Magnetization Orientation Dependence of the Quasiparticle Spectrum and Hysteresis in Ferromagnetic Metal Nanoparticles

PHYSICAL REVIEW B 66, 094430 ͑2002͒

Magnetization orientation dependence of the quasiparticle spectrum and hysteresis in ferromagnetic metal nanoparticles

A. Cehovin Division of Solid State Theory, Department of Physics, Lund University, SE-223 62 Lund, Sweden

C. M. Canali Department of Technology, Kalmar University, 391 82 Kalmar, Sweden and Division of Solid State Theory, Department of Physics, Lund University, SE-223 62 Lund, Sweden

A. H. MacDonald Department of Physics, University of Texas at Austin, Austin, Texas 78712 ͑Received 27 May 2002; published 30 September 2002͒ We use a microscopic Slater-Koster tight-binding model with short-range exchange and atomic spin-orbit interactions that realistically captures generic features of ferromagnetic metal nanoparticles to address the mesoscopic physics of magnetocrystalline anisotropy and hysteresis in nanoparticle-quasiparticle excitation spectra. Our analysis is based on qualitative arguments supported by self-consistent Hartree-Fock calculations for nanoparticles containing up to 260 atoms. Calculations of the total energy as a function of magnetization direction demonstrate that the magnetic anisotropy per atom fluctuates by several percent when the number of electrons in the particle changes by 1, even for the largest particles we consider. Contributions of individual orbitals to the magnetic anisotropy are characterized by a broad distribution with a mean more than two orders of magnitude smaller than its variance and with no detectable correlations between anisotropy contribution and quasiparticle energy. We find that the discrete quasiparticle excitation spectrum of a nanoparticle displays a complex nonmonotonic dependence on an external magnetic field, with abrupt jumps when the magnetization direction is reversed by the field, explaining recent spectroscopic studies of magnetic nanoparticles. Our results suggest the existence of a broad crossover from a weak spin-orbit coupling to a strong spin-orbit coupling regime, occurring over the range from approximately 200- to 1000-atom nanoparticles.

DOI: 10.1103/PhysRevB.66.094430 PACS number͑s͒: 75.75.ϩa

I. INTRODUCTION quantum states in the discrete many-body excitation spectrum of single ferromagnetic metal nanograins with sizes in Interest in the properties of magnetic nanoparticles has the range from 1 to 4 nm. As in bulk ferromagnetic metals, grown recently, partly because of advances in synthesis and the low-energy excitations of the nanoparticles that were measurement techniques and partly because of potential ap- probed in these experiments involve both particle-hole exci- plications for high-storage-density magnetic media and spin tations of the electronic quasiparticles and quantized collec- electronics. Ferromagnetic nanoparticles with diameters of a tive excitations of the magnetization-orientation field that ap- few nanometers containing of order of 1000 or fewer atoms pears in the classical micromagnetic theory. In an initial can now be reliably fabricated and studied with a variety of attempt to achieve an understanding of the novel physics 1–3 evident in the external field dependence of the experimental different methods. Small monodomain magnetic particles 8 have traditionally4,5 been described using classical micro- excitation spectra, two of us recently analyzed a simple magnetic theory, in which the total energy is expressed as a quantum model with long-range exchange interactions. We were able to demonstrate explicitly that the low-energy ex- function of the magnetization orientation. Shape and magne- citations of a ferromagnetic metal nanoparticle are specified tocrystalline magnetic anisotropy leads to a dependence of by the occupation numbers of its quasiparticle orbitals, as in energy on orientation, to barriers that separate minima that a Fermi liquid, and by the global orientation of the aligned 6 and to hysteretic occur at easy magnetization orientations, spins of all single-occupied ͑and therefore spin-polarized͒ discontinuous changes in orientation as a function of the orbitals. This model is not, however, able to account realis- strength of an external magnetic field. When the size of a tically for the influence of spin-orbit interactions, which play magnetic particle is only a few nanometers, the discrete na- the essential role in controlling the complex hysteretic be- ture of its quantum energy spectrum can be directly observ- havior seen in these experiments.1,7 Kleff, et al.9,10 have pro- able at low temperatures and starts to affect the magnetic posed that the single-electron tunneling spectroscopy experi- properties of the particle. ments can be explained by accounting for nonequilibrium A milestone in the experimental study of individual ferro- spin accumulation and by assuming that the magnetic anisot- magnetic metal nanoparticles was achieved recently by ropy energy of a small magnetic particle has surprisingly Gue´ron et al.1 Deshmukh, et al.7 using single-electron tun- large fluctuations as a function of the number of electrons on neling spectroscopy. By exploiting the Coulomb blockade the particle. This assumption leads to a nontrivial magnetic effect, these experimentalists were able to resolve individual field dependence of tunneling resonances that resembles ex-

0163-1829/2002/66͑9͒/094430͑15͒/$20.0066 094430-1 ©2002 The American Physical Society A. CEHOVIN, C. M. CANALI, AND A. H. MacDONALD PHYSICAL REVIEW B 66, 094430 ͑2002͒ perimental behavior. More critically, these authors point out tions of the micromagnetic energy functional which appears that if nonequilibrium spin and quasiparticle excitations both in the classical theory of a magnetic nanoparticle. occur,10 the low-energy spectra are characterized by many Our paper is organized in the following way. In Sec. II we closely spaced resonances, consistent with experiment.10 In introduce the model and describe the formalism. In Sec. III Ref. 11 we presented a possible approach toward achieving a we analyze the qualitative change in quasiparticle energy- unified and consistent quantum description of both collective level statistics induced by spin-orbit interaction, and discuss and quasiparticle physics in magnetic nanoparticles. the connection between the quasiparticle properties and mag- The attempts put forward in Refs. 8–10 to develop a netic anisotropy of a ferromagnetic nanoparticle. Fluctua- quantum description of ferromagnetic metal nanoparticles do tions of the magnetic anisotropy as a function of spin- not address the microscopic origin of the magnetic anisot- splitting field, atom number, and electron number are ropy and the fluctuations of this quantity that are a necessary investigated in Sec. IV. Magnetic hysteresis and the external consequence of spin-orbit interactions in mesoscopic sys- field dependence of quasiparticle energy levels are discussed tems. This article addresses these issues and, more in gen- in Sec. V. Finally, in Sec. VI we summarize our findings and eral, investigates the changes in magnetic properties of small present our conclusions. metallic nanoparticles that occur because of the finite spacing between quasiparticle levels near the Fermi energy. Our II. MODEL conclusions follow from qualitative arguments based on per- N turbation theory expressions for the influence of spin-orbit We model the nanoparticle as a cluster of a atoms lo- interactions on quasiparticle energy levels and on numerical cated on the sites of a truncated crystal. The numerical re- studies of a simplified model that we believe is sufficiently sults we present here are for a cobalt cluster whose truncated fcc crystal is circumscribed by a hemisphere whose equator realistic to describe generic aspects of the interrelated meso- 12 scopic physics of quasiparticle energy levels and magnetic lies in the xy plane of the fcc crystal. anisotropy energy in ferromagnetic metal nanoparticles. We are particularly interested in the variation of quasiparticle A. Tight-binding Hamiltonian and Slater-Koster parameters energies with fields on the scale of the coercive field, which The model we use is intended to qualitatively capture the is trivial in the absence of spin-orbit interactions, but entirely physics of a transition-metal, itinerant-electron ferromagnet. nontrivial in their presence. The model we study is based on We use a s-p-d tight-binding model for the quasiparticle a Slater-Koster tight-binding Hamiltonian and a mean-field orbitals, with 18 orbitals per atom including the spin degree treatment of exchange interactions. Our aim is to understand of freedom. Nine orbitals per Co atom are occupied for in the dependence of magnetic anisotropy and quasiparticle neutral nanoparticles. The full Hamiltonian is energy-level-spacing statistics on particle size and shape, external magnetic field, and, with single-electron-transistor sys- HϭH ϩH ϩH ϩH ͑ ͒ tems in mind, also on electron number. We find that, because band exch SO Zee . 1 of the absence of strong correlations between angular momentum operator matrix elements and orbital energy differ- H Here band is a one-body term describing the orbital motion ences, the sizes of spin-orbit-induced energy shifts in nano- of the electrons. In second quantization, it has the form particles and bulk perfect crystals are similar. On the basis of estimates for energy shifts and for avoided crossing gaps, we predict that a crossover from weak to strong quasiparticle H ϭ i, j † ͑ ͒ band ͚ ͚ ͚ t␮ ,␮ ,sci,␮ ,sc j,␮ ,s , 2 ␮ ␮ 1 2 1 2 spin-orbit scattering will occur over the range from approxi- i, j s 1 , 2 mately 200-atom to approximately 1000-atom nanoparticles. We find that for small particles the contribution of individual where c† and c are Fermion creation and annihilation opera- quasiparticle orbitals to the magnetocrystalline anisotropy tors for single-particle states labeled by i,␮,s. The indices energy has a wide distribution, characterized by a variance i, j are atomic site labels, and ti, j couples up to second near- comparable to the spin-orbit-scattering lifetime broadening Ϫ est neighbors. The indices ␮ ,␮ label the nine distinct energy, ប␶ 1 , and a mean that is smaller by more than two 1 2 SO atomic orbitals ͑one 4s, three 4p, and five 3d). The spin orders of magnitude. Surprisingly, we find no measurable degrees of freedom, labeled by the index s, double the num- correlation between the contributions to magnetic anisotropy ber of orbitals at each site. It is useful for us to vary the from quasiparticles that are close in energy in this limit. As a spin-quantization axis, which is specified by a unit vector result of the statistical properties of the quasiparticle mag- ⍀ˆ ⌰ ⌽ ⌰ ⌽ netic anisotropies, the total magnetic anisotropy per atom ( , ) where and are the usual angular coordinates fluctuates by several percent when the number of electrons in defined with respect to the fcc crystal axes. The parameters i, j 13 t␮ ␮ are Slater-Koster parameters obtained after per- the nanoparticle changes by 1, even for particles containing 1 , 2 ,s 260 atoms. Finally, in agreement with experiment,1,7 we find forming a Lo¨wdin symmetric orthogonalization procedure14 that the quasiparticle excitation spectrum exhibits a complex on the set of Slater-Koster parameters for nonorthogonal nonmonotonic behavior as a function of an external magnetic atomic orbitals of bulk spin-unpolarized Co.15 field, with abrupt jumps when the magnetization orientation The electron-electron interaction term in Eq. ͑1͒ is simpli- of a nanoparticle changes discontinuously in response to the fied by introducing explicitly16 only the ferromagnetic ex- H field. Our analysis provides insight into mesoscopic fluctua- change interaction exch between the electrons spins of d

094430-2 MAGNETIZATION ORIENTATION DEPENDENCE OF THE... PHYSICAL REVIEW B 66, 094430 ͑2002͒ orbitals on the same atomic site. These interactions are H ϭϪ␮ ␮ ͉͑ ជ ϩ ជ͉␮Ј Ј͒ largely responsible for magnetic order in transition-metal fer- Zee B͚ ͚ ͗ ,s L gsS ,s ͘ i romagnets: ␮,␮Ј,s,sЈ ជ † •Hext ci,␮,sci,␮Ј,sЈ

H ϭϪ ͚ ជ ជ ͑ ͒ ϭϪ␮ ជ ␮ ͉ ជ ͉␮ ͒ † exch 2Udd Sd,i Sd,i , 3 B͚ Hext ͭ ͚ ͗ ,s L Ј,s ͘c ␮ ci,␮Ј,s i • • i, ,s i ␮,␮Ј,s g ϩ s † ␶ជ ͑ ͒ where ͚ c ␮ s,sЈci,␮,sЈͮ . 6 i, ,s 2 ␮,s,sЈ

1 The extreme sensitivity of magnetization orientation to ជ ϭ ជ ϭ † ␶ជ ͑ ͒ Sd,i ͚ Si,␮ ͚ ͚ ci,␮,s s,sЈci,␮,sЈ . 4 external magnetic field is a combined effect of the collective ␮෈ ␮෈ 2 d d s,sЈ behavior of many electrons in a ferromagnetic nanoparticle and the smallness of the magnetic anisotropy energy relative ͑ ͒ to the overall energy gain associated with ferromagnetic or- The parameter Udd in Eq. 3 determines the strength of the exchange interaction and is set equal to 1 eV. This value of der. The most delicate physics of a ferromagnetic nanoparticle, and in our view the most interesting, is that associated Udd gives rise in our finite clusters to an average magnetic ␮ with magnetization direction reorientation by weak external moment per atom of the order of 2 B , which is larger than the bulk value in Co,15 in agreement with other calculations17 magnetic fields. Thus the interplay between the Zeeman term and experimental results18,19 for Co clusters. This value of and the magnetic anisotropy produced by spin-orbit interactions is at the heart of the physics we intend to address. Udd is also approximately consistent with a mean-field relationship, derived below, between band spin-splitting and magnetization, using bulk values15 for these two quantities. B. Mean-field approximation ជ ␣ In Eq. ͑4͒, ␶ is a vector whose components ␶ , ␣ϭx,y,z are We seek a ferromagnetic solution to the mean-field equa- the three Pauli matrices. tions for this model, decoupling the quartic term in the ex- ͑ ͒ H The third term in Eq. 1 , SO , is a one-body operator, change interaction using the ansatz essentially atomic in character, representing the spin-orbit interaction. It can be written as20 ជ ϭ ជ ϩ␦ ជ ͑ ͒ Sd,i ͗Sd,i͘ Sd,i , 7 ␦ ជ ignoring terms that are second order in Sd,i and determining ͗ ͘ H ϭ␰ ͚ ͚ ͗␮ ͉ ជ ជ͉␮Ј Ј͘ † ͑ ͒ ground-state expectation values ... self-consistently. This SO d ,s L•S ,s ci,␮,sci,␮Ј,sЈ 5 i ␮,␮Ј,s,sЈ standard procedure leads to a Hamiltonian that can be diago- nalized numerically and to a self-consistency condition that can be solved iteratively to determine the mean-field order The atomic matrix elements ͗␮,s͉Lជ Sជ͉␮Ј,sЈ͘ϵ͗i,␮,s͉Lជ parameters. For the present calculation we have simplified • this procedure further, by averaging the spin-splitting ex- Sជ͉i,␮Ј,sЈ͘ can be been calculated explicitly as a function of • change mean field, ⍀ˆ 21 ␰ the direction of the magnetization . The energy scale d , which characterizes the coupling between spin and orbital ជ Udd ជ degrees of freedom, varies in the range from 50 to 100 meV h ϵh⍀ˆ ϭ 2͗S ͘, ͑8͒ i g ␮ d,i in bulk 3d transition-metal ferromagnets22. For our calcula- s B ␰ ϭ tions we have used d 82 meV, taken from Ref. 23. Spin- over all sites. Our motivation for doing so is to simplify the orbit coupling gives rise to a dependence of the total energy magnetic anisotropy energy landscape discussed below, forc- of a ferromagnet on the direction of its spontaneous magne- ing all spins to change their orientations coherently. We rec- 24 tization, an effect known as magnetocrystalline anisotropy ; ognize that complicated noncollinear spin configurations28,29 with spin-orbit coupling the magnetization is partially orbital commonly occur in magnetic nanoparticles and that under 25,26 in character and is sensitive to orbital anisotropy due to the action of an external field small groups of atoms can crystal-field interactions on an atomic site, due to the spatial change their orientation relative to other parts of the nano- arrangement of the atoms neighboring that site to which elec- particle. Complex magnetization reorientation processes are trons can hop and due to the overall shape of the full nano- obvious in addition spectroscopy experiments.7 By forcing particle that becomes available to an electron after many atoms to change their magnetic orientations coherently, we 27 hops. We will see below that the hemispherical shape of the are restricting our attention to relatively large nanoparticles nanoparticles we have studied plays the dominant role in N Ͼ ( a 50) in which most atoms have parallel spins and to determining their magnetic anisotropy energy. nanoparticles with simple bistable hysteretic behavior in The last term in Eq. ͑1͒ is a local one-body operator, which the physics we address will be easier to study. The representing the Zeeman coupling of the orbital and spin beautiful series of detailed superconducting quantum inter- ជ ͑ ͒ degrees of freedom to an external magnetic field Hext : ference device SQUID magnetometry experiments by

094430-3 A. CEHOVIN, C. M. CANALI, AND A. H. MacDONALD PHYSICAL REVIEW B 66, 094430 ͑2002͒

30,3 22 Wernsdorfer and colleagues demonstrate that nanopar- than 10% of the d-band width Wd in bulk materials, allow- ticles can be prepared that do have simple coherent magne- ing the energy shifts they produce to be estimated perturba- tization reversal properties. tively. Because of angular momentum quenching in the ab- By using a simplified model Hamiltonian we are able to sence of external fields, the expectation value of HSO is zero, deal with larger nanoparticle systems than would be possible even in case of ferromagnets.22 The quasiparticle energy shift with a first-principles calculation17,23,28,31; since we are inter- due to spin-orbit interactions is given by second-order per- ested only in generic aspects of the ferromagnetic nanopar- turbation theory as ticle physics, there is little to gain from the additional realism ⑀ ϵ⑀ Ϫ⑀0 that could be achieved by performing self-consistent spin- SO n,s n,s density functional calculations. ជ ͑␰ ͒2 ͉͗␺0 ͉ ជ ͉␺0 ͘ ␶ជ ͉2 Given the averaged spin-splitting field h, the mean-field d m,sЈ L n,s sЈ,s ϭ ͚ • , ͑10͒ Hamiltonian is now a single-body operator 4 ⑀0 Ϫ⑀0 sЈ n,s m,sЈ mÞn hជ hជ H ជ ͒ϭH ϩH ϩH ϩ • ␮ ͒2 where ͉␺0 ͘ and ⑀0 are, respectively, the single-particle MF͑h band SO Zee ͑gs B Na n,s n,s 2Udd eigenstates and energies in the absence of a spin-orbit interaction. In small particles, the importance of the spin-orbit Ϫ ␮ ជ ជ ͑ ͒ gs Bh ͚ 2Sd,i . 9 interactions can be assessed by comparing the spin-orbit en- • i ⑀ ␦ ergy shift SO with the single-particle mean-level spacing . ជ The self-consistent spin-spitting field is also the field, de- In an infinite periodic solid only states at the same k are noted by hជ Ã below, at which the total ground-state energy coupled and these are separated energetically by an energy Ã comparable to the bandwidth W . In a nanoparticle a given function E(hជ )ϭ͗H (hជ )͘ is minimized. Notice that g ␮ h d MF s B state will be coupled to many other orbitals, but the coupling has the dimension of an energy. In fact, in absence of a ␮ Ã matrix elements will be reduced in accord with the following spin-orbit interaction, 2gs Bh can be identified with the ⌬ϭ⑀0 Ϫ⑀0 ⑀0 ⑀0 sum rule: band spin splitting Fa Fi , where Fa and Fi are the majority- and minority-spin quasiparticle Fermi energies.32 Diagonalization of the quadratic Hamiltonian yields a set of ͚ ͉͗␺0 ͉Lជ ͉␺0 ͘ ␶ជ ͉2ϭ͗␺0 ͉Lជ Lជ ͉␺0 ͘ϳ4. ͑11͒ m,sЈ n,s • sЈ,s n,s • n,s quasiparticle energies ͕⑀ ͖, nϭ1,2,..., which in the sЈ n,s Þ ground state are filled up to a Fermi energy determined by m n the number of valence electrons in the nanoparticle. Strictly The estimate for the right-hand side of Eq. ͑11͒ is based on speaking, because of the spin-orbit interaction, the spin char- the atomic character of the angular momentum in our model ͉␺ ͘ acter of the corresponding eigenstates n,s is not well de- and uses that Lជ Lជ ϳ͓5ϫ6ϩ3ϫ2ϩ1ϫ0͔/9, with the esti- fined. For small nanoparticles, most eigenstates have pre- • mate of the typical Lជ Lជ expectation value representing an dominantly spin-up or spin-down character and we • average over d, s, and p orbitals. It follows that, unless there sometimes use this property to assign spin labels sϭ↑ for are important correlations between angular momentum ma- spins along the order direction and sϭ↓ for reversed spins. trix elements and quasiparticle energy differences, the typical ⑀ shift in energy caused by the spin-orbit interaction is SO III. SPIN-ORBIT INTERACTIONS AND THE MAGNETIC ϳ ␰ 2 ( d) /Wd , in both bulk crystals and in nanoparticles, ANISOTROPY ENERGY which is in the range between 1 and 10 meV. For example, in ␰ ϭ ϳ The magnetic anisotropy energy of small ferromagnetic Co, using d 82 meV and Wd 5 eV, this rough estimate ⑀ particles3 has two fundamentally distinct origins: long-range gives for the magnitude of the spin-orbit energy shift SO ϳ magnetic-dipole interactions which cause a dependence on 1.3 meV. The sign of the shift might be expected to be overall sample shape and short-range exchange interactions sensitive to the spacing of nearby quasiparticle orbitals. The that, because of atomiclike spin-orbit interactions, are sensi- anisotropy energy—that is, the dependence of the total band tive to all aspects of the electron hopping network including energy on the magnetization orientation—is given to a good bulk crystal symmetry, facet orientations, and also overall approximation by a partial canceling sum of spin-orbit- sample shape. We concentrate here on spin-orbit-induced induced energy shift dependences on magnetic orientation. In magnetocrystalline anisotropy which gives rise to the most the approximation that the exchange field is orbital indepen- interesting physics in ferromagnetic nanoparticles. When dent, majority-spin and minority-spin orbitals are identical magnetostatic shape anisotropy is important, it can be added and differ only in their occupation numbers. In this approxi- as a separate contribution. We begin our discussion with mation there is no contribution to the anisotropy energy from some qualitative estimates of the effect of spin-orbit interac- doubly occupied orbitals. Because of the cancellations, the ⑀ tions that are based on perturbation theory.33,34 anisotropy energy per atom is much smaller than SO . For example, the zero-temperature anisotropy energy per atom in bulk is 60 ␮eV for hcp Co and Ϸ1␮ eV for bcc Fe and fcc A. Perturbation theory considerations Ni.35 In a finite or disordered system, there will always be In bulk 3d transition-metal ferromagnets, spin-orbit inter- perturbative coupling to quasiparticle states close in energy actions are relatively weak. Their coupling strength is less in Eq. ͑10͒, but the matrix elements, which satisfy the sum

094430-4 MAGNETIZATION ORIENTATION DEPENDENCE OF THE... PHYSICAL REVIEW B 66, 094430 ͑2002͒ rule of Eq. ͑11͒, will be distributed among many states and typical energy shifts in nanoparticles should generally be comparable to those in bulk perfect crystals. Typically net anisotropy energies per atom in small magnetic particles are larger than the bulk because of the loss of symmetry at the surface. An important quantity used to characterize the strength of spin-orbit interactions in bulk systems and large nanopar- ␶ 36–39 ticles is the spin-orbit scattering time SO . In the weak- coupling regime, it is given by Fermi’s golden rule

͑␰ ͒2 Ϫ d ប␶ 1ϭ ͚ ͉͗␺0 ͉Lជ ͉␺0 ͘ ␶ជ ͉2␦͓⑀0 Ϫ⑀0 ͔, FIG. 1. Single-particle energy shifts caused by spin-orbit inter- SO m,sЈ n,s • sЈ,s n,s m,sЈ 4 sЈ actions in a ferromagnetic cobalt nanoparticle of 143 atoms. The mÞn single-particle mean-level spacing at the Fermi energy is ␦ ͑12͒ Ϸ4.3 meV. ͑a͒ Magnetization in the z direction. ͑b͒ Magnetization where the ␦ function is understood to be broadened to a in the x direction. width much larger than the level spacing. Assuming that there is no correlation between angular momentum matrix netization is in the xy plane, consistent with the rough esti- elements and orbital energy differences, it follows from the mates above. We also note that there is not a strong correla- sum rule mentioned above that tion between the sign of the energy shift and the energy of the orbital. ␰2 Because of the spin-orbit interaction, each individual ប␶Ϫ1ϳ⑀ ϳ d ͑ ͒ SO SO . 13 eigenlevel has an energy dependence on the spin-splitting Wd field ͑or magnetization͒ direction ⍀ˆ . To illustrate typical The absence of strong correlations between energy differ- properties of these dependences, we plot in Fig. 2 the varia- ences and angular momentum matrix elements, a property tion of a few energy levels around the Fermi level for cases that we find somewhat surprising, has been verified numeri- in which the magnetization rotates in the zx and xy planes, cally as we discuss below. The character of the nanoparticle respectively. In the absence of spin-orbit interactions there quasiparticle energy spectrum changes when these intensive would be no dependence of any of these orbital energies on energy scales become comparable to the nanoparticle level magnetization orientation. spacing ␦. Notice that the angle dependence in the xy plane is con- siderably weaker than in the zx plane, for the hemispherical B. Numerical results for a Co nanoparticle nanoparticle we consider. This trend indicates that for our nanoparticles, it is the overall shape which dominates the The qualitative considerations of the previous section pro- spin-orbit-induced anisotropy physics. For this size cluster, vide a framework for thinking about the effects of spin-orbit there are many narrowly avoided level crossings, a point to interactions. Our microscopic model, on the other hand, al- which we return below. The difference in eigenvalue sensi- lows us to explore realistic magnetic nanoparticle systems in tivity to magnetization rotations in the two planes is clearly great detail. We have studied numerically nanoparticles con- visible in Fig. 3, where we plot the single-particle level taining up to 260 atoms. Most of the results presented below Ã Ã Ã Ã ⑀ ˆ Ϫ⑀ ˆ ⑀ ˆ Ϫ⑀ ˆ anisotropies n,s(hzˆ z) n,s(hxˆ x) and n,s(hxˆ x) n,s(hyˆ y) are for hemispherical 143-atom nanoparticles. The calcula- Ã Ã tions have been performed with a R12000 300-MHz proces- versus the eigenvalue index n. Here hxˆ and hzˆ are the mag- sor on an SGI Origin 2000 computer. Diagonalizations rely nitudes of the self-consistent spin-splitting field when its di- on LAPACK drivers. A single diagonalization of the Hamil- rection is along xˆ and zˆ, respectively. We note that the typi- tonian for a 143-atom cluster has a running time of approxi- cal change in orbital energy between xˆ and zˆ direction mately 1 h, and requires around 750 Mb of internal memory. In Fig. 1 we plot the energy shifts caused by a spin-orbit interaction for a hemispherical cobalt nanoparticle of 143 atoms with a fcc crystal structure. For this nanoparticle size, the minority and majority single-particle mean-level spac- ings at the Fermi level are ␦↓Ϸ4.9 meV and ␦↑Ϸ50 meV, respectively, when spin-orbit interactions are absent. The single-particle mean-level spacing averaged over all states ͑i.e., without distinguishing between majority and minority levels͒ is ␦Ϸ4.3 meV at the Fermi level.40 The spin-orbit- FIG. 2. Variation of a few quasiparticle energies of a 143-atom induced shifts are both positive and negative and their abso- Co nanoparticle as a function of the direction of the magnetization lute values go from 1 meV up to 10 meV. The average ab- ⍀ˆ . The Fermi level is the dotted line. ͑a͒⍀ˆ lies in the xy plane and solute value of the energy shifts is 2.7 meV when the ⌽ is the angle with the x axis. ͑b͒⍀ˆ lies in the zx plane and ⌰ is magnetization is in the z direction and 2 meV when the mag- the angle with the z axis.

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FIG. 5. Distribution function of single-particle anisotropies in ␦⑀ ϭ ⑀ Ãˆ Ϫ⑀ Ã ˆ the zx plane, n,s ͓ n,s(h z) n,s(h x)͔, for a 143-atom nano- ␦⑀ ϭ ␮ FIG. 3. Single-particle level anisotropies in the xy and zx planes particle. The mean value is ͗ n,s͘ 15 eV; the width of the dis- ͑ ͒ ⌬ ␦⑀ for a 143-atom nanoparticle. tribution enclosed by the vertical dashed lines is ( n,s) ϭ2.9 meV. magnetizations is only ϳ30% smaller than the typical total shifts induced by spin-orbit interactions. On the other hand, to the kϭ0 value of the correlation function displayed in Fig. ˆ ˆ the typical difference in orbital energy between x and y di- 4. Single particle anisotropies of groups of orbitals over rection magnetization is 5 times smaller than the correspond- within a specified energy range tend to be anticorrelated, ing spin-orbit-induced energy shift. All orbital energies are leading to typical averages smaller than the variance of the relatively insensitive to the magnetization orientation within distribution. The large difference between the distribution the xy plane. In both cases the correlation between position mean and variance will play an important role in Sec. IV B, within the band and the sign and magnitude of the energy where we discuss fluctuations of the anisotropy energy as a shift is weak. In addition, energy shifts at nearby energies are weakly correlated. That is, the correlation function function of electron number. ⌬ ␦⑀ The width of the distribution, ( n,s), gives a measure ␦⑀ ␦⑀ Ϫ ␦⑀ 2 of the average magnitude of the single-particle level anisot- ͗ n,s nϩk,sЈ͘ ͗ n,s͘ , ⌬ ␦⑀ ␦ ropy, and the ratio ( n,s)/ characterizes the strength of ␦⑀ ϭ⑀ ͑ Ã⍀ˆ ͒Ϫ⑀ ͑ Ã⍀ˆ ͒ ͑ ͒ n,s n,s h1 1 n,s h2 2 , 14 mixing between quasiparticle orbitals that results from spin- orbit interactions. As mentioned previously, this identifica- where the average ͗ ͘ is over the occupied levels n, drops ••• tion of weak and strong spin-orbit interaction regimes is to zero very rapidly with k, as clearly shown in Fig. 4. equivalent to the usual one36–39 based on a comparison of the It is useful to consider the distribution of the quasiparticle ␶ ␦ ␦⑀ spin-orbit scattering time SO and the mean-level spacing . anisotropies, P( n,s). As an example, in Fig. 5 we plot the ␦␶ បӷ distribution of anisotropies in the zx plane, ␦⑀ When SO / 1, a limit achieved for small enough particle n,s size for any value of ␰ , spin-orbit coupling is a relatively ϭ͓⑀ (hÃzˆ)Ϫ⑀ (hÃxˆ )͔, constructed with the Nϭ9ϫN d n,s n,s a weak effect and there is little mixing between spin-up and ϭ9ϫ143 occupied single-particle states of a 143-atom nano- spin-down states. As a consequence, the level crossings be- particle. The distribution has a width—characterized by the ⌬ ␦⑀ ϳ ␰2 ϳ tween states of predominantly opposite spins that occur as a root mean square—of ( n,s) 2.2 d/Wd 2.9 meV and a ជ Ã function of the magnitude and orientation of h will be only much smaller mean value of ͗␦⑀ ͘ϭ͗⑀ (h zˆ) n,s n,s weakly avoided. With increasing particle size, ␦ decreases Ϫ⑀ Ã ˆ ͘ϳ ␮ ⌬ ␦⑀ n,s(h x) 15 eV. Note that ( n,s) is exactly equal and we enter the regime of a strong spin-orbit interaction and strong level repulsion. The single-particle spectrum becomes relatively rigid, level crossing will be strongly avoided, and ␦ will limit the variation of individual levels as a function of the magnetization direction. Within our model we have found that the crossover between these two regimes is very broad, it starts for nanoparticles containing of order 200 atoms, and as we argue below, it will be completed when the nanoparticles contain approximately 1000 atoms. For a 143- atom nanoparticle, we are already approaching the crossover ⌬ ␦⑀ ϳ regime; we find ( n,s) 2.9 meV while the single-particle mean-level spacing at the Fermi level is ␦Ϸ4.3 meV. In this regime level crossings will be moderately avoided. FIG. 4. Correlation function of single-particle anisotropies as The typical size of avoided crossing gaps between oppo- defined in Eq. 14. The correlation function drops immediately to site spin orbitals in small nanoparticles can be understood by zero for kϾ0. Note that the kϭ0 value of the correlation function the following argument. We first note that the unperturbed is equal to the width of the anisotropy distribution, plotted in Fig. 5. orbitals satisfy the following sum rule:

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FIG. 6. Distribution function of spin-orbit matrix elements for a spin-polarized 143-atom nanoparticle. ͉n,s͘ and ͉nЈ,sЈ͘ are single- ͑ ͒ ͉ ͘ particle levels in the absence of a spin-orbit interaction. a n,s FIG. 7. Average squared matrix elements of the spin-orbit inter- ͉ Ј ͘ ͑ ͒ ͉ ͘ ͉ Ϫ ͘ and n ,s have like spins; b n,s and n, s have opposite action vs energy difference for a polarized 143-atom nanoparticle. spins. Solid and dotted lines are for the cases of nearby levels ͉n,s͘ and ͉nЈ,sЈ͘ are quasiparticle states without spin-orbit interac- ͉⑀0 Ϫ⑀0 ͉ ␦Ͻ ( n,s nЈ,sЈ / 3) and any pair of levels, respectively. tions. The four curves correspond to the four possible spin combi- nations. ͉͗ Ј ↓͉ ជ ជ͉ ↑͉͘2ϭ͗ ͉ ͉ ͘ ϳ ͑ ͒ ͚ n , L•S n, n LϪLϩ n )/4 2/3. 15 on external fields discussed below, to change in character nЈ when the typical matrix element becomes comparable to the level spacing, i.e., when The estimate for the right-hand side of Eq. ͑15͒ is based on ␰2 the same considerations leading to Eq. ͑11͒ and uses that 0.14 d ϭ␦. ͑16͒ ϳ ជ ជ ϳ ␦N LϪLϩ 2/3L•L 8/3. If angular momentum matrix ele- a ments between orbitals are not correlated with the energy Our numerical calculations of quasiparticle spectra are con- differences between these orbitals and the matrix elements Ϫ sistent with using the condition ប␶ 1ϳ⑀ ϭ␦ as a criterion are reasonably narrowly distributed, this sum rule can be SO SO for the start of the crossover to the strong-coupling limit and used to estimate the typical matrix element. Expectation val- the condition that the typical avoided crossing gap estimate ues of the angular momentum operators are zero because of be equal to ␦ as a criterion for completion of the crossover to angular momentum quenching, and a finite fraction of the strong-coupling relatively rigid spectra. For cobalt nanopar- matrix elements vanish because of symmetries present in our N ϳ rather regularly shaped nanoparticles. Aside from these fea- ticles the later condition is reached for a 2000; for tures, we find numerically that correlations between matrix smaller nanoparticles, the quasiparticles generally have elements and energy differences are too small to be clearly somewhat well-defined spin character, some Poisson charac- observable. Figure 6 shows the distribution function we have ter in their spectral statistics, and complicated evolution pat- obtained for matrix elements between opposite- and like-spin terns with external field and order parameter variations. For states. We have considered both the matrix element distribu- larger nanoparticles, which we are not, however, able to tion for closely spaced levels and the distribution for any pair study numerically, we expect that quasiparticles will have of levels, not necessarily nearby. The distributions are found strongly mixed spins and more rigid spectra with smoother to be very similar. Approximately 50% and 70% of matrix evolution patterns. All the ferromagnetic nanoparticles that elements are zero for opposite-spin and like-spin cases, re- we are able to study here and many nanoparticles studied spectively. Based on these numerical results and the sum experimentally, are in the crossover regime. Note that since ␦ϰ N ␰2 ␦N ϰប␶Ϫ1 rule, Eq. ͑15͒, we estimate the typical value of Wd / a ,(d/( a) SO ; the two conditions differ ͉͗ Ј ↓͉ ͉ ↑͉͘2 2 ␰2 quantitatively not parametrically. n , HSO n, as 3 d divided by half the total number of Ã N The total ground-state energy of the particle E(h ⍀ˆ ), s, p, and d orbitals, 9 a/2. This implies a typical nonzero ⍀ˆ ϳ␰ ͱ N matrix element equal to d 0.14/ a. In Fig. 7 we plot the obtained by summing over the lowest N occupied orbitals, average square matrix element versus energy difference for a depends on the direction of the self-consistent spin-splitting ⍀ˆ Ãˆ Ϫ Ã ⍀ˆ 143-atom cluster, obtaining remarkably precise agreement field . In Fig. 8 we plot E(hzˆ z) E(h⍀ˆ ) versus the angle with this estimate provided that the energy differences are ⌰ between ⍀ˆ and zˆ ͑when ⍀ˆ lies in the zx plane͒ and much smaller than the d-band width. The average square Ã Ã E(h xˆ )ϪE(h ⍀ˆ ) versus the angle ⌽ between ⍀ˆ and xˆ matrix element for like spin orbitals is approximately a factor xˆ ⍀ˆ of 2 smaller, consistent with the type of argument presented ͑when ⍀ˆ lies in the zx plane͒. From the figure we can see ͗ ͉ 2͉ ͘ that the xy plane is almost an easy plane for the model nano- above which would imply proportionality to n Lz n in that case. particle, except for a weak energy dependence which gener- We expect the dependence of quasiparticle energies on the ates four easy axes in the directions (Ϯxˆ Ϯyˆ )/2. Again this magnitude and orientation of the order parameter, and also property reflects the large significance of the overall sample

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FIG. 8. Magnetic anisotropy energies for a 143-atom hemi- Ã spherical nanoparticle. The dotted line represents ͓E(h zˆ) zˆ FIG. 9. Change of quasiparticle energies with increasing spin- Ϫ Ã ⍀ˆ ͔ ⌰ ⍀ˆ ˆ ⍀ˆ E(h⍀ˆ ) vs the angle between and z, when is in the zx splitting field ͉hជ͉ near hÃ for a 143-atom nanoparticle. Here 200 ͓ Ã ˆ Ϫ Ã ⍀ˆ ͔ plane. The black line represents E(hxˆ x) E(h⍀ˆ ) vs the angle levels are shown. The Fermi level is the thick black line. The ver- ⌽ between ⍀ˆ and xˆ , when ⍀ˆ lies in the xy plane. tical white dashed lines indicate the positions of the charge redis- tributions displayed in Fig. 10. The vertical black dashed line corresponds to the self-consistent spin-splitting field. shape. The four easy axis directions are remnants of the magnetic anisotropy symmetry in bulk fcc ferromagnets. Group IV. FLUCTUATIONS OF MAGNETIC ANISOTROPY theory considerations demand that a bulk fcc ferromagnet have an easy axis in one of the directions perpendicular to So far we have considered only average values of the the eight (Ϯ1,Ϯ1,Ϯ1) ͑Ref. 6͒ planes. In a hemispherical anisotropy constants. In ferromagnetic nanoparticles, how- nanoparticle the fcc symmetry is partially lifted and the mag- ever, these quantities are also characterized by large fluctua- netization is forced to lie in the xy plane. We can define the tions as a function of experimentally relevant parameters. following two anisotropy-energy-per-atom constants: Anisotropy fluctuations are the topic of this section.

Ã Ã A. Charge-induced fluctuations as a function of the E͑h ˆ zˆ ͒ϪE͑h ˆ xˆ ͒ ͑ˆ ˆ ͒ϵ z x Ϸ ͑ ͒ spin-splitting field k z,x N 0.13 meV, 17 a When spin-orbit interactions are included, all quasiparticle eigenstates have mixed majority-spin and minority-spin character. Our specific calculations, however, are for small ͑ Ã ˆ ͒Ϫ ͑ Ã ˆ ͒ E h ˆ x E h ˆ y enough particles such that most eigenstates still have pre- ͑ ˆ ˆ ͒ϵ x y Ϸ ͑ ͒ k x,y N 0.01 meV. 18 dominantly one spin character. It is convenient to use this a ‘‘predominant spin’’ to label the states when discussing their dependence on ͉hជ͉ or on an external magnetic field. In a As expected on the basis of the qualitative considerations of ជ Ãϭ Sec. III A, the anisotropy per occupied orbital is much paramagnetic system (h 0) in the absence of an external ⑀ ͑ field, there is a Kramer degeneracy that pairs up eigenstates smaller than the average single-particle level shift SO approximately 200 times smaller in both cases͒ due to cancel- with opposite predominant spin character. The degeneracy is lations between positive and negative shifts mentioned lifted in the ferromagnetic state. Majority-spin states will above. move down in energy while minority-spin states will move ͉ជ͉ We notice that k(zˆ,xˆ ) for our nanoparticle is larger than up as h increases. Because of spin-orbit-induced level re- ϭ ␮ pulsion, all of the level crossings are avoided; the levels the bulk anisotropy per atom kbulk 60 eV, as one would ជ expect because of the hemispherical shape of the sample. evolve continuously with ͉h͉, gradually changing their spin However, k(xˆ ,yˆ ) is actually smaller than k . This com- and orbital character. In Fig. 9 we plot the variation of 200 bulk ␮ ͉ជ͉ ␮ Ãϭ parison should be regarded with caution, since it is known levels as a function of gs B h near gs Bh 2.2 eV, for a that accurate theoretical estimates of the magnetic anisotropy 143-atom nanoparticle. The Fermi level is the thick black for bulk crystals are very delicate and agreement with experi- line lying in a region of predominantly minority-spin quasi- ment values even in the bulk is still not completely particles ͑lines with positive slope͒. By increasing ͉hជ͉, indi- satisfactory41; we have not evaluated the anisotropy energy vidual majority-spin quasiparticle energies ͑negative slopes͒ that results from the bulk limit of our nanoparticle model and come down from regions above the Fermi level, creating it may well not agree with experiment. Nonetheless, the avoided crossing gaps as they approach minority-spin quasi- small value that we find for the anisotropy in the xy plane particles moving in the opposite direction. Since a 143-atom might be connected to the puzzling finding, in both nanoparticle is still toward the weak spin-orbit coupling tunneling7 and switching-field3 experiments on single ferro- limit, the level crossings will be avoided only weakly. When- magnetic nanoparticles, of anisotropy energies per atom a ever one of the majority-spin quasiparticles crosses the factor of 5 smaller than bulk values. Fermi level, there is a change in the spin character of one of

094430-8 MAGNETIZATION ORIENTATION DEPENDENCE OF THE... PHYSICAL REVIEW B 66, 094430 ͑2002͒

B. Mesoscopic fluctuations as a function of electron number The analysis of the field dependence of tunneling resonances in experiments on magnetic nanoparticles suggests that the anisotropy energy fluctuates significantly from eigenstate to eigenstate.7,9 In Refs. 7 and 9 the effects of these fluctuations were mimicked by using two different an- ϭ ϩ␦ isotropy energy constants kN and kNϮ1 kN kϮ for N- and Ϯ ␦ (N 1)-electron states. By assuming kϮ /kN in the range of a few percent, it was possible to explain the nonmonotonic behavior of the tunneling resonances seen experimentally. Within our microscopic model, we have calculated kN and ͑ ͒ ͑ ͒ kNϮ1 as defined in Eqs. 17 and 18 . In light of results of FIG. 10. Relation between charge fluctuations and anisotropy in the type summarized in Fig. 8, it suffices to compute the total ͑ ͒ 143-atom nanoparticle. a Total charge fluctuation defined in Eq. energy for two different directions of the spin-splitting field ͑ ͒ 19 . Charge fluctuations of the order of one electron result in fluc- and take the difference. The total energy for an N-electron ͑ ͒ tuations in the magnetic anisotropy. b Anisotropy energy in the zx ⍀ˆ ϭ plane vs the magnitude of the spin-splitting field taken as a free system, for spin-splitting fields along directions i ,i 1,2, parameter. is given by

Ã the quasiparticle levels and the total spin of the nanoparticle N ͑h ͒2 ͑ Ã⍀ˆ Ã͒ϭ ⑀ ͑ Ã⍀ˆ Ã͒ϩ i N ϭ ͑ ͒ increases approximately by 1. Due to the exchange interac- E hi i ͚ n hi i a , i 1,2, 20 nϭ1 2U tion, such spin ﬂips bring about charge redistribution inside dd

the nanoparticle, which in turns gives rise to fluctuations in Ã the anisotropy energy. In Fig. 10͑a͒ we plot the total atomic where hi is calculated self-consistently for a given fixed di- site charge redistribution, when hϵ͉hជ͉⇒hϩ⌬h, for ⌬h rection. The anisotropy-energy-per-atom constant is then ϭ12.5 meV, 1 N ͑⍀ˆ ⍀ˆ ͒ϭ ͫ ⑀ ͑ Ã⍀ˆ Ã͒Ϫ⑀ ͑ Ã⍀ˆ Ã͒ͬ kN 1 , 2 N ͚ n h1 1 n h2 2 d␳ ͑h͒ a nϭ1 ␦␳͑ ͒ϭ ͉␳ ͑ ϩ⌬ ͒Ϫ␳ ͑ ͉͒Ϸ ͯ i ͯ⌬ h ͚ i h h i h ͚ h, Ã Ã i i dh ͑h ͒2Ϫ͑h ͒2 ϩ 1 2 ␮ ͒2 ͑ ͒ ͑19͒ ͑gs B , 21 2Udd ␳ where i(h) is the total charge at atom i. It is seen that where the subscript N emphasizes the fact that the constant ␦␳ ⌬ ϭ (h) changes by 1 in the small interval h 12.5 meV refers to an N-electron system. It turns out that the value of where majority-spin quasiparticles weakly avoid crossing the Ã h depends very weakly on ⍀ˆ . For example, in a 143-atom Fermi level. ͑Note that this is a charge redistribution, not a Ã Ã Ã nanoparticle ͉h Ϫh ͉/h Ϸ10Ϫ3. This property reflects the change in total charge.͒ This result is not unexpected, since 1 2 1 the majority- and minority-spin orbitals should have uncor- large ratio between the total magnetic condensation energy related spatial distributions. The charge redistribution is and the anisotropy energy, even for nanoparticles. The strong likely overstated by our simplified model, however, since it coupling between amplitude and orientation fluctuations does not account for long-range Coulomb interactions and mentioned above occurs only when a small change in h leads related screening effects. It is, however, still a relatively to a crossing between majority- and minority-spin orbitals. ⍀ˆ Ã ϭ Ãϭ Ãϩ␦ Ã small ϳ1/N effect. The anisotropy energy fluctuations asso- Evaluating E(h 2 )ath h1 h2 h , expanding in pow- ␦ Ã Ã Ã ⍀ˆ Ã ciated with the level crossings that occur as a function of h, ers of h /h1 , and remembering that h2 minimizes E(h 2 ) shown in Fig. 10͑b͒, are much larger in relative terms. The yields variation in h corresponds to the variation in the amplitude of the magnetic order parameter. In micromagnetic modeling of 1 N nanoparticles, it is implicitly assumed that the magnitude of ͑⍀ˆ ⍀ˆ ͒ϭ ͫ ⑀ ͑ Ã⍀ˆ Ã͒Ϫ⑀ ͑ Ã⍀ˆ Ã͒ͬ kN 1 , 2 N ͚ n h1 1 n h1 2 the order parameter is fixed and that this collective degree of a nϭ1 freedom can be ignored in modeling nanoparticle properties. 2 The substantial dependence of anisotropy energy on h that N ␦hÃ ϩOͫ ͩ ͪ ͬ ͑22͒ we find demonstrates that the amplitude and orientation fluc- N Ã a h tuations of the order parameter can be strongly coupled in 1 small magnetic nanoparticles. Related anisotropy fluctuations as a function of electron number are important in un- for any N. derstanding the addition-potential-spectroscopy single- Adding an electron to the system changes the magnitude 7 Ã→ Ã, ͉ Ã,Ϫ Ã͉ Ã electron-transistor experiments which are the topic of the of the spin-splitting field hi hi , but again hi hi /hi next section. Ӷ1. Thus we obtain

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FIG. 12. Anisotropy energy as a function of nanoparticle size ␦ ϵ Ϫ ␰ FIG. 11. Relative shift kϮ /kN (kNϩ1 kN)/kN of the anisot- for two values of the spin-orbit coupling d . A splitting field of 2 ˆ ˆ ropy energy constant kN(z,x), when the electron number increases eV is assumed for all sizes. by 1, N→Nϩ1, as a function of spin-splitting field. ͑a͒ 260-atom nanoparticle. ͑b͒ 143-atom nanoparticle. The self-consistent spin- atom and a 260-atom nanoparticle͒. For a 143-atom nanopar- splitting field is hÃϷ2.2 eV for both a 143-atom and a 260-atom ticle the calculated anisotropy energy fluctuations are of the nanoparticles. order of 20%. For a 260-atom nanoparticle the anisotropy energy fluctuations are smaller, but still close to 5%. ⍀ˆ ⍀ˆ ͒ We can understand the surprisingly large fluctuations in kNϩ1͑ 1 , 2 anisotropy energy with particle and electron number by the Nϩ1 ϩ ␦ Ã 2 ͑ 1 Ã Ã Ã Ã N 1 h following observations. Our numerical results see Fig. 3 and ϭ ͫ ͚ ⑀ ͑h ⍀ˆ ͒Ϫ⑀ ͑h ⍀ˆ ͒ͬϩOͫ ͩ ͪ ͬ ͒ n 1 1 n 1 2 Ã Fig. 5 indicate that the contribution of a given orbital to the N ϭ N a n 1 a h1 anisotropy energy is chosen essentially at random from a Ã Ã Ã Ã distribution which, for zx-plane anisotropy, has a width ⑀ ϩ ͑h ⍀ˆ ͒Ϫ⑀ ϩ ͑h ⍀ˆ ͒ ϭ ͑⍀ˆ ⍀ˆ ͒ϩ N 1 1 1 N 1 1 2 ⌬(␦⑀ )ϳ2.2␰2/W ϳ2.9 meV and a much smaller mean kN 1 , 2 N n,s d d a ␦⑀ ϭ ⑀ Ãˆ Ϫ⑀ Ã ˆ ϭN ˆ ˆ value ͗ n,s͘ ͗ n,s(h z) n,s(h x)͘ akN(z,x)/N Ã 2 ϳ ␮ ˆ ˆ ϳ Nϩ1 ␦h 15 eV, where kN(z,x) 0.13 meV. The relative change ϩOͫ ͩ ͪ ͬ. ͑23͒ N Ã in the anisotropy energy expected when one additional or- a h 1 ϳ ␰2 ͓ ˆ ˆ N ͔ bital is occupied is 4.3 d/Wd / kN(z,x) a , which can N р Replacing easily be larger than 1% for a 1500. We expect a strong level repulsion in the rigid spectrum Ã Ã Ã Ã ⑀ ϩ ͑h ⍀ˆ ͒Ϫ⑀ ϩ ͑h ⍀ˆ ͒ ⌬͓␦⑀ ϩ ͑⍀ˆ ,⍀ˆ ͔͒ of larger nanoparticles to be accompanied by a more regular N 1 1 1 N 1 1 2 Ϸ N 1 1 2 N N , behavior of the anisotropy energy per atom, with smaller a a ជ Ã ͑24͒ variations as a function h , and electron number. We note that fluctuations in the contribution of a given orbital to the ⌬ ␦⑀ ⍀ˆ ⍀ˆ where ͓ Nϩ1( 1 , 2)͔ is the width of the distribution of anisotropy energy should not exceed ϳ␦, which vanishes in the single-particle anisotropies in plane containing the two the limit of very large particles. We expect that the distribu- ⍀ˆ ⍀ˆ directions 1 and 2, we finally obtain tion function of contributions to anisotropy from orbitals near the Fermi energy to become narrow when ␦ is smaller ͚ ␮ ⌬͓␦⑀ ϩ ͑⍀ˆ ,⍀ˆ ͔͒ than the average anisotropy energy contribution 10 eV. ͑⍀ˆ ⍀ˆ ͒Ϸ ͑⍀ˆ ⍀ˆ ͒ϩ N 1 1 2 kNϩ1 1 , 2 kN 1 , 2 N . This condition is satisfied for particles containing more than a ϳ 5 ͑25͒ 10 atoms.

The ﬂuctuations of kN due to an additional electron are there- C. Anisotropy energy dependence on particle atom ⌬ ␦⑀ ⍀ˆ ⍀ˆ N fore of the order of ͓ Nϩ1( 1 , 2)͔/ a and are regu- number and shape lated by the mean-level spacing ␦, which suppresses the We conclude this section with a few remarks on the de- ⌬͓␦⑀ ⍀ˆ ⍀ˆ ͔ magnitude of Nϩ1( 1 , 2) at large nanoparticle sizes pendence of the magnetic anisotropy energy on nanoparticle as we discuss below. For a 143-atom dot, size and shape. In Fig. 12 we plot the anisotropy energy per ⌬ ␦⑀ ⍀ˆ ⍀ˆ Ϸ ͓ Nϩ1( 1 , 2)͔ 2.9 meV in the zx plane, where atom for nanoparticles of different sizes. For a small number ˆ ˆ Ϸ ␦ ˆ ˆ ͑ ͒ kN(z,x) 0.13 meV. Therefore kϮ /kN(z,x) of atoms below 60 the anisotropy per atom is very large and decreases rapidly with size. In this regime, nonextensive ϭ(1/N )⌬͓␦⑀ ϩ (⍀ˆ ,⍀ˆ )͔/k (zˆ,xˆ )Ϸ15%. These esti- a N 1 1 2 N surface contributions, which are present because of the mates are conﬁrmed by a direct calculation of k (zˆ,xˆ ) and N abrupt truncation of the lattice, are clearly playing a domi- ˆ ˆ kNϩ1(z,x) for 143-atom and 260-atom nanoparticles, shown nating role. By increasing the size of the nanoparticle, we ˆ ˆ ˆ ˆ in Fig. 11. Here kN(z,x) and kNϩ1(z,x) are plotted as a must eventually reach a regime where total magnetic anisot- ␮ ͑ function of gs Bh, taken as a free parameter the self- ropy becomes proportional to the nanoparticle volume. Fig- ␮ Ãϭ consistent value is close to gs Bh 2.2 eV for both a 143- ure 12 shows that this regime is not yet fully established

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Ã FIG. 14. ‘‘Uniaxial’’ anisotropy energy as a function of the di- FIG. 13. Magnetic anisotropy energy in the zx plane ͓E(h zˆ) zˆ rection of the spin-splitting field ⍀ˆ , for different magnitudes of the Ϫ Ã ⍀ˆ ͔ ⌰ E(h⍀ˆ ) for spherical nanoparticles of two sizes. is the angle external magnetic field. The external field is in the zx plane at an between zˆ and ⍀ˆ . By symmetry, the anisotropy energy in the xy angle ␪ϭ␲/4 with the zˆ axis. ⌰ is angle between ⍀ˆ ͑lying in the zx plane is now the same as in the zx plane. The spin-splitting field is plane͒ and zˆ. assumed to be 2 eV for both sizes. ϵជÃ ͉ជÃ͉ ជ even for nanoparticles with hundreds of atoms, and fluctua- h / h , for several values of Hext . Here we consider two tions are still pronounced. In this regime the anisotropy per simplified cases. In Fig. 14 we plot the total energy differ- Ã Ã atom is still 2–3 times larger than the bulk value for cobalt, ence E(hជ zˆ)ϪE(hជ ⍀ˆ ) as a function of an external magnetic which is 0.06 meV. There is no regime in which bulk ͑pro- zˆ ⍀ˆ N ͑ N 2/3 field Hជ . Here ⍀ˆ is allowed to rotate in the zx plane. In this portional to a) and surface proportional to a ) contribu- ext tions to the anisotropy can be cleanly separated. picture we recognize the familiar features of a micromag- It is clear that for small particles the shape of the nano- netic energy functional characterized by a uniaxial magnetic ជ ϭ particle plays an important role in the determination of the anisotropy: at Hext 0 there are two degenerate minima, magnetic anisotropy. For example, as seen in Fig. 13, for separated by an energy barrier, representing two equivalent spherical particles we find anisotropy energies which are one easy directions Ϯxˆ . When a magnetic field is applied in the order of magnitude smaller than the zx anisotropy of hemi- zx plane, the degeneracy is lifted and the minimum in the spherical particles, comparable instead to the xy anisotropy Ϫxˆ direction becomes a metastable local minimum, until a of hemispherical particles with the same number of atoms. A switching field is reached at which the local minimum dis- 140-atom spherical particle has an anisotropy energy per ͉ ជ ͉Ϸ atom of Ϸ0.01Ϫ0.02 meV. It is interesting to notice that appears. This happens at Hext 1 T for a 143-atom nano- both tunneling7 and switching-field3 experiments on single particle. This simple model captures the essence of classical ferromagnetic nanoparticles find anisotropy energies per hysteresis in ferromagnetic nanoparticles. Quantum me- atom which are of the order 0.01 meV, a factor of 5 smaller chanically, for sufficiently weak fields the system can still sit than the bulk value. The nanoparticles in tunneling experi- in the quantum state characterized by a magnetic moment ments are roughly hemispherical,1,7 whereas the nanoparticle pointing along the classically metastable direction until the shape in Ref. 3 is close to spherical. Further theoretical stud- switching field is reached, the system relaxes to its true ground state, and the magnetization orientation collective co- ies that focus on the relationship between nanoparticle shape 42 and anisotropy could be helpful to efforts to engineer ferro- ordinate changes discontinuously. With a further increase magnetic nanoparticles whose shape, size, and crystal struc- of the external field, the magnetization is gradually twisted ture are tuned to produce desired magnetic properties. from the easy x axis toward the direction of the magnetic ͉ ជ ͉Ͻ field, as shown in Fig. 15. At low fields ( Hext 1 T), the V. HYSTERESIS AND VARIATION OF SINGLE-PARTICLE LEVELS IN AN EXTERNAL MAGNETIC FIELD In the last section we shall investigate the effect of an external magnetic field on the total energy of the nanoparticle. This will allow us to make a connection between our microscopic model and more familiar classical micromagnetic energy functional expressions which are normally used to interpret the results of switching-field experiments.3 We shall also study the dependence of the quasiparticle energy levels on external magnetic fields; this complex behavior has been probed directly in recent tunneling experiments.1,7

A. Hysteresis FIG. 15. Variation of the magnetization direction of the stable ͉ ជ ͉ ⌰ Ideally one would like to study the three-dimensional en- minimum of Fig. 14, as a function of Hext . is the angle between ជ Ã ជ ⍀ˆ ⌰ ⌽ ⍀ˆ ˆ ជ ␲ ˆ ergy landscape E(h ,Hext) as a function of ( , ) and z. Hext is in the zx plane at /4 from the z axis.

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FIG. 16. Anisotropy energy in the xy plane for different mag- ⑀ ជ Ã ជ nitudes of an external magnetic field. The direction of the external FIG. 17. Variation of quasiparticle energy levels n,s(h ,Hext) around the Fermi level in an external magnetic field. The levels are field is (␪ϭ␲/4,␾ϭ␲/16). ⌽ is the angle between ⍀ˆ ͑lying in the calculated assuming that the ground state of the nanoparticle xy plane͒ and xˆ . changes in the external field in the way described in Fig. 14. direction of the magnetization corresponding to this local tations is partly obscured by the effect of spin-orbit minimum is very close to the xy plane. coupling.11,44 Sorting out this problem is particularly impor- Because our hemispherical nanoparticles have a nearly tant in order to interpret current tunneling experiments in isotropic easy xy plane, the magnetization will rotate in the 7 43 single-electron transistors. Here we examine only particle- xy plane in response to weak applied external fields. This is hole excitations around the Fermi level, which are immedi- exemplified in Fig. 16, where we plot the the total energy as ately available within our Hartree-Fock treatment of the ͉ ជ ͉ ⍀ ͉ ជ ͉ a function of Hext , when lies in the xy plane. Hext is many-body Hamiltonian. In Fig. 17 we plot the magnetic oriented in the direction (␪ϭ␲/4,␾ϭ␲/16). In the absence field dependence of a few single-particle energy levels of the external field, the ground state of the nanoparticle is around the Fermi energy. As a simplified illustration, the lev- fourfold degenerate, with the degeneracy corresponding to els are calculated assuming that the ground-state dependence one of the four magnetization directions Ϯxˆ Ϯyˆ , associated on the external field is as described in Fig. 14: at low field with the four local minima of Fig. 16. As the external field the ground-state magnetization is oriented around the Ϫxˆ increases, three of these minima will become classically axis until the switching field is reached, whereupon the mag- metastable with small barriers separating them from each netization is reversed along a direction around the xˆ axis. other and the true ground state. In general the different local The corresponding field dependence of the quasiparticle is minima will lose their metastability at different external field quite complex. In the small-field regime (͉Hជ ͉Ͻ1 T) there strengths. In the case we consider, the first switching field is ext is a hysteretic switching at a certain ͉Hជ ͉ϭH Ϸ1 T, due reached at ͉Hជ ͉Ϸ0.15T, when the minimum originally at ext sw ext to an abrupt change of the ground-state magnetic moment. Ϫ ˆ Ϫ ˆ x y disappears. If the system starts out in this local mini- There is basically no correlation between single-particle mum at zero external field, at the switching field it will jump states before and after reversal. Notice in particular that the to the ground state with a corresponding rotation of the mag- 45 levels can jump either up or down at Hsw . Furthermore, the netization in the xˆ ϩyˆ direction. Since everything takes place quasiparticle energies have continuous nonmonotonic varia- essentially within the xy plane, where the typical anisotropy tions, which seem to differ randomly from level to level. In energies are one order of magnitude smaller than in the zx ͉ ជ ͉ӷ the large-field regime ( Hext Hsw), the quasiparticle ener- plane, the scale of the coercivity is also much smaller than gies depend roughly linearly on ͉Hជ ͉ and their slopes al- the one in the zx plane. More complicated hysteretic behav- ext most all have the same sign. In the small-field behavior, the iors are also possible, in which, by starting from a different variation of the quasiparticle energies as a function of the minimum at zero external field, the system jumps first from external field is due to a combination of the rotation of the one metastable to another metastable state and only at a sec- particle’s magnetic moment direction ⍀ and Zeeman cou- ond switching field reaches its true ground state. pling. We have shown above ͑see Fig. 3͒ that the dependence of the quasiparticle energies on ⍀ˆ varies randomly from B. Dependence of quasiparticle levels on external fields level to level. Thus the complex nonmonotonic behavior at The hysteretic behavior that we have seen in the ground- small fields can be understood within our model. At large ͉ ជ ͉ӷ ⍀ˆ state properties of a ferromagnetic nanoparticle has profound fields ( Hext Hsw), is slowly twisted toward the field implications for the magnetic field dependence of its low- direction and the Zeeman coupling plays the dominating role energy elementary excitations. In ferromagnetic metals there in the field dependence. Moreover, because the Fermi level exist two kinds of elementary excitations: collective spin ex- lies in a region of predominantly minority-spin energy levels, citations associated with magnetization orientation degrees it is expected that almost all particle and hole excitations of freedom and particle-hole excitations. In a ferromagnetic around the Fermi level have the same high-field slope.8 In nanoparticle the distinction between these two kinds of exci- the discussion above we have assumed, out of simplicity, that

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model we are only taking into account Hartree-Fock quasiparticles, the expected level spacing of the low-lying excitation is approximately ␦↓ , which is a few meV in our 143- atom nanoparticle and should be of the order of 0.5 meV for a 1500-atom nanoparticle considered in the experiment. This ␦ р value is still larger than the value res 0.2 meV observed experimentally. We believe that a uniﬁed and consistent in- clusion of collective spin excitations, and possibly also the nonequilibrium transport effects proposed in Refs. 9 and 10, could resolve this confusion.

⑀ ជ Ã ជ FIG. 18. Variation of quasiparticle energy levels n,s(h ,Hext) VI. CONCLUSIONS around the Fermi level in an external magnetic field. ͑a͒ The levels are calculated assuming that the ground state of the nanoparticle In this article we have investigated the effects of spin- changes in the external field in the way described in Fig. 16. ͑b͒ For orbit interactions on the properties of ferromagnetic metal magnetic fields larger than 1 T, the magnetization starts to develop nanoparticles. In particular, we have focused on their novel a non-negligible component perpendicular to the xy plane, while its microscopic magnetocrystalline anisotropy physics and on ជ projection in the xy plane is parallel to the component of Hext in hysteresis in the quasiparticle excitations spectra of metallic that plane. nanomagnets. Our analysis, based on qualitative considerations backed up by numerical studies of a generic model for ferromagnetic transition metal nanoparticles, provides an un- the magnetization is arbitrarily constrained to rotate in the zx derstanding of some emergent properties of their quasi- plane. At weak external fields, however, the magnetization particle states. We find two regimes separated by a broad will stay close to the xy plane. For a weak external field in crossover and characterized by the comparison of several ␪ϭ␲ ␾ϭ␲ the ( /4, /16) direction, the actual dependence of characteristic energy scales. For small nanoparticles with the ground-state energy on the magnetization direction at dif- N ϳ fewer than a 200 atoms, the single-particle mean-level ferent field strengths will be the one represented in Fig. 16. ␦ ⑀ spacing is larger than spin-orbit induced energy shifts SO Consequently, the change of quasiparticle energies as a func- in the quasiparticle spectra. These shifts have the same typi- tion of the external field will reflect this more complicated cal size as the spin-orbit scattering lifetime broadening ener- behavior. We illustrate this point in Fig. 18͑a͒ where, as an ប␶Ϫ1 gies of very large particles, SO . The quasiparticle levels of example, we assume that at zero field the magnetization ␦␶ Ͼប small nanoparticles in which SO evolve in a compli- points in the Ϫxˆ ϩyˆ direction. As the magnetic field in- cated way as a function of magnetization orientation and creases, there will be a first jump in the quasiparticle-level external magnetic field, with relatively small avoided cross- Ϸ dependence at Hsw 0.15 T, when the magnetization reori- ing gaps and spin-orbit shifts of nearby orbitals that are ents itself in the (ϩxˆ ϩyˆ ) direction. There will be a second nearly uncorrelated. The size of the avoided crossing gaps is hysteresis jump in the quasiparticle energies at H Ϸ0.3 T, determined by matrix element of the spin-orbit coupling op- sw erator between quasiparticle energy levels that are adjacent ϩ ˆ when the magnetic moment finally switches to the ( x in energy. Surprisingly, even though expectation values of Ϫyˆ ) direction, corresponding to the only metastable configu- these matrix elements vanish because of angular momentum ration left at this field. For a different starting local minimum quenching, typical values between energetically adjacent or- ͉ ជ ͉ϭ bitals are comparable to those for orbitals at any position in at Hext 0, the variation of the quasiparticle energies as a function of the external field will be in general different. By the spectrum. Eventually, for nanoparticles with more than ϳ increasing the magnetic field beyond 1 T, the magnetization 1000 atoms, the typical avoided crossing gap estimated in will start to develop a non-negligible component perpendicu- this way becomes comparable to the level spacing and we lar to the xy plane, while its component in the xy plane will expect the crossover to the strong coupling limit is complete. N ϳ become essentially frozen along the direction of the compo- Nanoparticles with fewer than a 1000 atoms can easily be ជ prepared with current synthesis techniques and systems of nent of Hext in that plane. experimental interest are often in the middle of the crossover The quasiparticle-energy dependence on an external mag- between small particle and bulk ͑weak and strong spin-orbit netic field presented here, has striking similarities with the coupling͒ limits. For example, the nanoparticles investigated field dependence of the tunneling resonance energies in by electron tunneling experiments contain between 50 and 1,7 single-electron tunneling experiments. Notice that for the 1500 atoms.1,7 hysteretic behavior occurring as a result of the rotation of the For nanoparticles in the size range we are able to study N magnetization in the vicinity of the easy xy plane, the order numerically, a smaller than a few hundred, we find that the of magnitude of the coercivity that we find is close to the anisotropy energy per atom displays large changes of order experimental value. The only discrepancy between the re- several percent when the electron or atom number changes sults of our theoretical model and the experiment is the mean by 1. Our analysis allows us to make a connection between ␦ level spacing res of the low-energy excitations. Since in our the microscopic model of a metal nanomagnet and more fa-

094430-13 A. CEHOVIN, C. M. CANALI, AND A. H. MacDONALD PHYSICAL REVIEW B 66, 094430 ͑2002͒ miliar classical micromagnetic energy functional expressions siparticle configurations10 in the nanoparticle in order to un- which are normally used to interpret the results of switching- derstand the size of the quasiparticle level spacing. field experiments.3 The ground-state energy as a function of the magnetization direction is characterized by minima sepa- ACKNOWLEDGMENTS rated by energy barriers. The quasiparticle levels exhibit a complex nonmonotonic behavior and abrupt jumps when the It is a pleasure to thank Mandar Deshmukh and Dan magnetization direction is reversed by an external magnetic Ralph for several discussions about their experimental results field. The nanoparticles investigated by electron tunneling and Jan von Delft for stimulating and informative interac- experiments contain between 50 and 1500 atoms.1,7 The re- tions. C.M.C. would like to acknowledge helpful discussions sults that we have presented here are therefore particularly with L. Samuelson and with the participants of the 2001 relevant for the interpretation of these experiments. In par- workshop ‘‘Spins in Nanostructures’’ at the Aspen Center for ticular, we find that the anisotropy fluctuations inferred from Physics where part of this work was completed. A.H.M. ac- interpretations of these experiments are indeed to be ex- knowledges informative discussions with W. Wernsdorfer pected. Similarly, the dependence of the tunneling reso- and P. Brouwer. This work was supported in part by the nances on the external magnetic field is qualitatively similar Swedish Research Council under Grant No. 621-2001-2357 to the behavior of the quasiparticle excitations of our model, and in part by the National Science Foundation under Grant although it appears necessary to invoke nonequilibrium qua- DMR 0115947.

1 S. Gue´ron, M.M. Deshmukh, E.B. Myers, and D.C. Ralph, Phys. 19 I.M.L. Billas, A. Chtêlain, and W.A.D. Heer, J. Magn. Magn. Rev. Lett. 83, 4148 ͑1999͒. Mater. 168,64͑1997͒. 2 C.T. Black, C.B. Murray, R.L. Sandstrom, and S. Sun, Science 20 P. Bruno, Phys. Rev. B 39, 865 ͑1989͒. 287, 1989 ͑2000͒. 21 H. Takayama, K.P. Bohnem, and P. Fulde, Phys. Rev. B 14, 2287 3 M. Jamet, W. Wernsdorfer, C. Thirion, D. Mailly, V. Dupuis, P. ͑1976͒. Me´linon, and A. Pe´res, Phys. Rev. Lett. 86, 4676 ͑2001͒. 22 D. D. Koelling and A. H. MacDonald, in Relativistic Effects in 4 E.C. Stoner and E.P. Wohlfart, Philos. Trans. R. Soc. London 240, Atoms, Molecules and Solids, Vol. 87 of NATO Advanced Study 599 ͑1948͒. Institute, Series B: Physics, edited by E. E. Malli ͑Plenum, New 5 H. Zijlstra, in Ferromagnetic Materials, edited by E. P. Wohlfarth York, 1983͒. ͑North-Holland, New York, 1982͒, Vol. 3, p. 37. 23 J. Dorantes-Da´vila and G.M. Pastor, Phys. Rev. Lett. 81, 208 6 R. Skomski and J.M.D. Coey, Permanent Magnetism ͑Institute of ͑1998͒. ͒ Physics, Bristol, 1999 . 24 More precisely, this is the magnetoelectric contribution to the 7 M.M. Deshmukh, S.G.S. Kelff, E. Bonnet, A.N. Pasupathy, J. von magnetocrystalline anisotropy. Delft, and D.C. Ralph, Phys. Rev. Lett. 87, 226801 ͑2001͒. 25 8 O. Ericsson, B. Johansson, R.C. Albers, A.M. Boring, and M.S.S. C.M. Canali and A.H. MacDonald, Phys. Rev. Lett. 85, 5623 Brooks, Phys. Rev. B 42, 2707 ͑1990͒. ͑2000͒. 26 H.J. Jansen, Phys. Today 48 ͑4͒,50͑1995͒. 9 S. Kelff, J. von Delft, M.M. Deshmukh, and D.C. Ralph, Phys. 27 The additional anisotropy due to magnetic dipole interactions is Rev. B 64, 220401 ͑2001͒. also sensitive to the overall nanoparticle shape, but is not in- 10 S. Kleff and J. von Delft ͑unpublished͒. cluded here because it is generally less important for nanopar- 11 A.H. MacDonald and C.M. Canali, Solid State Commun. 119, ticles than for macroscopic samples. This contribution to the 253 ͑2001͒. 12 The choice of a hemisphere is motivated by the tunneling experi- anisotropy is additive to a good approximation, does not have ments of Refs. 1 and 7. The crystal structure could not be iden- substantial mesoscopic fluctuations, and can be simply added to tified in these experiments. High-resolution transmission elec- the effects discussed here when it is important. 28 tron microscopy measurements on Co clusters prepared in a T. Oda, A. Pasquarello, and R. Car, Phys. Rev. Lett. 80, 3622 different way ͑Ref. 3͒ have shown that the nanoparticles are well ͑1998͒. 29 crystallized in the fcc structure even though bulk Co has an hcp A. Cehovin ͑unpublished͒. structure. 30 W. Wernsdorfer, E.B. Orozco, A. Benoit, D. Mailly, O. Kubo, H. 13 J.C. Slater and G.F. Koster, Phys. Rev. 94, 1498 ͑1954͒. Nakamo, and B. Barbara, Phys. Rev. Lett. 79, 4014 ͑1997͒. 14 L.F. Mattheiss, Phys. Rev. B 2, 3918 ͑1970͒. 31 G.M. Pastor, J. Dorantes-Da´vila, S. Pick, and H. Dreysse´, Phys. 15 D. A. Papaconstantopoulos, Handbook of the Band Structure of Rev. Lett. 75, 326 ͑1995͒. Elemental Solids ͑Plenum, New York, 1986͒. 32 The use of two Fermi energies for majority and minority spins 16 Notice, however, that the one-body term Hband implicitly includes can generate some confusion: in this terminology, the energies a mean-field approximation to those portions of the interaction are pure kinetic ͑or band͒ energies; i. e., the exchange contribu- not captured by the exchange term. tion is not included. When the exchange energy is included, as 17 H.M. Duan and Q.Q. Zheng, Phys. Lett. A 280, 333 ͑2001͒. we do below in the context of mean-field theory, majority and 18 I.M.L. Billas, A. Chaˆtelain, and W.A. de Heer, Science 265, 1682 minority spins have the same Fermi level. ͑1994͒. 33 P. Bruno and J. Seiden, J. Phys. C 49, 1645 ͑1988͒.

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34 P. Bruno, G. Bayreuther, and P. Beauvillain, J. Appl. Phys. 68, ing them. At low fields and for nanoparticles of 1–4 nanometers 5759 ͑1990͒. in diameter, quantum tunneling is exponentially small. 35 M. B. Stearns, in 3d, 4d and 5d Elements, Alloys and Compounds, 43 Clearly, if the external field is not in the xy plane, the magneti- edited by H. Wijn, Landolt-Bo¨rstein, New Series, Group III, Vol. zation will not stay exactly in this easy plane. However, the 19 of pt. a ͑Springer, Berlin, 1986͒,p.34. displacement of the magnetization from easy to hard directions 36 ͑ ͒ W.P. Halperin, Rev. Mod. Phys. 58, 533 1986 . is negligible at the weak fields (͉Hជ ͉Ͻ1T) ͑see Fig. 15͒ and it 37 ext P.W. Brouwer, X. Waintal, and B.I. Halperin, Phys. Rev. Lett. 85, becomes relevant only at larger fields, when the component of 369 ͑2000͒. the magnetization in the xy plane is essentially frozen along the 38 K.A. Matveev, L.I. Glazman, and A.I. Lakin, Phys. Rev. Lett. 85, direction of the component of ͉Hជ ͉ in that plane. 2789 ͑2000͒. ext 44 ͑ ͒ 39 ͑ ͒ C. M. Canali, A. Cehovin, and A. H. MacDonald unpublished . J.R. Petta and D.C. Ralph, Phys. Rev. Lett. 87, 266801 2001 . 45 40 This is approximately the value of the single-particle mean level The quasiparticle energy difference on the opposite sides of the at the Fermi level also when the spin-orbit interaction is in- transition is of the order of the typical quasiparticle anisotropy cluded. energy. For the nanoparticles that we can study numerically this 41 M.D. Stiles, S.V. Halilov, R.A. Hyman, and A. Zangwill, Phys. energy is smaller than the mean-level spacing. However, for Rev. B 65, 104430 ͑2001͒. larger particles the energy difference at the transition can be 42 This scenario neglects the possibility of quantum tunneling be- larger than ␦. In this case the jumps at the switching fields will tween the two local minima through the energy barrier separat- cause a complete reshuffling of the quasiparticle levels.

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