Renormalized tunnel splitting with a finite distribution in the biaxial spin
model for molecular magnets is obtained by taking into account the dipolar
interaction of enviromental spins. Oscillation of the resonant tunnel splitting
with a transverse magnetic field along the hard axis is smeared by the finite
distribution which subsequently affects the quantum steps of hysteresis curve
evaluated in terms of the modified Landau-Zener model of spin flipping induced
by the sweeping field. We conclude that the dipolar-dipolar interaction drives
decoherence of quantum tunnelling in molcular magnets Fe&#8328;, which explains
why the quenching points of tunnel spliting between odd and even resonant
tunnelling predcited theoretically were not observed experimentally.Comment: 5 pages including 3 figure and 1 table. To appear in Physical Review
  

A. Cuccoli

A. Garg

B. Barbara

D.A. Garanin

E. Rastelli

E.M. Chudnovsky

J. Liu

J.-Q. Liang

J.Q. Liang

M. Ueda

M.N. Leuenberger

N.V. Prokof’ev

R.G. Palmer

S. Miyashita

Shun-Qing Shen

T. Ohm

W. Wernsdorfer

Y.H. Jin

Z.D. Chen

Zhi-De Chen

English

arXiv

Renormalized tunnel splitting with a finite distribution in the biaxial spin model for molecular magnets is obtained by taking into account the dipolar interaction of enviromental spins. Oscillation of the resonant tunnel splitting with a transverse magnetic field along the hard axis is smeared by the finite distribution, which subsequently affects the quantum steps of the hysteresis curve evaluated in terms of the modified Landau-Zener model of spin flipping induced by the sweeping field. We conclude that the dipolar-dipolar interaction drives decoherence of quantum tunneling in the molecular magnet Fe8, which explains why the quenching points of tunnel splitting between odd and even resonant tunneling predicted theoretically were not observed experimentally.published_or_final_versio

Chen, ZD

Liang, JQ

Shen, SQ

HKU Scholars Hub

Title Suppression of quantum phase interference in the molecularmagnet Fe8 with dipolar-dipolar interactionAuthor(s) Chen, ZD; Liang, JQ; Shen, SQCitation Physical Review B - Condensed Matter And Materials Physics,2002, v. 66 n. 9, p. 924011-924014Issued Date 2002URL http://hdl.handle.net/10722/43360Rights Creative Commons: Attribution 3.0 Hong Kong LicensePHYSICAL REVIEW B 66, 092401 ~2002!Suppression of quantum phase interference in the molecular magnet Fe8with dipolar-dipolar interactionZhi-De Chen,1 J.-Q. Liang,2 and Shun-Qing Shen3,*1Department of Physics and Institute of Modern Condensed Matter Physics, Guangzhou University, Guangzhou 510405, China2Institute of Theoretical Physics and Department of Physics, Shanxi University, Taiyuan, Shanxi 030006, China3Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China~Received 16 July 2002; published 12 September 2002!Renormalized tunnel splitting with a finite distribution in the biaxial spin model for molecular magnets isobtained by taking into account the dipolar interaction of enviromental spins. Oscillation of the resonant tunnelsplitting with a transverse magnetic field along the hard axis is smeared by the finite distribution, whichsubsequently affects the quantum steps of the hysteresis curve evaluated in terms of the modified Landau-Zenermodel of spin flipping induced by the sweeping field. We conclude that the dipolar-dipolar interaction drivesdecoherence of quantum tunneling in the molecular magnet Fe8, which explains why the quenching points oftunnel splitting between odd and even resonant tunneling predicted theoretically were not observed experi-mentally.DOI: 10.1103/PhysRevB.66.092401 PACS number~s!: 75.45.1j, 03.65.Yz, 73.43.Jn, 75.50.XxMacroscopic quantum phenomena in magnetic molecularclusters have been an attractive field of research in recentyears.1–8 Octanuclear iron~III! oxo-hydroxo cluster Fe8 is ofspecial interest because it shows not only regular steps in thehysteresis curve but also oscillation of the tunnel splittingdue to quantum phase interference.3 Oscillation of tunnelsplitting of the ground state with respect to the external fieldalong the hard axis was predicted theoretically by Garg4 as aconsequence of the quantum phase interference of tunnelpaths, and it was subsequently generalized to tunneling atexcited states and resonant tunneling for the quantum transi-tion between different quantum states with its x componentof the spin Sx5210 and 102n ~along the easy axis of Fe8)recently.6 The quenching points between even and odd nhave a shift p/2. However, a serious problem—why thetheoretically predicted shift of quenching points of tunnelsplitting between odd and even resonant tunneling was notobserved in the experimental hysteresis curves,3,6,9—remainsto be solved. This is the main motivation of this paper. Herewe use the Landau-Zener model3,8–10 to describe the spinflipping induced by the sweeping field with a modified baretunnel splitting considering the dipolar interaction with envi-ronmental spins. There are two basic interactions to be con-sidered: spin-phonon and spin-spin interactions. For the mo-lecular magnets Fe8 in the mK temperature region, the spin-phonon interaction8 can be safely ignored as the spin-latticerelaxation time is extremely long.11 The interaction betweenthe big spin and the environmental spins was considered asthe main source of decoherence of tunneling in magneticmacromolecules12 and recently it was shown that the nuclearspin plays an important role in magnetic relaxation.13,14 Inthis paper, starting from the mean-field approximation, thedipolar interaction is treated as a local stray field hW ~see thefollowing! with a Gaussian distribution. The tunnel splittingin the Landau-Zener transition rate should be considered asan average over the local stray field hW . In doing so we findthat the quenching of the tunneling due to quantum interfer-0163-1829/2002/66~9!/092401~4!/$20.00 66 0924ence is suppressed by the local stray field, and the steps inthe hysteresis curve corresponding to odd resonant tunnelingare understood.We start with the biaxial spin model for the molecularmagnets Fe8.3–5 The Hamiltonian is given by15H5K1Sz21K2Sy22gmBS~B1hW !, ~1!where K1.K2.0 and B is the external magnetic field. Theterm 2gmBShW is the dipolar-dipolar interaction betweenthe magnetic molecular cluster and the environmental spins,i.e., hW 5( jJ i jSj , where the summation runs over the neigh-boring clusters. Strictly speaking, this should be a many-body problem. In this paper, hW is treated approximately as alocal stray field, hW 5( jJ i j^Sj&. Both experimental14 and theMonte Carlo studies16,17 show that hW has a random distribu-tion with a distribution width in proportion to (12uM u) andits mean value proportional to M where M is the total mag-netization of the system. Here we assume that hW has a Gauss-ian distribution with an equal distribution width in alldirections:18P~hW !51~2ps2!3/2exp@2~hW 2hW 0!2/2s2# . ~2!To simulate the experimental setup,3 the external magneticfield is taken to be B5$Bx,0,Bz%: a uniform field Bz alongthe hard axis and the sweeping field Bx on the easy axis Bx5nDB6ct where n is an integer, DB is the field intervalbetween neighboring resonant tunneling, and c5dBx /dt . Inthe following calculation, we take K150.310 K, K250.229 K, and c50.1 T/sec for the molecular magnetsFe8.3Theoretically, quantum tunneling for a spin system with-out a local stray field can be understood in the instantonmethod,4–6 the Landau-Zener model,8–10 and by diagonaliz-ing the Hamiltonian numerically.3,19 The instanton methodcan give the tunnel splitting. When the field along the easy©2002 The American Physical Society01-1BRIEF REPORTS PHYSICAL REVIEW B 66, 092401 ~2002!axis satisfies the resonant condition Bx1hx5nDB , the tran-sition rate, while the field on the easy axis sweeps over theresonant point, is given by the Landau-Zener transitionformula,8–10 PLZ512exp(2pDn2/nn), where Dn is the tunnelsplitting and nn52gmB\(2s2n)c . It should be noted that inthis way we have assumed tacitly that the tunnel splitting forall spins inside the resonant window are the same and thusall spins tunnel with the same transition rate. However, whenthe local stray field due to the dipolar-dipolar interaction istaken into account, such a picture should be modified. Arandom distribution of local stray fields like Eq. ~2! with adistribution width s;0.05 T typical for the molecular mag-nets Fe8 will block the resonant tunneling of either theground state or the low-lying excited states.13 Nevertheless,such a problem can be circumvented by using a sweepingfield along the easy field. When Bx sweeps over the resonantpoint, it will make the spins with different hx’s satisfy theresonant condition and allows continuous relaxation. Sincethe tunnel splitting is very sensitive to the transverse localfields Bz1hz , and hy ,4–6 the spins tunnel with different tun-nel splitting while Bx is sweeping over the resonant point.Consequently, the spin transition rate observed in the experi-ment should be given by^PLZ&.12exp$2p^Dn2&/nn%, ~3!where ^& represents the average over the distribution ofthe local stray field, i.e., ^Dn2&5*Dn2(hW )p(hW )dhW . Accord-ingly, the tunnel splitting extracted from the measured tran-sition rate should be A^Dn2& but not Dn . In other words, thestarting point to understand the experimental observationshould be A^Dn2& instead of Dn . The two quantities are quali-tatively different from each other as we shall show in thefollowing.The instanton method4–6 is efficient and powerful toevaluate the tunnel splitting Dn . The Lagrangian for the bi-axial model, Eq. ~1!, isL~n!52s\~12cos u!f˙ 2^nuHun& , ~4!where un& is the spin-coherent state. With the help of themapping technique, (f ,p5s\ cos u) is regarded as a pair ofcanonical variables. To calculate the excited-state tunnelingor resonant tunneling, one needs to apply the Bohr-Sommerfeld quantization rule rpdf5n\ to define the clas-sical orbits (n is an integer!. Then a propagator with bothimaginary and real time will be used to describe the tunnel-ing between two degenerate states,K~n f ,T/2;ni ,2T/2!5^n f ueiHT/\uni&5E dVexpF i\E2T/2T/2 L~n!dtG . ~5!The tunnel splitting is found by integrating over two degen-erate classical orbits. In molecular magnets Fe8 the localstray field is rather weak, i.e., gmBuhW u/(K2s)!1. We calcu-late the tunnel splitting at the nth resonant tunneling pointwith the transverse field Bz1hz and hy and obtain that09240Dn’Qn2 e2Scn$ue2qhy1e22qhy12 cos@2~sp2np/22dnhz2dnBz!#u%1/2, ~6!where Scn is the instanton action,Scn5E0p2fndfA V~f!2EnK1~12lsin2f!1~gmBnDh/s !cos f,~7!V~f!5K2s2sin2f2gmBnDhs cos f2@gmB~hz1Bz!#22K1~12l sin2f!12~gmBnDh/s !cos f, ~8!En is the energy of the nth excited state, fn is the turningpoint determined by V(p2fn)5En , Qn is the prefactor,Qn.4p/AV9~0 !~2K11gmBnDh/s !, ~9!q5gmBpl1/2/2K2(12l)1/2, l5K2 /K1, anddn5gmB2K1E0p df12sin2f2gmB2K2snDh cos f. ~10!Using the parameters in Fe8, it is found that the contributionfrom hy and hz to Scn and thus Qne2Scnis very small underthe condition gmBuhW u/(K2s)!1. The average value of Dn2 isgiven by^Dn2&’Qn024 e22Sc0n$e2q2s2~e2qh01e22qh0!12e22dn2s2cos@2~sp2np/22dnh02dnBz!#%,~11!where Qn05Qn(hz5hy50), and Sc0n 5Scn(hz5hy50). Inthe absence of a stray field, i.e., s5h050, the above expres-sion reduces toA^Dn2&us5h0505Dn~hx5hy50 !5Qn0e2Sc0nucos~sp2np/22dnBz!u,~12!which indicates the oscillation of the tunnel splitting with thetransverse field and a shift p/2 of quenching point betweenthe odd and even resonant tunneling, recovering the resultsin the previous works.3–6 This is known as a result of thequantum interference of the tunneling along two differentpaths. The qualitative difference between A^Dn2& and Dn(hx5hy50) can now be seen by comparing Eq. ~12! with Eq.~11!. In the case of Bz50 and integer spin, Eq. ~12! predictsthat odd-n resonant tunneling quenches due to the quantuminterference, while in the presence of the stray field, Eq. ~11!,gives nonzero tunnel splitting,1-2BRIEF REPORTS PHYSICAL REVIEW B 66, 092401 ~2002!A^Dn2&q.A22 Qn0e2Sc0n Ae2q2s22e22dn2s2, ~13!for h050. The quenching due to the quantum interference issuppressed by the local stray field. In another word the quan-tum tunneling for odd n is decoherenced because of the di-polar interaction with the environmental spins. The tunnelsplittings of all six resonant tunnelings for the molecularmagnets Fe8 with and without a local stray field are shown inTable I. We see that A^Dn2& for an odd n increases from zerowhile the random field becomes stronger. The random fieldalso increases the tunnel splitting of even resonant tunneling.It increases about 2.7 times as s becomes as large as 0.08 T,which resolves the puzzle that the experimental observationis about 3.0 times larger than the numerical result for thetunnel splitting.3 A detailed evolution of the tunnel splittingwith the distribution width around the topological quenchingpoints is shown in Fig. 1. As the width of the distribution isproportional to (12uM u), the calculated results for differentM are shown in Fig. 1, which are in good agreement with theexperimental observation ~see Fig. 10 in Ref. 20!. One cansee from Eq. ~11! that the main effect of h0 is to provide aninitial phase and thus shifts the oscillation. For Fe8 , h0.s/4,16 and the effect of modification for nonzero h0 isalmost omissible.The oscillation of the tunnel splitting for s510 with vari-ous distribution width s’s is shown in Fig. 2. From Fig. 2, itTABLE I. Tunnel splitting A^Dn2& ~the unit is kelvin! for Fe8 inthe case of Bz5h050.n s50.0 T s50.02 T s50.05 T s50.08 T0 8.399310210 8.547310210 1.08731029 2.312310291 0.0 2.459310 29 3.266310 29 6.450310292 3.41431028 3.47331028 4.41831028 9.393310283 0.0 2.39931027 3.18731027 6.293310274 2.01531026 2.05031026 2.60831026 5.544310265 0.0 9.87831026 1.31231025 2.591310256 6.22431025 6.33331025 8.05531025 1.71331024FIG. 1. Illustration of A^Dn2& (n50,1) around topologicalquenching points due to quantum interference with different distri-bution width s’s.09240is shown that the oscillation of the tunnel splitting due toquantum interference is suppressed by the local stray field hW .For a distribution width s50.05 T which is estimated forFe8,9,13 the oscillation of tunnel splitting with respect to thefield along the hard axis is still visible, while the oscillationis suppressed completely for the width as large as 0.08 T. Infact, when the distribution width approaches the half oscilla-tion period, the oscillation due to quantum interference dis-appears and the classical behavior—i.e., tunnel splitting in-creases monotonously with Bz—is resumed. The aboveanalysis leads to a decoherence mechanism for quantum in-terference due to the dipolar-dipolar interactions between thespins without dissipation.12The magnetization jump from the spin flipping at resonanttunneling can be calculated from the modified Landau-Zenertransition rate given in Eq. ~3!. In principle the time evolu-tion of the spin system in Eq. ~1! can be obtained by solvingthe time-dependent Schro¨dinger equation i\(]/]t)uF&5HuF&, which contains a set of (2s11) coupled differen-tial equations for the model in Eq. ~1!. It was shown19 thatthe coupled differential equations can be reduced to that ofan effective two-level system with the effective Hamiltonian.Here we haveHeff~ t !5S 2~102n !gmBct A^Dn2&/2A^Dn2&/2 10gmBct D , ~14!near the resonant condition, and the time-dependent state isgiven by uFeff&5a210(t)u210&1a102n(t)u102n&. The tun-neling splitting in Eq. ~14! is A^Dn2& instead of Dn as wediscussed. Correspondingly, the magnetization jump from thenth resonant tunneling is obtained asDM n5^FeffuSxuFeff&u t51‘2^FeffuSxuFeff&u t52‘ .~15!Numerical results are shown in Fig. 3. It is worth emphasiz-ing that the resulting jump is modified as a rather smooth onedue to the local stray field. The hysteresis curves in Fig. 3 aredrawn with the initial condition of Sx5210, i.e.,a210(t52‘)51. As is shown in Fig. 3 the steps in the hysteresisFIG. 2. The oscillation of A^D02& with different distributionwidth s’s for s510. From top to bottom: s50.08 T, 0.05 T, 0.02T, and 0.0 T.1-3BRIEF REPORTS PHYSICAL REVIEW B 66, 092401 ~2002!curve are smeared gradually with increasing the distributionwidth of the local stray field and the observed curve inexperiment3,9 can be recovered from the present theory.In this paper, the local stray field is treated as a ‘‘frozen’’one inside the resonant window. Strictly speaking, both thewidth and mean value of the distribution of the field shouldvary with the time-dependent magnetization during the reso-nant tunneling. However, it should be noted that a ‘‘frozen’’distribution is based on the validity of the Landau-Zenermodel. If the spins ‘‘feel’’ the change of the local field due tospin flipping, the spin transition rate is no longer the one inEq. ~3!. In that case, one should consider the nonlinearLandau-Zener tunneling21 and the ‘‘hole-digging’’mechanism.13,14 This indicates that our result is valid whenthe field sweeping rate is not too small such that the evolu-tion of the local field is relatively slower than the sweepingfield. Namely, the overlap time of two levels in resonancet1;Dn /(2mBSc) should be less than the characteristic re-FIG. 3. Hysteresis curves with different distribution widths’s.09240laxation time t2 due to the dipolar-dipolar interaction. Thismeans that the field sweeping rate c.c0;Dn /(2mBSct2).In Fe8,14,16 the ground-state tunneling D0;1027 K, t2;1025 sec, and c0 is estimated to be 1023 T/sec., which isin good agreement with the experimental condition.3,20 Onthe other hand, a finite distribution of tunnel splitting due tothe local stray field has a deeper impact on the magneticrelaxation. If all the spins tunnel with the same tunnelingrate, the magnetic relaxation should obey the exponentiallaw, i.e. e2Gt where G52PLZc/A , where A is amplitude ofthe ac field used in the experiment.20 Instead, in the presentpicture, there is a finite distribution of tunnel splitting p(Dn)which will lead to a finite distribution of the relaxation ratep(G) characteristic of a complex system like spin glass.22Consequently the resulting relaxation will obviously deviatefrom the simple exponential law as observed in theexperiment.20 Further analysis will be provided elsewhere.We have studied the effect of dipolar interaction betweengiant spins in the molecular magnets Fe8 in the mean-fieldapproximation which leads to a Zimman term of the spin ina local stray field. Our main observation is that the topologi-cal quench due to the quantum phase interference of tunnelpaths is suppressed by the finite distribution of the local strayfield and the steps in the hysteresis curve corresponding toodd resonant tunneling are explained theoretically. Thus weconclude that the dipolar-dipolar interaction leads to the de-coherence of quantum tunneling in Fe8. Finally it is worthpointing out that the mechanism of decoherence may not bejust limited in Fe8, but can be generalized to other molecularmagnets such as Mn12 since the local stray field due to thedipolar-dipolar and hyperfine interactions always exists.This work was supported by the Nature Science Founda-tion of China under Grant No. 10075032, the Fund ‘‘Na-nomic Science and Technology’’ of Chinese Academy of Sci-ence, a RGC grant of Hong Kong, and a CRCG grant of theUniversity of Hong Kong.*Electronic address: sshen@hkucc.hku.hk1 L. Gunther and B. Barbara, Quantum Tunnelling of Magnetization~Kluwer, Dordrecht, 1995!.2 E.M. Chudnovsky and J. Tejada, Macroscopic Quantum Tunnel-ling of the Magnetic Moment ~Cambridge University Press,Cambridge, England, 1998!.3 W. Wernsdorfer and R. Sessoli, Science 284, 133 ~1999!.4 A. Garg, Europhys. Lett. 22, 205 ~1993!.5 J.Q. Liang et al., Phys. Rev. B 61, 8856 ~2000!; Y.H. Jin et al.,ibid. 62, 3316 ~2000!.6 Z.D. Chen, Phys. Rev. B 65, 085313 ~2002!.7 E.M. Chudnovsky and D.A. Garanin, Phys. Rev. Lett. 87, 187203~2001!; D.A. Garanin and E.M. Chudnovsky, Phys. Rev. B 65,094423 ~2002!.8 M.N. Leuenberger and D. Loss, Phys. Rev. B 61, 12 200 ~2000!.9 B. Barbara et al., J. Magn. Magn. Mater. 200, 167 ~1999!.10 S. Miyashita, J. Phys. Soc. Jpn. 64, 3207 ~1995!.11 M. Ueda et al., Phys. Rev. B 66, 073309 ~2002!.12 N.V. Prokof’ev and P.C.E. Stamp, J. Phys.: Condens. Matter 5,L663 ~1993!; J. Low Temp. Phys. 113, 1147 ~1998!; Rep. Prog.Phys. 63, 669 ~2000!.13 N.V. Prokof’ev and P.C.E. Stamp, Phys. Rev. Lett. 80, 5794~1998!.14 W. Wernsdorfer et al., Phys. Rev. Lett. 82, 3903 ~1999!.15 The higher-order terms such as C(S24 1S14 ) are omitted in Eq.~1!. It is already known ~Ref. 3! that the higher-order termsincrease both the tunnel splitting and the oscillation period, butwill not change the main conclusion.16 T. Ohm, C. Sangregori, and C. Paulsen, Eur. Phys. J. B 6, 195~1998!; J. Low Temp. Phys. 113, 1141 ~1998!.17 A. Cuccoli et al., Eur. Phys. J. B 12, 39 ~1999!.18 As shown by experiment ~Ref. 14! and Monte Carlo studies~Refs. 16 and 17! the random field distribution can be of otherkinds starting from different initial states. Here the choice of theGaussian distribution is just to facilitate the calculation. Otherdistributions such as the Poisson distribution do not affect quali-tatively the main conclusion.19 E. Rastelli and A. Tassi, Phys. Rev. B 64, 064410 ~2001!.20 W. Wernsdorfer et al., J. Appl. Phys. 87, 5481 ~2000!.21 J. Liu et al., Phys. Rev. B 65, 224401 ~2002!.22 R.G. Palmer et al., Phys. Rev. Lett. 53, 958 ~1984!.1-4

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Suppression of Quantum Phase Interference in Molecular Magnets Fe&#8328;
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