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Resonant spin tunneling in small magnetic particles
LETTER TO THE EDITOR
Journal of Magnetism and Magnetic Materials 185 (1998) L267—L273
Letter to the Editor
Resonant spin tunneling in small magnetic particles Eugene M. Chudnovsky!,",* ! Physics Department, Lehman College, The City University of New York, Bedford Park Boulevard West, Bronx, New York 10468-1589, USA " Service de Physique de l+Etat Condense& , CEA Saclay, 91191 Gif sur Yvette Cedex, France Received 12 September 1997; received in revised form 30 December 1997
Abstract A new macroscopic test of spin tunneling in single-domain magnetic particles is proposed. It is shown that quantization of spin, even in particles of a considerable size, may lead to a detectable zero-field maximum in the dependence of the rate of the magnetic relaxation on the magnetic field. This effect must be especially pronounced in antiferromagnetic particles. It is argued that recent observation by several groups of the non-monotonic field dependence of the blocking temperature in ferritin may, in fact, present the experimental evidence of this effect. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 75.45.#j; 75.50.Tt Keywords: Resonant spin tunneling; Nanoparticles
This paper considers the effect of spin quantization on magnetic properties of single-domain ferromagnetic (FM) and antiferromagnetic (AFM) particles. Despite of the vast literature on small magnetic particles, this question has received very little attention [1,2]. Meanwhile, as I am going to demonstrate, even in particles of a considerable size, the descrete nature of spin leads to the effect which can be tested in a macroscopic experiment. In the last years a number of experimental groups * Correspondence address: Physics Department, Lehman College, The City University of New York, Bedford Park Boulevard West, Bronx, New York 10468-1589, USA. Fax: #1 718 960 8627
attempted to observe quantum tunneling of the magnetic moment in single-domain magnetic particles. Besides the most recent effort to measure individual nanometer size particles [3,4], most of this research concentrated on measuring magnetic relaxation in systems containing many particles of different size and shape [5]. In the limit of low temperature, many such systems exhibit a nonthermal magnetic relaxation. Although a reasonable explanation of this behavior is provided by quantum tunneling of magnetic moments of particles [6—11], a detailed comparison between theory and experiment is hampered by the fact that even a small variation in the parameters of particles spreads their tunneling rates over many orders of
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 0 4 0 - 7
LETTER TO THE EDITOR
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E.M. Chudnovsky / Journal of Magnetism and Magnetic Materials 185 (1998) L267—L273
magnitude. In this paper I will demonstrate that the thermally assisted resonant spin tunneling, conjectured [12,13] and proved [14] in a molecular magnet Mn-12 acetate [15,16], should be also observable in systems of non-identical nanometersized particles. This provides an independent test of the quantum nature of the relaxation. I will show that the effect must be especially pronounced in antiferromagnetic particles and will argue that it may have been already observed in recent macroscopic measurements of ferritin [17—19]. The Hamiltonian of a uniaxial single-domain FM particle of total spin S in the magnetic field H, applied at an angle with the anisotropy axis Z, is H"!DS2!gk H S !gk H S z B z z B x x H 2 "!D S # z !gk H S #const., z 2*H B x x
C
D
(1)
where H *H, !/ 2S
(2)
and H "2DS/gk is the anisotropy field. Con!/ B sider first the case when the external field is applied along the anisotropy axis, that is, H "0. The x energy levels of H are then eigenstates of S , z S DmT"mDmT. At H"H "n *H and n"0, $1, z n $2,2, $(2S!1), there are degenerate pairs of levels, m and m@"n!m, separated by the energy barrier, as is illustrated by Fig. 1. In the presence of H @H , the degeneracy is removed due to the x !/ quantum tunneling between the levels. The rate, C(m, n), of tunneling from the mth level onto the m@"(n!m)th level, or, better to say, the tunneling splitting of these levels, *(m, n)"+C(m, n), can be calculated by extending the perturbative method of Ref. [20] to nO0. For an integer S the answer reads [21]
A B
H 2m~n S2m~n~1 x *(m, n)"gk H B !/ H [(2m!n!1)!]2 !/ ]
C
D
(S#m!n)!(S#m)! 1@2 . (S!m)!(S!m#n)!
(3)
Fig. 1. Thermally assisted resonant tunneling at S"10: (a) H"0, (b) H"*H. Vertical arrows show thermal processes, while tunneling is shown by horizontal arrows. The tunneling probability and the width of the levels increase drastically as one goes from the bottom to the top of the wells.
LETTER TO THE EDITOR
E.M. Chudnovsky / Journal of Magnetism and Magnetic Materials 185 (1998) L267—L273
The power of H in this formula reflects the order of x the perturbation theory in which the effect first appears, that is, the number of virtual steps along the staircase of levels in Fig. 1. At H @H the x !/ probability of tunneling from different levels decreases exponentially as one goes from the top to the bottom of the well in Fig. 1. For Mn-12 molecules of S"10, there is an internal H of a x few hundred gauss due to hyperfine interactions [22], while H &10¹. This means that for !/ n@S, that is, at H @H , the tunneling can be z !/ noticeable only if it occurs from high excited levels at the top of the barrier, that is, at sufficiently high temperature. Such tunneling effectively lowers the barrier and is responsible for the maxima observed in the spin relaxation of Mn-12 at H "H z n [14]. Let us now try to understand how much precision is needed in tuning the field to the resonance in order to observe this effect? Let w(m, n) be the maximal of the two widths: the width of the mth and the width of the (n!m)th levels with respect to both, tunneling across the barrier and decay down the well. Then the width of the resonance around H for that pair of levels must be of order n w(m, n)/gk . At ¹"0 only levels at the bottom of B the wells are occupied. At, e.g., H "0 their width is z solely due to tunneling and, thus, negligible if the tunneling occurs at an exponentially small rate. Thus, in the limit of a very low temperature and zero field bias, observation of tunneling may require a very precise tuning of levels by shielding the system from the detuning effect of stray fields. This situation cannot be achieved in Mn-12, where hyperfine interactions are huge, but can be the case in ferritin [23,24]. It is important to understand, however, that the condition of a very small bias, emphasized in the quantum theory of a two-state system [25] as the condition necessary to provide a significant tunneling rate, is only relevant to almost degenerate double well at temperature which is small compared to the energy of the first excited level. It is not relevant to experiments in which tunneling occurs from and/or onto excited levels. In magnetic systems these levels have large width for two reasons: due to the high probability of decay down to the bottom of the well and due to the exponential growth of the tunneling width
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with the height of the level, as is illustrated by Fig. 1. We now want to answer two questions: whether the above resonant effects can be observed in a magnetic particle of a mesoscopic size, and whether they can reveal themselves in macroscopic measurements of a large number of non-identical particles. To answer the first question, let us notice that while the Hamiltonian (1), that was used to explain experiments on Mn-12 [14], possesses equidistant resonances on H , at least a few approxiz mately equidistant resonances must be possessed by any reasonable spin Hamiltonian at SA1. This is because all physical potential wells are parabolic near the bottom. Consequently, the first few excited levels must be approximately equidistant on energy. Consider then a magnetic particle with the total spin S looking along its anisotropy axis and the magnetic field applied in the opposite direction. This is a metastable state that wants to decay by the underbarrier rotation of S towards the direction of the field. To conserve energy it must rotate to some excited state in which S precesses around H. Let the first excited level have the energy +u . Then the 0 conservation of energy requires 2gk HS"+u B 0 and the first resonance occurs at H "u /2cS, 1 0 where c"gk /+ is the gyromagnetic ratio. For B a FM particle at H@H (low-lying resonances) !/ u "cH and H coinsides with *H of Eq. (2). 0 !/ 1 Since the first few excited levels are approximately equidistant, the second resonance occurs when 2gk HS"2+u , that is, at H "2*H, and so on. B 0 2 ¹hus, a few resonances equally spaced by H /2S !/ must appear independently of the explicit form of the spin Hamiltonian. Observation of such resonances in a FM particle of an appreciable size requires the spacing *H to be greater than the width of the excited states. The width of the first excited state is the width of the FM resonance (FMR). Consequently, 2S must be at least smaller than Q, the quality factor of the FMR. The latter is defined as the ratio of the FMR frequency to the FMR width. For a FM particle of a considerable size the condition 2S(Q can be satisfied only if the particle is fabricated of a high-quality magnetic insulator. We shall now demonstrate that the condition for observing resonant spin tunneling in AFM particles is much more relaxed than it is in FM particles.
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Consider an AFM particle with a Hamiltonian (4) H"JS ) S !gk H Sz #gk H Sz . B !/ 2 1 2 B !/ 1 Here J is a positive constant of the AFM exchange interaction between sublattices S and S , and the 1 2 last two terms are taken into account via Kittel [26], the effect of the magnetic anisotropy. We shall assume that both, S and S , are very large (104 in 1 2 ferritin) and that there is a small non-compensation, s"S !S '0 due to the irregular shape of 2 1 the particle. The degeneracy of the energy levels in the absence of the external field is provided by the time-reversal symmetry of the Hamiltonian. (Note that H changes sign under the time-reversal trans!/ formation.) Compared to the Hamiltonian (1), the problem of the energy levels is now quite ugly but as long as the low-lying excited levels are concerned it can be solved analytically using the Holstein—Primakoff transformation. Extending the method of Ref. [26] to non-compensated sublattices one obtains in terms of magnon operators H"(e e )1@2(a a #a`a`)#(e #e )a`a 1 !/ 2 2 1 2 12 1 2 (5) #(e #e )a`a #const., 2 !/ 1 1 where e "JS and e "gk H . The diagonal1,2 1,2 !/ B !/ ization of this Hamiltonian yields the folowing equation for the energy of the low-lying excitation: e2#e(e !e )"e (e #e ). (6) 2 1 !/ 1 2 The characteristic energy of the problem is e " AFM [e (e #e )]1@2. Depending on the degree of the !/ 1 2 non-compensation, Eq. (6) gives e"e AFM
A B A B
e 1@2 "[gk H J(S #S )]1@2 at s@S !/ , B !/ 1 2 e %9 S #S e 1@2 2 at sAS !/ e"e "gk H 1 , (7) FM B !/ s e %9 where we denoted JS by e ; S being the spin of one %9 sublattice (S +S ). Because the anisotropy in 1 2 AFM is small compared to the exchange, a very small non-compensation is needed to switch from the AFM to the FM dynamics. The energy of the FM excitation is, however, comparable to the FMR energy, e "gk H , in a one-sublattice ferromag!/ B !/
net only when the non-compensation is of the order the sublattice spin S. At small non-compensation, both energies in Eq. (7) are large compared to the typical FMR energy in ferro and ferrimagnets. The corresponding factor is of the order of [e /e ]1@2&102—103 for small non-compensation, %9 !/ and of the order of S/s for large non-compensation. In nanometer scale AFM particles, like ferritin, both factors are of the order 100. With the typical value of H being &1 kOe the energy spacing !/ bewteen the low-lying spin levels must be of the order of a few kelvin, a huge value given that a particle like ferritin consists of a few thousand of magnetic atoms. To make sure that the approximation of the uniform AFM resonance is correct for nanoscale particles, one should estimate the effect of the spatial dispersion of the spin excitations. At large wave vectors the dispersion law for AFM magnons is [26] e +4J3e (qa) where a is the distance be%9 q tween magnetic atoms. This formula is valid at e Ae which is always the case for nanometer q AFM scale particles as we shall see in a moment. Indeed, because of the quantization of the wave number, the minimal value of q in an AFM particle of diameter d is of order 2p/d. This means that the first non-uniform spin-wave mode has energy e +8pJ3e (a/d). The ratio of this energy to the q %9 energy of the uniform AFM resonance is e /e +4pJ6(e /e )1@2(a/d). For ferritin parq AFM %9 !/ ticles this ratio is greater than 100. Consequently, the q-dependent spectrum must be separated from the spectrum of uniform resonances by a gap of a few hundred kelvin, which makes non-uniform spin modes irrelevant in the range of fields and temperatures where resonant spin tunneling has been or can be studied. Let us now consider the effect of the magnetic field applied along the anisotropy axis. Adding the Zeeman term, !gk H ) s, to Eq. (4) and repeating B the above calculation we find, as in the compensated case [26], that the field which is small compared to e/k simply adds a small term, gk H, to the B B spectrum of excitations. In the AFM case, e/k is B the spin-flop field, &10 T in ferritin. The effect of a small field on the energy of the excitation can, therefore, be neglected. The spacing between the field values which provide matching of the spin
LETTER TO THE EDITOR
E.M. Chudnovsky / Journal of Magnetism and Magnetic Materials 185 (1998) L267—L273
levels can, then be obtained using the same argument and the same equation, 2gk *H s"e, as B AFM in the case of a single-sublattice FM particle. Depending on the degree of the non-compensation, it gives
C
D
A B
H J(S #S ) 1@2 e 1@2 1 2 *H " !/ at s@S !/ , AFM 2s gk H e B !/ %9 e 1@2 H S #S 2 at sAS !/ . (8) *H " !/ 1 AFM s e 2s %9 Comparing Eqs. (2) and (8), we see that in AFM particles the field spacing between resonances is greater than the spacing in FM particles by a huge factor: (S/s)(e /e )1@2 for small non-compensation, %9 !/ and (S/s)2 for large non-compensation. In ferritin, e &104 e , S&104, and s&102, so that both %9 !/ factors are of the order of 104. This means that the field spacing which would be only about 0.1 Oe in a FM particle of S&104, must be of the order of 1 kOe in ferritin. There is, therefore, little danger that the effect of resonant spin tunneling from excited levels in ferritin and comparable AFM particles is smeared by the finite width of these levels. For the tunneling to occur, the Hamiltonian of an AFM particle must contain either transversal field or transversal anisotropy which does not commute with s [23,27—29]. If the longitudinal and transverz sal components of the anisotropy are comparable, which should be a common situation in small particles, the prefactor and the exponent of the tunneling rate at ¹"0 and not very high fields, H@e/k , B can be estimated as [11,30,31]
A B
C A B D A B A B
e e 1@2 e 1@2 C& AFM exp !S !/ , at s@S !/ + e e %9 %9 e e 1@2 C& FM exp(!s) at sAS !/ . (9) + e %9 Due to the condition e @e and smallness of s in !/ %9 weakly compensated AFM, particles like ferritin must exhibit tunneling on a mesoscopic scale, S&104. At ¹*¹ &e/+ (a few kelvin in ferritin) spin # transitions become dominated by thermally assisted tunneling from excited levels. We shall now argue that recent observations [17—19] of the nonmonotonic field dependence of the blocking tem-
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perature, ¹ (H), in ferritin can be due to the deB screte nature of spin. In these experiments the system of non-identical particles was cooled down at H"0 from high temperature, where particles were superparamagnetic, to a low temperature, where transitions across the anisotropy barrier were frozen. Then a small field was applied and the M(¹) curve was obtained. For a macroscopic sample, even an extremely weak field results in a measurable magnetization. To develop MO0 in the direction of the field, magnetic moments of the particles must tunnel through or jump thermally over the anisotropy barrier º. Thermal transitions occur with the probability &exp(!º/¹). This probability rapidly goes up as temperature increases, causing M(¹) to rise. At ¹&¹ , the transitions become B fast enough to bring particles in thermal equilibrium on the time scale of the experiment, so that the total M decreases as 1/¹ in accordance with the Curie law. The value of ¹ is determined by the B particles which contribute to M(¹) most. In classical physics, ¹ decreases with H because the field B lowers the barrier. Such a behavior of ¹ (H) has B been observed in many systems. However, in ferritin, the observed dependence of ¹ on H shows B initial increase until about 2.5 kOe to a maximum of 13.5 K, followed by the decrease of ¹ at higher B fields. In other words, transitions between different spin orientations in ferritin particles occur faster at H"0 than at HO0. This behavior of ¹ cannot B be attributed to the interactions between ferritin particles since, due to small moments of the particles, such interactions are negligible compared to the characteristic field of 2.5 kOe. The explanation can be obtained in terms of resonant spin tunneling between excited spin levels. At H"0 levels corresponding to $m match in all particles. Tunneling between these levels results in the effective reduction of the anisotropy barrier. Indeed, instead of going over the full barrier, the particle can be thermally activated to an excited level m, below the top of the barrier, and then tunnel to !m (see Fig. 1a). Consequently, at H"0 the effective barrier º , %&& which is the height of the mth level from which tunneling occurs on the time scale of the experiment, is lower than the full barrier º for all particles. For each particle there is a field *H at which the levels match again, Fig. 1b. This field, however,
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is different for different particles so that the universal resonance occurs only at H"0. If the system was at zero temperature, the width of that resonance on the magnetic field would be determined by a small tunneling splitting of the ground-state level, which would make it unobservable. At a few kelvin, however, it is determined by the width of the excited levels. The latter must be large given the fact that higher harmonics of FMR or AFMR are very difficult to observe. The upper limit on the field that destroys the resonant tunneling is, by order of magnitude, the distance between resonances *H. It is given by Eq. (2) for FM particles and by Eq. (8) for AFM particles. The field of 2.5 kOe is, therefore, consistent with our estimate of *H in ferritin. AFM The fact that all particles are different does not affect our estimate since particles measured within the experimental time window must have dimensions within the same order of magnitude. This is because the probabilty of spin relaxation, either thermal or quantum, depends exponentially on the parameters of the particles. A few aditional observations are in order. All the above considerations are applied to the particles of an integer spin. The total spin of a ferritin particle is formed by Fe3` ions of spin 5. Due to this fact and 2 to the non-compensation of sublattices, a system of a large number of ferritin particles must have statistically equal number of particles having integer and half-integer s. This may also be true for any system of ferromagnetic particles. For particles whose spin is a half-integer, the effect of the magnetic field is opposite: it unfreezes spins blocked by the Kramers degeneracy [7—9,32,33]. Within that subgroup of particles with various size and shape, ¹ must decrease monotonically with the field, as B the field reduces the barrier. For such particles, there is no single value of the field that would result in a large resonant tunneling for all particles simultaneously, as it happens for an integer-spin particles at H"0. One should expect, therefore, an H"0 minimum in ¹ (H) even for a mixture of B integer and half-integer spins. Our next observation is that the minimum in ¹ at H"0 discovered by several groups [17—19] B removes one of the most serious arguments [34—36] against tunneling interpretation of an earlier resonance experiment [23] in ferritin. That argument
was that particles which are blocked in the kelvin range cannot tunnel at a high rate in the millikelvin range. As we have seen, however, the new data, in accordance with millikelvin experiments, indicate that tunneling unblocks at HP0. Most recent results [19] also show that the width of the resonance on the magnetic field decreases as temperature goes to zero, in accordance with earlier measurements [23] and our understanding that at ¹"0 tunneling occurs between very narrow ground-state levels. Finally, I would like to comment on the possibility to observe the effect of spin quantization in relaxation experiments on FM particles. In the measurements of individual FM particles [3,4] the barrier can be lowered by the fine tuning of the magnetic field, so that tunneling can, in principle, occur in particles of spin as large as 105, In a system of non-interacting FM particles of random size and shape at ¹&¹ , tunneling, according to Eq. (9) # should generally fall within the experimental time window for S&25—30. These are usually the smallest particles in the system. They are responsible for the relaxation of a small part of the total magnetization. The quantization of spin must reveal itself in the enhanced resonant relaxation at H"0, which would be in apparent disagreement with the prediction of classical physics. A nonmonotonic field dependence of ¹ has been noticed B in nanoparticles of magnetite [37] and maghemite [38] but the origin of the effect was unclear. The experimental observation of the maximum in the magnetic viscosity at H"0 and ¹&¹ would be # a strong independent evidence of spin tunneling, complementary to the plateau in the viscosity which is often observed below ¹ [5]. For typical # values of the magnetic anisotropy, H & !/ 1—10 kOe, the width of that resonance on the field should be of the order of the FMR width but not greater than *H"H /2S, which should range !/ from a few tens to a few hundred gauss. Of course, the maximum in the viscosity will be observable only if the FMR width is smaller than *H.
Acknowledgements This work has been supported by the US National Science Foundation Grant No. DMR-9024250
LETTER TO THE EDITOR
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and was completed during the author’s stay at CEA Saclay, France. References [1] E.M. Chudnovsky, D.P. DiVinzenzo, Phys. Rev. B 48 (1993) 10548. [2] A. Garg, Phys. Rev. B 51 (1995) 15161. [3] W. Wernsdorfer et al., Phys. Rev. Lett. 77 (1996) 1873. [4] W. Wernsdorfer et al., Phys. Rev. Lett. 78 (1997) 1791. [5] See, e.g., J. Tejada, X. Zhang, in: L. Gunther, B. Barbara (Ed.), Quantum Tunneling of Magnetization, Kluwer, Dordrecht, 1995. [6] C.P. Bean, J.D. Livingston, J. Appl. Phys. 30 (1959) 120S. [7] M. Enz, R. Schilling, J. Phys. C 19 (1986) 1765. [8] M. Enz, R. Schilling, J. Phys. C 19 (1986) L711. [9] J.L. van Hemmen, A. Su¨to¨, Physica B 141 (1986) 37. [10] Europhys. Lett. 1 (1986) 481. [11] E.M. Chudnovsky, L. Gunther, Phys. Rev. Lett. 60 (1988) 661. [12] B. Barbara et al., J. Magn. Magn. Mater. 140—144 (1995) 1825. [13] M.A. Novak, R. Sessoli, in: L. Gunther, B. Barbara (Eds.), Quantum Tunneling of Magnetization, Kluwer, Dordrecht, 1995. [14] J.R. Friedman, M.P. Sarachik, J. Tejada, R. Ziolo, Phys. Rev. Lett. 76 (1996) 3830. [15] See also experimental confirmation in J.M. Hernandez et al., Europhys. Lett. 35 (1996) 301. [16] L. Thomas et al., Nature 383 (1996) 145.
[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]
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S. Gider et al., J. Appl. Phys. 79 (1996) 5324. J.R. Friedman et al., Phys. Rev. B 56 (1997) 10793. J. Tejada et al., Phys. Rev. Lett. 79 (1997) 1754. D.A. Garanin, J. Phys. A 24 (1991) L61. E.M. Chudnovsky, J.R. Friedman, unpublished. F. Hartmann-Boutron, P. Politi, J. Villain, Int. J. Mod. Phys. B 10 (1996) 2577. D.D. Awshalom et al., Phys. Rev. Lett. 68 (1992) 3092. S. Gider et al., Science 268 (1995) 77. A.J. Leggett et al., Rev. Mod. Phys. 59 (1987) 1. C. Kittel, Quantum Theory of Solids, Wiley, New York, 1963. E.M. Chudnovsky, J. Magn. Magn. Mater. 140—144 (1995) 1821. Ji-Min Duan, Anupam Garg, J. Phys.: Condens. Matter 7 (1995) 2171. A. Chiolero, D. Loss, Phys. Rev. B 56 (1997) 738. B. Barbara, E.M. Chudnovsky, Phys. Lett. A 145 (1990) 205. I.V. Krive, O.B. Zaslavskii, J. Phys.: Condens. Matter 2 (1990) 9457. D. Loss, D.P. DiVincenzo, G. Grinstein, Phys. Rev. Lett. 69 (1992) 3232. J. von Delft, C.L. Henley, Phys. Rev. Lett. 69 (1992) 3236. J. Tejada, Science 272 (1996) 424. A. Garg, Science 272 (1996) 424. S. Gider, D.D. Awschalom, D.P. Divincenzo, D. Loss, Science 272 (1996) 425. W. Luo, S.R. Nagel, T.F. Rosenbaum, R.E. Roeig, Phys. Rev. Lett. 67 (1991) 2721. R. Sappey et al., Phys. Rev. B 56 (1997) 14551.
Journal of Magnetism and Magnetic Materials 185 (1998) L267—L273
Letter to the Editor
Resonant spin tunneling in small magnetic particles Eugene M. Chudnovsky!,",* ! Physics Department, Lehman College, The City University of New York, Bedford Park Boulevard West, Bronx, New York 10468-1589, USA " Service de Physique de l+Etat Condense& , CEA Saclay, 91191 Gif sur Yvette Cedex, France Received 12 September 1997; received in revised form 30 December 1997
Abstract A new macroscopic test of spin tunneling in single-domain magnetic particles is proposed. It is shown that quantization of spin, even in particles of a considerable size, may lead to a detectable zero-field maximum in the dependence of the rate of the magnetic relaxation on the magnetic field. This effect must be especially pronounced in antiferromagnetic particles. It is argued that recent observation by several groups of the non-monotonic field dependence of the blocking temperature in ferritin may, in fact, present the experimental evidence of this effect. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 75.45.#j; 75.50.Tt Keywords: Resonant spin tunneling; Nanoparticles
This paper considers the effect of spin quantization on magnetic properties of single-domain ferromagnetic (FM) and antiferromagnetic (AFM) particles. Despite of the vast literature on small magnetic particles, this question has received very little attention [1,2]. Meanwhile, as I am going to demonstrate, even in particles of a considerable size, the descrete nature of spin leads to the effect which can be tested in a macroscopic experiment. In the last years a number of experimental groups * Correspondence address: Physics Department, Lehman College, The City University of New York, Bedford Park Boulevard West, Bronx, New York 10468-1589, USA. Fax: #1 718 960 8627
attempted to observe quantum tunneling of the magnetic moment in single-domain magnetic particles. Besides the most recent effort to measure individual nanometer size particles [3,4], most of this research concentrated on measuring magnetic relaxation in systems containing many particles of different size and shape [5]. In the limit of low temperature, many such systems exhibit a nonthermal magnetic relaxation. Although a reasonable explanation of this behavior is provided by quantum tunneling of magnetic moments of particles [6—11], a detailed comparison between theory and experiment is hampered by the fact that even a small variation in the parameters of particles spreads their tunneling rates over many orders of
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 0 4 0 - 7
LETTER TO THE EDITOR
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E.M. Chudnovsky / Journal of Magnetism and Magnetic Materials 185 (1998) L267—L273
magnitude. In this paper I will demonstrate that the thermally assisted resonant spin tunneling, conjectured [12,13] and proved [14] in a molecular magnet Mn-12 acetate [15,16], should be also observable in systems of non-identical nanometersized particles. This provides an independent test of the quantum nature of the relaxation. I will show that the effect must be especially pronounced in antiferromagnetic particles and will argue that it may have been already observed in recent macroscopic measurements of ferritin [17—19]. The Hamiltonian of a uniaxial single-domain FM particle of total spin S in the magnetic field H, applied at an angle with the anisotropy axis Z, is H"!DS2!gk H S !gk H S z B z z B x x H 2 "!D S # z !gk H S #const., z 2*H B x x
C
D
(1)
where H *H, !/ 2S
(2)
and H "2DS/gk is the anisotropy field. Con!/ B sider first the case when the external field is applied along the anisotropy axis, that is, H "0. The x energy levels of H are then eigenstates of S , z S DmT"mDmT. At H"H "n *H and n"0, $1, z n $2,2, $(2S!1), there are degenerate pairs of levels, m and m@"n!m, separated by the energy barrier, as is illustrated by Fig. 1. In the presence of H @H , the degeneracy is removed due to the x !/ quantum tunneling between the levels. The rate, C(m, n), of tunneling from the mth level onto the m@"(n!m)th level, or, better to say, the tunneling splitting of these levels, *(m, n)"+C(m, n), can be calculated by extending the perturbative method of Ref. [20] to nO0. For an integer S the answer reads [21]
A B
H 2m~n S2m~n~1 x *(m, n)"gk H B !/ H [(2m!n!1)!]2 !/ ]
C
D
(S#m!n)!(S#m)! 1@2 . (S!m)!(S!m#n)!
(3)
Fig. 1. Thermally assisted resonant tunneling at S"10: (a) H"0, (b) H"*H. Vertical arrows show thermal processes, while tunneling is shown by horizontal arrows. The tunneling probability and the width of the levels increase drastically as one goes from the bottom to the top of the wells.
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E.M. Chudnovsky / Journal of Magnetism and Magnetic Materials 185 (1998) L267—L273
The power of H in this formula reflects the order of x the perturbation theory in which the effect first appears, that is, the number of virtual steps along the staircase of levels in Fig. 1. At H @H the x !/ probability of tunneling from different levels decreases exponentially as one goes from the top to the bottom of the well in Fig. 1. For Mn-12 molecules of S"10, there is an internal H of a x few hundred gauss due to hyperfine interactions [22], while H &10¹. This means that for !/ n@S, that is, at H @H , the tunneling can be z !/ noticeable only if it occurs from high excited levels at the top of the barrier, that is, at sufficiently high temperature. Such tunneling effectively lowers the barrier and is responsible for the maxima observed in the spin relaxation of Mn-12 at H "H z n [14]. Let us now try to understand how much precision is needed in tuning the field to the resonance in order to observe this effect? Let w(m, n) be the maximal of the two widths: the width of the mth and the width of the (n!m)th levels with respect to both, tunneling across the barrier and decay down the well. Then the width of the resonance around H for that pair of levels must be of order n w(m, n)/gk . At ¹"0 only levels at the bottom of B the wells are occupied. At, e.g., H "0 their width is z solely due to tunneling and, thus, negligible if the tunneling occurs at an exponentially small rate. Thus, in the limit of a very low temperature and zero field bias, observation of tunneling may require a very precise tuning of levels by shielding the system from the detuning effect of stray fields. This situation cannot be achieved in Mn-12, where hyperfine interactions are huge, but can be the case in ferritin [23,24]. It is important to understand, however, that the condition of a very small bias, emphasized in the quantum theory of a two-state system [25] as the condition necessary to provide a significant tunneling rate, is only relevant to almost degenerate double well at temperature which is small compared to the energy of the first excited level. It is not relevant to experiments in which tunneling occurs from and/or onto excited levels. In magnetic systems these levels have large width for two reasons: due to the high probability of decay down to the bottom of the well and due to the exponential growth of the tunneling width
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with the height of the level, as is illustrated by Fig. 1. We now want to answer two questions: whether the above resonant effects can be observed in a magnetic particle of a mesoscopic size, and whether they can reveal themselves in macroscopic measurements of a large number of non-identical particles. To answer the first question, let us notice that while the Hamiltonian (1), that was used to explain experiments on Mn-12 [14], possesses equidistant resonances on H , at least a few approxiz mately equidistant resonances must be possessed by any reasonable spin Hamiltonian at SA1. This is because all physical potential wells are parabolic near the bottom. Consequently, the first few excited levels must be approximately equidistant on energy. Consider then a magnetic particle with the total spin S looking along its anisotropy axis and the magnetic field applied in the opposite direction. This is a metastable state that wants to decay by the underbarrier rotation of S towards the direction of the field. To conserve energy it must rotate to some excited state in which S precesses around H. Let the first excited level have the energy +u . Then the 0 conservation of energy requires 2gk HS"+u B 0 and the first resonance occurs at H "u /2cS, 1 0 where c"gk /+ is the gyromagnetic ratio. For B a FM particle at H@H (low-lying resonances) !/ u "cH and H coinsides with *H of Eq. (2). 0 !/ 1 Since the first few excited levels are approximately equidistant, the second resonance occurs when 2gk HS"2+u , that is, at H "2*H, and so on. B 0 2 ¹hus, a few resonances equally spaced by H /2S !/ must appear independently of the explicit form of the spin Hamiltonian. Observation of such resonances in a FM particle of an appreciable size requires the spacing *H to be greater than the width of the excited states. The width of the first excited state is the width of the FM resonance (FMR). Consequently, 2S must be at least smaller than Q, the quality factor of the FMR. The latter is defined as the ratio of the FMR frequency to the FMR width. For a FM particle of a considerable size the condition 2S(Q can be satisfied only if the particle is fabricated of a high-quality magnetic insulator. We shall now demonstrate that the condition for observing resonant spin tunneling in AFM particles is much more relaxed than it is in FM particles.
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Consider an AFM particle with a Hamiltonian (4) H"JS ) S !gk H Sz #gk H Sz . B !/ 2 1 2 B !/ 1 Here J is a positive constant of the AFM exchange interaction between sublattices S and S , and the 1 2 last two terms are taken into account via Kittel [26], the effect of the magnetic anisotropy. We shall assume that both, S and S , are very large (104 in 1 2 ferritin) and that there is a small non-compensation, s"S !S '0 due to the irregular shape of 2 1 the particle. The degeneracy of the energy levels in the absence of the external field is provided by the time-reversal symmetry of the Hamiltonian. (Note that H changes sign under the time-reversal trans!/ formation.) Compared to the Hamiltonian (1), the problem of the energy levels is now quite ugly but as long as the low-lying excited levels are concerned it can be solved analytically using the Holstein—Primakoff transformation. Extending the method of Ref. [26] to non-compensated sublattices one obtains in terms of magnon operators H"(e e )1@2(a a #a`a`)#(e #e )a`a 1 !/ 2 2 1 2 12 1 2 (5) #(e #e )a`a #const., 2 !/ 1 1 where e "JS and e "gk H . The diagonal1,2 1,2 !/ B !/ ization of this Hamiltonian yields the folowing equation for the energy of the low-lying excitation: e2#e(e !e )"e (e #e ). (6) 2 1 !/ 1 2 The characteristic energy of the problem is e " AFM [e (e #e )]1@2. Depending on the degree of the !/ 1 2 non-compensation, Eq. (6) gives e"e AFM
A B A B
e 1@2 "[gk H J(S #S )]1@2 at s@S !/ , B !/ 1 2 e %9 S #S e 1@2 2 at sAS !/ e"e "gk H 1 , (7) FM B !/ s e %9 where we denoted JS by e ; S being the spin of one %9 sublattice (S +S ). Because the anisotropy in 1 2 AFM is small compared to the exchange, a very small non-compensation is needed to switch from the AFM to the FM dynamics. The energy of the FM excitation is, however, comparable to the FMR energy, e "gk H , in a one-sublattice ferromag!/ B !/
net only when the non-compensation is of the order the sublattice spin S. At small non-compensation, both energies in Eq. (7) are large compared to the typical FMR energy in ferro and ferrimagnets. The corresponding factor is of the order of [e /e ]1@2&102—103 for small non-compensation, %9 !/ and of the order of S/s for large non-compensation. In nanometer scale AFM particles, like ferritin, both factors are of the order 100. With the typical value of H being &1 kOe the energy spacing !/ bewteen the low-lying spin levels must be of the order of a few kelvin, a huge value given that a particle like ferritin consists of a few thousand of magnetic atoms. To make sure that the approximation of the uniform AFM resonance is correct for nanoscale particles, one should estimate the effect of the spatial dispersion of the spin excitations. At large wave vectors the dispersion law for AFM magnons is [26] e +4J3e (qa) where a is the distance be%9 q tween magnetic atoms. This formula is valid at e Ae which is always the case for nanometer q AFM scale particles as we shall see in a moment. Indeed, because of the quantization of the wave number, the minimal value of q in an AFM particle of diameter d is of order 2p/d. This means that the first non-uniform spin-wave mode has energy e +8pJ3e (a/d). The ratio of this energy to the q %9 energy of the uniform AFM resonance is e /e +4pJ6(e /e )1@2(a/d). For ferritin parq AFM %9 !/ ticles this ratio is greater than 100. Consequently, the q-dependent spectrum must be separated from the spectrum of uniform resonances by a gap of a few hundred kelvin, which makes non-uniform spin modes irrelevant in the range of fields and temperatures where resonant spin tunneling has been or can be studied. Let us now consider the effect of the magnetic field applied along the anisotropy axis. Adding the Zeeman term, !gk H ) s, to Eq. (4) and repeating B the above calculation we find, as in the compensated case [26], that the field which is small compared to e/k simply adds a small term, gk H, to the B B spectrum of excitations. In the AFM case, e/k is B the spin-flop field, &10 T in ferritin. The effect of a small field on the energy of the excitation can, therefore, be neglected. The spacing between the field values which provide matching of the spin
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levels can, then be obtained using the same argument and the same equation, 2gk *H s"e, as B AFM in the case of a single-sublattice FM particle. Depending on the degree of the non-compensation, it gives
C
D
A B
H J(S #S ) 1@2 e 1@2 1 2 *H " !/ at s@S !/ , AFM 2s gk H e B !/ %9 e 1@2 H S #S 2 at sAS !/ . (8) *H " !/ 1 AFM s e 2s %9 Comparing Eqs. (2) and (8), we see that in AFM particles the field spacing between resonances is greater than the spacing in FM particles by a huge factor: (S/s)(e /e )1@2 for small non-compensation, %9 !/ and (S/s)2 for large non-compensation. In ferritin, e &104 e , S&104, and s&102, so that both %9 !/ factors are of the order of 104. This means that the field spacing which would be only about 0.1 Oe in a FM particle of S&104, must be of the order of 1 kOe in ferritin. There is, therefore, little danger that the effect of resonant spin tunneling from excited levels in ferritin and comparable AFM particles is smeared by the finite width of these levels. For the tunneling to occur, the Hamiltonian of an AFM particle must contain either transversal field or transversal anisotropy which does not commute with s [23,27—29]. If the longitudinal and transverz sal components of the anisotropy are comparable, which should be a common situation in small particles, the prefactor and the exponent of the tunneling rate at ¹"0 and not very high fields, H@e/k , B can be estimated as [11,30,31]
A B
C A B D A B A B
e e 1@2 e 1@2 C& AFM exp !S !/ , at s@S !/ + e e %9 %9 e e 1@2 C& FM exp(!s) at sAS !/ . (9) + e %9 Due to the condition e @e and smallness of s in !/ %9 weakly compensated AFM, particles like ferritin must exhibit tunneling on a mesoscopic scale, S&104. At ¹*¹ &e/+ (a few kelvin in ferritin) spin # transitions become dominated by thermally assisted tunneling from excited levels. We shall now argue that recent observations [17—19] of the nonmonotonic field dependence of the blocking tem-
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perature, ¹ (H), in ferritin can be due to the deB screte nature of spin. In these experiments the system of non-identical particles was cooled down at H"0 from high temperature, where particles were superparamagnetic, to a low temperature, where transitions across the anisotropy barrier were frozen. Then a small field was applied and the M(¹) curve was obtained. For a macroscopic sample, even an extremely weak field results in a measurable magnetization. To develop MO0 in the direction of the field, magnetic moments of the particles must tunnel through or jump thermally over the anisotropy barrier º. Thermal transitions occur with the probability &exp(!º/¹). This probability rapidly goes up as temperature increases, causing M(¹) to rise. At ¹&¹ , the transitions become B fast enough to bring particles in thermal equilibrium on the time scale of the experiment, so that the total M decreases as 1/¹ in accordance with the Curie law. The value of ¹ is determined by the B particles which contribute to M(¹) most. In classical physics, ¹ decreases with H because the field B lowers the barrier. Such a behavior of ¹ (H) has B been observed in many systems. However, in ferritin, the observed dependence of ¹ on H shows B initial increase until about 2.5 kOe to a maximum of 13.5 K, followed by the decrease of ¹ at higher B fields. In other words, transitions between different spin orientations in ferritin particles occur faster at H"0 than at HO0. This behavior of ¹ cannot B be attributed to the interactions between ferritin particles since, due to small moments of the particles, such interactions are negligible compared to the characteristic field of 2.5 kOe. The explanation can be obtained in terms of resonant spin tunneling between excited spin levels. At H"0 levels corresponding to $m match in all particles. Tunneling between these levels results in the effective reduction of the anisotropy barrier. Indeed, instead of going over the full barrier, the particle can be thermally activated to an excited level m, below the top of the barrier, and then tunnel to !m (see Fig. 1a). Consequently, at H"0 the effective barrier º , %&& which is the height of the mth level from which tunneling occurs on the time scale of the experiment, is lower than the full barrier º for all particles. For each particle there is a field *H at which the levels match again, Fig. 1b. This field, however,
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is different for different particles so that the universal resonance occurs only at H"0. If the system was at zero temperature, the width of that resonance on the magnetic field would be determined by a small tunneling splitting of the ground-state level, which would make it unobservable. At a few kelvin, however, it is determined by the width of the excited levels. The latter must be large given the fact that higher harmonics of FMR or AFMR are very difficult to observe. The upper limit on the field that destroys the resonant tunneling is, by order of magnitude, the distance between resonances *H. It is given by Eq. (2) for FM particles and by Eq. (8) for AFM particles. The field of 2.5 kOe is, therefore, consistent with our estimate of *H in ferritin. AFM The fact that all particles are different does not affect our estimate since particles measured within the experimental time window must have dimensions within the same order of magnitude. This is because the probabilty of spin relaxation, either thermal or quantum, depends exponentially on the parameters of the particles. A few aditional observations are in order. All the above considerations are applied to the particles of an integer spin. The total spin of a ferritin particle is formed by Fe3` ions of spin 5. Due to this fact and 2 to the non-compensation of sublattices, a system of a large number of ferritin particles must have statistically equal number of particles having integer and half-integer s. This may also be true for any system of ferromagnetic particles. For particles whose spin is a half-integer, the effect of the magnetic field is opposite: it unfreezes spins blocked by the Kramers degeneracy [7—9,32,33]. Within that subgroup of particles with various size and shape, ¹ must decrease monotonically with the field, as B the field reduces the barrier. For such particles, there is no single value of the field that would result in a large resonant tunneling for all particles simultaneously, as it happens for an integer-spin particles at H"0. One should expect, therefore, an H"0 minimum in ¹ (H) even for a mixture of B integer and half-integer spins. Our next observation is that the minimum in ¹ at H"0 discovered by several groups [17—19] B removes one of the most serious arguments [34—36] against tunneling interpretation of an earlier resonance experiment [23] in ferritin. That argument
was that particles which are blocked in the kelvin range cannot tunnel at a high rate in the millikelvin range. As we have seen, however, the new data, in accordance with millikelvin experiments, indicate that tunneling unblocks at HP0. Most recent results [19] also show that the width of the resonance on the magnetic field decreases as temperature goes to zero, in accordance with earlier measurements [23] and our understanding that at ¹"0 tunneling occurs between very narrow ground-state levels. Finally, I would like to comment on the possibility to observe the effect of spin quantization in relaxation experiments on FM particles. In the measurements of individual FM particles [3,4] the barrier can be lowered by the fine tuning of the magnetic field, so that tunneling can, in principle, occur in particles of spin as large as 105, In a system of non-interacting FM particles of random size and shape at ¹&¹ , tunneling, according to Eq. (9) # should generally fall within the experimental time window for S&25—30. These are usually the smallest particles in the system. They are responsible for the relaxation of a small part of the total magnetization. The quantization of spin must reveal itself in the enhanced resonant relaxation at H"0, which would be in apparent disagreement with the prediction of classical physics. A nonmonotonic field dependence of ¹ has been noticed B in nanoparticles of magnetite [37] and maghemite [38] but the origin of the effect was unclear. The experimental observation of the maximum in the magnetic viscosity at H"0 and ¹&¹ would be # a strong independent evidence of spin tunneling, complementary to the plateau in the viscosity which is often observed below ¹ [5]. For typical # values of the magnetic anisotropy, H & !/ 1—10 kOe, the width of that resonance on the field should be of the order of the FMR width but not greater than *H"H /2S, which should range !/ from a few tens to a few hundred gauss. Of course, the maximum in the viscosity will be observable only if the FMR width is smaller than *H.
Acknowledgements This work has been supported by the US National Science Foundation Grant No. DMR-9024250
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