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Non-adiabatic Landau-Zener transitions in low-spin molecular magnet V15
Journal of Magnetism and Magnetic Materials 221 (2000) 103}109
Non-adiabatic Landau-Zener transitions in low-spin molecular magnet V
I. Chiorescu *, W. Wernsdorfer , A. MuK ller, H. BoK gge, B. Barbara Laboratoire de Magne& tisme Louis Ne& el, CNRS, BP 166, 38042-Grenoble, France Faku( ltat fu( r Chemie, Universita( t Bielefeld, D-33501 Bielefeld, Germany
Abstract The V polyoxovanadate molecule is made of 15 spins with antiferromagnetic couplings. It belongs to the class of molecules with very large Hilbert space dimension (2 in V , 10 in Mn -AC). It is a low spin/large molecule with spin S". Contrary to large spins/large molecules of the Mn -AC type, V has no energy barrier against spin rotation. Magnetization measurements have been performed and despite the absence of a barrier, magnetic hysteresis is observed over a timescale of several seconds. This new phenomenon characterized by a `butter#ya hysteresis loop is due to the e!ect of the environment on the quantum rotation of the entangled 15 spins of the molecule, in which the phonon density of states is not at its equilibrium (phonon bottleneck). 2000 Elsevier Science B.V. All rights reserved. PACS: 75.50.Xx; 75.45.#j; 71.70.-d Keywords: Magnetic quantum e!ects; Relaxation phenomena; Dissipative two-level system
1. Introduction The recent evidence for quantum tunneling of the magnetization in big molecules with large spin S"10 [1}3] allows a link to be made between mesoscopic physics and magnetism [4]. In these systems, energy barriers are high and tunnel splittings between symmetric and antisymmetric quantum states are exponentially small (10\}10\ K). As a consequence, quantum relaxation is slow and is only related to the spin baths [5}11]. Resonant phonon transitions are irrelevant, unless between states at di!erent energies [12,13] or in the pres-
* Corresponding author. Tel./fax: #33-4-76-88-11-94/11-91. E-mail address: [email protected] (I. Chiorescu).
ence of a transverse "eld large enough to create a tunnel splitting of the order of the temperature energy scale [14]. Here we show that the reversal of an ensemble of non-interacting spins S" is irre versible as a consequence of di!erent couplings to the environment. Such an e!ect gives rise in our system to a `butter#ya hysteresis loop. Spin rotation in V may be viewed as an experimental realization of the theoretical problem of a two-level system with dissipation [14}17]. Spin}phonon transitions take place between the two splitted ground states at well de"ned energies u"D , & where D is the "eld dependent energy separation & of the two levels. The width Du of such transitions being relatively narrow, the number of available phonon states at energy u is much smaller than the number of molecules and a hole must be burned
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 3 8 5 - 1
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in the spin}phonon density of states [18}20]. This hole, moving with the sweeping "eld, leads to a non-equilibrium spin}phonon density of states, giving in the V complex the observed phonon bottleneck and hysteresis. A phonon `bottleneck plateaua, identi"ed on the measured hysteresis loop, gives a characteristic butter#y shape. These results are corroborated with numerical calculations based on this model.
2. The V15 magnetic moment evolution in low sweeping rates The V complex of formula K [V'4 As O (H O)] ) 8H O was prepared as described in Refs. [21}23]. It is made of a lattice of molecules with 15 V'4 ions of spin S", placed in a quasi-spherical layered structure formed of a triangle, sandwiched by two hexagons (Fig. 1). All the interactions being antiferromagnetic, the resultant (collective spin) S". The symmetry is trigonal (space group R3 c, a"14.029 As , a"79.263, <"2632 As ). The unit-cell contains two V clus ters and it is large enough so that dipolar interactions between spin molecules are negligible (few mK). Each hexagon contains three pairs of strongly coupled spins (J !800 K) and each spin at a corner of the inner triangle is coupled to two of those pairs (one belonging to the upper hexagon and the other belonging to the lower hexagon (J J !150 K, J J !300 K [22]). Thus, the V molecule can be seen as formed by three groups of "ve V'4 ions with the resultant spin and assembled on each corner of the inner triangle. These three spins interact with each other through two main paths, one passing by the upper hexagon and the other passing by the lower one. This is a typical example of frustrated molecule, where the exchange J between the spins is much smaller than the exchange interactions between two spins. Two types of single-crystal magnetization measurements have been performed: (i) characterization of the general thermodynamical properties of the system at equilibrium and (ii) study of the non-equilibrium to equilibrium transition. In the "rst case, a small dilution refrigerator allowing measurements above 0.1 K was inserted
Fig. 1. The geometry and the spin coupling scheme in V com plex. There are "ve antiferromagnetic exchange constants leading to a ground state S" in "elds below 2.8 T and S" for greater "elds (here such a "eld is pointing down).
into an extraction magnetometer providing "elds up to 16 T, with low sweeping rates. Below 0.9 K we observed one jump in zero "eld and two others at B $2.8 T (Fig. 2). They correspond to the ", !2", 2, and ", 2", 2 spin transitions, with respective saturations at 0.50$0.02l and 2.95$0.02l per V molecule, in agreement with the level scheme given in the inset of Fig. 2. For all temperatures the magnetization curves are reversible and do not show any anisotropy, showing that eventual energy barrier preventing spin reversal must be quite small (less than 50 mK). The magnetization curves measured at equilibrium (Fig. 2) are "tted using the Heisenberg Hamiltonian for three frustrated spins S": H"! J (S S #S S #S S ) ? ? ? ? ? ? ? ?6 7 8 !gk B (S #S #S ) (1) with S "S "S ", J "J (0, J (0 (anti 6 7 8 ferromagnetic coupling), g"2, B the applied "eld and k the Bohr magneton. In the isotropic case (J "J ) the con"guration S" becomes 6 7 8
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3. Spin}phonon coupling in V15 in low 5elds and high sweeping rates: a dissipative spin two-level system
Fig. 2. The agreement between the calculated curve at thermodynamical equilibrium (line) and the experimental magnetization data is quite good for an isotropic exchange J "!2.445 K. For J "J "!2.75 K, J "!2.2 K and 6 7 8 the "eld applied in the triangle plane (dashed line) one observes that the calculated second #ip is slightly displaced but its slope is still greater than the experimental one. The energy levels vs. applied "eld in the e!ective triangle approach in the inset are represented. When the applied "eld reaches !3J /(2gk ) the ground state switches from S" to S".
favourable against the S" one for B " !3J /(2gk ). The measured value B 2.8 T gives J !2.5 K. The comparison of the calculated curves (S "1i"S #S #S "i2 averaged over 8G 8 8 8 the di!erent eigenstates E , "i2, i"1}8) with the G measured ones is quite good at any temperature. The width of the transition at B $2.8 T is nevertheless broader in the experiments (Fig. 2). This broadening of about 0.7 T cannot be inferred to dipolar or hyper"ne "eld distributions (about 1 and 40 mT, respectively). The observed anisotropy of the electronic g-factor (g "g "1.95 and ? @ g "1.98) is also too small [23]. Antisymmetrical A Dzyaloshinsky}Moriya interactions (see Refs. [24}26] and references therein) allowed by symmetry and coupling the states S" and S", could explain this broadening. In low "elds, susceptibility measurements give the following e!ective paramagnetic moments: k "g(S(S#1)"1.75$0.02k corresponding to S" below 0.5 K, and k "3$0.02k corre sponding to three independent spins S" below 100 K. It con"rms clearly that the ground state of the molecule is S" below 0.5 K.
We showed that, between !2.8 and 2.8 T, the total spin of this molecule is S". Such a small spin leads to vanishingly small energy barrier and relatively large splitting in zero "eld (&10\ K). Resonant spin}phonon transitions between the symmetric and antisymmetric states at resonance are then possible in sub-Kelvin experiments, contrary to the case of big molecules in zero "eld, where these states are much too close. The sensitivity and time resolution of the microSQUID magnetometer [27] allowed to study very small V crystals in good contact with the thermal bath. Down to 50 mK non-equilibrium behavior was nevertheless observed at fast sweeping rate (up to dB /dt"0.7 T/s). Few hysteresis loops are rep resented in Figs. 3a and 4a (only the positive part, the other one being rigorously symmetrical). When the "eld increases, coming from negative values, the magnetization passes through the origin of the coordinates, reaches a plateau and then approaches saturation. This leads to a winged hysteresis loop characterized by the absence of irreversibility near zero "eld. The wings depend sensitively on temperature and sweeping "eld rate. As an example, in Fig. 3 where three hysteresis loops are presented (¹"0.1, 0.15 and 0.2 K) for a given sweeping rate (0.14 T/s), the plateau is higher and more pronounced at low temperature. The same tendency is observed at given temperature and faster sweeping rate (Fig. 4). At a given temperature, the equilibrium magnetization curve can be approximated by the median of the two branches of a low sweeping rate hysteresis loop (e.g. in Fig. 4 for dB /dt" 4.4 mT/s). When compared to the equilibrium curve, a given magnetization curve shows that (i) in negative "elds the system is colder than the bath, (ii) the spin system remains colder until the magnetization curve intersects the equilibrium curve, (iii) Then the spin temperature overpass as the bath temperature, and (iv) at su$ciently high "elds (about 0.5 T) the system reaches the equilibrium. In order to interpret this magnetic behaviour of the V molecules, we will analyse how the level occupation numbers vary in this two-level system (see Fig. 4a inset) when sweeping an external "eld.
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Fig. 3. Measured (a, top) and calculated (b, bottom) hysteresis loops for three temperatures and for a given "eld sweeping rate 0.14 T/s. The plateau is more pronounced at low T. The inset is a schematic representation of a two-level system S "$ with 8 repulsion due to non-diagonal matrix elements. In a swept "eld the switching probability P is given by the Landau}Zener formulae (see text). The two levels are broadened by the hyper"ne "elds and the absorption or the emission of phonons can switch the polarization state of spins.
In the bare Landau}Zener (LZ) model [28}32] the probability for the ", !2", 2 transition is pD . P"1!exp 1! (2) 4 k r
Taking the typical value r"0.1 T/s and the zero"eld splitting D 0.05 K [33], one gets a For an isolated spin , D "0, but di!erent couplings could generate a splitting, such as the hyper"ne one AI ' S, I", S", A+10 mK, D +10 mK [33]. Also, the molecule symmetry allows the Dzyaloshinsky}Moriya interactions that can generate D as large as 0.1 K [24]. A value D +50 mK is strongly sup ported by the experiment}to}model comparison in Figs. 3 and 4.
Fig. 4. Measured (a, top) and calculated (b, bottom) hysteresis loops for three "eld sweeping rates at ¹"0.1 K. The observed plateau is more pronounced at high sweeping rate. The equilibrium curve can be approximated by the median of the two branches of the low sweeping rate hysteresis loop (dotted curve). In the top inset is plotted the spin and phonon temperature ¹ "¹ for ¹"0.1 K and r"0.14 T/s, when the "eld is swept 1 from negative values. ¹ decreases until zero "eld and then 1 increases linearly within the plateau region. Then it overpasses the bath temperature to "nally reach the equilibrium. In the bottom inset the calculated number of phonons with u"D is & plotted vs. the sweeping "eld modulus (note the arrows) at equilibrium (¹ "¹ "¹, dashed line) and out-of-equilib 1 rium, (n "n , r"0.14 T/s, black line). The di!erence be2 22Q tween the two curves (thick segment Du) suggests the moving hole in the phonon distribution, while their intersection gives the plateau intercept of the equilibrium magnetization curve.
ground-state switching probability very close to unity: the spin occupancy of the two-level system is not modi"ed when the "eld is reversed. In such an LZ transition, the plateaux of Fig. 4 should decrease if the sweeping rate increases, which is contrary to the experiments. Not surprisingly, the
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S" spin system is not isolated: LZ transitions are non-adiabatic due to spin couplings to the environment. The mark of the V system is that this coupling is acting also near zero "eld because u+D is of the order of the bath temperature. The spin temperature ¹ is such that n /n "exp(D /k ¹ ), where 1 & 1 D is the two levels "eld-dependent separation, and & n (n ) the out of equilibrium (equilibrium) level occupation numbers. In the magnetization curves at 0.1 K (Figs. 3a and 4a), the spin temperature is signi"cantly lower than the bath temperature ¹ (n 'n , ¹ (¹) between !0.3 T 1 (when the magnetization curve departs from the equilibrium one) and 0.15 T (the "eld at which the magnetization curve intersects the equilibrium one). After this intercept, ¹ is larger than the bath 1 temperature (n (n , ¹ '¹), and at su$ 1 ciently high "elds (about 0.5 T) it reaches the equilibrium value (n "n , ¹ "¹). Note that the 1 magnetization curves measured between !0.7 and 0.02 T at fast sweeping rates (0.07 and 0.14 T/s) are nearly the same, suggesting weak exchange with the bath, i.e. nearly adiabatic demagnetization. Indeed, as we will show below, the silver sample holder is not su$cient to maintain the phonon temperature equal to the temperature of the bath. This is because in V below 0.5 K, the heat capa city of the phonons C is very much smaller than that of the spins C , so that the energy exchanged 1 between spins and phonons will very rapidly adjust the phonons temperature ¹ to that of the spin one ¹ . Thus, we can see the spin system and the 1 phonons as a single coupled system (quantum non-adiabatic LZ transitions) in weak exchange with the external bath (thermodynamical adiabatic demagnetization). The spins energy is transferred from the spins only to those phonons with
u"D (within the resonance line width). The & number of such lattice modes being much smaller than the number of spins, energy transfer between the phonons and the sample holder must be very di$cult, a phenomenon known as the phonon bottleneck [19,20]. Following Ref. [18], the number of phonons per molecule available for such resonant transitions is n " p(u) du/ 2 S (exp( u/k¹)!1), where p(u)du"3
107
u and u#du per molecule of volume V, v is the phonon velocity and Du is the transition linewidth due to fast hyper"ne "eld #uctuations (they broden both energy levels). Taking the typical values v+3000 m/s, ¹+10\ K and Du+10 MHz we "nd n of the order of +10\ to 10\ 2 phonons/molecule. Such a small number of phonons is very rapidly absorbed, burning a hole of width Du in the spin}phonon density of states at the energy u"D [19,20]. If this spin}phonon & density of states does not equilibrate fast enough, the hole must persist and move with the sweeping "eld, leading to a phonon bottleneck.
4. Quantitative approach and numerical calculations Now this description will be made quantitative. For a given splitting D , the time evolution of the & two-level populations n and of the phonon num bers n at ¹ obeys the set of two di!erential 2 equations [18]: !n "n "P n !P n , n "!(n !n )/q !P n #P n , (3) 2 2 2 where P are the transition probabilities be tween the two levels (they are themselves linear functions of n ) and q +¸/2v is the phonon2 bath relaxation time (L is the sample size). Using the notations x"(n !n )/(n !n ), y" (n !n )/(n #n/2) with n" p(u)du we get 2 2 S 2 x "(1!x!xy)/q , y "!y/q #bx , (4) where b"C /C and 1/q "P #P are the 1 direct spin-phonon relaxation time. By solving numerically this system for typical values, e.g. q "10\ s, q (10\ s, b'10, we can see that ¹ P¹ O¹ (phonon bottleneck) very rapidly, as 1 expected. This leads to y"1/x!1 and the second
The fast hyper"ne #uctuations are characterized by the transvers nuclear relaxation time ¹ associated with the dipolar internuclear interactions and the total spread of the energy is related to the hyper"ne coupling A [7}9].
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equation of the di!erential system becomes x "(x!x)/(1#bx)/q . In the limit b<1 (in our case b+10}10) this equation has the solution: !t/bq "x!x #ln((x!1)/(x !1)), (5) where x "x(t"0) and bq is the spin}phonon relaxation time (¹ "¹ P¹). When the system 1 is not far from equilibrium (x&1), we get an exponential decay of the magnetization, with the same time constant q "bq . For a spin system & [18] tanh(D /2k ¹) & , (6) D & with a"2p vNq /3Du (N is the molecule density). The dynamical magnetization curves calculated in this model are given in Figs. 3b and 4b. We started from equilibrium (x "1) in large negative "elds. Then we let the system relax for a very short time dt and we calculated x(dt) using Eq. (5). This value was taken as the initial value for the next "eld (the "eld step is rdt). The parameters have been chosen to mimic the measured curves of Figs. 3a and 4a. The obtained similarity supports the possibility of the phonon bottleneck e!ect at the timescale of a few 0.1 s. In Fig. 4a (inset), we show the variation of the calculated spin}phonon temperature ¹ for ¹"0.1 K and r"0.14 T/s. After 1 a cooling in negative "elds, we can note a linear variation in the plateau region (small positive "elds), where n /n +cst. The slope of this quasi adiabatic linear region gives the plateau position and varies with the bath temperature and sweeping rate. In Fig. 3b (inset) we show the calculated "eld evolution of the number of phonons at energy
u"D at equilibrium (¹ "¹ "¹, dashed & 1 line) and out-of-equilibrium (n "n , 2 22Q r"0.14 T/s, black line). The di!erence between the two curves (thick segment Du) suggests the moving hole in the phonon distribution, while their interq "a &
a"0.15 sK (2.9;10\ sJ), A+10 mK. Taking ¸& 30}50 lm, N&10 m\, Du&5;10 s\, one gets a phonon velocity v+2800}3600 m/s which is quite a reasonable value.
section gives the plateau intercept of the equilibrium magnetization curve (above which the hole disappears and ¹ "¹ '¹). Let us note that in 1 zero "eld the system is out}of}equilibrium even if magnetization passes through the origin of coordinates (without a barrier, the switch between # and ! follows the level structure shown in Fig. 3, inset). At larger "elds, in the plateau region, n /n + cst at timescales shorter than q "bq & (Eq. (6)), even after the plateau crosses the equilibrium curve. Equilibrium is reached when q be& comes small enough. Furthermore, we measured the relaxation of the magnetization of our crystal at di!erent "elds and temperatures, along the plateau region. The relaxation curves compared well with exponential decay and the obtained relaxation times are presented in Fig. 5a. The comparison with those calculated
Fig. 5. The relaxation times q , measured (a, top) and calculated & (b, bottom, same parameters as in Figs. 3 and 4b).
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(Fig. 5b) is acceptable. But we noted that a direct "t to Eq. (5) would necessitate larger values for a and D (+0.4!0.6 sK and +0.2}0.3 K).
[7] [8]
5. Conclusions [9]
In conclusion, this dynamical study of a single crystal of non-interacting V molecules with spin is an example of a non-adiabatic Landau}Zener model with dissipation. Due to the relatively large splitting D at the S" anticrossing in V mol ecules (i) the probability for ground-state adiabatic transitions is nearly equal to one and (ii) the spin system absorbs phonons during the Landau}Zener transition creating a hole in their distribution. The time and "eld evolution of this hole generates a `butter#ya hysteresis loop quite di!erent from the one of high-spin molecular magnets with large barrier and in"nitesimal D /k ¹ ratio (no phonons are available at the anticrossing). The e!ects presented in this paper seem to be a characteristic of molecules with low spin. Acknowledgements We are very pleased to thank P.C.E. Stamp, S. Myashita, I. Tupitsyn, A. K. Zvezdin, H. De Raedt for useful discussions and E. Krickemeyer, P. KoK gerler, D. Mailly, C. Paulsen and J. Voiron for on-going collaborations. References [1] L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, B. Barbara, Nature. 383 (1996) 145. [2] J.R. Friedman, M.P. Sarachik, J. Tejada, R. Ziolo, Phys. Rev. Lett. 76 (1996) 3830. [3] B. Barbara, L. Thomas, F. Lionti, I. Chiorescu, A. Sulpice, J. Magn. Magn. Mater. 200 (1999) 167. [4] B. Barbara, L. Gunther, Phys. World 12 (1999) 35. [5] A. Garg, in: Quantum Tunneling of Magnetization } QTM '94, NATO ASI, L. Gunther, B. Barbara (Eds.), Series E 301, Kluwer, Dordrecht, 1995, p. 273. [6] N.V. Prokof'ev, P.C.E. Stamp, in: Quantum Tunneling of Magnetization } QTM &94, NATO ASI, L. Gunther,
[10] [11]
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[28] [29] [30] [31] [32] [33]
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B. Barbara (Eds.), Series E 301, Kluwer, Dordrecht, 1995, p. 347. N.V. Prokof 'ev, P.C.E. Stamp, Phys. Rev. Lett. 80 (1998) 5794. P.C.E. Stamp, in: S. Tomsovic (Ed.), Tunneling in Complex Systems, Proc. Inst. for Nuclear Theory, vol.5, World Scienti"c, Singapore, 1998, pp. 101}197. G. Rose, P.C.E. Stamp, in: S. Tomsovic (Ed.), Tunneling in Complex Systems, Proc. Inst. for Nuclear Theory, vol. 5, cond}mat/9810350. I. Tupitsyn, B. Barbara, to be published. E.M. Chudnovsky, J. Tejada, Macroscopic Quantum Tunneling of the Magnetic Moment, Cambridge University Press, Cambridge, 1997. F. Hartmann}Boutron, P. Politi, J. Villain, Int. J. Mod. Phys. B 10 (1996) 2577. M. Leuenberger, D. Loss, EuroPhys. Lett. 46 (5) (1999) 692. G. Belessa, N. Vernier, B. Barbara, D. Gatteschi, Phys. Rev. Lett. 83 (2) (1999) 416. A.J. Leggett, S. Chakravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg, W. Zwerger, Rev. Mod. Phys. 59 (1987) 1. M. Grifoni, Peter HaK nggi, Phys. Rep. 304 (1998). 5}6. I. Chiorescu, W. Wernsdorfer, A. MuK ller, H. BoK gge, B. Barbara, Phys. Rev. Lett. 84 (2000) 3454. A. Abragam, B. Bleaney, Electronic Paramagnetic Resonance of Ions, Clarendon Press, Oxford, 1970 (Chapter 10). J.H. Van Vleck, Phys. Rev. 59 (1941) 724. K.W.H. Stevens, Rep. Prog. Phys. 30 (1967) 189. A. MuK ller, J. DoK ring, Angew. Chem. Int. Ed. Engl. 27 (1991) 1721. D. Gatteschi, L. Pardi, A.L. Barra, A. MuK ller, Mol. Eng. 3 (1993) 157}169. D. Gatteschi, L. Pardi, A.L. Barra, A. MuK ller, J. DoK ring, Nature 354 (1991) 465. M.I. Katsnelson, V.V. Dobrovitski, B.N. Harmon, cond}mat/9807176, July (1998). V.V. Dobrovitski, A.K. Zvezdin, private communication. B. Barbara, L. Thomas, F. Lionti, A. Sulpice, A. Caneschi, J. Magn. Magn. Mater. 177}181 (1998) 1324. W. Wernsdorfer, E. Bonet Orozco, K. Hasselbach, A. Benoit, D. Mailly, O. Kubo, H. Nakano, B. Barbara, Phys. Rev. Lett. 79 (1997) 4014. C. Zener, Proc. R. Soc. London, Ser. A 137 (1932) 696. L.D. Landau, E.M. Lifsitz, Quantum Mechanics, MIR, Moscow, 1966. H. De Raedt, S. Miyashita, K. Saito, D. Garcia}Pablos, N. Garcia, Phys. Rev. B 56 (18) (1997) 11761. S. Miyashita, J. Phys. Soc. Japan 64 (1995) 3207}3214. Y. Kayanuma, H. Nakayama, Phys. Rev. B 57 (20) (1998) 13099. G.C. Carter, L.H. Bennett, D.J. Kahan, Metallic Shifts in NMR, Progr. in Mat. Sci., Vol. 20, Part I, Pergamon Press, Oxford, 1977, p. 364.
Non-adiabatic Landau-Zener transitions in low-spin molecular magnet V
I. Chiorescu *, W. Wernsdorfer , A. MuK ller, H. BoK gge, B. Barbara Laboratoire de Magne& tisme Louis Ne& el, CNRS, BP 166, 38042-Grenoble, France Faku( ltat fu( r Chemie, Universita( t Bielefeld, D-33501 Bielefeld, Germany
Abstract The V polyoxovanadate molecule is made of 15 spins with antiferromagnetic couplings. It belongs to the class of molecules with very large Hilbert space dimension (2 in V , 10 in Mn -AC). It is a low spin/large molecule with spin S". Contrary to large spins/large molecules of the Mn -AC type, V has no energy barrier against spin rotation. Magnetization measurements have been performed and despite the absence of a barrier, magnetic hysteresis is observed over a timescale of several seconds. This new phenomenon characterized by a `butter#ya hysteresis loop is due to the e!ect of the environment on the quantum rotation of the entangled 15 spins of the molecule, in which the phonon density of states is not at its equilibrium (phonon bottleneck). 2000 Elsevier Science B.V. All rights reserved. PACS: 75.50.Xx; 75.45.#j; 71.70.-d Keywords: Magnetic quantum e!ects; Relaxation phenomena; Dissipative two-level system
1. Introduction The recent evidence for quantum tunneling of the magnetization in big molecules with large spin S"10 [1}3] allows a link to be made between mesoscopic physics and magnetism [4]. In these systems, energy barriers are high and tunnel splittings between symmetric and antisymmetric quantum states are exponentially small (10\}10\ K). As a consequence, quantum relaxation is slow and is only related to the spin baths [5}11]. Resonant phonon transitions are irrelevant, unless between states at di!erent energies [12,13] or in the pres-
* Corresponding author. Tel./fax: #33-4-76-88-11-94/11-91. E-mail address: [email protected] (I. Chiorescu).
ence of a transverse "eld large enough to create a tunnel splitting of the order of the temperature energy scale [14]. Here we show that the reversal of an ensemble of non-interacting spins S" is irre versible as a consequence of di!erent couplings to the environment. Such an e!ect gives rise in our system to a `butter#ya hysteresis loop. Spin rotation in V may be viewed as an experimental realization of the theoretical problem of a two-level system with dissipation [14}17]. Spin}phonon transitions take place between the two splitted ground states at well de"ned energies u"D , & where D is the "eld dependent energy separation & of the two levels. The width Du of such transitions being relatively narrow, the number of available phonon states at energy u is much smaller than the number of molecules and a hole must be burned
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 3 8 5 - 1
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in the spin}phonon density of states [18}20]. This hole, moving with the sweeping "eld, leads to a non-equilibrium spin}phonon density of states, giving in the V complex the observed phonon bottleneck and hysteresis. A phonon `bottleneck plateaua, identi"ed on the measured hysteresis loop, gives a characteristic butter#y shape. These results are corroborated with numerical calculations based on this model.
2. The V15 magnetic moment evolution in low sweeping rates The V complex of formula K [V'4 As O (H O)] ) 8H O was prepared as described in Refs. [21}23]. It is made of a lattice of molecules with 15 V'4 ions of spin S", placed in a quasi-spherical layered structure formed of a triangle, sandwiched by two hexagons (Fig. 1). All the interactions being antiferromagnetic, the resultant (collective spin) S". The symmetry is trigonal (space group R3 c, a"14.029 As , a"79.263, <"2632 As ). The unit-cell contains two V clus ters and it is large enough so that dipolar interactions between spin molecules are negligible (few mK). Each hexagon contains three pairs of strongly coupled spins (J !800 K) and each spin at a corner of the inner triangle is coupled to two of those pairs (one belonging to the upper hexagon and the other belonging to the lower hexagon (J J !150 K, J J !300 K [22]). Thus, the V molecule can be seen as formed by three groups of "ve V'4 ions with the resultant spin and assembled on each corner of the inner triangle. These three spins interact with each other through two main paths, one passing by the upper hexagon and the other passing by the lower one. This is a typical example of frustrated molecule, where the exchange J between the spins is much smaller than the exchange interactions between two spins. Two types of single-crystal magnetization measurements have been performed: (i) characterization of the general thermodynamical properties of the system at equilibrium and (ii) study of the non-equilibrium to equilibrium transition. In the "rst case, a small dilution refrigerator allowing measurements above 0.1 K was inserted
Fig. 1. The geometry and the spin coupling scheme in V com plex. There are "ve antiferromagnetic exchange constants leading to a ground state S" in "elds below 2.8 T and S" for greater "elds (here such a "eld is pointing down).
into an extraction magnetometer providing "elds up to 16 T, with low sweeping rates. Below 0.9 K we observed one jump in zero "eld and two others at B $2.8 T (Fig. 2). They correspond to the ", !2", 2, and ", 2", 2 spin transitions, with respective saturations at 0.50$0.02l and 2.95$0.02l per V molecule, in agreement with the level scheme given in the inset of Fig. 2. For all temperatures the magnetization curves are reversible and do not show any anisotropy, showing that eventual energy barrier preventing spin reversal must be quite small (less than 50 mK). The magnetization curves measured at equilibrium (Fig. 2) are "tted using the Heisenberg Hamiltonian for three frustrated spins S": H"! J (S S #S S #S S ) ? ? ? ? ? ? ? ?6 7 8 !gk B (S #S #S ) (1) with S "S "S ", J "J (0, J (0 (anti 6 7 8 ferromagnetic coupling), g"2, B the applied "eld and k the Bohr magneton. In the isotropic case (J "J ) the con"guration S" becomes 6 7 8
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3. Spin}phonon coupling in V15 in low 5elds and high sweeping rates: a dissipative spin two-level system
Fig. 2. The agreement between the calculated curve at thermodynamical equilibrium (line) and the experimental magnetization data is quite good for an isotropic exchange J "!2.445 K. For J "J "!2.75 K, J "!2.2 K and 6 7 8 the "eld applied in the triangle plane (dashed line) one observes that the calculated second #ip is slightly displaced but its slope is still greater than the experimental one. The energy levels vs. applied "eld in the e!ective triangle approach in the inset are represented. When the applied "eld reaches !3J /(2gk ) the ground state switches from S" to S".
favourable against the S" one for B " !3J /(2gk ). The measured value B 2.8 T gives J !2.5 K. The comparison of the calculated curves (S "1i"S #S #S "i2 averaged over 8G 8 8 8 the di!erent eigenstates E , "i2, i"1}8) with the G measured ones is quite good at any temperature. The width of the transition at B $2.8 T is nevertheless broader in the experiments (Fig. 2). This broadening of about 0.7 T cannot be inferred to dipolar or hyper"ne "eld distributions (about 1 and 40 mT, respectively). The observed anisotropy of the electronic g-factor (g "g "1.95 and ? @ g "1.98) is also too small [23]. Antisymmetrical A Dzyaloshinsky}Moriya interactions (see Refs. [24}26] and references therein) allowed by symmetry and coupling the states S" and S", could explain this broadening. In low "elds, susceptibility measurements give the following e!ective paramagnetic moments: k "g(S(S#1)"1.75$0.02k corresponding to S" below 0.5 K, and k "3$0.02k corre sponding to three independent spins S" below 100 K. It con"rms clearly that the ground state of the molecule is S" below 0.5 K.
We showed that, between !2.8 and 2.8 T, the total spin of this molecule is S". Such a small spin leads to vanishingly small energy barrier and relatively large splitting in zero "eld (&10\ K). Resonant spin}phonon transitions between the symmetric and antisymmetric states at resonance are then possible in sub-Kelvin experiments, contrary to the case of big molecules in zero "eld, where these states are much too close. The sensitivity and time resolution of the microSQUID magnetometer [27] allowed to study very small V crystals in good contact with the thermal bath. Down to 50 mK non-equilibrium behavior was nevertheless observed at fast sweeping rate (up to dB /dt"0.7 T/s). Few hysteresis loops are rep resented in Figs. 3a and 4a (only the positive part, the other one being rigorously symmetrical). When the "eld increases, coming from negative values, the magnetization passes through the origin of the coordinates, reaches a plateau and then approaches saturation. This leads to a winged hysteresis loop characterized by the absence of irreversibility near zero "eld. The wings depend sensitively on temperature and sweeping "eld rate. As an example, in Fig. 3 where three hysteresis loops are presented (¹"0.1, 0.15 and 0.2 K) for a given sweeping rate (0.14 T/s), the plateau is higher and more pronounced at low temperature. The same tendency is observed at given temperature and faster sweeping rate (Fig. 4). At a given temperature, the equilibrium magnetization curve can be approximated by the median of the two branches of a low sweeping rate hysteresis loop (e.g. in Fig. 4 for dB /dt" 4.4 mT/s). When compared to the equilibrium curve, a given magnetization curve shows that (i) in negative "elds the system is colder than the bath, (ii) the spin system remains colder until the magnetization curve intersects the equilibrium curve, (iii) Then the spin temperature overpass as the bath temperature, and (iv) at su$ciently high "elds (about 0.5 T) the system reaches the equilibrium. In order to interpret this magnetic behaviour of the V molecules, we will analyse how the level occupation numbers vary in this two-level system (see Fig. 4a inset) when sweeping an external "eld.
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Fig. 3. Measured (a, top) and calculated (b, bottom) hysteresis loops for three temperatures and for a given "eld sweeping rate 0.14 T/s. The plateau is more pronounced at low T. The inset is a schematic representation of a two-level system S "$ with 8 repulsion due to non-diagonal matrix elements. In a swept "eld the switching probability P is given by the Landau}Zener formulae (see text). The two levels are broadened by the hyper"ne "elds and the absorption or the emission of phonons can switch the polarization state of spins.
In the bare Landau}Zener (LZ) model [28}32] the probability for the ", !2", 2 transition is pD . P"1!exp 1! (2) 4 k r
Taking the typical value r"0.1 T/s and the zero"eld splitting D 0.05 K [33], one gets a For an isolated spin , D "0, but di!erent couplings could generate a splitting, such as the hyper"ne one AI ' S, I", S", A+10 mK, D +10 mK [33]. Also, the molecule symmetry allows the Dzyaloshinsky}Moriya interactions that can generate D as large as 0.1 K [24]. A value D +50 mK is strongly sup ported by the experiment}to}model comparison in Figs. 3 and 4.
Fig. 4. Measured (a, top) and calculated (b, bottom) hysteresis loops for three "eld sweeping rates at ¹"0.1 K. The observed plateau is more pronounced at high sweeping rate. The equilibrium curve can be approximated by the median of the two branches of the low sweeping rate hysteresis loop (dotted curve). In the top inset is plotted the spin and phonon temperature ¹ "¹ for ¹"0.1 K and r"0.14 T/s, when the "eld is swept 1 from negative values. ¹ decreases until zero "eld and then 1 increases linearly within the plateau region. Then it overpasses the bath temperature to "nally reach the equilibrium. In the bottom inset the calculated number of phonons with u"D is & plotted vs. the sweeping "eld modulus (note the arrows) at equilibrium (¹ "¹ "¹, dashed line) and out-of-equilib 1 rium, (n "n , r"0.14 T/s, black line). The di!erence be2 22Q tween the two curves (thick segment Du) suggests the moving hole in the phonon distribution, while their intersection gives the plateau intercept of the equilibrium magnetization curve.
ground-state switching probability very close to unity: the spin occupancy of the two-level system is not modi"ed when the "eld is reversed. In such an LZ transition, the plateaux of Fig. 4 should decrease if the sweeping rate increases, which is contrary to the experiments. Not surprisingly, the
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S" spin system is not isolated: LZ transitions are non-adiabatic due to spin couplings to the environment. The mark of the V system is that this coupling is acting also near zero "eld because u+D is of the order of the bath temperature. The spin temperature ¹ is such that n /n "exp(D /k ¹ ), where 1 & 1 D is the two levels "eld-dependent separation, and & n (n ) the out of equilibrium (equilibrium) level occupation numbers. In the magnetization curves at 0.1 K (Figs. 3a and 4a), the spin temperature is signi"cantly lower than the bath temperature ¹ (n 'n , ¹ (¹) between !0.3 T 1 (when the magnetization curve departs from the equilibrium one) and 0.15 T (the "eld at which the magnetization curve intersects the equilibrium one). After this intercept, ¹ is larger than the bath 1 temperature (n (n , ¹ '¹), and at su$ 1 ciently high "elds (about 0.5 T) it reaches the equilibrium value (n "n , ¹ "¹). Note that the 1 magnetization curves measured between !0.7 and 0.02 T at fast sweeping rates (0.07 and 0.14 T/s) are nearly the same, suggesting weak exchange with the bath, i.e. nearly adiabatic demagnetization. Indeed, as we will show below, the silver sample holder is not su$cient to maintain the phonon temperature equal to the temperature of the bath. This is because in V below 0.5 K, the heat capa city of the phonons C is very much smaller than that of the spins C , so that the energy exchanged 1 between spins and phonons will very rapidly adjust the phonons temperature ¹ to that of the spin one ¹ . Thus, we can see the spin system and the 1 phonons as a single coupled system (quantum non-adiabatic LZ transitions) in weak exchange with the external bath (thermodynamical adiabatic demagnetization). The spins energy is transferred from the spins only to those phonons with
u"D (within the resonance line width). The & number of such lattice modes being much smaller than the number of spins, energy transfer between the phonons and the sample holder must be very di$cult, a phenomenon known as the phonon bottleneck [19,20]. Following Ref. [18], the number of phonons per molecule available for such resonant transitions is n " p(u) du/ 2 S (exp( u/k¹)!1), where p(u)du"3
107
u and u#du per molecule of volume V, v is the phonon velocity and Du is the transition linewidth due to fast hyper"ne "eld #uctuations (they broden both energy levels). Taking the typical values v+3000 m/s, ¹+10\ K and Du+10 MHz we "nd n of the order of +10\ to 10\ 2 phonons/molecule. Such a small number of phonons is very rapidly absorbed, burning a hole of width Du in the spin}phonon density of states at the energy u"D [19,20]. If this spin}phonon & density of states does not equilibrate fast enough, the hole must persist and move with the sweeping "eld, leading to a phonon bottleneck.
4. Quantitative approach and numerical calculations Now this description will be made quantitative. For a given splitting D , the time evolution of the & two-level populations n and of the phonon num bers n at ¹ obeys the set of two di!erential 2 equations [18]: !n "n "P n !P n , n "!(n !n )/q !P n #P n , (3) 2 2 2 where P are the transition probabilities be tween the two levels (they are themselves linear functions of n ) and q +¸/2v is the phonon2 bath relaxation time (L is the sample size). Using the notations x"(n !n )/(n !n ), y" (n !n )/(n #n/2) with n" p(u)du we get 2 2 S 2 x "(1!x!xy)/q , y "!y/q #bx , (4) where b"C /C and 1/q "P #P are the 1 direct spin-phonon relaxation time. By solving numerically this system for typical values, e.g. q "10\ s, q (10\ s, b'10, we can see that ¹ P¹ O¹ (phonon bottleneck) very rapidly, as 1 expected. This leads to y"1/x!1 and the second
The fast hyper"ne #uctuations are characterized by the transvers nuclear relaxation time ¹ associated with the dipolar internuclear interactions and the total spread of the energy is related to the hyper"ne coupling A [7}9].
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equation of the di!erential system becomes x "(x!x)/(1#bx)/q . In the limit b<1 (in our case b+10}10) this equation has the solution: !t/bq "x!x #ln((x!1)/(x !1)), (5) where x "x(t"0) and bq is the spin}phonon relaxation time (¹ "¹ P¹). When the system 1 is not far from equilibrium (x&1), we get an exponential decay of the magnetization, with the same time constant q "bq . For a spin system & [18] tanh(D /2k ¹) & , (6) D & with a"2p vNq /3Du (N is the molecule density). The dynamical magnetization curves calculated in this model are given in Figs. 3b and 4b. We started from equilibrium (x "1) in large negative "elds. Then we let the system relax for a very short time dt and we calculated x(dt) using Eq. (5). This value was taken as the initial value for the next "eld (the "eld step is rdt). The parameters have been chosen to mimic the measured curves of Figs. 3a and 4a. The obtained similarity supports the possibility of the phonon bottleneck e!ect at the timescale of a few 0.1 s. In Fig. 4a (inset), we show the variation of the calculated spin}phonon temperature ¹ for ¹"0.1 K and r"0.14 T/s. After 1 a cooling in negative "elds, we can note a linear variation in the plateau region (small positive "elds), where n /n +cst. The slope of this quasi adiabatic linear region gives the plateau position and varies with the bath temperature and sweeping rate. In Fig. 3b (inset) we show the calculated "eld evolution of the number of phonons at energy
u"D at equilibrium (¹ "¹ "¹, dashed & 1 line) and out-of-equilibrium (n "n , 2 22Q r"0.14 T/s, black line). The di!erence between the two curves (thick segment Du) suggests the moving hole in the phonon distribution, while their interq "a &
a"0.15 sK (2.9;10\ sJ), A+10 mK. Taking ¸& 30}50 lm, N&10 m\, Du&5;10 s\, one gets a phonon velocity v+2800}3600 m/s which is quite a reasonable value.
section gives the plateau intercept of the equilibrium magnetization curve (above which the hole disappears and ¹ "¹ '¹). Let us note that in 1 zero "eld the system is out}of}equilibrium even if magnetization passes through the origin of coordinates (without a barrier, the switch between # and ! follows the level structure shown in Fig. 3, inset). At larger "elds, in the plateau region, n /n + cst at timescales shorter than q "bq & (Eq. (6)), even after the plateau crosses the equilibrium curve. Equilibrium is reached when q be& comes small enough. Furthermore, we measured the relaxation of the magnetization of our crystal at di!erent "elds and temperatures, along the plateau region. The relaxation curves compared well with exponential decay and the obtained relaxation times are presented in Fig. 5a. The comparison with those calculated
Fig. 5. The relaxation times q , measured (a, top) and calculated & (b, bottom, same parameters as in Figs. 3 and 4b).
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(Fig. 5b) is acceptable. But we noted that a direct "t to Eq. (5) would necessitate larger values for a and D (+0.4!0.6 sK and +0.2}0.3 K).
[7] [8]
5. Conclusions [9]
In conclusion, this dynamical study of a single crystal of non-interacting V molecules with spin is an example of a non-adiabatic Landau}Zener model with dissipation. Due to the relatively large splitting D at the S" anticrossing in V mol ecules (i) the probability for ground-state adiabatic transitions is nearly equal to one and (ii) the spin system absorbs phonons during the Landau}Zener transition creating a hole in their distribution. The time and "eld evolution of this hole generates a `butter#ya hysteresis loop quite di!erent from the one of high-spin molecular magnets with large barrier and in"nitesimal D /k ¹ ratio (no phonons are available at the anticrossing). The e!ects presented in this paper seem to be a characteristic of molecules with low spin. Acknowledgements We are very pleased to thank P.C.E. Stamp, S. Myashita, I. Tupitsyn, A. K. Zvezdin, H. De Raedt for useful discussions and E. Krickemeyer, P. KoK gerler, D. Mailly, C. Paulsen and J. Voiron for on-going collaborations. References [1] L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, B. Barbara, Nature. 383 (1996) 145. [2] J.R. Friedman, M.P. Sarachik, J. Tejada, R. Ziolo, Phys. Rev. Lett. 76 (1996) 3830. [3] B. Barbara, L. Thomas, F. Lionti, I. Chiorescu, A. Sulpice, J. Magn. Magn. Mater. 200 (1999) 167. [4] B. Barbara, L. Gunther, Phys. World 12 (1999) 35. [5] A. Garg, in: Quantum Tunneling of Magnetization } QTM '94, NATO ASI, L. Gunther, B. Barbara (Eds.), Series E 301, Kluwer, Dordrecht, 1995, p. 273. [6] N.V. Prokof'ev, P.C.E. Stamp, in: Quantum Tunneling of Magnetization } QTM &94, NATO ASI, L. Gunther,
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