100 runs fort⬎1200.
heat exchange rates that are comparable to⌫are found, then detailed balance should be enforced in MC simulations. The results of doing this lead to the plots shown in Fig. 3 forT
= 40 and 300 mK.
Results obtained from theory, in Sec. II A for very disor-dered systems are shown in Fig. 2共b兲 for ˜n= 0.1 and hw = 0.02 and for ˜n= 0.03 and hw= 0.006. Monte Carlo data points are also shown for the same values of˜n and of hw. Data points forn˜= 0.1 andhw= 0.02 fall on top of data points for˜n= 0.03 and hw= 0.006. This is as expected, since ⬀˜n, for full spatial disorder, implies that/hwhas the same value in both cases and theory predicts independence from any other parameter. The fitting function from Eq.共2兲falls right on top of the curve for theory in Fig. 2共b兲, and cannot there-fore be shown separately.
Finally, we consider size effects. For a 16⫻16⫻16 lattice and˜n= 0.1, for instance, approximately 410 spins make up the system. Of these, only approximately a fraction 2p共0兲hw are within the tunnel window. That is, approximately 820hw/h0spins, which only amounts to some 12 spins, are within the tunnel window. For this reason, we also simulated 32⫻32⫻32 and 64⫻64⫻64 lattices for˜n= 0.1 and 0.03, respectively. Monte Carlo data points are shown in Fig. 2. Clearly, no significant size effects are observed.
III. TIME EVOLUTION OF THE LINE SHAPE
In this section we first derive some results for f共h,t兲that are valid wheneverm⬀tpholds. We know from Ref. 14 and from the previous section thatm⬀tp holds in the time span 1ⱗ⌫tⱗ共/hw兲1/p for fully occupied sc, fcc, bcc, and Fe8 lattices, but we now know it is not so for spatially random systems. The results we derive below are applied to fully occupied Fe8 and fcc lattices and are compared to results from experiment共for Fe8兲and from MC simulations.
A. Theory
The starting points for the derivation are the following two equations, from Ref. 14:
g共h,t兲 ⯝
冕
0 t
ddm共兲
d G共h+H,t−兲 共9兲
and
G共h,H,t−兲 ⯝ u共t−兲
关共h+H兲2+u共t−兲2兴, 共10兲 where g共h,t兲⬅f共h, 0兲−f共h,t兲, u共t−兲⬅h0n共t−兲, and
n共t−兲is the fractional number of spins that flip at least once in timet−. The rationale for these two equations is given next, but关if兩h+H兩Ⰶand⌫tⰆ共/hw兲1/p兴Eqs.共9兲and共10兲 also follow from Eqs.共13兲and共14兲of Ref. 14, respectively. FIG. 2.共Color online兲 共a兲Magnetization versus⌫t. Lines are for
theory and symbols are for data from MC simulations of L⫻L
⫻L sc lattices with a fraction˜nof their sites occupied. For˜n= 0.6 the dashed line is for theory,〫and⽧are for MC data forL= 32 and 16, respectively. Forn˜= 1 the full line is for theory,䉭and 䉱 are for MC data for L= 32 and 16, respectively. All symbols are from averages of 800 MC runs while⌫t⬍103 and 40 MC runs when⌫t⬎103.共b兲Same as in共a兲but for˜n= 0.1 andh
w= 0.02, and
for˜n= 0.03 andhw= 0.006. The full line is for theory for both values
of˜. Forn ˜n= 0.03,䊊 and쎲stand for MC data for L= 64 and 32, respectively. For˜n= 0.1,䉯and䉴stand for MC data forL= 32 and 16, respectively. As in共a兲, all symbols are from averages of 800 MC runs while⌫t⬍103and 40 MC runs when⌫t⬎103.
FIG. 3.共Color online兲Magnetization versus time in hours. Units formare such thatm= 1 for full polarization.〫,䊐, and䊊are for experimental data taken from Ref. 6 for H= 3.92, 2.24, and 1.12 mT, respectively. Full lines are from our MC simulations for kTⰇw, and dashed lines are for theoretical predictions, that is, from Eqs.共6兲and共7兲. Data points that follow from MC simulations forH= 3.92 mT are also shown forT= 40共⽧兲and 300 mK共䉭兲. We assumed共Refs. 6, 14, and 21兲hw= 0.8 mT and used the values of andh0that are given in Table I. In the simulations, the initial state was prepared atT= 2 K. At this temperature we allowed the simu-lation to proceed in time up to the point when the energy of the system reached −36 mK, which is 0.07 of the ground-state energy. This is the value ofa we used in order to relate x1 and m, just above Eq.共6兲. We treat one MC sweep as⌫t= 1 and assume共Ref. 14兲⌫= 0.04 s−1in order to convert MC sweeps to hours. Note that
Assume that, between timestand, a fractionn共t−兲 of all spins flip at least once and thatn共t−兲Ⰶ1. This can later be checked to be satisfied if tⰆw, where w⬅⌫−1共/h
w兲1/p. Then, Eq. 共10兲 gives the probability density G共h,H,t−兲 that, at timet, the field is h+H at a site where the field at time was 0.26 To understand Eq. 共9兲, note first that the definition of g共h,t兲 implies that g共h,t兲 must satisfy 兰dhg共h,t兲=m共t兲−m共0兲. Equation 共9兲 does give m共t兲−m共0兲 =兰ddm/d, since兰dhG共h,H,t−兲= 1. Similarly, a variation in the magnetization共dm/d兲dcoming from some spin flip-ping between timesand+d, when the fieldh+Hacting on them was within the tunnel window, contributes tog共h,t兲
with a Lorentzian curve whose width is h0n共t−兲, where
n共t−兲/ 2 is approximately the fraction of the total number of sites where spins at timetpoint opposite to the way they did at time.14Furthermore, the area under the Lorentzian must be given by共dm/d兲d. That explains Eq.共9兲.
We make use ofn共t−兲⯝˜m共t−兲,14and of
m˜共兲 ⯝1.1hw 共⌫兲
p
, 共11兲
which has been shown to hold for all fully occupied cubic and Fe8 lattices in Ref. 14 and in Sec. II B, respectively, while 1ⱗ共⌫兲pⱗ/hw. We then divide byf共h, 0兲, which, for
hⰆ is approximately given by m0/
冑
2.14 Finally, the change of variable x⬅/t brings, if 1Ⰶ⌫tⰆ共/hw兲1/p andhwⰆ兩h+H兩Ⰶ, Eq. 共9兲into
f共h,t兲 f共h,0兲⯝1 −
sinp
冕
0 1dxxp−1 共1 −x兲
p
␣2+共1 −x兲2p, 共12兲
where␣⯝0.8共/h0兲2. Both Eqs.共3兲and共12兲show that the fieldh scales astpin the hole line shapes.
Finally, to obtain Eq. 共3兲 from Eq. 共12兲, note first that
f共0 ,t兲/f共0 , 0兲→0 as t→⬁ in Eq. 共12兲, since the integral therein equals/ sinpif= 0.30Then, breaking up the in-tegration interval into two pieces, 共1兲 from 0 to 共␣2兲1/2p, and共2兲from共␣2兲1/2pto 1, and expandingxp/共␣2+x2p兲in powers of⑀and 1 /⑀in the first and second integration inter-vals, respectively, where ⑀⬅x2p/␣2, gives兩兩1/p−1 for the leading term, which is the desired result, that is, Eq.共3兲.
B. Comparison with experiments and simulations
In this section we test our results, that is, the validity of Eqs.共3兲 and共12兲, against experiments6and against our MC simulations.
We first apply Eqs. 共3兲 and 共12兲 to Fe8. From Table I, /h0= 0.66 follows. Substitution of this number into Eq.共5兲 givesp= 0.58. Knowing the value ofp enables us to plot the data points for Fe8shown in Fig. 4共a兲. Unfortunately, data for holes in Fe8have only been published fort艋40 s, that is, for
⌫tⱗ1.6, a time which falls short of the validity range for Eqs.共3兲 and共12兲. Still, one can appreciate in Fig. 4共a兲how the data points seem to approach the theory curve for
f共h,t兲/f共h, 0兲 astincreases up to ⌫tⱗ1.6.
In order to see how this would go for longer times, we have used our model to simulate an experiment on Fe8. The
results are shown in Fig. 4共b兲. We have let one MC sweep equal⌫t= 1, which, by the argument given above, impliest
⯝25 s for Fe8. The agreement with theory is remarkable. This is better appreciated in the log-log plot shown, with the same data, in Figs. 5共a兲and 5共b兲. On the other hand, rescal-ing these plots, usrescal-ing p= 1 / 2 gives rise to some data point scatter, but not sufficiently large to convincingly rule out p
= 1 / 2. This is not too surprising, given the small difference between p= 1 / 2 and the value p= 0.58 that is given by Eq. 共5兲. Still, one might have hoped that these data would have been sufficient to discriminate between Eq. 共3兲, where is raised to the 1 /p− 1 power and the Lorentzian curve of Ref. 15. Again, data for smaller values ofwould be required for this. We know of no other experimental results for hole dig-ging we can make use of. As far as we know, all other re-ported experiments for SMM systems start from strongly po-larized initial states.20
Consequently, we decided to do simulations of SMM’s in fcc lattices, because Eq. 共5兲 gives then a value 0.73 for p, which differs significantly from 1 / 2. We are now at liberty to choose the value ofhw. In order to be able to obtain hole line shapes down to rather small values of, and still meet the validity criterion for Eqs.共3兲and共12兲, we lethwtake values down to 0.01.
We show how m varies with t in Fig. 1 in a simulated experiment in which all spins are up and down with prob-FIG. 4. 共Color online兲 共a兲fversus, defined in Eq.共4兲, for the times shown. All symbols stand for Wernsdorfer’s experimental data共Ref. 6兲for Fe8for兩h兩⬎2hw. Thin lines are for data points for
兩h兩⬍hw, to which Eq.共3兲does not apply. The full line stands for Eq. 共12兲, usinghw= 0.8 mT andp= 0.58, from Eq.共5兲, and/h0= 0.66, from Table I.共b兲 Same as in共a兲 but for MC simulations of Fe8. Points from within the tunnel window have been excluded. Times are given in seconds, after converting each MC sweep into 25 s, which is the value we assign to⌫−1. All symbols stand for averages over 1.6⫻105MC runs of systems of 16⫻16⫻16 spins. The full line stands for Eq.共12兲.
abilities 0.6 and 0.4 in the initial state, and evolve thereafter with no applied field. For⌫tⱗ1 共not shown兲,m˜⬀⌫t. Note that m˜⬀共⌫t兲p up to ⌫t⬃共/h
w兲1/p, which for hw= 0.01, for instance, ⌫t⬃9⫻103, as predicted.14 Note also the good agreeement with the value 0.73 which Eq.共5兲gives for p.
Hole line shapes obtained from MC simulations are shown in Figs. 6共a兲and 6共b兲. The nice agreement with theory is reassuring. Similar plots but usingp= 0.5 give unsatisfac-torily wide data point scatter. The data clearly follow Eq.共3兲 forⰆ1 and deviate sharply from a Lorentzian line shape. We do not exhibit results for sc or bcc lattices, but we have found them to follow our predictions equally well.
IV. CONCLUSIONS
Results that follow from our theory for the relaxation of the magnetization of interacting SMM’s are reported. They are in fair agreement with the experimental relaxation of the magnetization observed in Fe8 as well as with our own MC results for Fe8and for other lattices. Furthermore, we make some predictions for Fe8that can be checked experimentally. Experiments following the lead of Ref. 6 would be made feasible by the application of a transverse field H⬜ of ap-proximately 0.3 T, since ⌫, which increases as ⌬2, would then increase by a factor of roughly 50 ifH⬜is applied along the easier magnetization direction on thexyplane共see Figs. 2 and 3 of Ref. 4兲. This would in effect approximately reduce the time scale in Fig. 3 from hours to minutes. Comparison of experimental results with MC data shown in Fig. 3 would
be interesting. It would, for instance, show whether heat ex-change takes place readily, as inferred in Ref. 29 from nuclear magnetic resonance experiments, or not.
We have also shown, from theory and MC simulations, that the magnetization of spatially disordered SMM’s re-laxes, not as any power of time, but approximately as given by Eq. 共2兲. A counterintuitive prediction that follows from our theory and from MC simulations can be gathered from Figs. 2共a兲 and 2共b兲. One might have thought that dilution would lead to weaker dipole interactions and, consequently, to unhindered, faster relaxation. Instead, the opposite effect takes place for 0.1ⱗ˜n艋1 after some time.
Line shapes that develop in crystals of Fe8 clusters have been obtained from our theory. We have shown thatf共h,t兲is only a function of h/tp for all 兩h兩⬎hw, that is, for all h outside the tunnel window. This is the main content of Eq. 共12兲. Furthermore, we have shown that data points from ex-periments on Fe8, taken from Ref. 6, as well as results from MC simulations we have performed for the same system, follow this rule. Scaling also ensues for the data from our MC simulations of SMM’s on fcc lattices for p= 0.73, as given by Eq.共5兲, but not for the otherwise predicted13,15,16
p= 1 / 2 value that is supposed to hold universally. We have also shown that f共h,t兲⬃兩h/tp兩共1/p兲−1 if h
w/
ⰆⰆ1 andhis outside the tunnel window. Again, this is in agreement with experimental and MC results for Fe8 关see FIG. 5. 共Color online兲 共a兲 Everything is as in Fig. 4共a兲, except
that a log-log scale is used here. 共b兲 Same as in 共a兲 but for MC simulations, instead of experiments. Everything else is as in Fig. 4共b兲. The dashed line stands for1/p−1, as predicted by Eq.共3兲for p= 0.58, for Fe8.
FIG. 6.共Color online兲 共a兲f共h,t兲/f共h, 0兲versus, defined in Eq.
共4兲, for the shown times. Times are in MC sweeps.hw= 0.01.
Sym-bols stand for averages over 1400 MC runs for 65 536 spins in a fcc lattice. The full line is from Eq. 共12兲. As in Fig. 4, initially, all 65 536 spins are randomly up or down with probabilities 0.6 and 0.4, respectively.共b兲Same as in共a兲but in a log-log scale. The solid line stands for Eq.共12兲 and the dashed line is for the best-fitting Lorentzian curve. The dash-dotted line stands for 1/p−1, as pre-dicted by Eq.共3兲, forp= 0.73, given by Eq.共5兲for fcc lattices.
Figs. 5共a兲 and 5共b兲兴, fcc关see Figs. 6共a兲 and 6共b兲兴, and 共not shown兲sc and bcc lattices. A rough argument that explains why holes line shapes are not Lorentzian follows. Note first that while field distributions from dilute systems of dipoles are indeed Lorentzian,26only spin flips that take place after timetcontribute to the diffusion of a hole that was “dug” at timet. Since a full hole is only dug gradually in the course of time, a sum of Lorentzian functions of h 关see Eq.共12兲兴 of various widths is expected. Not surprisingly, a Lorentzian function does not ensue for f共h,t兲 关see, Eq. 共3兲兴. Here,
ex-perimental data for holes in the hundreds of seconds time range, over which the magnetization has already been ob-served experimentally, would be helpful.
ACKNOWLEDGMENT
Financial support from Grant No. BFM2003-03919/FISI, from the Ministerio de Ciencia y Tecnología of Spain, is gratefully acknowledged.
*Email address: [email protected]; URL: http://Pipe.Unizar.Es/˜jff †Email address: [email protected]
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