PHYSICAL REVIEW B VOLUME 58, NUMBER 21 1 DECEMBER 1998-I Spin-flip exchange scattering of low-energy electrons in ferromagnetic iron M. Plihal Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08855 D. L. Mills Department of Physics and Astronomy, University of California Irvine, Irvine, California 92697 Received 20 May 1998 We present a theory of spin-flip exchange scattering of low-energy electrons, directed at the ferromagnetic transition metals, with application to Fe. The model used employs a tight-binding description of the paramag- netic spd bands. Ferromagnetic exchange splitting of the bands is achieved by including on-site Coulomb repulsion between electrons in 3d orbitals, which is treated in a mean-field approximation. The low-energy electron interacts with the metal electrons via the Coulomb interaction, and the magnetic excitations in the metal are treated within the random-phase approximation. Both spin waves and Stoner excitations contribute to the energy loss of the low-energy electron. We show that the relative importance of these two loss mechanisms is influenced very importantly by the degree of localization of the 3d orbital. We also present results based on the use of accurate wave functions. These show that spin-wave loss peaks should be prominent features in spin-polarized electron energy-loss spectra. S0163-1829 98 03745-X I. INTRODUCTION the experiments to date have energy resolution insufficient to address these features. An earlier estimate of the cross In itinerant electron ferromagnets such as Fe, Co, or Ni, section,5 based on a simple model, and the calculations pre- there are two classes of magnetic excitations. One has low- sented in the present paper suggest one should be able to lying collective excitations, the spin waves. In addition, there observe these modes, with presently available spectrometers. is a continuum of particle-hole excitations in which the spin We comment next on the issue which motivates the cal- of the electron is flipped. These are referred to commonly as culations here. First, the ability to interpret SPEELS spectra Stoner excitations. in more than a qualitative manner has been inhibited by the There have been remarkably few experimental studies of absence of quantitative theory for real materials. Activity in the magnetic excitations in the classic materials just men- the field has declined as a consequence, in our view. This has tioned, particularly at large wave vectors where our recent stimulated theoretical efforts to generate a quantitative theoretical studies1 show the spin-wave spectrum to be re- theory of the loss spectra; the theory of the magnetic excita- markably sensitive to details of the electronic band structure, tions in itinerant ferromagnets is, of course, a central issue. at least in ferromagnetic Fe. Neutron scattering has been em- While very substantial efforts have been devoted to the study ployed to study spin-wave excitations, but in a material such of ground-state properties of magnetic surfaces and ultrathin as Fe, the spin-wave exchange stiffness is so large one can- films,6 very little has been directed toward their excitation not explore them beyond 25% of the way from the Brillouin- zone center, to the zone boundary. Through use of a spalla- spectra. Our recent paper1 is a publication which addresses tion source, Perring et al.2 have explored aspects of the such questions, within the framework of a model based on a excitation spectra of Fe at large wave vectors, though the realistic electronic structure. data are limited. It is striking to us that so little is known One must also develop a description of the scattering pro- about the magnetic excitations of this well-known ferromag- cess itself. Within the framework of a relativistic multiple- net. scattering theory, this issue was addressed recently.5 The Electron energy-loss spectroscopy EELS , or its spin- physical picture which underlies the scattering event is that polarized version SPEELS offers the possibility of probing the beam electron sees an array of spins disordered by fluc- magnetic excitations in such materials, over a wide range of tuations, and is deflected away from the specular or Bragg energy and wave vector. Here the incident electron has en- directions by an inelastic event in which energy is exchanged ergy which may range from 20 eV to a few hundred electron with the fluctuations. It was assumed in Ref. 5 that as the volts. In this energy range, the electrons have very short substrate moments fluctuate, they do so as rigid entities, un- mean free paths, so electron loss spectroscopy proves a pow- changed in magnitude and shape. A consequence is that the erful probe of excitations at surfaces, and in ultrathin films. SPEELS spectrum is described by the wave-vector and Experiments which use spin-polarized electron beams, and frequency-dependent transverse spin susceptibility spin detectors can isolate contributions to the loss spectrum (q , ) encountered frequently in discussions of the re- in which the spin of the beam electron is flipped. Such sponse of magnetic materials to external probes.7 We have ``complete experiments'' have explored Stoner excitations in recently completed detailed studies of (q , ) for bulk Fe and other magnetically ordered metals.3,4 However the Fe, and its counterpart for ultrathin films of Fe.1 Spin waves, much lower energy spin waves have yet to be detected since broadened by Landau damping, appear as strong features as 0163-1829/98/58 21 /14407 9 /$15.00 PRB 58 14 407 ©1998 The American Physical Society 14 408 M. PLIHAL AND D. L. MILLS PRB 58 they have in earlier theoretical studies of bulk Fe.8 However, tron scattering in bulk Fe, to find strong Stoner contributions. the Stoner excitations appear only as very weak features in Use of this matrix element allows us to generate quantitative this response function, while they show most clearly in the SPEELS spectra, for model descriptions in which the elec- experimental SPEELS spectra. If we combine the formalism tronic band structure is realistic. We find, as discussed be- of Ref. 5 with the results for (q , ) in Ref. 1, the pre- low, that the ratio of the spin wave to Stoner strength is dicted Stoner spectrum is far too weak. We remark that in the sensitive to the nature of the wave functions employed for bulk, in the limit q 0, considerations of spin rotation in- the substrate electrons. We illustrate this with model calcu- variance requires the Stoner contribution to vanish identi- lations. Our final set of studies employs realistic wave func- cally. We have found that even at large values of q , it is tions, for the case of Fe, and provides us with accurate esti- surprisingly weak. mates. In this paper, we address the issue of the origin of the Our interest ultimately resides in a realistic study of the strong Stoner contributions to SPEELS spectra, with the in- SPEELS spectra of ultrathin films of Fe. All calculations tention of providing a quantitative theory of the strength of presented here explore the losses experienced by a plane- the spin-wave feature relative to the Stoner continuum. In wave ``hot electron'' in bulk Fe. The theory for the ultrathin physical terms the assumption in Ref. 5 that the substrate film is under development presently. moments rotate rigidly as they participate in the thermal fluc- tuations must be reexamined. As they fluctuate, we must take due account of the fact that they change shape and magni- II. THEORETICAL MODEL tude. Stated otherwise, we require a more realistic exchange matrix element to couple the beam electron to the spin exci- In our previous paper,1 our theoretical analysis of the dy- tations. namic spin susceptibility of bulk Fe, and of ultrathin Fe We may see that this is so from an earlier study of spin- films, was based on a model of electronic structure provided flip electron scattering put forth by Vignale and Singwi.11 by the empirical Slater-Koster parametrization scheme. One These authors present a theory of SPEELS in bulk itinerant associates five 3d Wannier orbitals with each lattice site, ferromagnets within the framework of a very simple picture along with three 4p and one 4s orbital. Empirical values for of the electronic structure of the substrate. They address neu- the various hopping integrals between first through second tron scattering as well. The electrons all reside in parabolic neighbors are chosen to reproduce energy bands of paramag- energy bands, with wave functions of plane wave form. A netic Fe generated by ab initio calculations. We employed phenomenological, rigid exchange splitting is introduced for values from the literature, and developed a multiband exten- substrate electrons. The neutrons couple to the substrate sion of the Hubbard model to describe the intra-atomic Cou- through lomb interactions which produce ferromagnetism. In our pic- (q , ), in their picture, while use of a micro- scopic exchange matrix element in the descriptions of ture, electrons interact when they reside within the 3d SPEELS leads to a more complex response function. In their orbitals associated with one particular lattice site. We have studies of the neutron spectra, one sees the Stoner spectrum three adjustable parameters in this picture of the Coulomb is indeed weak compared to the spin-wave features as in our interaction; these are chosen to reproduce features of the fer- recent studies, while it is strong in the SPEELS calculations. romagnetic ground state of Fe. Full details are given in Ref. In their model, for high beam electron energy, the expression 1. for the SPEELS cross section becomes proportional to Within this scheme, we do not need explicit forms of the wave functions of the electrons, to explore the dynamic spin (q , ), with the consequence that the Stoner spectrum weakens relative to the spin-wave portion at high impact susceptibility. These are, in fact, form factors of wave func- energy. tions which enter, but we find the results rather insensitive to As we discussed earlier1 and mentioned above, the weak their precise form, so long as one takes care to endow the Stoner structure in model forms with proper symmetry. Thus, in the results re- (q , ) has its origin in a fundamental theorem. Spin-rotation invariance requires that as q 0, all ported in Ref. 1, we employed rather crude models of the the oscillation strength resides in the spin-wave pole at form factors. 0. Evidently, as we see from the work of Vignale and For reasons outlined in Sec. I, to calculate realistic Singwi11 and our recent studies, even at large wave vectors SPEELS spectra, we require explicit forms for the wave the Stoner spectrum remains weak. No such theorem applies functions of the valence electrons of the ferromagnetic metal. to the response function that is relevant when a full micro- As we shall demonstrate, the results are sensitive to the spa- scopic exchange matrix element is employed. tial structure of the Wannier functions used in the evaluation The theoretical situation in the theory of SPEELS thus of the exchange matrix element. In this section, we discuss differs substantially from that of phonon losses in electron- our form for the matrix element, along with our means of energy-loss spectra. In the latter case, theories based on the generating the SPEELS spectrum once this is in hand. notion that the potential of an ion shifts rigidly when the To begin our discussion, we illustrate the basic exchange nucleus is displaced provide very quantitative accounts of scattering process in Fig. 1 a . We imagine a beam electron, the measured loss spectra.12 or ``hot electron'' with momentum p i and spin down. It en- In the present paper, we develop a description of a micro- gages in a Coulomb scattering with a valence electron of scopically based exchange matrix element suitable for the spin up, wave vector k , and which resides in energy band n. calculation of SPEELS spectra, when an empirical tight- The valence electron is excited to state p f , and becomes the binding description of the substrate band structure is em- final-state electron in this exchange scattering, while the ployed. We then explore SPEELS spectra for spin-flip elec- electron p i is deexcited into an empty minority-spin state. PRB 58 SPIN-FLIP EXCHANGE SCATTERING OF LOW-ENERGY . . . 14 409 V * ex nk , p i , p f ,mk q d3x d3y x e ip i*y nk v x y eip f*x mk q y . 4 In this expression, v( x y ) is the electron-electron interac- tion. The form of this interaction appropriate to the present analysis is discussed shortly. It will be convenient to introduce operators which de- scribe the act of particle-hole creation, weighted by suitable exchange matrix elements associated with each pair. Thus, we define R q ,p i V* n1k q p f ;p in2k n1n2k FIG. 1. a The basic spin-flip exchange scattering process ex- c n2k c n1k q . 5 plored in this paper. Here (p i ) and (p f ) are the initial and final- state beam electron, while (nk ) and (mk q ) reside in the va- The operator R (q ,p i) is the Hermitian conjugate of lence bands of the ferromagnetic host. b A schematic illustration R (q ,p i). Central to our analysis will be a two-particle of the diagrams which contribute to the screening of the process Green's function formed from these operators. We define delineated in a . G q ,p i ; T R q ,p i ; R q ,p i ;0 , 6 In the end, we have spin-flip scattering of the beam electron from state p where T is the time ordering operator on a complex time i to state p f . The beam electron will be described by a wave function contour. that is a simple plane wave. The Bloch function associated Through use of the scattering amplitude defined in Eq. with the substrate electron is written 3 , it is a standard matter to derive the cross section (d2 /d d ) where (d2 /d d )d d is the probability the incident electron is scattered into solid angle d , with nk x a nk eik *R x R . 1 energy loss between and ( d ), accompanied by a R spin flip. We omit details here, since by now such deriva- Functions tions are standard. We find (x R ) are normalized so the integral of their square over all space is unity. We are interested in excita- d2 V2 m2 p 1 tions of the d-electron system, so in the sum over , we f retain five d orbitals d d 4 2 4 pi e 1 Im q p i ; , 7 (r R ); two have eg character, and three have t where g character. The admixture coefficients a (nk ) are generated from the empirical band-structure scheme dis- cussed above. We shall require explicit forms for the orbitals q p i ; dtei tGR q p i ;t 8 (r R ) in what follows. We write and GR(q p i ;t) is the retarded real-time Green's function associated with that defined in Eq. 6 . We remind the reader R2 A mY2m , 2 m that, with our convention, the spectral density is negative, and thus, of course, the cross section is positive. where A m is the transformation matrix from the spherical As mentioned in Sec. I, the earlier theory of SPEELS5 harmonics basis to the basis of eg ,t2g orbitals. Our choice assumed that the beam electron scattered inelastically from of R2( ) is discussed below. magnetic moments in each unit cell which rotate as rigid entities, as they participate in magnetic fluctuations. This A. Description of the exchange scattering process earlier analysis incorporated a full multiple-scattering de- We can imagine the beam electron encounters the sub- scription of the beam electron's propagation through the strate in an initial state M , and excites it to a final state crystal. If we apply the description of the inelastic event N , through the process depicted in Fig. 1 a . We describe given in Ref. 5 to the plane-wave electron considered pres- this by introducing a scattering amplitude we write as ently, the inelastic cross section would be given by an ex- pression virtually identical to Eq. 7 , but with one very cru- cial difference. In place of Im (q p A i ; ) we will have the p V i ,nk p f ,mk q ex * nk ,p i ,p f ,mk q spectral density associated with transverse fluctuations of n,m,k spin density of wave vector q . That is, let S (x ,t) N c c S mk q nk M . 3 x(x ,t) iSy(x ,t) be the second quantized operator which describes the spin density in the d electron at time t, with The exchange matrix element in Eq. 3 is given by S (x ,t) its Hermitian adjoint. Upon taking the Fourier trans- 14 410 M. PLIHAL AND D. L. MILLS PRB 58 form of these objects, one forms S (q ,t) and S (q ,t). 1 Within our Bloch representation, suppressing explicit refer- F nk q e i q k *R a nk f q , ence to the time t, we have N R 13 where S q c nk c mk q nk e iq *x mk q . nmk 9 f q d3xe iq *x x . 14 The two-particle propagator form from S (q ,t) and The sum over R in Eq. 13 may be carried out to give us S (q ,t) analogous to Eq. 6 is the dynamic transverse sus- ceptibility (q ,t). In previous work, we presented de- tailed studies of the spectral density of this function for bulk Fnk q N q ,k G a nk f k G . 15 Fe, and the associated generalization for ultrathin Fe films. G We found a prominent spin-wave feature in these spectral If Vi is the volume of the unit cell, then we find densities for both cases, but there was very little integrated strength in the Stoner region of the spectrum even at large Vex nk ,p i ,p f ,mk q wave vectors. The Stoner excitations show clearly in the ex- perimental data.3,4 We argued earlier1 that the weak Stoner contribution to a * nk a mk q (q ,t) is a consequence of spin-rotation invariance in the Hamiltonian, for the following reason. In the bulk material, and at q 0, spin-rotation invariance requires all the oscilla- v k p f G f * k G f k q G . 16 G tor strength in the spectral density to reside in the spin-wave pole, which is at 0 in this limit. The Stoner spectrum is As discussed earlier, the coefficients a (nk ) are provided completely absent. Our calculations show the Stoner spec- self-consistently from the multiband generalization of Hub- trum remains weak even at large wave vectors; there is a bard model with the hopping integrals given by the Slater- remnant of spin-rotational invariance even at short wave- Koster empirical band-structure scheme. We need the ex- length, evidently. plicit form of the orbitals to generate the form factors f (q ). When a proper microscopic exchange matrix element is We discuss our choices below. incorporated into the analysis, one encounters the Green's We conclude this section by discussing the appropriate function defined in Eq. 6 . Even at q 0, we shall realize the form for v(Q ), the matrix element of the electron-electron structure in the Stoner region, since considerations of spin- interaction. rotational invariance place no restraint on this function. A Since the electrons interact inside the metal, quite clearly consequence is that even in this limit, one has contributions the interaction will be screened. If we were to describe the from Stoner excitations. We shall see that for wave-vector screening within the framework of the random-phase ap- transfers such as those realized in SPEELS experiments proximation, the relevant diagrams are illustrated in Fig. found in the literature, the Stoner bands appear prominently. 1 b . The bare Coulomb interaction of Fig. 1 a should be We conclude this section with a discussion of the explicit replaced by the screened form, as illustrated. form of the exchange matrix element. We proceed as fol- A full account of the screening, within the framework of lows. We imagine a large crystal of volume V, with periodic our multiband picture is a formidable undertaking. However, boundary conditions applied to all quantities. Thus, for the for the exchange scattering process of interest here, we argue electron-electron interactions, we write screening effects are very modest and may be ignored. Note that the frequency argument of each bubble in Fig. 1 b is 1 ( v x y a b). We have in mind beam electrons energy a) at V v Q eiQ * x y . 10 least several electron volts above the vacuum level, while the Q final-state electron created in the excitation process energy Then b) resides within the 3d band complex of the substrate, 5­10 eV below the vacuum level. The wave-vector and 1 frequency-dependent dielectric response function involved in V the screening is thus evaluated at a frequency well above the ex nk , p i , p f ,mk q V v Q Fnk P f Q Q plasma frequency. In addition, the wave-vector transfer in- volved in the exchange scattering event is substantial, the F* P mk q i Q , 11 order of 108 cm 1 or higher. We may thus safely ignore screening, and use the bare Coulomb interaction in the analy- where sis. Thus, we choose v(Q ) 4 e2/Q2. The next subsection is devoted to our means of generating the Green's function defined in Eq. 6 . As remarked earlier, Fnk q d3xe iq *x nk x . 12 the random-phase approximation RPA will provide the ba- sis for our analysis. We note that in our earlier study, the Use of the form in Eq. 1 allows us to write RPA provides values for the spin-wave exchange stiffness in PRB 58 SPIN-FLIP EXCHANGE SCATTERING OF LOW-ENERGY . . . 14 411 excellent accord with experimental data on bulk Fe, and a q , 1,1 q , 1,0 q , large wave vectors, it accounted nicely for the features re- 0 0 ported in Ref. 2. U 1 0,0 0,1 0 q , U 1 0 q , , 20 where the multiplication in Eq. 20 has the character of B. RPA solution matrix multiplications in the space , of 3d orbital sym- To proceed, we define an auxiliary two-particle Green's metry labels. To derive Eq. 20 , one expresses all quantities function in Eq. 18 in the orbital basis, and carries out the formal solution in terms of matrix multiplication. A key to one's G ability to achieve this solution is our assumption that the 3d 2 nm,k q p i ; T c nk ; c mk q ; electrons interact only when they reside on the same lattice R q ,p i ;0 . 17 site. The kernel in Eq. 18 is then separable. In Eq. 20 , the various quantities that enter are We may generate an equation of motion for this function, and decouple the resulting form within the RPA scheme. The M,N 0 , ;q procedure is very similar to that described in Ref. 1. After the decoupling procedure, we find the auxiliary function de- 1 f nk fmk q fined in Eq. 17 obeys the integral equation N nmk mk q nk i f nk f mk q F k ,q ,p f Ma * nk a mk q G2 nm,k q p i ;i i mk q nk a* * mk q a nk F k ,q ,p f N. 21 Here, F V mk q ,p (k ,q , p f ) is an exchange matrix element form fac- f , p i ,nk tor, defined as U mk q ,n F k ,q ,p f v k p f G f 1p ,n2p * k G f k q G . G n1n2k 22 q ,nk G In Sec. III, we present numerical results based on the 2 n1n2 , p q p i ;i . formalism presented in this section, and we discuss the physical content of the various contributions to Eq. 20 , 18 which is the central result of the present paper. In this expression (nk ) and (nk ) are the energies of up- and down-spin electrons in the ground-state energy bands, III. RESULTS AND DISCUSSION and f (nk ), f (nk ) are the associated Fermi-Dirac func- As just remarked, Eq. 20 contains the central result of tions. the present paper. Formally, the response function described In Eq. 18 , we have a Bethe-Salpeter equation, which by this equation is structurally rather similar to the dynamic describes the repeated scattering of the excited down-spin transverse susceptibility studied earlier. However, as we electron (mk q ) against the up-spin hole (nk ) produced have remarked earlier, the explicit appearance of the ex- in the excitation event depicted in Fig. 1 a . The quantity change matrix element plays a crucial role in the results that U(mk q ,n follow, by breaking down the strong influence of spin- 1p ,n2p q ,nk ) is the matrix element of the Coulomb interaction responsible for this scattering. This rotation invariance. Before we present the results of our nu- is provided by our multiband generalization of the Hubbard merical studies, we discuss the content of this equation. model, as given in Ref. 1. If we were to retain only the first term in Eq. 20 in the We transform the various quantities which enter the analysis, and use this to evaluate the SPEELS cross section analysis to a representation labeled by the symmetry indices in Eq. 7 , then we would be describing the exchange scat- , of the d electron orbitals. For example, we introduce tering event depicted in Fig. 1 a . The electron and the hole created in the SPEELS excitation process are regarded as freely propagating noninteracting entities at this level of ap- G2 ,q p i ;i proximation. In the literature, it is commonly assumed that one may interpret SPEELS data within this simple scheme.13 1 The second set of terms in Eq. 20 recognize that after N a * nk a nk q G2 nm,k q p i ;i . the electron-hole pair have been created in the scattering pro- nmk 19 cess, in fact they interact with each other. In our RPA scheme, we have retained repeated scatterings between these The analysis may then be phrased entirely in terms of an entities, described by the ladder graph diagrams of many- ``exchange scattering response function'' ( , ;q ) body theory. which, in the basis set provided by the atomic orbitals, may The interaction between the particle and hole influences be expressed as follows: the SPEELS spectrum in a qualitative manner. Such final- 14 412 M. PLIHAL AND D. L. MILLS PRB 58 magnetic iron. These are the Stoner excitations, which ``ac- tivated'' in the exchange response function, for the reasons just discussed. Our earlier studies of the dynamic transverse susceptibility showed that at small wave vectors, the Stoner feature was absent entirely. The dotted line in Fig. 2 is just the spectral density associated with the first term in Eq. 20 . Recall this is the spectrum associated with noninteracting electron-hole pairs. We see that the final-state particle-hole interactions do not shift the maximum of this feature. Clearly, one sees that when the interactions are incorporated, oscillator strength is transferred from the Stoner feature, to the collective spin-wave excitation. The origin of the width of this structure is interesting. At q 0 we have only vertical transitions which contribute to the Stoner spectrum. Thus the width has its origin in the wave-vector dependence of the exchange splitting. In the calculations just described, and those reported be- low, we have employed a finite value of the parameter FIG. 2. The solid line shows the SPEELS spectrum calculated in with the exchange response function described by Eq. 20 . The the energy denominator in Eq. 21 . At q 0, the spin-wave dashed line is the spectrum at q 0 provided by the first term only, feature in the spectral density should be a Dirac function, in Eq. 20 . We see the particle-hole interactions shift oscillator suitably weighted, at zero frequency. The width of the spin- strength from the Stoner excitation region near 2.5 eV, down to the wave feature in Fig. 2 has its origin in our use of a finite spin wave at zero frequency. value of . We have chosen 35 meV, in this particular case. state interactions are responsible for the spin-wave loss fea- As remarked earlier, we find that the SPEELS spectrum is ture in the SPEELS spectrum for example. For a fixed value quite sensitive to the radial wave function R2( ) in Eq. 2 . of the wave vector q , the spin-wave excitation appears as a Of primary interest to us is the strength of the spin-wave pole in the denominator 1 (0,0) 0 (q , )U 1 which lie very feature, relative to the Stoner spectrum. We wish here to close to the real axis in the complex plane. These poles lie asses the relative strength of the spin-wave loss peak, as a close to, but not on the real axis, because in itinerant electron guide to future experiments. magnets, the spin waves are Landau damped: they have a We illustrate this point in Fig. 3, where we present a finite lifetime, since they may decay to particle-hole pairs. In series of theoretical SPEELS spectra, for the choice the transverse dynamic susceptibility studied in Ref. 1, the same spin-wave poles enter. There we demonstrated that we obtain an excellent account of the spin-wave dynamics of R2 8 7 45 2e . 23 bulk Fe with our model, including an account of the short- wavelength features studied in Ref. 2. and several values of . This function may be used to repre- The dynamic transverse susceptibility (q , ) may be sent crudely the wave function of the 3d electron in atomic expressed in a form quite identical to Eq. 20 , except in Fe. The wave-vector transfer q has been fixed at q (M,N) 0 we find orbital form factors in place of the exchange (2 /a)(0.325x 0.25z ) and the angle and energy of the matrix elements described by F in the present discussion. incoming electron beam at i 55° and Ei 31.5 eV, com- As remarked above, a general theorem based on spin-rotation patible with the SPEELS data in Fe 100 reported in Ref. 9. invariance of the underlying Hamiltonian enters importantly For small values of , say 0.5 bohr 1, we see a in the analysis of (q , ). This theorem states that as Stoner spectrum rather similar to that calculated from the q 0, the only contribution to (q , ) is that from the first term of Eq. 20 , shown as a dashed line in the topmost spin-wave pole, which in this limit is at 0. In the Stoner panel of the figure. The spin-wave peak is almost completely region, the first term in the analog of Eq. 20 is cancelled absent. As is increased, with the consequence that R2( ) is precisely by the second term. This theorem is exact in the spatially more compact, we see the spin-wave peak develop limit q 0, and is obeyed by our approximate theory based quite nicely. By the time we reach rather large values of , on the RPA. The calculation reported in Ref. 1 show that the spin-wave peak is the dominant feature in the calculated even at large wave vectors q , the Stoner structure is very loss spectrum, and the Stoner band becomes very weak. We weak indeed. The response function in Eq. 20 , however, see this in the calculation for 4. In this bottom most has no constraints placed on it even at q 0 by such consid- panel, we compare the SPEELS spectrum calculated from erations. the exchange response function in Eq. 20 , with that pro- We illustrate this in Fig. 2, where for q 0 the solid line vided in a picture where the dynamic susceptibility shows the spectral density associated with the exchange re- (q , ) controls the spectrum. Even at this rather large sponse function in Eq. 20 . To evaluate the exchange matrix value of wave-vector transfer, we see virtually no hint of the elements, we have used radial orbitals discussed below. One Stoner bands, as discussed above. sees a prominent spin-wave peak, centered at zero frequency. The trends in Fig. 3 may be understood from the follow- At the same time, we see a broad feature centered around 2.5 ing argument. When is large, and R2( ) is spatially com- eV, which is the average exchange splitting present in ferro- pact, in the exchange matrix element, the only process which PRB 58 SPIN-FLIP EXCHANGE SCATTERING OF LOW-ENERGY . . . 14 413 TABLE I. Expansion coefficients for the Gaussian representa- tion of the 3d orbitals of paramagnetic iron, Eq. 24 . The coeffi- cients 1.75 j are given in terms of G j as j 20480.25G j /() 0.75). All quantities are in atomic units. j Gj Aj 1 127.0130 0.003946568 2 50.5179 0.01909717 3 20.092900 0.09034960 4 7.991720 0.2971454 5 3.178610 0.5665444 6 1.264260 0.6360391 7 0.502843 0.4632799 8 0.200000 0.3317246 with the expansion coefficients given in Table I. The q 0 calculation reported in Fig. 2 was carried out with the wave functions just described. It is of interest to compare the wave functions generated by the electronic structure analysis, with the empirical form in Eq. 23 . We do this in Fig. 4, where the radial wave function generated by the full calculation is displayed as the bold, solid line. Its peak lies closer to the nucleus than that with 4. However, the proper solid-state wave function has a long tail that extends well out into the far reaches of the unit cell. Thus, we have a substantial probability of exciting FIG. 3. SPEELS spectra calculated using the atomic radial or- a particle-hole pairs at appreciable separations. The prob- bitals given in Eq. 23 for different values of . The top panel also ability of encountering the electron in a spherical shell of shows the spectral density of the noninteracting transverse dynamic thickness dr is r2 R2(r) 2dr, so the factor of r2 enhances susceptibility 0 dotted line . The dotted line in the bottom panel the importance of relatively large separations. shows the spectral density of the transverse dynamic susceptibility In Fig. 5 a , we show a calculated SPEELS spectrum, (RPA) . through use of the wave functions supplied by the electronic structure calculations. The parameter in the denominator of contributes significantly is that in which the excitation pro- Eq. 21 has been chosen to be 35 meV. Both the spin-wave cess is highly localized; the electron and hole are initially loss peak, and the Stoner region appear prominently. localized at the same atomic site. In this limit, the electron Roughly 35% of the integrated strength of the full spectrum and hole interact very strongly, and the spin-wave feature resides in the spin-wave feature. The small structures just dominates the loss cross section. When is small, and R2( ) is spatially extended, there is a large probability that in the initial excitation process, the electron and hole are separated considerably. They then interact only weakly, and the calcu- lated spectrum resembles that calculated for noninteracting entities. Because of the sensitivity of the calculation to the choice of d orbitals, our final set of calculations employ forms of R2( ) generated by electronic structure calculations for bulk Fe. These orbitals were obtained by Pickett et al.10 in the process of reformulating the ``LDA U'' method for local orbital basis. Their formulation is based on the standard lin- ear combination of atomic orbitals expansion of the Bloch basis functions in which the numerical representation of the atomic orbitals was fit to a sum of Gaussians. The choice of the 3d orbitals used in this paper is identical to the Fe03d orbitals Pickett et al.10 used for Fe in paramagnetic FeO. These are expressed in terms of eight Gaussians FIG. 4. The radial wave functions used in the SPEELS calcula- tions. The bold line corresponds to the radial orbital generated by 8 electronic structure calculations and the thin lines show the atom- R iclike orbitals of iron Eq. 23 for two different values of param- 2 jAje Gj 2 24 j 1 eter . 14 414 M. PLIHAL AND D. L. MILLS PRB 58 ized incident beam, and which utilize spin analysis of the scattered electrons.3,4,9 In such studies, the spin-flip contribu- tion to the electron-loss spectrum can be isolated. It is our view that the calculations which lead to the prominent spin- wave loss peak in Fig. 5 are quantitatively reliable, and es- tablish that the spin-wave loss peak should be observable, in a suitable experiment. We should remark here that our re- sults possibly suppress the spectral weight of the spin-wave peak somewhat. This is because we use the unscreened Cou- lomb interaction for the beam electrons, whereas we assume that the interaction between substrate electrons is strongly screened and only on-site Coulomb matrix elements need be considered. The presence of off-site interactions between substrate electrons, as well as some screening for the beam electrons should enhance the correlation between the final- state electron-hole pair, and consequently enhance the spin- wave mode in the SPEELS spectrum. However, we feel FIG. 5. SPEELS spectra calculated through use of the wave these effects should prove to be only minor corrections to the functions generated by electronic structure calculations with a presented results. 35 meV and b 35 meV in the spin-wave region and While the calculations reported here are for an electron 200 meV in the Stoner region. which scatters from spin excitations in bulk Fe, we have shown earlier1 that standing spin waves appear as clear, well- below 1 eV loss energy are produced by low-lying Stoner defined excitations in ultrathin films similar to those em- excitations; we find these structures also in the spectral den- ployed in past SPEELS experiments. In the film, or at mag- sity of the first term of Eq. 20 , which describes the nonin- netic surfaces, the spin-wave loss feature should appear with teracting electron-hole pair. intensity, relative to that of the Stoner spectrum, comparable We see clear and prominent structure in the Stoner region, to that found in the present study. We shall direct attention to in the SPEELS spectrum displayed in Fig. 5 a , while the the SPEELS spectrum of ultrathin films in the future. experimental spectra are rather featureless. Of course, the One may inquire why the spin-wave loss peak has not experiments are carried out on surfaces, under conditions been reported in experiments carried out to date. Since the where the electron mean free path is in the range of two or Stoner feature is very broad, all experiments to date have three interatomic spacings. Our calculations apply to bulk employed rather low resolution, in the range of 300 meV. Fe. It is the case, however, that the experimental studies of Under these circumstances, the spin-wave loss feature is ob- the Stoner region are carried out with rather poor energy scured by the broad, quasielastic peak always present. In resolution, in the range 200­300 meV. In Fig. 5 b , we off-specular studies of surface phonons, energy resolution in present a theoretical spectrum where is retained to be 35 the range of 3 meV is employed in numerous experiments meV in the spin-wave loss regime, but is increased to 200 carried out at higher resolution, as the quasielastic peak is meV in the Stoner band. By this means, we may simulate the confined to the loss region well below the spin-wave fre- relatively poor resolution employed in these experiments. quency domain. Such an experiment will prove a challenge, We see the structure is now washed out, and the theoretical since as the energy resolution is improved, the signal that result bears a strong resemblance to the data. may be realized degrades substantially.14 It is difficult to envision a ``complete'' experiment with energy resolution in IV. FINAL REMARKS the 3 meV range used in the surface phonon studies, since as discussed earlier, the absolute cross section for exciting spin In this paper, we have presented realistic calculations of waves is substantially smaller than that for exciting surface the SPEELS spectrum of an electron propagating in the bulk phonons.5 Also, spin detectors are highly inefficient. We note of ferromagnetic Fe. These are based on a realistic electronic that it is not necessary to use spin analysis in the final state to structure, and an exchange matrix element generated by a observe the spin-wave loss feature, since it resides in a loss microscopic analysis. For reasons outlined above, use of a regime well above the phonon spectrum. proper exchange matrix element is essential, if quantitative It would be intriguing to perform an ``on/off'' experiment results are to be achieved. Past discussions have been based as follows. As we have seen, if the spin of the beam electron on either greatly oversimplified, schematic models of the is in the minority-spin direction, the spin-wave loss peak is a electronic structure,11 or simple density of states arguments13 prominent feature in the loss spectrum. This feature is com- which overlook the important role played by final-state inter- pletely absent, if the spin of the beam electron is in the actions between the electron and the hole. Such interactions majority-spin direction. This is a consequence of angular are essential, since they are responsible for the appearance of momentum conservation. Emission of a spin wave decreases the spin-wave loss peak in the SPEELS spectrum. the z component of the angular momentum of the substrate Our calculations have been undertaken for several rea- spins by . This angular momentum must be transferred to sons. One key issue is the following. As we have discussed, the beam electron. If the beam electron spin is parallel to the the Stoner region of the SPEELS spectrum has been explored minority-spin direction, this is accomplished by the spin flip in ``complete'' SPEELS studies which employ both a polar- depicted if Fig. 1 a . The beam electron cannot absorb the PRB 58 SPIN-FLIP EXCHANGE SCATTERING OF LOW-ENERGY . . . 14 415 angular momentum if its spin is parallel to the majority-spin probes of magnetics surfaces in films, for electrons photo- direction, as spin-wave emission is not allowed in this case. emitted from such materials, and for secondary electrons.15 By using a polarized beam, and comparing the SPEELS However, it is the case that quantitative calculations have spectrum for the case where the beam electron has spin first shown16 that spin-dependent elastic scatterings can provide parallel and then antiparallel to the majority-spin direction in adequate accounts of phenomena that some have argued17 the substrate, one should be able to isolate the spin-wave loss provide evidence for the presence of spin dependence of the feature, and discriminate against the non-spin-flip back- inelastic mean free path. Quantitative calculations of the spin ground, without detecting the spin of the scattered electron. dependence of the inelastic mean free path, as opposed to It is our view that such a study, carried out with most reso- simple phenomenological models, will provide a basis for lution 25­50 meV should be feasible with presently avail- assessing the relative importance of these two sources of spin able spectrometers. Such an experiment, if successful, would asymmetry in electron propagation. The analyses described allow the first access to the short-wavelength collective spin in Ref. 1, and the formalism developed in the present paper excitations of an important class of materials. provide the basis for such a study. The present analysis provides us with a theoretical base from which another basic question may be explored. This is ACKNOWLEDGMENTS a quantitative assessment of the energy dependence and magnitude of the spin asymmetry of the electron mean free The authors are indebted to Professor Warren E. Pickett, path in the ferromagnetic metals. The spin-flip scattering who kindly supplied us with the wave functions employed to processes examined here control the spin asymmetry in the generate the results displayed in Figs. 2 and 5. This research ``hot electron'' mean free path, in the ferromagnetic transi- was supported by the U.S. Department of Energy, through tion metals. It has been argued that this controls a number of Grant No. DE EC03-84EB4583 and by NSF Grant No. key spin-dependent phenomena; for electron beams used as DMR-97-08499. 1 H. Tang, M. Plihal, and D. L. Mills, J. Magn. Magn. Mater. 187, 8 J. F. Cooke, J. W. Lynn, and H. L. Davis, Phys. Rev. B 21, 4118 23 1998 . 1980 . 2 T. G. Perring, A. T. Boothroyd, D. McK. Paul, A. D. Taylor, R. 9 T. G. Walker et al., Phys. Rev. Lett. 69, 1121 1992 . Osborn, R. J. Newport, J. A. Blackman, and H. A. Mook, J. 10 W. E. Pickett, S. C. Erwin, and E. C. Ethridge, Phys. Rev. B 58, Appl. Phys. 69, 6219 1991 . 1201 1998 . 3 D. Venus and J. Kirschner, Phys. Rev. B 37, 2199 1988 . 11 G. Vignale and K. S. Singwi, Phys. Rev. B 32, 2824 1985 . 4 D. L. Abraham and H. Hopster, Phys. Rev. Lett. 62, 1157 12 See, for instance, S. Lehwald, F. Wolf, H. Ibach, Burl M. Hall, 1989 . and D. L. Mills, Surf. Sci. 192, 131 1987 . 5 M. Gokhale, A. Ormeci, and D. L. Mills, Phys. Rev. B 46, 8978 13 For example, H. Hopster, in Ultrathin Magnetic Structures I, ed- 1992 . ited by B. Heinrich and J. A. C. Bland Springer-Verlag, Heidel- 6 For a review of early pioneering studies, see A. J. Freeman, C. L. berg, 1994 , p. 123. Fu, S. Ohnishi, and M. Weinert, in Polarized Electrons in Sur- 14 H. Ibach and D. L. Mills, Electron Energy Loss Spectroscopy and face Physics, edited by R. Feder World Scientific, Singapore, Surface Vibrations Academic, San Francisco, 1982 , Chap. 2. 1985 , p. 3; a more recent review is given by Ruqian Wu and A. 15 H. C. Siegmann, in Selected Topics in Electron Physics, edited by J. Freeman unpublished . M. Campbell and H. Kleinpoppen Plenum, New York, 1996 . 7 For example, R. M. White, Quantum Theory of Magnetism 16 M. P. Gokhale and D. L. Mills, Phys. Rev. Lett. 66, 2251 1991 . McGraw-Hill, New York, 1970 , Chaps. 1­3. 17 D. P. Pappas et al. Phys. Rev. Lett. 66, 504 1991 .