Magnetic relaxation in molecular magnets under a sweeping field is studied by taking into account local stray fields. It is found that the randomness of local stray field leads to a distribution of the relaxation rate which subsequently makes the relaxation deviate from the exponential law as predicted by the Landau-Zener model such that the Landau-Zener method needs to be revised to deduce an exact tunneling splitting. The tunneling splitting and distribution width of local stray fields are derived from the experimental data for molecular magnets Fe8.published_or_final_versio

Chen, ZD

Shen, SQ

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Title Nonexponential relaxation and quantum tunnel splitting in themolecular magnet Fe8Author(s) Chen, ZD; Shen, SQCitation Physical Review B - Condensed Matter And Materials Physics,2003, v. 67 n. 1, p. 124081-124084Issued Date 2003URL http://hdl.handle.net/10722/43373Rights Creative Commons: Attribution 3.0 Hong Kong LicensePHYSICAL REVIEW B 67, 012408 ~2003!Nonexponential relaxation and quantum tunnel splitting in the molecular magnet Fe8Zhi-De Chen1,2 and Shun-Qing Shen11Department of Physics, University of Hong Kong, Pokfulam Rd., Hong Kong2Department of Physics and Institute of Modern Condensed Matter Physics, Guangzhou University, Guangzhou 510405, China~Received 19 September 2002; published 29 January 2003!Magnetic relaxation in molecular magnets under a sweeping field is studied by taking into account localstray fields. It is found that the randomness of local stray field leads to a distribution of the relaxation ratewhich subsequently makes the relaxation deviate from the exponential law as predicted by the Landau-Zenermodel such that the Landau-Zener method needs to be revised to deduce an exact tunneling splitting. Thetunneling splitting and distribution width of local stray fields are derived from the experimental data formolecular magnets Fe8.DOI: 10.1103/PhysRevB.67.012408 PACS number~s!: 75.45.1j, 75.50.XxMagnetic relaxation by quantum tunneling in high spinmolecular magnets has become an attractive field of researchin recent years.1–15 One of the well-studied systems is octa-nuclear iron ~III! oxohydroxo clusters Fe8, which has a well-defined temperature-independent region below 0.36 K.2–6Theoretically the key to understanding magnetic relaxationby the quantum tunneling is the tunnel splitting. The tunnelsplitting and the magnetization relaxation are connected viathe Landau-Zener model.9,16,17 Since the Landau-Zener tran-sition rate is explicitly related to the tunnel splitting, themeasurement of the change of the magnetization after onesweep over the resonant point can give the tunnel splitting.17Usually the tunnel splitting is very small ~like the ground-state tunneling of Fe8), and the transition rate due to tunnel-ing is also very low. One sweep cannot lead to an observablemagnetization relaxation. In this case, the tunnel splitting isdetermined by the Landau-Zener method:3–5 multisweeps aredone and the tunnel splitting is deduced from the experimen-tal data in a short time region. Up to now, the Landau-Zenermethod3–5 has served as a basic tool to study quantum tun-neling in molecular magnets and many other interesting phe-nomena including the oscillation of tunnel splitting with re-spect to the field along the hard axis and the parity effect forodd and even resonance.3,18,19 The tunnel splitting from theLandau-Zener method is found to be sweeping-rate indepen-dent and agrees with the result found by using a square-rootdecay method.3–5 However, there is still a puzzle that mag-netization relaxation under a sweeping field shows a cleardeviation from the exponential behavior as predicted by theLandau-Zener model.4,5 Such a consequence may lead to aquestion of how the tunnel splitting is deduced from theexperimental data in short time region exactly. This is themain motivation of the present paper.In the present paper, we start with the biaxial spin modelwith a local stray field to study the magnetization relaxationbehaviors. It is shown that the ‘‘uncompensated’’ transversecomponent of the local stray field leads to a distribution ofrelaxation rates which makes the relaxation in molecularmagnets follow a different mechanism as in some complexsystems like spin glass.20,21 Our analysis shows that the re-laxation is determined by two independent quantities: thetunnel splitting and the distribution width of the local strayfield. Although the magnetization relaxation deviates the ex-0163-1829/2003/67~1!/012408~4!/$20.00 67 0124ponential law we can derive the two quantities from experi-mental data of molecular magnets Fe8 successfully.The biaxial spin model for the molecular magnets Fe8with a local stray field is written as18,19,22H5K1Sz21K2Sy22gmBSB1h, ~1!where K1.K2.0, and B is the applied magnetic field. h isthe local stray field which may originate from the interac-tions between the giant spin and the environmental spins~including other giant spins or nuclear spins!. To simplify theproblem we assume that h has a Gaussian distribution withan equal distribution width in all directions,23P~h!51~2ps2!3/2exp@2~h2h0!2/2s2# . ~2!Since our main interest is the magnetization relaxation undera sweeping field, the external magnetic field is taken to beB5$Bx,0,0%: Bx5nDB6ct where n is integer, DB is thefield interval between neighboring resonant tunneling, andc5dBx /dt . In the following calculation, we take K150.321 K, K250.229 K for the molecular magnets Fe8.3When the field along the easy axis is sweeping over theresonant field nDB , the biased local stray field hx will becompensated by the sweeping field which brings spins intothe resonant tunneling and leads to a continuous relaxation.If we omit the transverse component of the local stray field,then all the spins will have the same tunneling rate inside theresonant window. The resulted relaxation is the simple expo-nential decay according to the Landau-Zener model,4,5M ~ t !5M 0e2Gt, ~3!where G5kPLZc/A and PLZ is the Landau-Zener transitionrate, PLZ512exp(2pDn2/nnc), Dn is the tunnel splitting, nn52gmB\(2s2n), k52 for n50, k51 for n51,2,3, . . . ,and A is the amplitude of the ac field used in the experiment.In the low transition rate limit, i.e., PLZ!1, which holds forthe Fe8 system,4,5 the above equation leads tolnM ~ t !M 052kpnnADn2t112 S pDngnc D2 kcA t1 .©2003 The American Physical Society08-1BRIEF REPORTS PHYSICAL REVIEW B 67, 012408 ~2003!In the Taylor series expansion, one hasM 02M ~ t !M 05kptnnADn2212 F S kptnnA D 21 p2ktgn2cAGDn41 .~4!In a short time region, we can keep the first term on the rightside of Eq. ~4!, and then the tunnel splitting Dn can be de-duced from the magnetization relaxation of the molecularmagnets.However, the tunnel splitting strongly depends on thestrength of the transverse local stray field. In other words, thelocal stray field will leads to a distribution of the tunnelingsplitting, and furthermore a distribution of the relaxationrates. Consequently, the resulted relaxation is modified asM ~ t !5M 0E dhP~h!e2G(h)t, ~5!whereG~h!5$12exp@2pDn2~h!/nnc#%kc/A . ~6!The tunnel splitting Dn2(h) can be calculated by the instantonmethod,19,23Dn2~h!.g6~h!Dn02, ~7!where the renormalized factor is caused by the local strayfield, g6(h)5@cosh(2qhy)6cos(2dnhz)#/2 and 6 stands forthe even and odd resonant tunneling.dn5gmB2K1E0p df12K2K1sin2f2gmBnDB2sK2cosf, ~8!q5gmBp/2(K1K22K22)1/2, and Dn0 is independent of thetransverse field. In the low transition rate limit,4,5 we haveM ~ t !.M 0E dhP~h!exp$2g6~h!G0t%, ~9!where G05kpDn02 /(nnA).The above equation shows that, in the low transition ratelimit, the magnetic relaxation is sweeping-rate independent,provided that the sweeping rate is large enough, namely, fordBx /dt*1.0 mT/s.4,23 In the absence of the transverse localstray field, i.e., P(h)5d(h), we have M (t).M 0e2G0t for aneven resonant tunneling and M (t)5M 0 for an odd resonanttunneling as expected. The effect of the transverse local strayfield can be observed by doing the integration in Eq. ~9!.Numerical results24 as shown in Fig. 1 show that the magne-tization relaxation deviates apparently from the exponentialdecay as the distribution width becomes larger than 0.03 T,which is in qualitative agreement with the experimental mea-surement. The nonexponential decay indicates the Landau-Zener model cannot be applied to measure the tunnelingsplitting explicitly. In fact, for a system with a distribution ofrelaxation time, the resulted relaxation can be a large varietyof shapes of decay.20,21 The above analysis shows that wecannot deduce the tunnel splitting according to Eq. ~4! when01240the relaxation deviates strongly from the exponential law. Onthe other hand, consider the local stray fields one obtainsM 02M ~ t !M 05kptnnA^Dn2&212 F S kptnnA D 21 p2ktgn2cAG ^Dn4&1 ,~10!where ^X&5*dhP(h)X . Comparing with Eq. ~4!, one cansee that the tunnel splitting determined from @M (t)2M 0#/M 0 under such an approximation is A^Dn2& instead ofDn . From Eq. ~7!, one hasA^Dn2&5Q6Dn0 , ~11!where the averaging renormalization factor due to the localstray fields isQ651A2@e2q2s2cosh~2qh0!6e22dn2scos~dnh0!#1/2.~12!Dependence of Q6 on the distribution width s is plotted inFig. 2. It is seen that Q6 increase rapidly with increasing sFIG. 1. Short time relaxation behavior for both odd and evenresonant tunneling with s50.08 T and h05s/4.FIG. 2. Illustration of the dependence of Q6 on the distributionwidth s . The dashed and solid lines represent for Q1 and Q2 ,respectively.8-2BRIEF REPORTS PHYSICAL REVIEW B 67, 012408 ~2003!and become coincident as s.0.08 T. For an even reso-nance, Q1>1, and A^Dn2& is always larger than the exacttunnel splitting Dn0, while for an odd resonance, the tunnelsplitting A^Dn2& is not always quenched once the local strayfield appears.Now we are ready to derive the tunneling splitting fromthe experimental data of Fe8. The experimental data of theground-state tunneling in Fe8 were provided byWernsdorfer.4,5 To fit the experimental data we choose G056.531024/sec and s50.05 T. Both theoretical and ex-perimental results are plotted in Fig. 3. One can find that ouranalytic result fits the experimental curve quite well. Theestimated value s50.05 T is consistent with the linewidth ofthe resonance for Fe8.8 However, the hole-digging method5,6gives a value of 0.03 T, which is smaller than what we esti-mated. The tunnel splitting from the chosen G0 is D057.8531028 K, and the renormalization factor is Q1(s50.05 T).1.292. Therefore we have A^D02&.1.0131027 K, which is very closed to 1.031027 K in the con-ventional Landau-Zener model by Wernsdorfer et al.4,5 Sowhat measured in the Landau-Zener model is A^D02& insteadof D0 in the short time limit. Clearly, D0 and A^D02& are twodifferent concepts. For Fe8, since the averaging renormaliza-tion factor is as large as 1.292, we should consider the effectfrom the local stray field. Our result shows that the magne-tization relaxation is determined by the tunnel splitting andthe distribution width of the local stray field. The later quan-tity leads to the relaxation deviating from the exponentialdecay. On the other hand, after the local stray field is intro-duced, the main modification to the kinetic equation of therelaxation is to replace Dn0 with A^Dn2& in the low transitionrate limit.It should be noted that the tunnel splitting from the ex-perimental data is sweeping-rate independent since the mag-netic relaxation is independent of the sweeping rate when c.1.0 mT/sec. From Eq. ~6! we found that, rigorously speak-ing, the magnetization relaxation depends on the sweepingrate c via the expression for G(h). However, the larger cleads to a smaller transition rate. When the low transitionFIG. 3. Relaxation curve of ground state tunneling in the Fe8molecule using a sweeping field with A57.249531022 T anddBx /dt51.4 mT/s.01240rate approximation becomes valid for a sufficiently large c,the transition rate is independent of c. This is consistent withthe experimental measurement. Another problem is that theabove analysis is based on the adiabatic approximation. It isassumed that all the spins have sufficient time to tunnel nomatter how small the tunnel splitting is. When the distribu-tion width of the local stray field is taken into account, thedistribution of tunnel splitting becomes quite large. Approxi-mately there will be a cutoff tunnel splitting Dnc.nnc/psuch that the spins with the tunnel splitting lower than Dncwill have no sufficient time to tunnel.15,25 To find out thedistribution of the tunnel splitting due to the local stray field,we have made Monte Carlo simulation for Fe8 systems withhigher-order terms in the Hamiltonian. The resulted distribu-tion of tunnel splitting is plotted in Fig. 4 for s50.05 T. Thetunnel splitting spreads over about 2 to 3 orders which ismuch narrower than that in Mn12 molecules due todislocation.15,25 As the local stray field from dipolar-dipolaror hyperfine interaction in Mn12 is stronger than that in Fe8molecules and dislocation will lead to a wider distribution ofthe tunnel splitting. The Mn12 system will have a much widerdistribution of relaxation time than Fe8 system. It impliesthat the magnetization relaxation in Mn12 system will deviatefar away from the exponential decay.In conclusion we have studied the effect of the local strayfield on the magnetic relaxation under a sweeping field inFe8 molecules. The uncompensated transverse local strayfield leads to a distribution of the transition rate such that therelaxation deviates from the exponential law. The interplayof the quantum tunneling and the distribution of the localstray field determines the magnetization relaxation. Based onthis picture we proposed a revised scheme to deduce thetunnel splitting Dn from experimental measurement insteadFIG. 4. Distribution of ground-state tunnel splitting for a Fe8model with higher-order term C(S14 1S24 ) in Hamiltonian under alocal stray field with a Gaussian distribution. (C522.9531025 K).8-3BRIEF REPORTS PHYSICAL REVIEW B 67, 012408 ~2003!of the conventional Landau-Zener method. Our conclusioncan be generalized to other molecular magnets since the localstray field due to the dipolar-dipolar or hyperfine interactionexists extensively.01240The authors thank W. Wernsdorfor for his providing theexperimental data in Fig. 3. The work was supported by agrant from the Research Grants Council of Hong Kong,China, and a CRCG grant of the University of Hong Kong.1 L. Gunther and B. Barbara, Quantum Tunnelling ofMagnetization—QTM’94 ~Kluwer, Dordrecht, 1995!.2 C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, and D. Gatteschi,Phys. Rev. Lett. 78, 4645 ~1998!.3 W. Wernsdorfer and R. Sessoli, Science 284, 133 ~1999!.4 W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia,and D. Mailly, J. Appl. Phys. 87, 5481 ~2000!.5 W. Wernsdorfer, Adv. Chem. Phys. 118, 99 ~2001!.6 W. Wernsdorfer, T. Ohm, C. Sangregorio, R. Sessoli, D. Mailly,and C. Paulsen, Phys. Rev. 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B 65,094423 ~2002!.16 L. Landau, Phys. Z. Sowjetunion 2, 46 ~1932!; C. Zener, Proc. R.Soc. London, Ser. A 137, 696 ~1932!.17 S. Miyashita, J. Phys. Soc. Jpn. 64, 3207 ~1995!; 65, 2734 ~1996!.18 A. Garg, Europhys. Lett. 22, 205 ~1993!.19 Z.D. Chen, Phys. Rev. B 65, 085313 ~2002!.20 R.G. Palmer, D.L. Stein, E. Abrahams, and P.W. Anderson, Phys.Rev. Lett. 53, 958 ~1984!.21 P. Erhart, A.M. Portis, B. Senning, and F. Waldner, J. Phys.: Con-dens. Matter 6, 2881 ~1994!; P. Erhart, B. Senning, and F. Wald-ner, ibid. 6, 2893 ~1994!.22 E.M. Chudnovsky and L. Gunther, Phys. Rev. Lett. 60, 661~1988!.23 Z.D. Chen, J.-Q. Liang, and S.Q. Shen, Phys. Rev. B 66, 092401~2002!.24 The numerical result shows that the resulted relaxation curve isnot sensitive to the distribution center. In the present paper, wepresent results for h050 or h05s/4 according to the experi-mental result ~Ref. 10!.25 K.M. Mertes, Y. Suzuki, M.P. Sarachik, Y. Paltiel, H. Shtrikman,E. Zeldov, E. Rumberger, D.N. Hendrickson, and G. 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Nonexponential relaxation and quantum tunnel splitting in the molecular magnet Fe8

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Nonexponential relaxation and quantum tunnel splitting in the molecular magnet Fe8

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