Intra-atomic aspects of magnon-plasmon interactionsR. Skomski and P. A. Dowben Citation: Journal of Applied Physics 99, 08F508 (2006); doi: 10.1063/1.2177070 View online: http://dx.doi.org/10.1063/1.2177070 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/99/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Plasmons in an AtomicallyThin Ag Monolayer AIP Conf. Proc. 1393, 79 (2011); 10.1063/1.3653618 Above-room-temperature ferromagnetism in half-metallic Heusler compounds NiCrP, NiCrSe, NiCrTe, andNiVAs: A first-principles study J. Appl. Phys. 98, 063523 (2005); 10.1063/1.2060941 Magnetothermopower and magnon-assisted transport in ferromagnetic tunnel junctions Appl. Phys. Lett. 81, 3609 (2002); 10.1063/1.1519730 Nonlinear dynamics of spin-injected magnons in magnetic nanostructures J. Appl. Phys. 91, 8046 (2002); 10.1063/1.1450819 Anomalies of the magnon spectrum of a bounded antiferromagnet with a center of antisymmetry. II. Effects of theinhomogeneous exchange interaction Low Temp. Phys. 27, 130 (2001); 10.1063/1.1353704

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Intra-atomic aspects of magnon-plasmon interactionsR. Skomskia� and P. A. DowbenCenter for Materials Research and Analysis, 111 Brace Laboratory, P.O. Box 880111,University of Nebraska, Lincoln, Nebraska 68588and Department of Physics and Astronomy, 116 Brace Laboratory, P.O. Box 880111,University of Nebraska, Lincoln, NE 68588

�Presented on 1 November 2005; published online 28 April 2006�

Magnon-plasmon interactions are modeled by considering the spin-dependent dielectric response ofatoms placed in crystalline environment. Hund’s exchange rules favor parallel spin alignment, butthe strength of the exchange depends on the displacement of the centers of gravity of the atomicspin-up and spin-down electron charge clouds. The intra-atomic exchange is modeled byconsidering a Hubbard-type interaction, and interatomic interaction then yields a k-space dispersion.The eigenmodes of the plasma are a mixture of spin-up and spin-down degrees of freedom,described by a 2�2 interaction matrix. Minority and majority bands yield different plasmonfrequencies. However, these modes are not orthogonal but coupled by intra-atomic exchange andobtained by explicit matrix diagonalization. The effect is largest for small wave vectors, inagreement with experiment. © 2006 American Institute of Physics. �DOI: 10.1063/1.2177070�

I. INTRODUCTION

Interactions between magnetic and electric degrees offreedom have been investigated very extensively, butmagnon-plasmon interactions, without any coupling throughphonons, are almost never considered.1–3 The main reason isthat the effect is quite small. Magnon energies tend to bemuch smaller than plasmon energies, so that the experimen-tal investigations of the interactions are a challenge. An ex-ception is dilute magnetic semiconductors, such asEu1−xGdxO,2 where low carrier densities yield low plasmonenergies. Recently, spin-polarized electron energy loss spec-troscopy �SPEELS� has been used to investigate magnon-plasmon interactions in thin films of strained Gd�0001� onMo�112�.4 In this technique, a spin-polarized electron is scat-tered off the surface at remanence, and the energy of thescattered electrons is analyzed to determine the lossstructure.5 The experiment indicates spin-polarized plas-mons, in spite of the high concentration of 5d and 6s con-duction electrons in Gd.

In this paper, we investigate spin-dependent plasmonmodes in nonsemiconducting systems with high plasmon fre-quencies. The considered excitations refer to longitudinalspin degrees of freedom, as contrasted to ordinary magnons,which are transversal. The difference is that longitudinalmodes involve intra-atomic exchange energies larger thanabout 1 eV, whereas transversal magnon modes are due tointeratomic exchange of less than about 0.1 eV. In otherwords, the present excitations involve changes in the magni-tude of the local moment, as compared to moment rotationsthat change Sz but keep S�S+1� fixed.

II. BASIC MECHANISM

In the simplest case, the plasmon is created by an inho-mogeneous magnetic field acting on a spin-polarized mag-

netic body. If the position of the magnet is fixed, the fieldcreates a plasmon mode and electric surface charges. Figure1 illustrates the phenomenon for an inhomogeneous mag-netic field H=H�x�ez characterized by a positive magnetiza-tion gradient �H /�x. Since the Zeeman interaction of a mo-ment m with the external field is equal to −�0m ·H, themoment-carrying electrons move towards regions of higherfield intensity. This corresponds to a plasmon mode of somemagnitude or electron displacement. For laboratory-scalemagnetic fields, the effect is relatively small, because theZeeman energy competes against much stronger energies ofelectrostatic origin. The relative strength of the magnetic in-teractions is of the order of �2,6 where �=1/137 is Sommer-feld’s fine-structure constant. By contrast, SPEELS experi-ments correspond to much larger local exchange fields.

In a free-electron picture, a spin-polarized electron gasgives rise to different plasmon frequencies �↑�n↑

1/2 and �↓�n↓

1/2, which can be probed experimentally. However, freeelectrons are a poor starting point, because atomic magneticmoments are centered around the atomic cores, even in itin-

a�Electronic mail: rskomski@neb.rr.com

FIG. 1. Plasmon created by an inhomogeneous magnetic field H acting on aspin-polarized magnetic body. If the position of the body is fixed, the fieldcreates a plasmon mode and electric surface charges.

JOURNAL OF APPLIED PHYSICS 99, 08F508 �2006�

0021-8979/2006/99�8�/08F508/3/$23.00 © 2006 American Institute of Physics99, 08F508-1

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erant magnets. In Gd, the magnetic moment is caused bylocalized 4f electrons, but the Gd 4f electron creates an ex-change field that slightly polarizes the 5d and 6s conductionelectrons of the Gd and gives rise to different electron den-sities n↑ and n↓.

III. CALCULATION AND RESULTS

Let us now investigate the intra-atomic aspect of themagnon-plasmon interactions. Each atom �index i� has twosubshells, ↑ and ↓, and these shells may be displaced withrespect to the position Rio of the atomic core. Let us denotethe displacements by Ri↑ and Ri↓. Figure 2 illustrates thatthere are two types of excitations, out-of-phase excitationswhere Ri↑ and Ri↓ point in different directions �b� and in-phase excitations where Ri↑ and Ri↓ point in the same direc-tion �c�. We treat these excitations in a harmonic approxima-tion and assume that the shape of the ↑ and ↓ shells remainsunchanged �rigid orbitals�. Next, we expand the magneticenergy into powers of the small vectors �i↑=Ri↑−Rio and�i↓=Ri↓−Rio. Here we ignore anisotropic effects, whichwould mix different displacement directions, and considerthe longitudinal modes �i↑ and �i↓ shown in Fig. 2.

The procedure is illustrated by considering the Coulombenergy EC, which is the origin of the intra-atomic exchange.In the Hubbard formulation, EC=Un↑n↓, where n↑ and n↓ areelectron densities,7,8 but in the present case, a more generalformulation of this expression must be used,

EC =� U�r − r���↑�r − Ri↑��↓�r� − Ri↓�drdr�. �1�

Here �↑�r�=�↑* �r��↑�r� and �↓�r�=�↓* �r��↓�r� are elec-tron densities. This expression can be evaluated by expan-sion into powers of �i↑ and �i↓. The same is true for otherenergy contributions, such as the attractive interaction be-tween the atomic core and the ↑ and ↓ shells. The resultingtotal on-site energy is

E = E0 +u↑2

�↑2 + um�↑�↓ +

u↓2

�↓2. �2�

In addition to this on-site energy, there are wave-vector de-pendent contributions. Figure 3 illustrates this by comparingtwo modes with �a� k=0 and �b� k=� /a. The energy of Fig.4�b� is higher than that of Fig. 4�a�, because in �b� the ↑orbitals nearly overlap. The wave-vector dependence of themodes shown in Fig. 4 refers to the ↑ electrons. In the figure,the ↓ electron clouds are all shifted to the right direction.However, there may also be other modes, such as ↓ and

mixed modes, where ↑ and ↓ electrons on different atomsinteract with each other.

The eigenfrequencies � of the excitations are obtainedby diagonalizing the 2�2 matrix expression

��2 0

0 �2 � = ��↑2�k� �m

2 �k��m

2 �k� �↓2�k�

� . �3�

In the simple case of a chain of magnetic atoms, the wave-vector dependence is given by 1−cos ka. Figure 4 shows atypical k dependence for this one-dimensional example. Ageneral feature, not restricted to the one-dimensional model,is some admixture of ↓ character to the ↑ modes. However,this mixing refers to the origin of the mode. The magnitudeof the local magnetization changes, because the electronclouds of opposite spin are shifted to different positions, butthe mxing does not indicate spin flipping. In fully spin-polarized magnets, the dielectric response is provided by ↑electrons, which then behave similar to ordinary electrons.

IV. DISCUSSION AND CONCLUSIONS

The modes described in Sec. III mean that ↑ and ↓ elec-tron clouds have different properties. Equation �2� and �3�parametrize this difference, but no attempt is made in thispaper to calculate the parameters from first principles. Forexample, �↑, �↓, and �m depend on the interatomic hopping

FIG. 2. Magnetic phonon-electron interaction model: �a� ground state, �b�out-of-phase excitation, and �c� in-phase excitation. The small gray circle isthe atomic core �nucleus�.

FIG. 3. Wave-vector dependence: �a� one k=0 mode and �b� one k=� /amode.

FIG. 4. Wave-vector dependent modes.

08F508-2 R. Skomski and P. A. Dowben J. Appl. Phys. 99, 08F508 �2006�

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integrals, which determine, in combination with Eq. �1�, theferromagnetic moment of itinerant magnets. The harmonicapproximation used in Eq. �2� is reasonable, because the ef-fect is not very large.4 Another relatively unessential ap-proximation is the assumption of rigid charge clouds.

An experimental challenge is the small magnitude of theeffect, and there is no hope to observe excitations such asthose shown in Fig. 2 in laboratory-scale field gradients ofthe order of 1 T/cm. The SPEELS experiments outlined inRef. 4 involve much higher energies, because the spin-polarized electrons interact directly with the ↑ and ↓ electronclouds. As emphasized in Sec. II, this corresponds to muchlarger effective local magnetic fields �exchange fields�. Fig-ure 4 shows that the excitations are accessible most easily atsmall wave vectors, in agreement with the preliminary ex-perimental findings. Naturally, the relatively simple model ofSec. III captures only the most basic features of the phenom-enon, and refined materials-specific calculations of this inter-esting effect remain a challenge for future research.

In conclusion, we have analyzed magnetic excitations ofintra-atomic origin. They are related to ordinary magnons butinvolve longitudinal magnetic degrees of freedom and in-volve much higher energies. This makes it possible to excitethem in metallic ferromagnets, as contrasted to magnetic

semiconductors. The effect is wave vector dependent and canbe investigated experimentally by spin-polarized electron en-ergy loss spectroscopy.

ACKNOWLEDGMENTS

This work was supported by the National Science Foun-dation through the NSF “QSPINS” MRSEC �DMR-0213808� and by the Center for Materials Research andAnalysis at the University of Nebraska, which is funded bythe Nebraska Research Initiative.

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5Spin-dependent plasmons and small intraband excitations across the Fermilevel do not involve a spin flip scattering, so that spin detection after thescattering event is not necessary if a spin polarized source is used.

6R. Skomski, H.-P. Oepen, and J. Kirschner, Phys. Rev. B 58, 3223 �1998�.7W. Jones and N. H. March, Theoretical Solid State Physics �Dover, NewYork, 1985�, Vol. 1.

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08F508-3 R. Skomski and P. A. Dowben J. Appl. Phys. 99, 08F508 �2006�

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