Exchange interactions and full magnetization process of multispin nanoclaster Mn4
EExchange interactions and full magnetization process of multispin nanoclaster Mn A.K. Zvezdin and D.I. Plokhov
A.M. Prokhorov General Physics Institute of Russian Academy of Sciences,38 Vavilov Str., 119991, Moscow, Russia (Dated: November 27, 2018)Full magnetization process of magnetic nanocluster Mn , including all its actual multiplets S =9 / S = 11 / S = 13 /
2, and S = 15 /
2, is theoretically investigated. The formulas needed todetermine the exchange constants of cluster Mn from experimental data are obtained. It is shownthat quantum jumps of magnetization of this nanocluster are in area of megagauss magnetic fields.Remarkable feature of considered nanocluster Mn differing it from other multispin molecules isthat its exchange spin Hamiltonian supposes exact diagonalization. This provides a reliable basisfor comparison of theoretical predictions with experimental data and is important for developmentof new experimental techniques of high-spin molecule research. PACS numbers: 75.50.Xx — molecular magnets
I. INTRODUCTION
Presently, magnetic molecular nanoclusters draw greatattention . The clusters are metal-organic molecules con-taining a number of ions, usually ions of transitive d -metals such as Fe, Mn, Co, Ni, etc. or the rare-earthions, for example Ho, Y, La, etc. They are shortly de-noted as X n , where X is Fe, Mn, etc., n is the number ofX-ions in the molecule, although this denotation is notcomplete and unambiguous. The most known and in-vestigated nanoclusters are Fe , Fe , Fe , Mn , V ,etc. Magnetic nanoclusters are also called molecular,or mesoscopic, magnets because they occupy intermedi-ary position between microscopical objects such as atomsand ions, which possess spin moments, and macroscopicalmagnetic bodies characterized by magnetization, averagemagnetic moment per volume unit.Such mesoscopic magnets possess interesting quantumproperties : a) presence of a quantum hysteresis loop ofa single molecule, b) macroscopic quantum tunneling ofthe full magnetic moment of a molecule, c) effects of aquantum interference at magnetic reversal in a field per-pendicular to an easy axis (the effects are related with theBerry-phase) . Their properties are rather interesting inthe range of submillimeter frequencies .From practical point of view molecular magnets are in-teresting as building blocks for construction of new ma-terials and nanostructures. They are also considered asperspective objects for quantum computer science andmagnetic refrigerating tequnics .Among this wide variety of such mesoscopic magnetsX n nanoclusters with formula Mn O Br(OAc) (dbm) ,where OAc is the acetate ion, and dbm is the dybenzoyl-methane (see fig. 1) are of special interest. It is known ,that this molecule behaves at low temperatures as an iso-lated magnet with an easy axis along C axis and heightof a power barrier of 1.25 meV.The point symmetry group of molecule Mn is (approx-imately) C v . According to , antiferromagnetic interac-tions of three ions Mn ( S = 2) and Mn ( S = 3 / C (passing through Br ions), form fer- rimagnetic spin structure with spin of S = 9 / S = 7 / E ≈
22 meV.Magnetic properties of cluster Mn in the ground state(multiplet) of S = 9 / . In thepresent paper the full magnetization process of Mn , in-cluding all its actual multiplets S = 9 / S = 11 / S = 13 /
2, and S = 15 /
2, is theoretically considered, andthe formulas needed to determine exchange constants ofthe cluster from experimental data are obtained. Re-markable and rare feature of considered here nanoclusterMn , differing it from other multispin molecules, is thatits exchange spin Hamiltonian supposes an exact diago-nalization. This provides a reliable basis for comparisonof theoretical predictions with experimental data and itis important for development of new experimental tech-niques of research of high-spin molecules. II. SPIN HAMILTONIAN
A spin Hamiltonian of considered molecule Mn withinthe framework of Heisenberg model reads H = H E + H A , (1)where exchange Hamiltonian H E = J ( S + S + S ) S ++ J ( S S + S S + S S ) + (2)+ µ B B ( g ( S + S + S ) + g S ) . Here S , S , S , and S are the spins of ions Mn ( S = S = S = 2) and Mn ( S = 3 / g and g are their g -factors, J > J > B is the external magnetic field.Term H A in Eq. (1) describes the anisotropy energythat for the ground multiplet ( S = 9 /
2) can be intro- a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t duced as H A = D (cid:20) S z − S ( S + 1) (cid:21) ++ B ˆ O + E ( S x − S y ) , (3)where D = − · − eV, B = − . · − eV, and | E | = 2 . · − eV . Comparison of these figures withabove mentioned value of energy ∆ E ≈
22 meV of thefirst excited multiplet shows that (cid:107) H A (cid:107) << (cid:107) H E (cid:107) . Thismeans that it is possible to be limited to the exchangespin Hamiltonian for calculation of a full magnetizationcurve. If necessary, the influence of magnetic anisotropycan be taken into account by means of the perturbationtheory. III. ADDITION OF SPINS
To find the energy spectrum of exchange spin Hamil-tonian H E one can use the following scheme of the spin moments addition: S + S = S , S + S = S , (4) S + S = S t , where S t is the total spin of a molecule and S is the totalspin of ions Mn . The chosen scheme of spin additionis not unique. It is possible to choose other ways, say S = S + S or S = S + S . It is known, that sucharbitrariness does not influence final results because forany addition scheme the space of the states forms a fullset . It is necessary to notice that it is usual to use fourmoment pared addition scheme, for example, S or S .Due to natural reasons the above-stated scheme (3 + 1)is more convenient. Transition between various schemesof addition can be made by means of Racah factors orWigner 9 j -symbols (Fano factors).The corresponding wave functions being eigenfunctionsof operators S , S , S , S , S , S , S t , and S tz are | S S ( S ) S ( S ) S S t M (cid:105) = (cid:88) m m (cid:88) m m (cid:88) m m C S t MS m S m C S m S m S m C S m S m S m | S m (cid:105)| S m (cid:105)| S m (cid:105)| S m (cid:105) , (5) TABLE I: Addition of the moments S + S = S . S S S where C SmS (cid:48) m (cid:48) S (cid:48)(cid:48) m (cid:48)(cid:48) are the Clebsch-Gordan coefficients.According to the addition scheme S takes all integervalues from | S − S | to S + S , i.e. from 0 to 4. Indexes m i ( i = 0, 1, 2, 3, 4, 12) run all integer (or half-integerfor i = 4) values from − S t to S t . The addition rule of m -indexes should be taken into account: m + m = m ,etc. Allowable values S are shown in Table I.From this table one sees that some values S appearin different combinations of S and S , i.e. various wavefunctions correspond to them. This fact is characterizedby the factor of degeneration K ( S ) = 2 S + 1 for S ≤ K ( S ) = 7 − S for S ≥
2. Addition of the moments S and S is described by Table II. The states with given S t have degeneration factors Q ( S t ) shown in Table III. TABLE II: Addition of the moments S + S = S t . S S S t / /
21 3 / /
2, 3 /
2, 5 /
22 3 / /
2, 3 /
2, 5 /
2, 7 /
23 3 / /
2, 5 /
2, 7 /
2, 9 /
24 3 / /
2, 7 /
2, 9 /
2, 11 /
25 3 / /
2, 9 /
2, 11 /
2, 13 /
26 3 / /
2, 11 /
2, 13 /
2, 15 / S t . S t / / / / / / / / Q ( S t ) 2 4 4 4 4 3 2 1 IV. ENERGY SPECTRUM
Using the considered scheme of spin addition, it is easyto find the energy spectrum of exchange Hamiltonian H E : E ( S t , S , M ) = J S t ( S t + 1) − J − J S ( S + 1) ++ 2 µ B M B − J S ( S + 1) − J S ( S + 1) , (6)where it is supposed that g = g = 2. If necessary, dis-tinction between g and g is easy for taking into accountunder the perturbation theory. It should be remindedthat quantum number M accepts all half-integer valuesfrom − S t to S t . Assuming B = 0 and normalizing energyof levels on J scale one obtains E J = (cid:15) ( S t , S ) = 12 S t ( S t + 1) − γ S ( S + 1) (7)i.e. the spectrum consists of 48 levels completely definedby dimensionless parameter γ = ( J − J ) /J .At this stage it is expedient to address to experimentaldata. Using Table II and Eq. (7), it is easy to be con-vinced that the observable ground state ( S t = 9 /
2) canbe realized in the given model only if γ > /
4. Indeed,at increase of γ level (cid:15) ( S t = 9 / , S = 6) consequentlycrosses levels (cid:15) ( S t = 1 / , S = 2), (cid:15) ( S t = 3 / , S = 3), (cid:15) ( S t = 5 / , S = 4), and (cid:15) ( S t = 7 / , S = 5) at points γ = 2 /
3, 7 /
10, 8 /
11, and 3 /
4. As it is supposed that J > J >
0, the area of γ change is limited tointerval 3 / < γ < S = 7 / S = 5), we receive an additional condition for de-termining of exchange parameters J and J (or J and γ ): ∆ EJ = (cid:15) (cid:18) , (cid:19) − (cid:15) (cid:18) , (cid:19) = 6 γ − , (8)∆ E = 22 meV. V. FULL MAGNETIZATION PROCESS
Now we consider a curve of magnetization of moleculeMn at T = 0. We should take into account the abovementioned data on the power spectrum. From Eq. (6)follows that the basic condition of system correspondsto M = − S t . It is obvious that there is a consecutivecrossing of levels S t = 9 / → /
2, 11 / → /
2, and13 / → / S = 6 does not change. The values of magnetic fields atwhich there are these crossings are easily calculated fromEq. (6). They are consequently equal to92 →
112 : B = 114 J µ B →
132 : B = 134 J µ B (9)132 →
152 : B = 154 J µ B At each point of crossing there is the correspondingchange of the ground state accompanied by jump ∆ M =2 µ B of the magnetic moment of the molecule. The curveof magnetization is shown in Fig. 2. Having measured the value of the field correspondingto the first jump of the magnetic moment it is possibleto determine value J and after that the value of J fromEqs. (8) and (9). J = 411 µ B B (10) J = 14 J − ∆ E γ are in interval ( , B , B and B : B > Eµ B = 231 TB > Eµ B = 273 T (11) B >
56 ∆ Eµ B = 315 T Thus, the full magnetization problem have to be solvedby megagauss magnetic field technique . VI. ISOTHERMS OF MAGNETIZATION
Isotherms of magnetization M ( B, T ) at any tempera-tures can be calculated in a standard way by means ofthe partition function Z = (cid:88) S K ( S ) (cid:88) S t Q ( S t ) (cid:88) M exp (cid:18) − (cid:15) − bMτ (cid:19) (12)where normalized energy (cid:15) ( S t , S ), factors of degener-ation K ( S ) and Q ( S t ), and parameter γ are definedabove, τ = T /J , b = 2 µ B /J . The magnetic moment ofthe system is then determined as M = TZ ∂Z∂B = gµ B Z ∂Z∂b . (13)Of course, practical calculations of isotherms M ( B, T )in actual region of temperatures can be reduced essen-tially if one uses properties of the power spectrum of amolecule. The corresponding phase diagram is shown inFig. 3.
VII. MOTIVATION
At discussion of motivation of works on molecular mag-netism of nanoclusters as one of arguments frequently re-sults the thesis that nanoclusters, being mesoscopic ob-jects, allow to throw light on not quite clear questions oftransition from quantum laws, characteristic for atomsand molecules, to classical ones, characteristic for macro-scopical bodies. The model of a multispin nanoclusterMn investigated in this work is represented rather per-spective in this respect since it, being rather simple andadmitting an exact diagonalization of its Hamiltonian, issubstantially interesting and rich for the description ofquantum transformations having well investigated ana-logues in macroscopical magnetism. For instance, classi-cal analogue of the quantum jumps of the magnetic mo-ment of a molecule considered above (fig. 2) is effectof a turn magnetic sublattices (spin flip) in ferrimagnet-ics, induced by an external magnetic field . The role ofsublattices in a case of Mn is played by ions Mn andMn or their total spins S and S . The analogue of fullmagnetization of ferrimagnetic is the magnetic momentof a molecule equal to 2 µ B S t .The Hamiltonian in Eq. (2) supposes generalizationfor any spins S i ( i = 1, 2, 3) and S . Limiting transi-tion to classics ( S ∼ S ∼ N → ∞ ) is naturally per-formed at the additional assumption J N = const . Thelast can be interpreted as the requirement of indepen-dence of N the exchange field between sublattices. Itis natural to normalize the magnetic moments so that m i = µ B S i N = const were analogues of magnetizationsublattices. Then the picture of transition from ”ferri-magnetic” spin structure with S t = | S − S | to ”ferro-magnetic” ( S t = S + S ) looks the same as previouslydescribed (fig. 222), but the number and amplitude ofjumps aspire, accordingly, to ∞ and 0 with the increaseof N , i.e. the transition becomes quasi-continuous. Toeach jump ( S t → S t + 1) there corresponds the value of afield at which there is a crossing of E ( S t ) and E ( S t + 1)levels, i.e. B S t ,S t +1 = J ( S t +1)2 µ B . It also follows that crit-ical fields to which the beginning and the end of spinreorientation correspond are equal B c = J ( S − S + 1)2 µ B = λ | m − m | + O (cid:18) N (cid:19) , (14) B c = J ( S + S )2 µ B = λ ( m + m ) , (15)where λ = J N (2 µ B ) is a constant of a molecular field knownin the theory of ferrimagnetism.Eqs. (14) and (15) coincide with those from the theoryof spin-flip in macroscopical ferrimagnetics (the secondequation coincides precisely, and the first one coincides toan order of 1 /S ). It is natural, that this quantum amend-ment aspires to the classical formula to 0 with growth ofa spin value. But for nanoclusters with small value of aspin in the ground state distinction between the quan-tum and classical description in this respect is essential.It is possible to tell that this distinction is caused byquantum fluctuations of spins in a vicinity of the criticalfields. They are maximal in area of B c and close to zeroin area B c .Quantum fluctuations are important also for consider-ation of a question on value of the magnetic moments ofspin subsystems (ions Mn and Mn in this case). In the classical theory it is supposed that the magnetic mo-ments sublattices are constants. Quantum mechanicalconsideration shows that quantum reduction of sublat-tices, especially strong near B c , takes place. In work it is shown for a nanocluster Mn on the basis of the per-turbation theory. It is desirable to do this for Mn , whichwill be given in the future publications.Let us note one more interesting difference betweenspin transformations in ”ferrimagnetic” nanocluster andthose in macroscopical ferrimagnetics. The last are con-sidered as cooperative phase transitions of second kind with corresponding critical anomalies, while the firstare essentially quantum transformations occuring in onemolecule. Possibly, the role of thermal fluctuations of aparameter in a quantum case is played by quantum spinfluctuations attention was paid on (for a case of nanoclus-ters such as Mn and Fe ) in works , which howeverare executed in a model of giant spin, i.e. in stronglyreduced Hilbert space. The model of a multispin systemconsidered in this work with magnetic transformation isinteresting and in this respect, as admitting the exactsolution in full Hilbert space of the system.At last, we note one more important question which de-serves studying within the framework of the given model.It is known, that energy of chemical binding in moleculesdepends on a spin state. In this case the magnetic field,changing the spin state, influences also chemical bind-ing of a molecule and thus its configuration. In its turn,change of a configuration influences critical fields of spintransformations. In molecular magnetism the situationcan be more various and rich. In this connection we notepaper in which the effect of downturn of configurationsymmetry of a nanocluster in a situation with the spin-degenerated ground state is investigated. VIII. ACKNOWLEDGES
This research was financed by the Russian Foundationfor Basic Research (project 08-02-01068).
IX. FIGURES
Figure 1. (a) Molecular structure of the title complexMn . View along the approximate C3 axis. For claritythe H atoms are omitted. Mn ions are drawn as largeblack spheres, C and O atoms as small black and greyspheres, respectively. The large grey sphere representsthe Br − ion, which obscures the Mn ion just behind.(b) Schematic view of the core of Mn , with III and IVrepresenting Mn ( S = 2) and Mn ( S = 3 / − ionschematically represent the situation at ambient and highexternal pressure, respectively. Figure 2.
A curve of full magnetization and quantumsteps in nanocluster Mn . Figure 3.
Phase diagram. For both figures b =2 µ B B/J , γ = 0 . τ = T /J = 0 .
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