Macrospin approximation and quantum effects in models for magnetization reversal
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Macrospin approximation and quantum effects in models for magnetization reversal
Mohammad Sayad, Daniel G¨utersloh and Michael Potthoff
I. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany
The thermal activation of magnetization reversal in magnetic nanoparticles is controlled by theanisotropy-energy barrier. Using perturbation theory, exact diagonalization and stability analysisof the ferromagnetic spin- s Heisenberg model with coupling or single-site anisotropy, we study theeffects of quantum fluctuations on the height of the energy barrier. Opposed to the classical case,there is no critical anisotropy strength discriminating between reversal via coherent rotation and vianucleation/domain-wall propagation. Quantum fluctuations are seen to lower the barrier dependingon the anisotropy strength, dimensionality and system size and shape. In the weak-anisotropy limit,a macrospin model is shown to emerge as the effective low-energy theory where the microscopic spinsare tightly aligned due to the ferromagnetic exchange. The calculation provides explicit expressionsfor the anisotropy parameter of the effective macrospin. We find a reduction of the anisotropy-energybarrier as compared to the classical high spin- s limit. PACS numbers: 75.60.Jk,75.10.Jm
I. INTRODUCTION
Nanosystems of interacting magnetic moments haveattracted much interest due to ongoing technological ad-vances in the field of magnetic data-storage devices. Withdecreasing system size, ferromagnetic particles on solidsurfaces or ferromagnetic molecular magnets like Fe and Mn , for example, are found in a single-domainstate. The question of the stability of the ferromagneticstate against different kinds of thermal and quantum fluc-tuations is a crucial technological issue which at the sametime provides serious challenges to a theoretical modelingand understanding.The magnetic properties of a ferromagnetic nanopar-ticle can be described by a Heisenberg model H J = − X ij J ij s i s j (1)with positive exchange coupling J ij > s i and s j . The spins give rise to magneticmoments which are assumed to be localized at the sites i = 1 , ..., L . The latter may constitute a d -dimensionallattice of finite size with L sites in total and constant ex-change J = J ij between nearest neighbors i and j only.In a ground state of the model all spins are perfectlyaligned, and the particle is ferromagnetic. However, thisstate is unstable against thermal fluctuations. Strictlyspeaking, for a system of finite size, the ferromagneticstate is destroyed at any finite temperature by fluctu-ations originating from a coupling of the system to anexternal heat bath.On the other hand, anisotropic contributions to theHamiltonian, such as a single-site anisotropy of the form H D = − D X i s iz , (2)which for D > z -axis as the easy axis,stabilize the ferromagnetic state. Inclusion of H D leads to a model with only two degenerate ground states inwhich all spins are pointing into the + z or − z direction,respectively. The anisotropic ferromagnetic Heisenbergmodel H = H J + H D in fact represents the simplestmodel to study magnetization reversal, i.e. the thermallyactivated transition between two ground states acrossan energy barrier induced by the anisotropy. The rateof magnetization switching and thus the magnetic sta-bility obviously crucially depends on the height of theanisotropy-energy barrier ∆ E .For weak anisotropy D ≪ J , it is self-evident to con-sider the spins as tightly coupled by the exchange inter-action and forming a huge macrospin, S = X i s i , (3)the dynamics of which can be approximated by an effec-tive model H macro = − D macro S z + const. . (4)with an anisotropy barrier∆ E = D macro S . (5)In this macrospin model, magnetization reversal takesplace at zero temperature by suppression of the barrierdue to an external magnetic field as described by theStoner-Wohlfarth model. At finite temperature T , re-versal may be caused by thermal activation. For low T , AB FIG. 1: Two different models for magnetization reversal. A:Coherent rotation. B: Nucleation and domain-wall propaga-tion. Figure inspired by Ref. 4. the switching frequency is exponentially small and a fer-romagnetic state is stable over macroscopically relevanttimes while for high T , in the so-called superparamag-netic state, the magnetization vanishes on the time scaleof the measurement. Classical theory for superparam-agnetic dynamics based on the Landau-Lifshitz equation and including a stochastic Langevin field to simulate athermal bath leads to an Arrhenius law for the thermalactivation rate Γ ∝ exp( − β ∆ E ). This Neel-Brown lawagain emphasizes the role of the energy barrier ∆ E forthe magnetic stability of the particle.For stronger anisotropy, the macrospin approximationis no longer applicable as magnetization reversal prefer-ably takes place by nucleation and domain-wall propa-gation (see Fig. 1). Here, classical many-spin models are frequently used to study the reversal mechanism andin particular the transition from coherent rotation tonucleation and domain-wall propagation with increasingstrength of the anisotropy. In particular, classical Monte-Carlo simulations are employed where Monte-Carlo up-date steps are related to physical time propagation. Asa function of the model parameters, reversal mechanismsdifferent from coherent rotation, can be identified and“phase boundaries” can be found in this way. The stronginfluence of the shape of the nanoparticles on the switch-ing rate, as observed experimentally, can be explainedby theoretical analysis of domain-wall propagation. The macrospin has a high spin quantum number S and may be treated to a good approximation classically.There are, however, different types of quantum effects tobe considered: For example, quantum tunneling as a re-versal mechanism is competing with thermally inducedreversal. In case of not too small systems, tunnelingis significant only for systems extremely isolated fromany dissipative environment at ultralow temperatures.For a macrospin coupled to a conduction band, Kondoscreening is another issue. Since
S > / As thequantum character of the macrospin is not accounted forby simple Langevin dynamics, different approaches have been suggested to treat the macro- or many-spin dy-namics in contact with bosonic baths by means of quan-tum master-equation approaches and to determine, e.g.,the blocking temperature. The importance of a many-spin model, for example, is demonstrated by studies ofthe probability density function of the macrospin ob-tained by replacing thermal with Markov processes. Differences between the classical-spin limit and largequantum spins have been discussed. Here we reconsider the question of competing reversalmechanisms by taking a quantum many-spin model asthe starting point. The competition between coherent ro-tation and nucleation can be addressed in a less ambitiousway by studying the static properties of an anisotropicquantum Heisenberg model H = H J + H ani . As thesingle-site anisotropy (2), which is frequently studied in the classical context, cannot be used for the lowest spin s = 1 / s i = const,we also consider a coupling anisotropy of the form H ∆ = − X ij ∆ ij s iz s jz , (6)i.e. H ani = H D or H ani = H ∆ . This model will be inves-tigated by different complementary techniques includingperturbation theory in H ani , exact diagonalization andthe Lanczos approach for systems with small size L andclassical stability analysis in the high-spin limit.There are different goals of the present paper: As thestatus of the macrospin approximation is less clear inthe quantum case, we aim at a strict derivation of themacrospin model in the weak-anisotropy limit. In thisway it should be possible to relate the effective anisotropystrength of the macrospin to the parameters of the un-derlying many-spin model. The dependence on the spinquantum number s should exhibit quantum effects forsmall s and recover the classical result for s → ∞ . Fur-thermore, beyond the weak-anisotropy limit, the break-down of the macrospin approximation and the competi-tion between the different reversal mechanisms, depend-ing on system size, coordination, dimensionality and on s ,will be investigated. Our goal is to understand whetheror not it makes a qualitative difference for the transi-tion from coherent rotation to nucleation with increasinganisotropy strength if starting from a classical or froma quantum spin model. Finally, the high-spin limit isaddressed as a reference and for comparison with thequantum-mechanical calculations.The paper is organized as follows: In the next sec-tion II we briefly introduce the model, the basic conceptsand notations by referring to exact-diagonalization re-sults. Sec. III present the results of first-order pertur-bation theory for weak anisotropy and the derivation ofthe macrospin model. Its limitations are discussed inSec. IV on the basis of calculations employing the Lanc-zos technique. The dependence of the anisotropy-energybarrier on the spin-quantum number s and classical sta-bility analysis in the high-spin limit is addressed in Sec.V. Finally, Sec. VI concludes the paper. II. EXACT-DIAGONALIZATION RESULTS FORTHE ANISOTROPY BARRIER
Due to the SU(2) spin-rotation symmetry of theHeisenberg model, Eq. (1), the total (or macro) spin, Eq.(3), is a conserved quantity: [ S , H ] = 0. We consider theeigenstates of H J : H J | n, S, M i = E n ( S ) | n, S, M i . (7)The states are classified according to their total spinquantum number S = S min , ..., S max − , S max and theirtotal magnetic quantum number M = − S, ..., S − , S .Here, S max = Ls and S min = 0 for s even or L even,while S min = 1 / L for the sake of sim-plicity. Further, n labels the eigenstates in the invariantsubspace with fixed S and M . For a given S , the SU(2)invariance implies the eigenenergies E = E n ( S ) to be2 S + 1-fold degenerate and independent of M .For the present case of ferromagnetic exchange cou-pling J ij >
0, the total spin quantum number is maxi-mal, S = S max = Ls , for a ground state. It is easy tosee that the fully polarized state with all spins pointinginto + z direction, | F i ≡ | m = s i ⊗ · · · ⊗ | m L = s i , isa ground state. Assuming all exchange interactions tobe equal between nearest neighbors, J ij = J >
0, theground-state energy is given by E ( S max ) = − ( J/ Ls z where z = P Li =1 z i /L is the average of the coordinationnumbers z i .The 2 S max +1 = 2 Ls +1-fold degeneracy of the ground-state energy will partly be lifted by the anisotropy H ani ,Eq. (2) or Eq. (6). H ani breaks the SU(2) symmetry,[ S , H ani ] = 0, but preserves the rotational symmetrywith respect to the easy axis ( z axis): [ S z , H ani ] = 0.The anisotropic eigenstates | m, M i are therefore charac-terized by M and by an additional quantum number m labeling the states in the subspace with fixed M . Thecorresponding eigenenergies E m ( M ) will depend on M with E m ( M ) = E m ( − M ) , (8)resulting in a symmetric anisotropy barrier with a height∆ E given by the respective ground-state energies of thefull Hamiltonian in the invariant subspaces with M = 0and M = S max :∆ E = E ( M = 0) − E ( M = S max ) > . (9)For systems with a moderately large Hilbert space, i.e.for small s and small number of lattice sites L , the energyeigenvalue problem can be solved numerically. In case ofthe anisotropic problem H J + H ani , conservation of S z can be exploited only. The dimension of the invariantsubspace H ( M ) is given by: dim H ( M ) = [( S max − M ) / (2 s +1)] X k =0 ( − k (cid:18) Lk (cid:19) × (cid:18) L − S max − M − (2 s + 1) kL − (cid:19) (10)where the binomial coefficients (cid:18) ab (cid:19) = 0 for b > a , andwhere [ x ] denotes the greatest integer with [ x ] ≤ x . For s = 1 / L = 14 spins.A simple example for a spin- s = 1 / L = 10sites with open boundaries is given by Figs. (2), (3)and (4). The perturbed (∆ >
0) ground-state energy -5 -4 -3 -2 -1 0 1 2 3 4 5M-2024 ene r g y / J FIG. 2: Energy eigenvalues of the isotropic s = 1 / L = 10 sites andopen boundaries as obtained numerically by exact diagonal-ization. Eigenvalues are classified according to the total mag-netic quantum number M = − S max , ..., S max with S max = 5.The nearest-neighbor exchange coupling J = 1 sets the energyscale. in the one-dimensional M = S max subspace is easily cal-culated as E ( M = S max ) = − ( J/ / Ls z . The M = ± ( S max −
1) subspaces are reached by a single spinflip, i.e. excitation of a magnon. As there must be asmany magnons as lattice sites, the dimension of the sub-spaces is given by L each. Their discrete energy spectrumcan be seen in Figs. (2) and (4) for ∆ = 0 and ∆ = J ,respectively.The low-energy sector of the isotropic spectrum forarbitrary M is shown in Fig. 3: Ground states have themaximal total spin S = S max . The ground-state energyis 2 S max + 1-fold degenerate. Excited states with M = ± ( S max − S = S max − -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6M-2.2-2.1-2-1.9 ene r g y / J S=5S=4S=3S=2 S=4S=1
FIG. 3: The same as Fig. 2 but on enlarged energy scale closeto the ground-state energy. The total spin quantum numbersof the different spin multiplets are indicated. -4 -2 0 2 4M-4-20246 ene r g y / J FIG. 4: The same as Fig. 2 but with the coupling-anisotropyterm, Eq. (6), included. A constant anisotropy ∆ ij = ∆between nearest neighbors is assumed. Calculations have beenperformed for ∆ = J . perpositions of states with different S (but the same M )in this case, their energy becomes M dependent. Conse-quently, an anisotropy barrier develops which, for a verystrong anisotropy ∆ = J , is large as compared to thefinite-size gap between the ground states and the firstexcited states. Still, there is a residual degeneracy ofthe two ground states | F i and | − F i in the subspaces M = ± S max which implies that thermal fluctuations de-stroy ferromagnetic order. However, switching between | F i and | − F i now requires that thermal fluctuationsmust overcome the barrier. The anisotropy thus leads toa superparamagnetic stabilization of the magnetic stateon a certain time scale as described by the N´eel-Brownmodel. III. PERTURBATIVE DERIVATION OF THEMACROSPIN MODEL
In the weak-anisotropy limit, the different microscopicspins are tightly bound together by the ferromagneticexchange coupling and form a huge macrospin which ro-tates from + z to − z direction, i.e. the microspins rotatecoherently. To make this intuitive argument rigorous andquantitative, we derive the macrospin model as an effec-tive low-energy model by means of first-order perturba-tion theory in H ani in the following. This will result ina linear dependence of ∆ E on the anisotropy strength.Non-trivial dependencies, however, may be expected withrespect to L , s and the matrix J ij of exchange-couplingconstants.First-order perturbation theory requires to computethe matrix element h S max , M ′ | s iz s jz | S max , M i where thestates | S, M i form an orthonormal common eigenbasis of S and S z . Note that this includes the case of coupling( i = j ) and single-site ( i = j ) single-site anisotropy. To find the action of s iz s jz onto the state | S max , M i we write | S max , M i = p ( S max + M )! p ( S max − M )!(2 S max )! S S max − M − | F i (11)in terms of the fully polarized state with all spins point-ing in + z direction, | F i ≡ | S max , M = S max i = | m = s i · · · | m L = s i . Here S − = S x − iS y . Eq. (11) is obtainedby straightforward spin algebra. Using the commutator[ s iz s jz , S k − ] = − kS k − − ( s i − s jz + s j − s iz )+ kδ ij S k − − s i − + k ( k − S k − − ( s i − s j − + s j − s i − ) (12)where s i ± = s ix ± is iy , we are left with matrix elementsof the form h S max , M ′ |O| F i where O contains microspinoperators s ... on the right-hand side acting on | F i as wellas macrospin operators S ... on the left-hand side whichare considered to act on h S max , M | . The effect of theformer operators is trivial. The effect of the latter onesis obtained with the help of the relation S k + | S max , M i = p ( S max + M + k )!( S max − M )! p ( S max + M )!( S max − M − k )! × | S max , M + k i . (13)Using h S max , M | s i − | F i = 2 s √ S max (14)for M = S max − h S max , M | s i − s j − | F i = 8 s p S max (2 S max −
1) ( i = j )= 4 s (2 s − p S max (2 S max −
1) ( i = j )(15)for M = S max −
2, a straightforward calculation leads to h S max , M | s iz s jz | S max , M ′ i = δ MM ′ (cid:20) s + (2 s − δ ij ) M − L s L s − L (cid:21) . (16)This is a remarkably simple result which may be usedto compute the anisotropy energy barrier E ( M ) in dif-ferent situations. To give an example, we consider themodel H = H J + H ani where a constant exchange interac-tion J ij = J between nearest neighbors is assumed (andlikewise for ∆ ij in case of a coupling anisotropy). Thisimplies that geometrical properties enter via the latticetopology only. For the case of the single-site anisotropy(2), s = 1 /
2, we get: E ( M ) = − JLs z − D (cid:18) s − Ls − M + s L ( L − Ls − (cid:19) + O ( D ) , (17)while for the coupling anisotropy (6) with non-zero andconstant ∆ ij = ∆ between nearest neighbors only, wefind: E ( M ) = − JLs z − ∆ z (cid:18) s Ls − M − Ls Ls − (cid:19) + O (∆ ) , (18)where z = P i z i /L is the average coordination number.This corresponds to a macrospin model Eq. (4) with D macro = 2 s − Ls − D (19)for the single-site anisotropy, and D macro = zs Ls − E = D macro S . Eqs. (19)and (20) provide an explicit and quantitative relationbetween the anisotropy parameter of the effective low-energy macrospin model and the microscopic model pa-rameters.Quite generally we note that this relation is non-trivial.In the limit L → ∞ we find D macro ∝ /L formally.However, for fixed D or ∆, the macrospin approxima-tion becomes less accurate with increasing system size L :Eventually the finite-size gap, which controls the valid-ity of the perturbative treatment, is of the same orderof magnitude as the anisotropy strength, i.e. the expan-sion parameter. The classical limit is reached for s → ∞ .Here, D macro = D/L or D macro = ( z/ /L , respec-tively. The validity of the macrospin approximation forthe classical model is discussed below. IV. LIMITATIONS OF THE MACROSPINMODEL
It is easily seen that the limits of weak and of stronganisotropy correspond to two qualitatively different waysof how to overcome the barrier. Consider the model sys-tem discussed above but with a large coupling anisotropy∆ ≫ J . In the extreme case J = 0, we recover the Isingmodel, H = H ∆ , which exhibits a highly degenerate en-ergy spectrum. It is displayed in Fig. 5. Still there aretwo degenerate ground states in the M = ± S max sec-tors which are separated by an energy barrier. However,the barrier is entirely flat and thus qualitatively differentfrom the quadratic trend given by the macrospin model.The tendency towards a flat barrier is already seen inFig. 4 for ∆ = J .The energy eigenstates are constructed trivially in theIsing limit. In the ground state with M = S max the spinsare aligned ferromagnetically, i.e. | S max , S max i = | ↑ , ↑ , ..., ↑i . Its energy is E ( M = S max ) = − (∆ / Ls z = − .
25 for the example shown in the figure. The groundstate with M = S max − -5 -4 -3 -2 -1 0 1 2 3 4 5M-2-1012 ene r g y / ∆ FIG. 5: The same as Fig. 2 but for J = 0 and ∆ = 1. spin, i.e. | S max , S max − i = | ↓ , ↑ , ..., ↑i . The correspond-ing excitation energy is ∆ E = 2∆ s = 0 .
5, resulting in E ( M = S max −
1) = − .
75. Shifting the “domain wall”to the right, i.e. | S max , M i = | ↓ , ..., ↓ , ↑ , ..., ↑i , is possi-ble without further energy cost. Hence, nucleation anddomain-wall propagation is the apparent mechanism formagnetization reversal in the Ising limit.The breakdown of the macrospin approximation hap-pens for still weak anisotropies but is gradual ratherthan abrupt. This is demonstrated by the exact-diagonalization results for the anisotropy-energy barrier E ( M ) shown in Fig. 6. For strong anisotropy ∆ = J (corresponding to Fig. 4), the macrospin approximationEq. (18) completely fails and strongly overestimates thebarrier height ∆ E . With decreasing ∆, the macrospinapproximation becomes more and more reliable. Still,at ∆ = J/
10 deviations from the exact-diagonalizationresults are visible on the scale of Fig. 6. If interpretedwithin the classical theory, this is still significant as theenergy barrier affects the thermal activation rate expo-nentially strong.The fact that the breakdown of the macrospin modelwith increasing strength of the anisotropy is gradualrather than abrupt also implies a gradual change be-tween the corresponding reversal mechanisms. Within aquantum model, there are no well-defined phase bound-aries separating coherent rotation from nucleation anddomain-wall propagation. This becomes obvious againin Fig. 7 where the anisotropy energy barrier ∆ E isshown as a function of ∆. The macrospin and the exact-diagonalization results start to deviate from each otherat arbitrarily weak ∆, i.e. already at second order in theanisotropy strength.Nevertheless, a typical anisotropy strength can beidentified that marks the smooth crossover from coher-ent rotation to nucleation or the qualitative breakdownof the macrospin approximation. Fig. 7 shows that thiscrucially depends on the system size L . For L = 10spins (red lines), the macrospin model is valid up to muchstronger ∆ as compared to the case with L = 24 spins -5 -4 -3 -2 -1 0 1 2 3 4 5-4.5-4.0-3.5-3.0-2.5-2.0 ∆ =J ∆ =J/100micro ∆ =J/10macro ene r g y / J total spin projection M FIG. 6: Anisotropy barrier E ( M ) as obtained from themacrospin model (Eq. (18), lines) compared to the exact-diagonalization result (points) for three different anisotropystrengths as indicated. Results for an open chain of L = 10spins with s = 1 / (blue lines). Generally, with increasing L , substantialdeviations from the linear trend of ∆ E (∆) are found forweaker and weaker ∆. In the strong-anisotropy limit,∆ E (∆) approaches a linear trend again which is givenby the Ising limit (black line). Note that for the one-dimensional model considered here, the domain-wall en-ergy and thus ∆ in the strong-anisotropy limit is indepen- I s i n g li m i t m a c r o s p i n ∆ / J m a c r o s p i n ene r g y ba rr i e r / J s=1/2 FIG. 7: Height of the anisotropy energy barrier ∆ E as a func-tion of the (coupling) anisotropy strength ∆ for open chainsof L = 10 and L = 24 spins with s = 1 / E as a function of ∆ for the Isinglimit with J = 0 for both, L = 10 and L = 24. f u ll y c onne c t ed micro m a c r o ∆ / JL=24 s=1/2 ene r g y ba rr i e r / J FIG. 8: Height of the anisotropy energy barrier ∆ E as afunction of the (coupling) anisotropy strength ∆ for systemsof L = 24 spins with s = 1 / E in the full modelis always smaller). Results for different planar geometriesas indicated. ∆ E for the fully connected model, where themacrospin approximation is exact, is shown for comparison(black line). dent of the system size while in the macrospin or weak-anisotropy limit ∆ E is roughly proportional to L (see Eq.(18)).Calculations for L = 24 spins with s = 1 / This approximates the ground-state en-ergy E ( M ) of H in the invariant subspace H ( M ) by theground-state energy of H in a Krylov space K n ( M ) = span {| i, M i , H | i, M i , ..., H n − | i, M i} ⊂ H ( M )(21)of dimension n ≪ dim H ( M ) that is spanned bythe states | i, M i , H | i, M i , ..., H n − | i, M i where | i, M i ∈H ( M ) is an arbitrary initial state. Typically, n ≈ z . This is demonstrated with Fig. 8where again macrospin and exact-diagonalization (Lanc-zos) results for the ∆ dependence of the energy barrier arecompared. Here, the system size is kept fixed to L = 24while the system geometry varies from a one-dimensionalchain to a compact two-dimensional array, i.e. the aver-age coordination number z increases. The correspondingincreasingly better agreement of the macrospin approxi-mation with the exact-diagonalization results is easily un-derstood by consideration of the extreme case (see blackline in Fig. 8): For z i = L − z , i.e. for the fullyconnected model with J ij = J and ∆ ij = ∆ for arbitrarypairs i, j , the Hamiltonian reduces to H = − J S − ∆ S z + const (22)and thus E ( M ) = − ∆ M + const , (23)i.e. in this limit the macrospin approximation is triviallyexact. V. CRITICAL ANISOTROPY STRENGTH INTHE CLASSICAL LIMIT
The limit of the classical anisotropic Heisenberg modelshould be recovered for spin quantum number s → ∞ .Fig. 9 shows the anisotropy barrier ∆ E as a function ofthe coupling anisotropy strength ∆ as obtained from theLanczos technique for a small chain of L = 4 sites. Thispermits a study of the spin model with quantum numbersup to s = 10. To compare the results for different s , weconsider rescaled parameters, J ij J ij /s ( s + 1) , ∆ ij ∆ ij /s ( s + 1) , (24)which corresponds to a rescaling s i s i / p s ( s + 1) ofthe spin variables. Disregarding quantum fluctuations,this amounts to a classical spin of unit length | s i | = 1 inthe limit s → ∞ .For s = 1 / s = 1, the range of anisotropy strengths where analmost linear trend is found, increases while the crossoverto the Ising limit is seen to be delayed. With increasingspin quantum number s , these changes continue in a sys-tematic way and become more and more pronounced.From the result for s = 10, one can easily anticipate the s → ∞ limit: Here the anisotropy barrier is strictly linearin ∆ for weak anisotropies up to a certain critical value∆ c /J = 1. At ∆ c we find a cusp, i.e. a discontinuousjump of ∂ ∆ E/∂ ∆ . Beyond this point, a fairly broadcrossover region follows until finally a linear-in-∆ trendis established again in the ∆ → ∞ limit.The strong-anisotropy limit is easily understood, asquantum fluctuations are suppressed anyway for all s .For weak anisotropy, on the other hand, the macrospinapproximation is found to be exact in a finite range upto ∆ = ∆ c , i.e. perturbative corrections of second and s=1/2 3/2s=1 s=10s=1/2 s=10 2 ∆ / J de r i v a t i v e L=4 ∆ / J ene r g y ba rr i e r / J FIG. 9: ∆ dependence of energy barrier ∆ E for openchains with L = 4 spins and different spin quantum num-bers s as indicated. Exact results for the full model H J + H ∆ after rescaling s i → s i / p s ( s + 1) for s =1 / , , / , , / , , / , , ,
10. The inset shows the firstderivative of ∆ E with respect to ∆. higher order in ∆ are decreasing with increasing s andfinally vanish exactly in the classical limit s → ∞ . Forall ∆ < ∆ c , magnetization reversal takes place by a co-herent rotation of completely correlated spins in the clas-sical case. For any finite s , on the other hand, quantumfluctuations break up this perfect correlation. The mag-netic moments h s i i are still aligned to the z axis duringthe reversal – we have h s i i ∝ e z since S z is a conservedquantity – but the effect of fluctuations can be seen inthe spin-spin correlation functions: In the M = 0 sector,for example, we find h s iz s jz i = 0 classically ( s → ∞ ) and h s iz s jz i = s (1 − / (1 − / Ls )) in the quantum case forsufficiently small ∆ (see Eq. (16)), i.e. a small quantity ofthe order O ( L − ) which is independent of i and j . Withincreasing ∆ at finite s , however, h s iz s jz i gradually be-comes more and more site dependent. As a function of j and fixing i to one of the chain edges, h s iz s jz i is monoton-ically decreasing in the M = 0 sector. Spins at the chainedges are oriented antiferromagnetically: h s iz s jz i < i = 1 and j = L . Hence, for any finite ∆ in the quan-tum case there are already indications for domain-wallformation while in the classical limit there is a clear-cutdistinction, given by a critical coupling ∆ c , between thetwo reversal mechanisms.To understand the origin of the “phase transition”in the classical limit and to find a means to compute∆ c for arbitrary system size and geometry, we firstperformed numerical minimizations of the total energy E ( { s , ..., s L } ) of the classical model H = H J + H ani forboth, H ani = H ∆ and H ani = H D . Arbitrary spin con-figurations { s , ..., s L } with | s i | = 1 and satisfying theconstraint P Li =1 s iz = M are considered. For M = 0 this D
FIG. 10: Sketch of the different magnetization-reversalmechanisms in the classical model for anisotropy strength D smaller (macrospin model, coherent rotation) or larger (nucle-ation) than the critical value D c and for infinite D (nucleation,vanishing domain-wall width) as verified by calculations for L = 4 spins. provides the classical barrier height and the correspond-ing optimal spin configuration. Calculations have at firstbeen done for different small system sizes L , typically L = 4, to reproduce the critical point anticipated fromthe results of Fig. 9.We found the optimal spin configuration with mini-mal total energy for a given M to be coplanar in allcases, i.e. there is a coordinate frame with s iy = 0 for allspins. As dictated by the symmetry of the model, con-figurations that transform into each other by rotationsaround the z axis are degenerate. For a chain geometry,there is also degeneracy between configurations obtainedby mirror transformations, s i → s L +1 − i , and nucleationcan thus start from both edges with equal probability.Furthermore, for M = 0 we have s i,z = − s L − i +1 ,z in theoptimal configuration. The results can be summarized bythe sketch of the optimal spin configurations at a given M in Fig. 10. Results for coupling and for single-siteanisotropy are found to be qualitatively the same. Asis seen from Fig. 10, the different mechanisms for mag-netization reversal in the two limits of weak and stronganisotropy are reproduced from the calculations for small L .More important, however, the physical origin of thecritical point can be identified easily: At ∆ = ∆ c (or at D = D c , respectively), and approaching the critical pointfrom the weak-anisotropy side, the optimal spin configu-ration exhibits an instability towards a non-collinear or-dering. As sketched in Fig. 10, the instability developssimultaneously for any M .This observation can be exploited to calculate the criti-cal anisotropy strength for arbitrary system size and sys-tem geometry in the following way: Using s iy = 0, weparameterize the spin variables as s i = ( p − m i , , m i )with − ≤ m i ≤
1. This yields the energy functional inthe form E ( { m , ..., m L } ) = − X ij D ij m i m j − X ij J ij (cid:18)q − m i q − m j + m i m j (cid:19) , (25) which comprises both, the coupling and the single-siteanisotropy case. To find the critical point, it is sufficientto concentrate on M = 0 and to compute the magneticresponse of the system in the M = 0 state with spinconfiguration m = · · · = m L = 0. Close to the criti-cal point, we can assume m i ≪
1. Expanding the totalenergy in powers of m i then yields E ( { m , ..., m L } ) = const. + 12 X ij χ − ij m i m j + O ( m i )(26)with the inverse susceptibility matrix χ − ij = − J ij + J i δ ij − D ij , (27)and where we have defined J i ≡ P j J ij . We assume themodel parameters as real and symmetric, J ij = J ji and D ij = D ji .Note that approaching the isotropic limit withanisotropy strength D →
0, the susceptibility matrix ex-hibits an eigenvalue approaching zero. This correspondsto an instability of the state towards ferromagnetic order-ing with all m i > M .For finite D , this eigenvalue is negative and thus im-plies an indefinite susceptibility. This indicates that, byconstruction, χ refers to the thermodynamically unstableexcited state with M = 0 and m i = 0. The local insta-bility of this state towards a non-collinear alignment ofthe magnetic moments at the critical anisotropy strength D c shows up as a zero of another eigenvalue of χ . Fora short chain of L = 4 spins with coupling anisotropythis happens exactly at ∆ c = J , in agreement with theLanczos results displayed in Fig. 9. The eigenvector cor-responding to the zero eigenvalue describes the criticalspin profile which agrees with the M = 0 spin configura-tion for D > D c sketched in Fig. 10.Results for the critical anisotropy strength as a func-tion of the system size are shown in Fig. 11 for both,coupling and single-site anisotropy and for systems of dif-ferent dimensions d . In all cases we considered systemswith open boundaries. Generally, the anisotropy strengthup to which the macrospin approximation is found tobe valid, decreases rapidly with system size but dependsonly weakly on the type of the anisotropy term. There isalso a weak dependence on the form of the particle as isobvious from the comparison of cubic arrays and spheresin d = 3 dimensions.For small nanosystems, the dimension and shape de-pendence of D c is somewhat irregular. For a larger num-ber of classical spins, on the other hand, the criticalanisotropy strength for which the macrospin model canbe applied, is the stronger the higher the dimension of thenanostructure. In fact, in the limit L → ∞ , the differentresults are found to follow a simple power law, D c ( L ) ∝ L − /d = l − , (28)where d is the dimensionality, L the number of spins and l = L /d the linear extension of the system.
100 200 300 400 5000.00010.0010.010.11 c r i t i c a l an i s o t r op y s t r eng t h D c / J linear chains square arrayscubic arrays system size L spherescoupling anis.single-site anisotropy FIG. 11: System-size dependence of the critical value forthe anisotropy strength D c in units of J for one-dimensionalopen chains, for two-dimensional square arrays and for three-dimensional simple cubic arrays and spheres with open bound-aries. Blue lines: single-site anisotropy. Red lines: couplinganisotropy. This can easily be explained by comparing the en-ergy barrier for coherent rotation ∆ E cor . rot . with the bar-rier for nucleation and domain-wall propagation ∆ E nucl. .While the volume of the system ∝ l d is relevant for∆ E cor . rot . ∝ D l d , the d − σ ( D ) ∝ p J/D essentially determines ∆ E ∝ D σ ( D ) l d − . The critical anisotropy is then obtainedfrom D c l d ∝ D c p J/D c l d − , resulting in the scaling law(28). VI. CONCLUSIONS
Anisotropy terms coupled to the ferromagnetic Heisen-berg model of many-spin nanosystems give rise to anenergy barrier ∆ E that crucially determines the ther-mal activation of magnetization reversal and thus themagnetic stability of nanosystems. Employing perturba-tion theory, exact diagonalization and Lanczos as wellas classical total-energy minimization and stability anal-ysis, different quantum effects have been revealed thatsubstantially affect the barrier height.Quantum effects are absent in the limit of stronganisotropy D . For D/J → ∞ the Ising model is ap-proached, i.e. a classical model where there are no quan-tum fluctuations. The magnetization-reversal mecha-nism in the Ising limit is given by nucleation at theedge or surface of the system followed by the propaga-tion of a domain wall with minimal (unit) width. Withdecreasing anisotropy strength, the domain-wall widthincreases. Finally, a critical value D = D c is reachedwhere the reversal mechanism changes to a coherent ro-tation of the spins that are tightly aligned by the ex-change coupling J and form a huge macrospin. This critical anisotropy strength is well defined in the clas-sical limit s → ∞ (with model parameters rescaled bythe factor s ( s + 1)) and can be computed for arbitrarydimension, system size and shape by an analysis of thelocal stability of the susceptibility matrix in the globally(thermodynamically) unstable excited state at the topof the barrier. For D < D c , i.e. for coherent rotation,the classical anisotropy-barrier height is simply given by∆ E class = E ( m = · · · m L = 0) − E ( m = · · · = m L = 1),i.e.: ∆ E class = L D . (29)This is the same as the barrier of a classical macrospinof length S = L s , composed of microscopic spins ofunit length | s | = 1, as described by a macrospin model H = D macro S z with a barrier ∆ E = D macro L if D macro = D/L . Classically, and in the regime wherecoherent rotations takes place, the macrospin approxi-mation is exact.The first quantum effect manifests itself in the absenceof the a well-defined critical anisotropy strength. Ascan be seen in the spin-spin correlation function h s iz s jz i ,there is rather a smooth crossover from nucleation anddomain-wall propagation to coherent rotation with de-creasing D . This crossover becomes less and less sharpwith decreasing s , i.e. as the importance of quantum fluc-tuations increases.Second, for any finite anisotropy strength, themacrospin approximation is no longer correct in thequantum case. Only within perturbation theory to firstorder in D , the macrospin model emerges as the effectivelow-energy theory of the anisotropic quantum Heisenbergmodel. Comparing the true anisotropy-energy barrier∆ E of the many-spin system with the one of the cor-responding effective macrospin model, we find∆ E < ∆ E macro . (30)In the quantum case, there is virtually never a reversalby pure coherent rotation but always some “admixture”of nucleation and domain-wall propagation. The latterleads to the decrease of ∆ E as compared to ∆ E macro .This quantum effect becomes more and more pronouncedwith increasing anisotropy strength, with increasing sys-tem size, with decreasing dimensionality and with de-creasing average coordination number.Third, the quantum derivation of the macrospin modelby means of first-order perturbation theory provides anexplicit expression for the anisotropy parameter [see Eqs.(19) and (20)]. This leads to an energy barrier that islower than the classical one:∆ E macro < ∆ E class . (31)Consider a single-site anisotropy case as an example: Af-ter the rescaling D → D/s ( s + 1) we have ∆ E macro =[(2 s − / (2 Ls − L s /s ( s + 1)] D . Therewith, theclassical result (29) is recovered in the high-spin limit:lim s →∞ ∆ E macro = ∆ E class . In the low-spin case, e.g. for0 s = 1, which is the extreme case for single-site anisotropy,however, we have ∆ E macro = ( L/ D . This is smaller bya factor of 4 than the classical barrier ∆ E class . Even inthe limit of large systems L → ∞ , there is substantialdifference,∆ E macro = 12 2 s − s + 1 LD < LD = ∆ E class , (32)compared to the classical limit which is approachedrather slowly, i.e. ∆ E macro / ∆ E class = 1 − s/s + O ( s − ).We conclude that, in addition to quantum many-spin ef- fects, there is a quantum correction of the macro-spinmodel itself which, for low spin quantum numbers, canbe as important as the former one. Acknowledgments
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