PHYSICAL REVIEW B VOLUME 59, NUMBER 9 1 MARCH 1999-I Long-period oscillation in the magnetic coupling through chromium in a magnetic multilayer: Bulk issues Dale D. Koelling Materials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439-4845 Received 10 September 1998; revised manuscript received 4 November 1998 The long period oscillation of magnetic coupling through Cr as the spacer layer of especially Fe/Cr magnetic multilayers is examined. It is shown that a reasonable empirical adjustment of the d-band position does bring calipers on the N-centered ellipses into agreement with experiment. The lens surface becomes too small and more anisotropic providing further evidence that it is not involved. However, the neck of the jack surface does reach the proper size and so accounts for the photoemission observations. A mechanism of mode coupling to enhance the strength of the long period oscillation is examined and rejected. A thorough examination of the Cr/ V,Mn alloy spacer data strongly suggests instead a predilection for mode exclusion. S0163-1829 99 00309-4 I. INTRODUCTION being related to the same Fermi-surface nesting as the bulk spin wave-but such is not the case for the long-period. Among the artificial magnetic multilayer systems that ex- Hence, the desire to understand the cutoff of longer-ranged hibit giant magnetoresistance, the Fe/Cr system is very spe- periods gives added significance to the question why the Cr cial. One of the earliest discovered1 and most thoroughly long period is so robust. One would like to use the exception examined,2­5 this system comes closest to perfect matching to prove the rule, so the long-period oscillation of Fe/Cr of atomic spacing between the individual layers (aCr 5.44 becomes quite interesting on this basis alone. It is not com- au 2.88 Å vs aFe 5.41 au 2.86 Å . Deleterious lattice pletely unique: Co/Ag exhibits a 16 Å repeat distance,17 for strain effects are thereby minimized. Alas, there is, as usual, example. But it is a member of a very restricted class. a price to pay: interdiffusion easily occurs at the interface - And second, the origins of this long period are only be- requiring careful management.6 Fe/Cr is also very special ginning to become clear. There have been multiple hypoth- because it exhibits at least two periods repeat distances : eses offered for its occurrence.18­23 However, it is apparent One of the shortest, at two monolayers; and one of the long- that the Cr long period can indeed be associated with a mac- est, at about 18 Å. The short period arises from Fermi- roscopic Kohn anomaly from Cr bulk metal. Because of our surface nesting also associated with the spin-density wave in interest in the bulk issues, this effect is viewed as providing the bulk. The short period quickly disappears if the interface the coherent coupling through the Cr spacer layer which quality is not carefully maintained7­10 as a result of phase must then amplified by the reflection properties at the Fe cancellations.11­13 interfaces16,24,20­22 quantum-well considerations . For our purposes, this is actually a benefit, as it permits There are several very important experimental observa- direct examination of the long period-which is the focus tions to consider: 1 the Cr long period's existence in mul- here. There are two good reasons to carefully examine that tiple directions;25 and 2 the observation of possibly re- period. First is that it has very significant implications: lated quantum well states in the 100 direction by Nearly all observed periods fall in the range of 9­11 Å,14 so photoemission;26 and 3 persistent existence of the period long periods acquire general interest. The most credible ex- with vanadium and manganese alloying;27 and 4 its robust planation for this narrow range is that it is merely a reflection character. of a window of opportunity. The cutoff at short range is quite The Cr long period exists at the same repeat distance of easily understood. Periods smaller than the observed range 18 Å for both the 100 and the 211 directions.25,28 Al- correspond to only a repeat distance of only 2 or 3 monolay- though experimental problems exist for the 110 ers. Coupling is strongly suppressed by surface roughness or direction,29,30 a significantly long, albeit somewhat shorter, interdiffusion which generally occurs over such distances. period occurs in that direction as well.14,27 The long period Note that this is another way in which Fe/Cr is special: it also persists in polycrystalline Fe/Cr samples with an 18 Å does exhibit such a short repeat distance. A window-of- repeat distance which, if the polycrystals do not exhibit a few opportunity type explanation also requires that the range be favored orientations, would imply a very isotropic origin. cut off at the larger distances. The natural assertion is that The existence of this long period in multiple directions rules long periods are difficult to observe because the coupling out numerous hypotheses concerning its origin. The very effect decays with distance. Assuming a point Fermi-surface near constancy between 100 and 211 puts some pretty caliper or Kohn anomaly, the coupling should decay as the severe restrictions on a Kohn-anomaly interpretation. Given inverse square of the spacer thickness d.15,16 The coupling standard band-structure results, this similar occurrence in all will decay more slowly as d 1) if there is actual Fermi- three directions suggested the possibility that the spanning surface nesting-which is the case for the Cr short period, vectors occur on a small surface occurring along the 100 0163-1829/99/59 9 /6351 17 /$15.00 PRB 59 6351 ©1999 The American Physical Society 6352 DALE D. KOELLING PRB 59 directions known as the lens.19 This lens is a small Fermi- eliminate interest in the lens regime since similar spanning surface piece arising from the interaction of a large vectors occur in that region. -centered piece jack body with a modest sized ellipse Just what should be made of the quantum-well states ob- occurring along the 100 directions jack knob . These two served in that region by the photoemission experiments?26 surfaces interact to form the famous Fermi-surface jack to- Are they to be abandoned as superfluous artifacts?20,21 Per- gether with the small lens occurring within the juncture of haps that would be a bit too hasty. Since they have been seen the jack body and its knob. In standard local-density- to exist, is it possible they could act in concert to enhance the approximation LDA calculations, the lens has nearly the strength of the coupling? Normally, the oscillating behavior, correct size and shape to account for the observed behavior. in the asymptotic regime, is a sum of terms: However, it has a very low local joint density of states mass factor suggesting very weak intensity. States on this surface I I I d , 1a are of almost pure d-wave function character; a concern be- o d q cause s-p character only p character is available at the Cr Fermi surface is favored for observable coupling both be- Zm * cause these states are more robust against scattering losses19 I q d F d,T sin q d , 1b in the bulk and because they exhibit the best surface reflec- d2 tion with the iron.20­22 Nonetheless, photoemission does ob- serve quantum-well states in the vicinity of the lens.26 And, 1/2v m x y z , 1c obviously, d-state based coupling cannot be casually ruled * vzvz out in Cr since it occurs in the short period oscillation. The long period is unlikely to have the short period's benefit of d/L full surface nesting, certainly not for the lens as an origin, so F sinh , 1d d/L this cannot be a serious proposal for an alternative. It will be seen shortly that the lens is also just too small, although v other nearby possibilities exist. Still, one must proceed with L z T . 1e caution: one cannot ``cavalierly'' discard the lens in spite of 2 T T* all its deficiencies. In Co/Cr multilayers,5 the repeat distance is increased to 21 Å more nearly what is found for the lens q is the extremal caliper connecting two points on the in the adjusted calculations of Sec. II. Fermi surface: i.e., dq /dk 0 for k along the layers. Note A more consistent picture assumes that spanning vectors that any quantity labeled with an involves both Fermi- on the N-centered ellipses are the origins of the long surface points. v's are velocities with period.20­22 The N ellipse Fermi-surface pieces arise from v 1 v 1 v hybridized p-d wave function character and so satisfies cri- z z z 1 /2, 2 teria for robustness and good reflection properties. In a and x/y are the eigenvalues of the curvature matrix perpen- straightforward local-density approximation LDA calcula- dicular to q . The effective mass m * is a local joint density tion, the N-centered ellipses are unfortunately too large. of states about the caliper that is often referred to as the However, it is known that the relative positioning of the s-p geometrical factor. A slight improvement in its determina- and d states is slightly in error when applying the LDA for tion is outlined in the Appendix. A Dingle-Robinson tem- elements occurring in the center of the transition series, and perature has been incorporated into the coherence length L especially the 3d series. By examining the experimental to account for scattering in the bulk of the spacer. The same data31 for the size of these surfaces, it is found20 that the N scheme cannot incorporate the interface effects. Both a ellipses are somewhat smaller than calculated and can indeed Ruderman-Kittel-Kasuya-Yosida RKKY -type treatment15 yield calipers of the correct size in all three directions. This and a quantum-well-type treatment37­39 arrive at this form in is consistent with improvements found for Nb Ref. 19 us- the asymptotic regime although the quantum-well formula- ing an empirical adjustment for relative positions of the p tion will exhibit a more complex form for the decay than and d orbitals. The adjustment employed was developed ex- d 2. In the RKKY formulation, a 1/d arises from the inte- plicitly for Fermi-surface studies32,33 but then found to im- gration along q and this is the factor that differs. Two fac- prove other properties as well.34 One might well ask whether tors of d 1/2 arise from each of the perpendicular integrals as the same procedure will improve the results for Cr. So, an a component of how rapidly phase coherence is lost in that empirically adjusted calculation similar, but not identical, to direction. So it will not appear for each direction in which that employed for Nb is presented in Sec. II. Calculations of there is nesting. Hence, d 3/2 drop off for line nesting and this type are known in other contexts since 1 such adjusted 1/d drop off for full planar nesting. The factor Z involves the calculations are closely related to the constrained variational interfaces and may be written in terms of reflection coeffi- calculations that are performed as part of ``LDA U'' cients there. Actually, neglecting the content of Z is much analyses,35 and 2 in the form used here, it becomes a sim- like performing a generalized susceptibility calculation plified form of the state-dependent-potential model.36 Sec- which focuses on phase-space effects alone. tion II first examines how well the ellipses serve as possible When calculating the strength of the interactions, one in- origins of the observed repeat distances spanning vectors cludes the geometric multiplicity number of equivalent and then looks to see what occurs in the lens/jack-neck re- spanning vectors in determining the strength of the cou- gion. The ellipses prove out reasonably well, although some pling. That multiplicity factor actually represents the cou- improvement could be desired as always . But this does not pling together of degenerent interactions. It is easily accom- PRB 59 LONG-PERIOD OSCILLATION IN THE MAGNETIC . . . 6353 modated since the surface reflection terms, at least for the ideal surfaces, are also identical and this becomes a simple extension of the discussion for the rates of drop off. But what about degenerent modes even when the spin asymmetry of the reflection is not so favorable? Even for the perfect model, they are involved in the relevant component of the suscepti- bility. And if one adds possible coupling channels due to rough, imperfect surfaces, to mismatch strains and disloca- tions, and so on, then the possibility to couple in these addi- tion modes is significant. This can be expected to increase the strength of the coupling. The possibility that one could have mode coupling is especially interesting since one issue is the strength of the long period. V and Mn alloyed spacer systems are examined in Sec. III. Vanadium alloying experiments give the strongest evi- dence against the lens as the sole origin of the long period: As V is alloyed into Cr, the lens actually vanishes, yet the long period persists at least in the 110 direction. So al- though the lens-jack regime can be pretty well ruled out as the sole cause of the long period for Fe/Cr, it is further ex- amined here as a possible enhancing mechanism through mode coupling. FIG. 1. Intersection of the Fermi surface with the bounding The effect of the interfaces is considered with a somewhat planes of the Brillouin zone. The Fermi surface shown as the different view in the Sec. IV. Of course, part of the motiva- heavier line results when the optimal adjustment (V 0.05 Ry) tion is the strong effect that surface roughness is seen to have has been incorporated. See text. The unadjusted surface is shown as on the calculated amplitudes.27,22 But a more interesting the lighter line. All three cross sections of the small hole ellipses question arises from the observation that a CsCl structure located at the N points are seen. Note that these ellipses are not might be significant for analyzing this long period.23 The spherical. The octahedron centered at H reaches out and almost original observation pointed to the possible existence of an- touches the -centered jack along the -H line. The separation is tiferromagnetic couplings. However, for the 100 direction actually a result of the spin-orbit coupling. The lens is clearly seen considered, the boundary surfaces can induce a similar along that line lying just at the constriction of the jack. This is symmetry-breaking coupling. This intriguing possibility gen- mandated by the obvious anticrossing separation that converts what would otherwise be an octahedron about and an elliptical surface erally has not been incorporated in analyses performed and is at ``X,'' half way between and H, into the jack and the lens. examined in Sec. IV. Because the p character arises as an admixture from II. ADJUSTED MODEL CALCULATIONS higher lying bands remnant of the plane-wave band , it seems most natural to adjust the position of the p orbitals. In this section, multilayer systems involving Cr spacer That is the choice made previously for Nb.32,33,19 However, layers will be analyzed in terms of Kohn anomaly spanning because the Cr 3d states are almost completely contained vectors, but incorporating an empirical correction for slight within the nonoverlapping spherical muffin tin region that inaccuracies of the LDA. A similar adjustment has already surrounds each atomic site, the adjustment might be easier to been seen to improve the results for Nb.19 Because the cal- understand if it is instead applied to the d orbitals, which is culations presented are bulk calculations, effects of the sur- what is done here. Certainly, it is easier to relate it to the face properties20 and the quantum-well states34 that can re- constrained variation calculations used for LDA U. While sult must be considered separately. The focus in this section the comparison to LDA U has been omitted for brevity, it is is the examination of the changes induced on the available interesting to note two results that do appear when such a Kohn anomalies by the correction being applied. comparison is made. First, the d-count within muffin-tin The empirical adjustment applied is based on the obser- spheres that results at the optimally adjusted shift is 3 for V vation that the most sensitive deficiency of the LDA is rela- and 4 for Cr. In fact, the adjustment could have been done tive band placement. Bands derived from different atomic about as well by just imposing the d-count restriction in- angular momentum l character are shifted relative to one stead. Interesting as it is, that observation requires serious another due to the minor inadequacies of the LDA poten- thought. Second, from the process, it is possible to extract tial. Because of Fano antiresonance, the only wave function screened U values for the model where only the d orbitals are character found at the Cr Fermi surface derives from d orbit- correlated in the sense of having a nonzero U in a model als with small, but crucial, p orbital admixture, primarily on Hamiltonian. The values resulting are 3.5 eV for V and 5.1 the N-centered ellipses see Fig. 1 . Thus we are concerned eV for Cr. about the relative placement of the p and d orbitals. Limiting Choosing to incorporate the empirical correction by ad- our interest to the Fermi surface, energy shifts of these or- justing the placement of the d orbitals, one ends up perform- bitals are the only two possibilities for a simple adjustment. ing calculations on a simplified form of the state-dependent- Either, but not both, can be adjusted to see if we can accom- potential model specifically introduced for V and Cr,36 which plish our goal. also focused on the N ellipse size and shape. The state- 6354 DALE D. KOELLING PRB 59 dependent-potential model has been extensively explored for the bcc transition metals V,36,40­42 Cr,36,41,43 Mo,43 and Nb.40,44 Calculations were performed using the X model in a muffin-tin shape approximation with the addition of a po- tential of 0.05 Ry for the t2g their d ) orbitals and of 0.09 Ry for the eg their d ) states. The results were adjusted using the de Haas­van Alphen data for V and applied to both Cr and V. X-ray form factors were examined and found to be dramatically improved although large discrepancies re- mained for V. Those results definitely harbinger success. To shift the d-derived states relative to the remaining es- pecially p) states, we introduce an extra empirical ``semilocal'' potential which, in its simplest form, is Va r V lm lm , 3 m where lm (r R) Yml(r ) and the projection is defined using a three-dimensional 3D integral over a sphere of ra- dius R. Clearly, when R is set equal to the muffin-tin radius RMT in any band-structure method that uses augmenting or- bitals within that sphere, implementation becomes trivial: One merely carries through the calculation in a completely FIG. 2. Variation of the half principle axes of the ellipses due to standard fashion except that, for the given l, the empirical adjustment value V . The short principal axis (NH) l , the orbital energy to solve for the augmenting function, is replaced by achieves the experimental value for an adjustment of 0.03 Ry. The ( longest principal axis (NP) value crosses its experimental value l V ) in the construction of the Hamiltonian matrix. In the calculations reported here, the linear-augmented plane- between 0.04 and 0.05 Ry while the (N ) value crosses at 0.06 Ry. The smaller slope of the (NH) curve makes 0.05 Ry a reasonable wave LAPW method is used, so this potential is easily compromise. Lengths extracted from experiment are 0.173, 0.234, accommodated. Implementation only requires a minor exer- and 0.268 Å 1 or 0.158, 0.214, and 0.246pi/a Ref. 31 . cise in bookkeeping plus having to reconverge the self- consistency process-for each parameter choice. Calcula- tions are performed in the warped muffin-tin WMT dimension of 16a. This means that the data points must be approximation: the density and potential are spherically av- separated by a distance of /16a or the procedure will ex- eraged within the muffin-tin sphere but fully described out- perience singular matrices resulting from inconsistency with side. One must consider that this has consequences which the requirement that it be able to pass the curve through all appear below. data points no matter what the value. The choice of cubic Several other aspects of calculational technique are also truncation is motivated by the desire to have all three direc- worthy of note. All calculations, including the self- tions independently resolved to this precision. consistency iterations, incorporated spin-orbit coupling via a While the incorporation of V into the calculation is quite second variational treatment45 because it results in a separa- easy, the actual assignment of its value, and the interpreta- tion of the lens and the jack - an important issue in this tion of that value, are a bit more involved. The approach study. Calculations were performed with tighter tolerances taken adjusts V so that, as best possible, the resultant band appropriate for examination of the smaller features relevant structure produces the same ellipsoidal axes as found for the to the long period. Self-consistency was pushed to the limit experimentally-derived31 N-centered ellipses. That band that no change occur within the print format normally used: structure is then examined to see whether the observed Kohn 0.01 for the radial density within the spheres and 0.1 for anomalies repeat distances are indeed found and how they all Fourier components within the interstitial region multi- might arise. To determine V , a simple search was per- plied by the unit-cell volume . The irreducible wedge of the formed by trying values that varied by increments of 0.01 Ry Brillouin zone was sampled at 506 points-a cubic grid with in the range between zero and 0.10 Ry. The most appropriate a /10a linear spacing. Thus, since 18 Å represents value was found to be 0.05 Ry, as can be seen from Fig. 2. 3.2 ( /10a), all results are thus extremely closely tied to Obviously, the adjustment does not solve all problems and so the almost ab initio calculation. This same grid of points results in a spread of optimal V values. To put this in con- was used as the basis for a Fourier spline46­48 interpolation. text, the experimental semimajor axes were not plotted in That interpolation utilized a total of 910 star functions the Fig. 1 because they could not be clearly distinguished from plane waves corresponding to all lattice vectors interrelated the adjusted surface: being slightly outside larger along by the symmetry operations of the cubic group are incorpo- N-H and N-P and inside smaller along N- . rated into a single symmetrized star function . The interpo- Adjustment to the vanadium data is a more reliable pro- lation scheme is constrained to pass through all data points cedure in that it is real data: the Cr calipers are actually and the extra freedom arising from surplus of functions is extracted data from Cr measurements in the presence of the used to smooth the variation between points. The star func- spin-density wave requiring the use of an ellipsoidal shape tions represent all plane waves within a cube with a half-side approximation which we will see to be weak . Table I PRB 59 LONG-PERIOD OSCILLATION IN THE MAGNETIC . . . 6355 TABLE I. Adjustment of V for vanadium by comparison of will be reported as a wavelength in Ångstroms: Vector calculated extremal areas for the N-centered ellipses with the ex- lengths are given as 2 /q ( 2a/q with q expressed in perimental values of Ref. 49. Orbits are identified by their field /a units . It has the advantage of keeping focus on the direction. The 100 orbits are distinguished by their degeneracies: target repeat distance but requires care in discussion since the twofold degenerate occurs in the basal plane whereas the other the wavelength has a reciprocal relation to the quantities ac- occurs perpendicular to that plane in the zone face. The 110 orbits tually calculated. It is necessary to be explicit which is being are the fourfold degenerate, or canted, orbit that does not fall in the considered, since different choices invert the meaning of basal plane, the zone face ZF orbit with its field along the H-N smaller and larger. line, and the straight out SO field along the -N line. Units used First, how well does the adjustment actually work? As are (atomic units) 2 for consistency with Ref. 36 and 50. noted above, the agreement for the ellipse calipers is good Orbit No Adj. V enough to be difficult to exhibit graphically. That agreement 0.04 V 0.05 V 0.06 Expt. can be better probed by comparing the ``dHvA'' column of 100 2 0.202 0.177 0.168 0.160 0.161 Table II with the two calculated repeat distances i.e., 100 1 0.171 0.149 0.140 0.133 0.141 (0.00) and (0.05) . The adjustment has clearly improved 110 4 0.172 0.152 0.144 0.137 0.144 the size of the ellipses, and a finer adjustment of V could do 110 ZF 0.225 0.197 0.187 0.178 0.179 even better. But what about the anisotropy? In the basal 110 SO 0.183 0.163 0.155 0.148 0.149 plane, the experimental ratio is NH / N 1.36 while the unadjusted ratio is NH / N 1.62 and the adjustment has only brought the ratio down to NH / N 1.55. In reciprocal shows the results of current calculations. Clearly the best space, the ellipse is is too large perpendicular to the zone adjustment is between V 0.05 Ry and V 0.06 Ry. It is face (N- ) and too small along the basal plane in that face on this basis that the value of 0.05 will be used across the the (N-H). Also, the interpretation of the de Haas­van Alphen alloy series in the calculations for the next section. dffvA data suggests that the ellipse should be slightly elon- The current results can be seen to be consistent with those gated: a slightly longer, by about 25%, semimajor axis along of the state-dependent-potential model as follows. A degen- N-P than perpendicular to the zone face (N- ). The calcu- eracy weighted average of the two shifts used in the state- lation has these two axes essentially identical in the unad- dependent-potential is 0.066 Ry. A pretty good guess for the justed calculation and reaching a ratio increased by only 6% effective to use in an X treatment to match the current in the adjusted calculation. So, while the adjustment has im- exchange-correlation calculation is 0.72. Increasing proved the size, it has done but little for the anisotropy. To will pull the d states down relative to the sp states. Thus the improve the anisotropy, one would either have to incorporate average upward shift must be larger for the state-dependent- the relative t2g-eg splitting the ellipses are still roughly half potential calculation than for the current one. The 0.04 Ry d character so it would have a direct effect discussed above splitting found in the state-dependent-potential model is a bit or, perhaps, incorporate the remaining nonspherical terms. harder to reconcile - the value of zero has been used here, One would really prefer to examine the issue of further ad- which is clearly too small but consistent with what is being justment on top of a general potential calculation rather than sought. Further, this splitting of the d's should have no re- the WMT approximation used here. The approximate WMT duction from the size of the potential to the eigenvalue effect used here is a good deal simpler and adequate to our pur- as in the case of the d-sp shift. Consequently it is a really poses. large effect. How much of it is due to the shape approxima- One should note a fairly reasonable agreement for the tions used and how much due to orbital effects?51 For the observed repeat distances for all three directions. Interest- case of the state-dependent-potential calculation using a ingly, that match occurs for the ellipse caliper with the larg- muffin-tin potential, one expects a very strong component of est m* value: the largest phase-coherent joint density of the effect to be due to the shape approximation. Yet the states, i.e., phase, in another sense space. In Table II, the form-factor calculations42 would accommodate an even derivative of the repeat distance with shift potential is tabu- larger splitting. But, examination of Table I would suggest lated. The matching calipers also coincide with the larger another scenario. Note that the various orbit areas are derivative - or greater sensitivity, if preferred. To better matched either for V 0.05 or V 0.06 suggesting that, for appreciate these results for the ellipses, it is helpful to com- the current calculations, an applied splitting of 0.01 Ry pare them to those resulting from a simple ellipsoidal model: would be quite adequate. The warped muffin-tin approxima- tion employed here incorporates variations ``asphericity'' o F o k1/0.234 2 k2/0.158 2 kz/0.268 2 , in the interstitial region outside the muffin-tin spheres but 4 retains the spherical approximation inside those spheres that enclose the atoms. The interstitial region has been shown to where k1 is the separation from the N point along 110 and contribute the larger effect and the remaining difference is k2 is the separation along (11¯0). Choosing o as the calcu- near what would be expected from the nonspherical terms. It lated top of the band yields m* values which are slightly too is perhaps a bit large but the question of what level contri- small but cannot affect the values of found. The results are bution really arises from the orbital effect should not be ad- tabulated in the column labeled ``Ellips.'' in Table II. Note dressed without switching to full potential calculations. that while this simple surface matches the initial data by Table II lists the resulting Kohn-anomaly spanning vec- construction, it yields noticeably poorer results in the 100 tors that could be germane to the long period oscillation. For and 211 directions. It really does pay to consider the real simplicity, all information about reciprocal space distances band structure. On the other hand, it does also raise some 6356 DALE D. KOELLING PRB 59 TABLE II. Kohn-anomaly repeat distances. For the three separate directions, repeat distances in Å are compared for the spanning vectors found in the unadjusted calculation (V 0.00) and the adjusted calcula- tion (V 0.05 Ry). The local-joint-density-of-states strength parameter m* is given for a single spanning vector - no multiplicity factor has been incorporated. The calipers derived from interpretation of the deHaas­vanAlphen data are listed in the column labeled dHvA Only in the case of the ellipse (11¯0) do any of these correspond to an observed multilayer repeat distance. The results for a simple geometrical ellipsoid are tabulated in the column labeled Ellips. Also tabulated are derivatives of the repeat distance with respect to V and lattice constant. 100 (0.00) m*(0.00) (0.05) m*(0.05) dHvA Ellips. d /dV d /da Ellipse 110 13.9 0.82 16.5 0.67 15.6 8.1 17 Ellipse 011 9.9 0.51 11.9 0.45 10.7 10.7 5.5 13 Lens 100 30.1 0.67 37.8 0.69 13.8 53 Lens 010 20.2 0.44 24.5 0.42 7.9 10 Lens-jack 17.7 1.19 20.7 0.91 5.0 24 Jack-jack 16.6 0.63 19.1 0.58 4.9 28 211 (0.00) m*(0.00) (0.05) m*(0.05) dHvA Ellips. d /dV d /da Ellipse 110 11.0 0.54 13.4 0.49 12.8 6.7 19 Ellipse (11¯0) 15.1 0.92 17.9 0.76 16.7 8.6 19 Ellipse 011 10.3 0.51 12.3 0.45 11.3 6.0 17 Ellipse (01¯1) 12.2 0.58 14.6 0.55 13.7 7.3 17 Lens 100 27.5 0.70 34.0 0.68 10.3 40 Lens 010 22.7 0.54 27.6 0.51 8.4 28 Lens-jack 11.9 0.53 14.5 0.32 4.6 12 Lens-jack 11.7 0.33 13.4 0.47 2.4 Jack-jack 11.1 0.66 12.9 0.67 2.9 11 Jack-jack 10.7 0.65 12.5 0.71 2.4 12 110 (0.00) m*(0.00) (0.05) m*(0.05) dHvA Ellips. d /dV d /da Ellipse 110 10.2 0.37 12.6 0.39 13.4 13.3 7.1 17 Ellipse (11¯0) 16.6 1.06 19.4 0.83 18.2 18.3 8.9 18 Ellipse 011 11.9 0.61 14.4 0.57 12.3 7.4 18 Lens 100 26.0 0.68 32.0 0.63 10.5 25 Lens 001 20.6 0.50 24.9 0.46 7.2 17 Lens-jack 17.0 0.75 20.1 0.72 5.2 18 Jack-jack 16.2 0.57 18.8 0.53 4.6 22 question as to the uncertainty in the extraction of the calipers The data to be taken from these calculations is that the from the dHvA data, although not too severe a question since empirically corrected calculation does bring the ellipse cali- the same planes were used in that case. pers in line with the experimentally observed long repeat Next, focusing on the lens-jack region, the computational distances; that those calipers are the large m* ones; and that improvements already result in the lens being too small - jack neck-lens structure also produces similar calipers in at repeat distance too large - to be responsible for the long least two directions. period except for Co/Cr where the repeat distance is too large to arise from the ellipses . The d-p shift corrections then act to increase the difference so that the lens can be III. ALLOYS definitely ruled out as a source for the long period in the Fe/Cr multilayers. The change in this region of k space is Alloying of the spacer material proves a very convenient more due to the much finer sampling of the Brillouin zone probe, in no small part because the Cr-V and Cr-Mn alloys and the closer Fourier fit rather than to the shifts. Even are amenable to very simple theory treatments.27 Adequacy though extra points were incorporated near the lens in the of alloy representation is revisited through careful compari- previous calculations,19 this proved inadequate. Making the son of a supercell calculation and the virtual crystal approxi- more careful examination for the smaller spanning vectors, mation VCA for a 50-50 Cr-V alloy in the bulk. Thereafter, calipers near the observed value do occur in the 100 and the discussion continues using the much simpler VCA. The 110 directions by bridging from the lens to the jack and van Schilfgaarde­Herman vSH model, originally proposed across the narrow neck of the jack. The situation as regards in Ref. 52 and used to examine the magnetic coupling in the 211 direction is less clear although, looking at the ge- these alloy spacer materials, is most useful to the delibera- ometry, one can easily suspect that no caliper will be found tions here: the density within the layer interiors is maintained in the right range. at the bulk values for the respective pure materials; only at PRB 59 LONG-PERIOD OSCILLATION IN THE MAGNETIC . . . 6357 the interface is the electronic structure allowed to relax to- calipering. An alternate approach can be had at the price of wards self-consistency. The vSH model was originally moti- using a Slater-Koster tight-binding representation for the vated by the efficiencies it offered. By restricting the free- bands. Within the limitations of that model, one can use the dom of the calculation easily accomplished using the analytic Green's function method allowing one to focus upon atomic-sphere approximation that is normally used with the the appropriate calipers with a knowledge of the associated linear muffin-tin orbital LMTO technique inside the lay- surface matrix elements within a hierarchy of limiting ap- ers, one gains efficiency by avoiding unnecessary explora- proximations. This has been carried out for pure Cr.21,22 A tion of a lot of phase space the ``charge sloshing'' . The major advantage of so doing is that one gets around the precision lost is partially offset by exploiting a Harris- difficult problem of having to obtain the repeat distances by Foulkes functional approach,53,54 to improve the precision of a fitting procedure. That is a tremendous advantage since the the total energy - which is what is used in the analysis. The fitting process must also account for differing decay rates vSH model does, however, provide insufficient screening be- and suffers from a limited range of data due to restrictions as tween the layers so the charge transfer effects are overem- to the largest system that can be calculated. phasized. Examination of Table I of Ref. 52 indicates that An overall observation should be made here. The vSH this effect will not be too large for the short repeat distances calculations found only two repeat distances to be observed ( 0.1 Å) but can be much more noticeable for the long except very near a transition. That only very few periods are repeat distances ( 2 Å). For our purposes, the vSH com- observed even in this computational model system highlights putational model provides us with a simple picture: The the interesting question of why the many other Kohn singu- treatment of the layer interiors represents a linear-response larity pairs are not observed. Scattering and surface rough- approach, without the non-linear terms which are sometimes ness effects present in real world samples that might further postulated to be significant. Successes of this model can give diminish these oscillations19 are simply not present in this credence to the view that at least some multilayer properties computational model. This situation is quite consistent with do relate to bulk properties - if one can disentangle the that for Kohn anomalies in phonon spectra, where only a them. That disentangling is no small task especially when small fraction of Kohn singularities are actually observed. In trying to use a Kohn-anomaly-type analysis as is done here. the case of the multilayer systems, the surface introduces the The author's own initial view of the process was that the matrix-element-like effects that can be characterized as re- exercise would merely present too many possibilities for any flection coefficients. These are indeed found to be strong transition element. The wisdom of that early view haunts the selection factors.56 It is easy to assume that that is the answer discussion to follow. The model does allow Fermi-level but is it perhaps only one factor? The differentiation between shifts, which would not normally be considered in a bulk Cr and isoelectronic Mo Ref. 57 whose repeat distance cali- Fermi-surface analysis. The interfaces used in the calcula- perings involve the same Fermi-surface pieces but in rather tions are greatly oversimplified compared to the real world different combinations should provide a good probe of this - a point that will reoccur often here. Nonetheless, the in- issue. Further, there is the appearance that one has a selection terface relaxation built into the model at least partially incor- factor, i.e., on/off, rather than just varying relative strengths. porates the scattering properties of the interfaces. It is rea- This could be due to only probing the asymptotic regime but sonable to view the interface scattering calculations,20 which the switch over to be seen below casts doubt on such an suggest that s-p states are the most strongly reflected states, explanation. as a model system dissecting the interface component of the Returning to the alloys, we briefly reexamine the utility of vSH model and the Kohn anomaly analysis here as a dissec- the VCA - which is not a particularly favored approxima- tion of the bulk component. Alternately, the Fermi-surface tion for the description of alloy properties. That V-Cr and analysis is precisely a parallel of a generalized susceptibility Cr-Mn are adjacent elements in the periodic table and are calculation with all the same issues while the focus on the also located in the central portion of the 3d transition-metal surfaces is the parallel of a matrix element analysis. series greatly favors the applicability of such a simplified The approach taken here is to examine the vSH model treatment. Supercell results for the ordered compound, calculations specifically, Fig. 2 of Ref. 27 as an ``experi- coherent-potential approximation results, and VCA results ment'': which must be disassembled for insight. Here, the were all compared in the VSH model calculations27 and objective is a closer look at the bulk effects. By examining found to yield very similar results. We can test this in a way the computational model, we can observe results without the specifically targeted to our preference for examining the sys- extra complications of strains, defects, rough surfaces, etc. tem using the bcc Brillouin zone: the results of a CsCl struc- that do occur in the real experiment. This is important be- tured ordered-alloy supercell calculation for CrV are com- cause it was found that surface roughness effects alone27 pared to the virtual crystal approximation for the 50-50 alloy were responsible for an order of magnitude difference in i.e., setting Z 23.5) by ``unfolding'' them onto the bcc coupling amplitude relative to the model assuming perfect Brillouin zone. Note: one wants to ``unfold'' the bands - surfaces. Also the surface treatment selects a different origin precisely the inverse operation to the one normally applied to for the long repeat distance in the 100 direction55 - clearly estimate an ordered compound from the bands structure of a a point to be discussed below. The computational experiment single constituent. This inverse problem is the harder one is all the more interesting since the vSH model calculations requiring some knowledge of the wave functions. were carried out, as needs be, by calculating total energies Slater-Koster58 provide a very clear description of the re- and then fitting them to an oscillatory behavior-no Kohn- lation of the simple cubic CsCl Brilloin zone to that of the anomaly-style analysis was made other than to compare to larger - with twice the volume - bcc zone. The CsCl zone the Fermi-surface caliper results55 for V and Cr to assay the is carved from the bcc zone by inserting a new set of 100 6358 DALE D. KOELLING PRB 59 FIG. 3. The Fermi surfaces that result from the virtual crystal approximation for the range of alloy compositions from V to 30­70 Mn-Cr. In g for pure V, the heavier lines represent the results at the lattice constant for V and the lighter lines for the Cr lattice constant. These differences represent the changes other than the simple geometric lattice dilation effect. In h , the appropriate approximate Fermi energies are shown on the Cr band structure. planes passing through the N and P points to cut out a cube. resentation of the LAPW basis set. Given a wave-function Incidentally, these planes will pass through the 100 axes expansion calculated for the CsCl structure, one divides the somewhat outside the lenses near the end of the ball on the basis set of the expansion into contributions from each of the jack. As Figs. 1 and 3 are set up, this is a true folding right two k points from a bcc lattice. This is easily done precisely down the middle line connecting N and P. The remaining because we know the associated k for each basis function. It volumes cut off in each direction are square-based pyramids could also be done for other bases sets using a projection. with the H points for their apex. This remaining volume is One thus breaks the CsCl wave function into the sum of two mapped onto the central cube by translating each of these basis functions coming from the two k points and formulates pyramids back using a primitive translation of the CsCl re- the 2 2 secular equation. What one finds is that, as long as ciprocal lattice. The process maps precisely two bcc Bril- one stays away from the CsCl zone faces, the wave function louin zone points onto a single point of the CsCl zone which is 95% from one k point making the unfolding an easy pro- is the folding operation. The object here is precisely the in- cess. Further, the off-diagonal terms give the scattering be- verse: to take the solutions at a single CsCl zone point and tween the two k's which directly reveals very weak scatter- distribute them appropriately back onto the two correspond- ing. It is perhaps this very weak scattering that precludes a ing bcc points. To do this, one exploits the plane-wave rep- predicted38 exponential decay in coupling with impurity con- PRB 59 LONG-PERIOD OSCILLATION IN THE MAGNETIC . . . 6359 TABLE III. Pseudoalloy Kohn-anomaly repeat distances for the 50-50 Cr-V Alloy. For the three separate directions, repeat distances in Å are tabulated for the spanning vectors on the ellipses of an ``unfolded'' CsCl structure CrV calculation. The same information is then also tabulated for a virtual crystal approxima- tion VCA with Z 23.5. The Expt. and vSH columns are the experimentally observed period and multilayer calculated periods from Fig. 2 of Ref. 27. The and ``X'' pocket calipers have been included as a simple indication of their size. 100 (CsCl) m* (VCA) m* Expt. vSH Ellipse 110 8.0 1.69 8.1 1.58 13.7 Ellipse 011 5.9 0.95 6.0 0.78 211 (CsCl) m* (VCA) m* Expt. vSH Ellipse 110 6.9 1.08 7.0 1.17 Ellipse (11¯0) 8.7 2.26 8.6 1.34 Ellipse 011 6.6 0.84 6.8 1.41 Ellipse (01¯1) 7.1 1.05 7.1 0.95 110 (CsCl) m* (VCA) m* Expt. vSH Ellipse 110 6.5 0.77 6.5 0.77 Ellipse (11¯0) 9.4 1.82 9.4 1.32 9.6 10.2 Ellipse 011 7.1 1.10 7.2 1.19 band 3 26.3 4.00 n.p. band 4 32.1 0.48 n.p. ``X'' pocket 011 band 2 56. 0.60 71 0.60 centration although then one must ask what happens in the ture as we are here. The Vegard law dependence observed Ni-Cu alloys. for the lattice constants i.e., linear in concentration has In Table III, the results for the VCA and supercell calcu- been included in these calculations for Cr-V. The lattice con- lations are presented together with the observed repeat dis- stant has been approximated as constant Invar-like for the tances and the calculated vSH model results. The shift V of Cr-Mn alloys. The inclusion of the lattice change in the self- the previous section is not incorporated into those calcula- consistency process incorporates relaxation effects beyond tions of this section which are used to compare with other the simple geometric lattice dilation incorporated in the con- calculations. None of the other calculations incorporate the version from monolayers to Ångstroms. These can be seen adjustment. Agreement between the ordered-alloy calcula- from a comparison of the V Fermi surface calculated at both tion and the virtual-crystal calculation is quite remarkable. the V and Cr lattice constants included in Fig. 3. Being plot- The supercell and the VCA Fermi surfaces are very similar, ted for a Brillouin zone scaled to the same size, the differ- although the ordered compound calculation does contain two ences observed are only those other than the straightforward small -centered pieces not present in the VCA. The com- geometric dilation. pensating volume comes from within the surface around Across the alloy series, there are several topological Lif- H-which is predominantly of d character-which is the ex- shitz transitions. See Fig. 3. The most pronounced transition planation for the similarity of the ellipse sizes. The quite occurs for V admixtures between 10 and 20%. It correlates small difference between the two results that occurs does so well with the break in Fig. 2 of Ref. 27. It is at this concen- because the d states at are being pulled through the Fermi tration that the anticrossing structure along in the bands is energy - a Lifshitz transition. Of course, very near the CsCl depopulated. That anticrossing structure in the bands is what zone face, the splitting of the ordered supercell has been connects the balls onto the vertices of the -centered octa- artificially eliminated consistent with a disordered material. hedron to produce the ``jack'' surface and simultaneously Comparisons near that boundary would be unreasonable and produces the lens surface within that connection. Removal of we are fortunate they are not necessary. the connection between these two large pieces of surface The VCA incorporates relative shifts of the average band results in a simpler structure consisting of a -centered oc- position in addition to the simple band filling of a rigid-band tahedron and a centered ellipsoid set much as was origi- treatment. The rigid-band picture is already an excellent first nally envisioned for Cr as its Fermi surface was being sorted picture for these materials and the band shifts further im- out . A second transition occurs very near the 50% V con- prove the reality of the description. Progressing from V to centration where the centered octahedron is lost. Nearer Cr, the d states as measured by the simple plus H index pure V, the H-centered octahedra bridge across to form the for their centroid drop by about 0.5 eV. And they drop yet so-called jungle gym, while a -centered structure is uncov- another 0.07 eV by addition of 30% Mn. The VCA does not, ered. In the opposite direction, where Mn is admixed instead of course, incorporate any broadening effects-which can be of V, the famous transition to commensurate nesting occurs a very significant factor when considering singularity struc- with dramatic effect. With further Mn admixture, the balls on 6360 DALE D. KOELLING PRB 59 the jack surface bridge across the N-H lines just beyond the range of Fig. 3 between 20 and 30% Mn. In the presence of such drastic changes, it is somewhat startling that the periods found in Ref. 27 vary rather smoothly - though the same cannot be said for the amplitudes. Although the main focus is the long periods, insight can be gained by first digressing to the short periods. In the 110 direction, the short period vascilates only slightly about 2.5 monolayers roughly 5 Å) rising slightly towards the V end of the series. It should be remembered that this short period in the 110 direction represents a computer experiment since this period has not been observed. In light of the strong sur- face microstructure occurring for this configuration,29,30 the prospects of actual experimental observation is poor. Al- though experiments with Cr-Ag spacer bilayers59 between Co have been performed in order to examine enhanced electron-electron effects, they can also be examined to at- tempt to provide some evidence here. If one assumes that the FIG. 4. Comparison of the short period found for the van coupling between the Cr and the Ag layer is only through Schilfgaarde­Herman vSH model in the 110 direction with se- magnitude - the surface having wiped out phase or deriva- lected Fermi-surface calipers. The results for the vSH model are tive information and that the period in the Ag is extremely indicated by the heavier curve. Data referred to in the key as ``H long (16 Å) as observed for Ag alone,17 then the Ag layer octahedron edges'' represents a caliper between the nearest edges has almost no effect on the repeat distance and the observed of two adjacent octahedra. That connection remains the same as the 5 Å repeat distance can be attributed to the Cr layer alone. octahedra actually bridge through the point to form the jungle Then, the 5 Å repeat distance, which is for the total bilayer gym so the distinction is not made. That denoted ``H octahedron thickness, must be scaled to the fractional Cr sublayer thick- edge 2 '' spans a singe octahedron edge to edge along a canted diagonal near the rounding for the vertices. This ceases to be pos- ness 1/2.35 to get a repeat distance of 2.1 Å. As this is sible as the bridging occurs. Data denoted ``H octahedron edge c '' quite close to the monolayer separation of 2.04 Å in the spans a single octahedron from edge center to edge center. The 110 direction, the natural assumption is that one is seeing faces also persist when the octahedra bridge to form the jungle gym. the basic antiferromagnetism consistent with the low tem- Data denoted ``Oct.- Ell.'' couples the -centered surface to the peratures employed. Of particular interest, however, is the N-centered ellipses. This is again a jargon since only from 0.5 to very strong harmonic modulation also observed at a roughly 0.2 is the -centered surface truly an octahedron. It then couples two monolayer repeat distance. One must be a little careful with the so-called X-centered ellipses to form the jack and lens about its interpretation. A long Ag repeat distance17 can offer surfaces. However, the octahedron like character is retained except at least a partial explanation of these extreme amplitude for the 100 directions so the term octahedron is retained for the variations within the assumed model. That size repeat dis- body of the jack. tance would modulate the amplitude appearing on the Ag-Cr interface at nearly the correct rate. It would not, however, Strictly as an easy index to facilitate the discussion in what produce an effect nearly as strong as the observed variations. follows, the alloy composition will henceforth be represented Subject to this concern, it is still quite reasonable to assume by its VCA effective charge. Define vanadium as z 1.0 a superposed variation due to the couplings we are examin- and Cr as z 0.0 so the full range runs from 1.0 to 0.3. ing. This would suggest 4.2 Å repeat distances with a large Across the range from 1.0 to 0.5, the vSH result corre- error bar due to few oscillations and the superposed antifer- sponds well with calipers arising from H-centered octahe- romagnetic component. It is not a lot, but it does suggest one dron edge to neighboring H-centered octahedron edge. One might be able to discover something about the 110 short is using sloppy language here since the octahedra actually period by using Ag, or other, coupling layers at the inter- bridge across to form an open jungle gym surface in the faces. V-rich alloys. However, the main body remains the same and When attempting to compare the Kohn-anomaly analysis the language carries the flavor so it will be used as a jargon. calculations to the results obtained from the vSH model, one Actually, these points are roughly line - rather than point is again impressed by the fact that this is indeed not the way - singularities; which would suggest a slower d 3/2 decay one wants to do things when dealing with a complex Fermi as well as the greater phase-space involvement. This appar- surface: it entails long hours of searching for the reasonable ently does not happen in the vSH calculations as their fits are result among numerous possibilities, none of which match consistent with a d 2 drop off. In the Mn alloy range from exactly. And then one spends lots of time trying to reassure Cr 0.0 to 0.30 Mn, there is a good correspondence of the oneself that whatever is seen actually is significant. Nonethe- vSH model results with calipers spanning a single less, a pattern does appear for the 110 short period as H-centered octahedron from edge to edge. Again, one is very shown in Fig. 4: a broad band of possible calipers can be close to a line nesting broken up by a slight variation-ether associated with the vSH result the heavier line as read from real or artifact-as can be seen in Figs. 1 and 3 from the Ref. 27 across most of the alloy range. Agreement is found nearly straight line on the H octahedron in the basal plane. In for the pure element cases V and Cr , as observed before.27 Fig. 4, two point singularities are plotted, one from center to PRB 59 LONG-PERIOD OSCILLATION IN THE MAGNETIC . . . 6361 center along the H-N line and one from the point where the edge starts to turn over forming the rounded vertex. Again, if a full line nesting were to occur, one would expect a d 3/2 decay. The correspondence is excellent except for 10% Mn where the vSH data jumps to the data denoted ``Oct.- Ell.'' That caliper stretches from the -centered piece to the N-centered ellipse. At greater than 10% V, this -centered piece is roughly an octahedron. However, in Cr and the Mn alloys, the so-called X-centered ellipse has been joined onto the vertices of that octahedron to form the jack surface. Since only in the 100 directions has there been a change, ``Octa- hedron'' is retained as a term to indicate the body of the jack. The error from inaccuracies reading the vSH model data off plots in Ref. 27 is probably at least 1/4 Å, but, if the fitting process faithfully represents a property of the vSH model - which is assumed for this discussion, then something more is happening than just the variation of the H octahedra and this shift should be taken as real. One possible explanation is the appearance of magnetic effects since z 0.1 is roughly where the vSH calculations find a commensurate Q vector in the 100 direction. Were this the case, one would have to un- derstand why the effect does not continue on to higher Mn concentrations as the magnetic character continues. A more viable explanation can be constructed from available phase- space considerations. The Oct.-Ell. caliper exhibits a very FIG. 5. The short period in the 100 direction. The van sharp peak precisely at 10% Mn which is so strong as to Schilfgaarde­Herman data heaviest curve marked is obtained suggest a small area of nesting. It should not go unnoticed by reading their Fig. 2. The break between z 0.25 and z that the ellipse involves sp character as well. Additionally, 0.1 is true to their results but the small differences seen at z two much smaller effective mass calipers, one from lens to 0.75 and z 0.20 probably are not. The line marked with filled lens and the other from jack-knob to jack-knob, nicely squares represents the mean nesting vector that should produce the bracket the result. Although no analysis has been made of the spin-density wave in Cr. This caliper vanishes at or somewhat be- possibility of ``mode coupling'' through the surface layer, it fore z 0.5 due to the depopulation of the -centered surface that is at one end of this calipering. The line marked with the open is appealing to consider it a possibility in this case. A posi- squares follows a set of calipers where both ends are on the piece of tive reinforcement between the mean period of the two cali- surface centered at H, whether it be the jungle gym, which bridges pers on each side of the observed distance and the third cali- across through , or the octahedron that results when the bridging per precisely at the distance could produce an enhancement. is closed off. Reminding ourselves that we are in some danger of creating science fiction rather than analysis, we nonetheless note that this explanation is consistent with the wild amplitude varia- caliper distances are all significantly shorter, actually appear- tions seen in the vSH results which peak significantly at 10% ing nearer the harmonic of the near-neighbor antiferromag- and 30% Mn. Even with some improvement in the calcula- netic repeat distance. One might argue that the variation is tion of masses see the Appendix , correlation between cou- hard to detect on the original plots from which the data was pling strength and effective mass is weak - because other taken and that the amplitude is strongly suppressed there. factors enter. Still, it is significant that the average masses for However, although the error bars are admittedly large, the all three curves peak somewhere near z 0.1 and perhaps are optical scanning does give some resolution and, most signifi- even indicating nesting line or area tendency. This would cant, it is clear that the two data points rise above a smooth explain the maximum at 10%, but what of the second rise at curve, while the Fermi surface calipers would fall below. 30% Mn? It is significant there that the two octahedron span- This small feature, much exaggerated in Fig. 4, does have a ning edge calipers the c and the 2 calipers collapse to reasonable probability of being reality. the same distance probably indicating a long line nesting. The short repeat distance for the 100 direction is shown Certainly the data from a paramagnetic calculation has done in Fig. 5. This case appears somewhat simpler. The vSH well for these magnetic systems. results very explicitly exhibit the break between z 0.25 The range between 30 and 10% V raises serious question. and z 0.1 that has been mentioned above. The alloy re- Of course, this is the range where the jack is being broken gime from 10 V to 30% is quite simple. Excellent agreement up, where amplitude for the short period in the vSH model is is seen to exist in this regime between the vSH curve and the nearly vanishing, and where the fit is suggesting possibility curve of the mean nesting vector spanning from the of more than one short repeat distance. However, in this H-centered octahedron to the -centered part body of the range there also are no calipering pairs for repeat distances jack. As above, this piece that forms the main body of the anywhere near the vSH results: the vSH model results are jack is identified using the name -centered octahedron. This larger while the calipers are at a minimum. This discussion is reasonable since the surface remaining after the Fermi en- is for real space, not reciprocal space. The Fermi surface ergy has sunk below the bridging energy is precisely a 6362 DALE D. KOELLING PRB 59 -centered octahedron. It is, of course, this nesting vector that is associated with the spin-density wave. In both the calculations for the vSH model and those here, that nesting vector becomes coherent at 10% Mn. It occurs sooner at about 5% experimentally, but that is in part due to pulling or lock-in effects.60­62 Nothing is to be made of the small dif- ference at z 0.2 because it appears to be an interpolation error not reflected in the original figure. The nesting vector curve does continue on below z 0.1 down to around z 0.5 where the -centered piece becomes fully depopu- lated. It is still very strong at z 0.25 where it would ap- pear to again agree well with the vSH results. At z 0.3, it begins to broaden - wider range of spanning vectors about the mean - and to weaken: i.e., fewer calipers found each with significantly smaller masses. In the regime from z 1.0 V to z 0.5, the vSH results agree very well with a different set of calipers that involve the H-centered piece as jungle gym or octahedron at both ends. These H-H calipers exhibit a strong local joint density of states, being multiply occurring with high mass values actually hinting of a pos- sible nesting, near the V-rich side, but become of negligible strength beyond z 0.4 where the calipering jumps to a single point of significantly shorter repeat distance and low FIG. 6. Comparison of the vSH model results with ellipse cali- mass. Obviously, such an analysis cannot be expected to tell pers for the long period in the 110 direction. Labels used in the us much about the switch over occurring between z 0.5 key indicate the following: vSH long and exp. indicate the results of and z 0.25. It does, however, suggest a closer look at this the vSH model and experimental results as read from Ref. 27; Straight Out indicate the ellipse caliper along the -N line; Zone region might prove interesting. Further, the fact that the vSH Surface is the ellipse caliper along the H-N line; Canted is the one model gives a point on the -H nesting curve beyond the caliper that extends out of the basal plane. Calipers were calculated gap indicates an important puzzle: There is nothing in the without the adjustment of the previous section for comparison with Kohn-anomaly analysis that would suggest the gap should the vSH model calculations. The arrows at z 0.0 illustrate the occur. That it might be an earlier switch over is denied by the effects of the adjustment, which is incorporated in Fig. 7 for com- existence of that point beyond the gap. It is perhaps possible parison with actual experiment. that, as the Fermi-surface breakup occurs, the transitioning k-space regime generated around the critical point produces thing new and quite interesting. In the alloy range from about greatly increased scattering from the states on the faces of 50-50 V/Cr to pure Cr, the experimental data coincides well the octahedron. with the zone surface caliper along the H-N line through the Ending the digression to the short periods, focus returns to ellipse. But, beyond 50% V, it switches to agree well with the long periods. In the 110 direction, initial presumption is the ``canted'' caliper one in the 101 direction for an el- that the long periods arise from the N-centered ellipses. A lipse along the 110 direction . This result causes regret that comparison of the results for the vSH model and the three there are no data between 50 and 75% V. The same switch Kohn-anomaly calipers is shown in Fig. 6. Although the does not occur in the vSH model - at least not with the real experimental data is presented there as well, its con- simplest version of the interface. The calculated caliper ef- sideration is delayed until considering the results incorporat- fective mass parameters do cross over near z 0.7 with the ing the empirical shifts while first continuing the computa- canted vector caliper having the largest mass on the V-rich tional experiment without them. For z 0.5 the vSH side where it is apparently being seen. Note that apparently results and the zone surface caliper occurring along the H-N only one or the other caliper is being observed, not a com- line across the ellipse agree reasonably well. The difference bination with varying amplitudes. And that this is occurring observed is somewhat larger than the difference obtained where the strengths of the two channels must be very nearly when comparing bulk APW and bulk LMTO calculations equal - the most favorable case to see both contributions. which suggests another small effect is present. Greater de- The most complicated situation occurs for the 100 long viation occurs above z 0.1, which is only mentioned but not period results shown in Fig. 8. A conclusion quickly drawn discussed here. The zone face ellipse caliper appears to cor- from this figure is that the vSH results have very little to do respond to the vSH model all across the alloy series. Of the with calipers on the ellipses. The sole weakly credible ex- other two N-ellipse calipers, the ``straight-out'' vector along ception is at z 0.25 where the ellipse caliper in the basal the -N line is too small in real space, meaning too large in plane is the closest caliper to the vSH curve. The closest reciprocal space, to be a possible factor in the response. The caliper on the upper side is a coupling between the cen- same is mostly true for the ``canted'' vector the one that tered octahedron and an ellipse. It is marked by a small as- does not occur in the basal plane except for alloy composi- terisk. Neither can be considered reasonably close. In the tions near the V-rich end. V-rich region, the vSH results are closely represented by a Comparison of the empirically corrected calculations to caliper pair coupling an ellipse to the body of the H-centered the real experimental data, shown in Fig. 7, reveals some- surface. It is the interconnected jungle gym at V converting PRB 59 LONG-PERIOD OSCILLATION IN THE MAGNETIC . . . 6363 FIG. 7. Comparison of experimental results with the ellipse cali- FIG. 8. Comparison of the vSH model results with selected pers for the long period in the 110 direction. Key labels are the calipers for the long period in the 100 direction. Key labels indi- same as in Fig. 6. Note that the experimental data apparently cate the following: vSH Calc. is the results of the vSH model for the switches from matching with the zone face caliper to matching the long period; Base Plane is the ellipse caliper occurring in the basal canted vector caliper near the V-rich end. plane two are equal by symmetry ; Vertical is the ellipse caliper perpendicular to the basal plane; H- ellipse is a caliper between the surface centered at the H point and an ellipse. The sole experi- to a closed piece roughly at the 50% alloy composition. Re- mental data point is indicated by a large X for Cr. The arrows give fer to Fig. 3. On the other side of the break, the dramatic the same information as in Fig. 6. singular curve seen for the vSH model in the Cr-Mn alloys arises from the same nesting as the short period. The reduced component of the problem to what is done by caliper calcu- symmetry of the slab, or aliasing due to discrete sampling, lations for the bulk response. Thicker Fe layers are needed produces an effective repeat vector in reciprocal space to achieve strong reflection for the calipers on the N ellipses, which is the difference from 2 /a of the short period vector. a not surprising result because of the p character admixture The long period vector goes through zero as the material in the wave functions for the states on that surface. Tight- becomes commensurate short vector goes through 2 /a) so binding calculations specifically studying effects due to the the repeat distance, which is the reciprocal, diverges. This degree of confinement63 find that the amplitude and phase result, correct for the vSH model calculation, does not rep- are dramatically affected, but not the repeat distance by in- resent the experimental situation. But it is clearly reflected in terface roughness. Actually, the repeat distance can be the Fermi-surface calipering. So this part of the computer changed, but through selection of a different caliper. Modi- experiment does not give further information. fication of relative amplitudes by changes in the magnetic Examining the cause is more interesting. Part of the an- layer thickness has also been seen for Co/Cu multilayers.64 swer can be gleaned from studies examining the effects of In that case, the variation in thickness actually changed surface roughness. In a computational experiment to deter- which of two differing experimental results were matched by mine sensitivity to interface roughness,27 a checkerboard pat- the theoretical calculations. The explanation tendered in- tern of Cr and Fe was created at the interface. The resulting volved the differing sample preparations. effect was that the amplitudes of both the long and the short Further confirmation that the root cause for the vSH result period were reduced by very similar factors 5.8 and 6.1 : is the thinness of the Fe layer can be gotten from actually a significant improvement to the order-of-magnitude calculations23 for a model quite similar to the vSH model - theoretical overestimate for the amplitude found using the being based on LMTO calculations with subsequent fits to perfect interface. Experimentally, however, the long period variation of the total-energy difference between ferro- and persists even for quite rough interfaces. So the precipitous antiferromagnetic alignment of the layers to extract repeat diminution of the long period is not consistent with the ex- distances - but with two major differences: retaining full perimental findings and is just one more piece of evidence self-consistency in the spacer layer; and utilizing semi- that the nesting alias is not the origin of the long period. infinite Fe slabs. The self-consistent response appears to af- Instead, model calculations for interface reflection fect the repeat distances very little, indicating a minimal ef- properties20 indicate that the Fe layers used in the vSH model fect of the incipient, or present, antiferromagnetism. Of calculation were too thin: Fe layers only two monolayer primary interest to the present consideration is that the semi- thick were used. One would like to view the reflection cal- infinite Fe slabs eliminate the appearance of the nesting alias culations as doing a similar decomposition for the interface as the short period. A short period of 2.07 monolayers 6364 DALE D. KOELLING PRB 59 2.98 Å is found which decays as 1/d consistent with a and with the experimental results but they are different. In nested surface. A long period of 11.98 monolayers 17.2 the 100 direction, this is attributed to using Fe layers that Å is also found which decays as 1/d2 consistent with a point were too thin in the calculations for the vSH model. Another calipering. Found in addition are two very rapidly decaying observation to be taken concerns possible intermode cou- terms. The long period, however, is interpreted not as arising pling. In the analysis for the short period in the 110 direc- from the N-centered ellipses but from an unusual coupling tion of the Mn alloys, it was suggested that one possible between H octahedra. It is argued that enough persists of the explanation would involve several calipers acting in concert. Cr antiferromagnetic character for the system to be analyzed That suggestion would seem at odds with a more general using a CsCl unit cell. In that structure, the octahedra would behavior involving switching between modes, as apparently be folded in to the point. One could then get a corner to seen relative to the H octahedron calipers and also for the corner of the octahedron in the next zone transition which long period in this same direction for the V-rich alloys, for comes out to be just the correct size. Present calculations an example. yield a repeat distance of 16.4 Å for such a ``CsCl caliper'' when the empirical adjustment is not applied and 17.4 Å IV. MORE SURFACE EFFECTS when it is. Most likely the reason that the correspondence is to the adjusted calculation is that the atomic-sphere shape Motivated by the lack of corresponding Fermi-surface approximation is simulating the effect. The effective mass calipers for the short period in the 110 direction at around for this calipering is fairly small-being about a half-and 20­25% V, the analysis of Mirbt et al.23 can be examined a the coupling is d to d so this interpretation is subject to all bit further. It can be extended under a somewhat different view which will prove instructive but not resolve the issue. the criticism applied to a lens/jack-type interaction. Such an Instead of focusing on the spin-density-wave character, it is interpretation also would encounter problems when examin- useful to instead reconsider the fact that the Cr-Fe interfaces ing the 211 direction. But more seriously, such an expla- lower the symmetry of the system. For interfaces perpen- nation would prove inconsistent with the V-alloy behavior dicular to the 100 direction, a symmetry lowering is intro- for the 100 direction. At 10% V, the repeat distance due to duced that would be properly represented by analyzing the such a calipering increases to 30 Å whereas the experimen- problem using a CsCl-type lattice. That would not, however, tal result65actually decreases. Also, photoemission ``sees'' be the case for other directions. Rigorously, once the inter- no evidence of quantum-well states forming along the direct faces have been introduced, one no longer has periodic 100 direction. It thus seems unlikely that this interaction boundary conditions in the normal direction and no proper occurs in the real world, although it apparently does exist reciprocal space in that direction. For large enough spacer within the model. It is important to take note that the inter- material and ``reasonable'' interfaces, one can get by in the pretation of this calculation does not point to an ellipse cali- analysis simply continuing as though all was in order. Such per. Whether it might exist in the calculation is not clear is the nature of most of the the Fermi-surface singularity because an ellipse caliper is not given. As an aside, the analysis being utilized here. So one knows that one can at observation can be made that the concerns of Mirbt et al.23 least ``almost get by'' and the natural question is what will about the possible appearance of a bcc caliper Q 4 from the be the first effect s to appear next. Most reasonable is that lens to the jack-knob/octahedron-tip is unfounded. That cali- all 2D reciprocal-lattice vectors of the interface plane should per would have to arise not from the H side of the lens but start to appear. Generally, the plane is more open than the from the side in order for the velocities to be antiparallel. bulk solid so this will introduce reciprocal lattice vectors In that case, the repeat distance would be far too short about which are some fraction of the bulk reciprocal-lattice vec- 9 Å) to be considered as a candidate for the long period. tors. See Sec. II B of Ref. 19 for a discussion. This will One learns somewhat more about the short period ampli- introduce a very specific new coupling within the Brillouin tude from Ref. 23. The amplitude obtained is larger than zone because the interfaces truncate the summation implied experiment by three orders of magnitude compared to the by the discussion of a generalized RKKY formalism66 that single order of magnitude found for the vSH model. A sig- brings in the reciprocal-lattice vectors precluding obtaining nificant part of this effect could be the greater effective con- the full bulk crystal orthogonality. Another way to get to this finement, so one cannot simply point to the greater response result is to realize that a supercell treatment for the ideal incorporated in the bulk without more careful consideration. system would result in a wafer-thin Brillouin zone based on It is certainly suggestive, though, since this oscillation in- the plane 2D zone. If one then assumes one can do an un- volves only d states and so should not require thick Fe lay- folding like that used to test the VCA above, then the first ers. In this model, even more is required from surface rough- new effect is the presence of the plane vectors. Accordingly, ness to bring the calculated amplitude down to what is one can look for a new set of spanning vectors except that the observed experimentally. two Fermi-surface spots associated with a repeat vector q are Several observations should be taken from this section. also displaced by a reciprocal-lattice vector of the interface. The primary result is that reasonable identification of cali- The ideas are easiest to see by considering the the original pers can be found throughout almost all of the alloy range. case where the interfaces have a 100 normal. In this case, This makes the exceptions much more interesting although the planes are simple square lattices which stack with each they are usually found at places of transition with small am- plane above the square center of the adjacent plane. The 2D plitudes. For the long period in the 110 direction in the primitive translations are (0a0) and (00a) with the associ- V-rich alloys and in the 100 direction generally, calipers ated reciprocal-lattice vectors are 2 (010)/a and can be successfully associated with the vSH model results 2 (001)/a. Clearly, the normal is in the x direction. These PRB 59 LONG-PERIOD OSCILLATION IN THE MAGNETIC . . . 6365 are the extra vectors in the perpendicular direction for a CsCl small for the long repeat distance, and increases the anisot- Brillouin zone to apply. These are the only directions needed ropy, thereby adding more evidence against the hypothesis for the effect observed. However, in the 100 direction, that it might be the responsible surface. The empirical cor- aliasing or coupling with one of the bulk reciprocal-lattice rection of the d-band energy then only acts to increase the vectors will produce a comparable effect. Hence, the inter- difference. At the same time, the caliper across the neck of faces do introduce the use of a CsCl Brillouin zone and are at the -centered jack surface comes to be nearly the correct least as probable a basis as the explanation used.23 size and the caliper from the lens to the jack neck becomes When the interface normal is in the 110 direction, the the correct size for Co/Cr multilayers - where the ellipses planes are face-centered rectangular 2D lattices which are apparently do not fit. So, for the 100 direction, one has the stacked above edge centers. Using the 3D vectors, the primi- at least three other features of the right size: jack neck, alias tive translations of the planes are a(1, 1,1)/2 and a(1, 1, of the short period, and coupling of the H octahedron in a 1)/2 and the reciprocal-lattice vectors are (1, 1,2)/a CsCl structure. The fact that the jack neck caliper is seen in and (1, 1, 2). The main effect of incorporating the photoemission26 coupled with the question of why the long surface-derived displacement vectors is to give an out-of- period is so robust suggests that one might ask whether these plane H-centered octahedron a virtual position allowing it to additional channels might not couple into the response to interact in the 110 direction with the ellipse and a basal enhance it. The first evidence that this may not be the case plane H octahedron. The major result was an interoctahedron actually comes from the fact that the amplitude and repeat caliper at 2.1 Å and an ellipse-octahedron/interoctahedron distance are essentially identical for both the 100 and the caliper at 6.2 Å. There does not appear to be anything 211 direction. The 211 direction is geometrically much closer to the vSH model result of 5.1 Å than the 6.2 Å more complex but it does not appear other possible channels which merely brackets the value on the high side by an about are to be found there - raising serious doubts about such a equal amount to what had been found on the low side using hypothesis. Much stronger evidence comes from the com- standard calipering. puter experiment on the Cr/ V,Mn alloys. In the one case Note that this simple extension does not introduce a where it might be tempting to appeal to mode coupling en- 2 (001)/a-type vector for the 110 direction unless some- hancement short period in the 110 direction for 10% Mn , thing acts to remove the face centering in the plane. Or, one it is actually much more dramatic that that result occurs be- could return to the magnetic response arguments . . . ) How- cause of a mode exclusion. And there are quite a few cases ever, if a 2 (001)/a vector is assumed to be present, then where one sees mode switching as a function of alloy com- 110 caliperings are introduced between the H octahedra position which would naturally suggest a predilection for and between the and H octahedra. One of these calipers mode exclusion. Understanding such an exclusion would be between the octahedron and an H octahedron has a repeat very useful to explain why there are so many more Fermi- distance of 4.9 Å and effective mass parameter of 2.2. So surface calipers than actual terms in the response: it is im- these two somewhat questionable results prove illusive probable that simple amplitude arguments are enough. within the models examined here. And what of the robust character exhibited by the long period? Is it a magnetic enhancement effect? Certainly com- parison of the amplitudes found with23 and without27 en- V. DISCUSSION hancement find a huge difference for the short period. Per- The results presented have discredited several of the hy- haps this really does extend to the long period but it still potheses under which this study was begun, and also focused needs checking since that is a piece of Fermi surface with several additional issues. By matching the size of the N el- very different wave function character. lipses to that inferred from dHvA data, through an empirical correction of the d-band energy, one indeed finds that Fermi- surface caliperings can be found on the ellipses20­22 appro- ACKNOWLEDGMENTS priate to the long period. This is clearly much better than the This work was supported by the U.S. Department of En- previous situation where only very special cases could be ergy, Basic Energy Sciences, Division of Materials Sciences found. It is also not a trivial result since the ellipse data is under Contract No. W-31-109-ENG-38 and by a grant of actually derived information from the actual experiment: it computer resources and the National Energy Research Su- must be backed out from the data by working around the percomputer Center. The author is grateful to Greg McMul- effects of the spin-density wave. The data provided is then lan for performing the series of LMTO calculations that per- the principal axis dimensions of an assumed ellipsoidal sur- mitted a detailed comparison of the results obtained by the face. It was shown that the ellipsoidal assumption is deficient two different approaches. for the purposes here although probably adequate to the original extraction. For the 100 direction, the primary cali- per arises from the basal plane as found previously . This APPENDIX: FURTHER MINOR IMPROVEMENT caliper does have the higher mass factor of the 2 occurring ON THE MASS LOCAL JOINT DENSITY OF STATES... on the ellipse, as does the zone-face caliper appropriate for EXPRESSION the 110 direction. This selection among the subset of el- lipse calipers is only slightly further emphasized by the The local joint density of states, or mass parameter, has higher spin antisymmetric reflection factors21,22 representing proven a poor criterion for the strength of any particular the surface effects. Fermi-surface caliper pair because of the other factors in- Improved calculations yield a lens surface which is too volved. A minor further improvement will be presented here 6366 DALE D. KOELLING PRB 59 which has an effect on the mass parameter calculated for The quadratic matrix is constructed to be symmetric. This some caliper pairs. expression is rotated to an orientation along the spanning The mass parameter is defined assuming a point extre- vector (kz) and the two-dimensional vector perpendicular to mum or saddle point.15,67,68 When there is actual nesting it, identified as the vector ``parallel'' to the surface (k along a line as for the H-centered octahedron in the basal ): plane for a 110 vector the definition of the mass parameter 1 1 diverges. When this happens, the joint density of states in- v *k 2 2k *DJ*k kz vz z *DJ*k 2 Dzzkz . volved is much greater and the strength of the coupling falls A3 off as d 3/2. When the nesting occurs for a full plane the 2 classic -centered jack to H-centered octahedron associated If the Dzzkz term can be neglected-one type of ``thin with the Cr spin-density wave, for example , the joint density Fermi shell,'' then one has an easy, approximate inversion: of states is even stronger and the strength of the coupling falls off as d 1. It is an unfortunate feature of the Fourier v *k k k *D J *k /2 series star function fitting technology used that line or area z vz z *DJ*k nesting is not found directly. Because the fit contains small nonphysical oscillations-remnants of Gibbs ringing, the fit 1 1 always exhibits point calipers. Nesting is then detected by z *DJ*k . A4 v v *k k *DJ*k /2 z vz multiple calipers of differing type and high mass that occur closely spaced in the same region. These require manual This is the expression used in Ref. 19. One can improve on identification by careful examination of the results. But they this by making a linear expansion about k z : are important to identify because extending nesting is so more significant than point calipering. It is most likely that k 2 2 z k z 2k z kz k z k z 2kz k z . A5 these small improvements in the determination of the mass Reinserting this approximation retains a linear solution in k parameter have, as their most useful consequence, helping to z and permits an improved approximation spot in spotting nesting. With the above proviso, improvement can be made even v *k 2D on the slightly improved version of Ref. 19. That notation k k *D J *k /2 k z zz/2 z , A6 will be followed here due to familiarity. We return to the vz z *DJ*k Dzzk z expression for the intensity which, however, is not linear in because of the k z in the Z f f denominator and its square in the numerator. This nonlinear- I d2k dk dk . ity in would preclude the straightforward contour integra- 2 2 4 z z eiz kz kz tion technique used for and . However, because those A1 integrations result only in the slowly varying In this expression, the factor Z is an approximate matrix (z/L)/sinh(z/L) term, this change would not be an important element which is assumed slowly varying - the arguments correction and is ignored. What is important is the correction expressing its dependence on the two calipering vectors have to the 0 surface associated with the steepest descent k merely been suppressed for clarity. That dependence, though integral. What is being done then is to modify the curvatures important, is not the focus here. It is where the often very of the approximate Fermi surfaces used in the steepest de- important surface interactions are buried. Only one k scent integral. For that, one sets to zero and expands the k inte- z gration appears because a two-dimensional function has and kz expressions to obtain the required quadratic expres- already been evaluated. The f 's are Fermi occupation- sion used in the steepest descent. The result can be achieved number functions. For simplicity of notation, the presence or by a simpler and more transparent approach if one notes that, nonpresence of a prime on the is used to associate it with to the level of expansion being utilized, the k z associated the appropriate k vector and band index arguments. The pro- with the quadratic term reduces to the simple tangent term cedure to evaluate the kz integrals is to convert them to en- ergy integrals using a linearized in kz) quadratic expansion k z v *k /vz . A7 about the calipering vectors. This is straightforwardly ac- complished so long as a linear relation is an adequate ap- Thus, one can evaluate the quadratic term for the vector proximation over the thin region where f ( ) f ( ) has a k significant value. The energy integrals can be evaluated by o k , v *k /vz A8 continuing into the complex plane and summing the residues. in the 0 expression and solve: Moving to the caliper vectors as an origin, the exponential contains an overall phase factor dependent on the spanning 1 1 vector the difference between the two calipering vectors kz . A9 v v *k z 2k o*DJ*k o which can merely be moved out in front of the integrals. One now works with the quadratic expansion for both vectors the Since the velocities at the two calipering vectors are antipar- primes are inserted when needed : allel, the linear term will cancel out. Because of the tangent relation, all terms in the bilinear product contain precisely 1 two factors of k v *k so it can be rewritten as a bilinear product 2k *DJ*k . A2 of k and a rotation performed to achieve a diagonal tensor. PRB 59 LONG-PERIOD OSCILLATION IN THE MAGNETIC . . . 6367 The resulting diagonal matrix elements are the inverse of the when the second-order terms are zero. These nesting effects mass elements needed for the local joint density of states are very important but, as noted above, the expansions used expression. This is believed to be about the best one can do do not permit direct evaluation without very elaborate ef- within the limits of the integration techniques used. One can forts. While generally less precise, the tight-binding go somewhat farther using alternate techniques to deal with representation20 does provide a more direct indication of the integrations and even analytically examine weak nesting because it does not contain the short-wavelength os- nesting69 by incorporating the next terms in the expansion cillations. 1 P. Gru¨nberg et al., Phys. Rev. Lett. 57, 2442 1986 . 36 S. Wakoh and J. Yamashita, J. Phys. Soc. Jpn. 35, 1394 1973 . 2 S. F. Alvarado and C. Carbone, Physica B 149, 43 1988 . 37 J. Mathon, M. Villeret, and D. Edwards, J. Phys.: Condens. Mat- 3 M. Baibich et al., Phys. Rev. Lett. 61, 2472 1988 . ter 4, 9873 1992 . 4 J. Krebs, R. Lubitz, A. Chaiken, and G. Prinz, Phys. Rev. Lett. 38 J. d'Albuquerque e Castro, M. Ferreira, and R. Muniz, Phys. Rev. 63, 1645 1989 . B 49, R16 062 1994 . 5 S. Parkin, N. More, and K. Roche, Phys. Rev. Lett. 64, 2304 39 J. Mathon et al., Phys. Rev. B 56, 11 797 1997 . 1990 . 40 S. Wakoh, Y. Kubo, and J. Yamashita, J. Phys. Soc. Jpn. 38, 416 6 A. Davies, J. A. Stroscio, D. Pierce, and R. Celota, Phys. Rev. 1975 . Lett. 76, 4175 1996 . 41 S. Wakoh, Y. Kubo, and J. Yamashita, J. Phys. Soc. Jpn. 40, 1043 7 D. Stoeffler and F. Gautier, Phys. Rev. B 44, 10 389 1991 . 8 1976 . D. Stoeffler and F. Gautier, J. Magn. Magn. Mater. 104-107, 1819 42 S. Wakoh and Y. Kubo, J. Phys. F 10, 2702 1980 . 1992 . 43 9 N. Shiotani et al., J. Phys. Soc. Jpn. 43, 1229 1977 . J. Unguris, R. Celotta, and D. Pierce, Phys. Rev. Lett. 67, 140 44 1991 . S. Wakoh, T. Fukamachi, S. Hosoya, and J. Yamashita, J. Phys. 10 J. Wolf et al., J. Magn. Magn. Mater. 121, 253 1993 . Soc. Jpn. 38, 1601 1975 . 45 11 Y. Wang, P. Levy, and J. Fry, Phys. Rev. Lett. 65, 2732 1993 . A. H. MacDonald, W. E. Pickett, and D. D. Koelling, J. Phys. C 12 Z.-P. Shi, P. M. Levy, and J. L. Fry, Phys. Rev. Lett. 69, 3678 13, 2675 1980 . 46 1992 . D. G. Shankland, in Computational Methods in Band Theory, 13 P. M. Levy et al., J. Magn. Magn. Mater. 121, 357 1993 . edited by P. Marcus, J. Janak, and A. Williams Plenum, New 14 S. Parkin, Phys. Rev. Lett. 67, 3598 1991 . York, 1971 , p. 362. 15 P. Bruno and C. Chappert, Phys. Rev. B 46, 261 1992 . 47 D. D. Koelling and J. H. Wood, J. Comput. Phys. 67, 253 16 P. Bruno, J. Magn. Magn. Mater. 116, L13 1992 . 1986 . 17 S. Araki, J. Appl. Phys. 73, 3910 1993 . 48 W. E. Pickett, H. Krakauer, and P. B. Allen, Phys. Rev. B 38, 18 M. van Schilfgaarde and W. A. Harrison, Phys. Rev. Lett. 71, 2721 1988 . 3870 1993 . 49 R. Parker and M. Halloran, Phys. Rev. B 9, 4130 1974 . 19 D. D. Koelling, Phys. Rev. B 50, 273 1994 . 50 M. R. Norman and D. D. Koelling, Phys. Rev. B 28, 4357 1983 . 20 M. D. Stiles, Phys. Rev. B 54, 14 679 1996 . 51 S. Wakoh, J. Phys. F 7, L15 1977 . 21 L. Tsetseris, B. Lee, and Y.-C. Chang, Phys. Rev. B 55, R11 586 52 M. van Schilfgaarde and F. Herman, Phys. Rev. Lett. 71, 1923 1997 . 1993 . 22 L. Tsetseris, B. Lee, and Y.-C. Chang, Phys. Rev. B 56, R11 392 53 J. Harris, Phys. Rev. B 31, 1770 1985 . 1997 . 54 W. Foulkes and R. Haydock, Phys. Rev. B 39, 12 520 1989 . 23 S. Mirbt, A. M. N. Niklasson, B. Johansson, and H. L. Skriver, 55 M. D. Stiles, Phys. Rev. B 48, 7238 1993 . Phys. Rev. B 54, 6382 1996 . 56 B. Lee and Y.-C. Chang, Phys. Rev. B 52, 3499 1995 . 24 P. Bruno, J. Magn. Magn. Mater. 121, 248 1993 . 57 25 Z. Qui, J. Pearson, and S. Bader, J. Appl. Phys. 73, 5765 1993 . E. E. Fullerton et al., Phys. Rev. B 48, 15 755 1993 . 58 26 J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 1954 . D. Li et al., Phys. Rev. Lett. 78, 1154 1997 . 59 27 F. Aliev et al., Phys. Rev. Lett. 78, 134 1996 . M. van Schilfgaarde, F. Herman, S. S. P. Parkin, and J. 60 Kudrnovsky´, Phys. Rev. Lett. 74, 4063 1995 . E. W. Fenton, Solid State Commun. 32, 195 1979 . 61 28 E. E. Fullerton, the Cr periods in the 100 and 211 directions E. W. Fenton and C. R. Leavens, J. Phys. F 10, 1853 1980 . 62 differ by less than 6% unpublished . R. S. Fishman and S. H. Liu, Phys. Rev. B 48, 3820 1993 . 29 63 W. Folkerts and F. Hakkens, J. Appl. Phys. 73, 3922 1993 . J. Mathon, M. Villeret, D. Edwards, and R. Muniz, J. Magn. 30 A. Kamijo, J. Magn. Magn. Mater. 156, 137 1996 . Magn. Mater. 121, 242 1993 . 31 64 J. E. Graebner and J. A. Marcus, Phys. Rev. 175, 659 1968 . L. Nordstro¨m, P. Lang, R. Zeller, and P. Dederichs, Phys. Rev. B 32 G. W. Crabtree, D. Dye, D. P. Karim, and D. D. Koelling, Phys. 50, R13 058 1994 . Rev. Lett. 42, 390 1979 . 65 C.-Y. You et al., J. Appl. Phys. to be published . 33 G. W. Crabtree et al., Phys. Rev. B 35, 1728 1987 . 66 F. Herman and R. Schrieffer, Phys. Rev. B 46, R5806 1992 . 34 R. Blaschke et al., J. Phys. F 14, 175 1984 . 67 P. Bruno and C. Chappert, Phys. Rev. Lett. 67, 1602 1991 . 35 V. I. Anisimov, F. Aryasetiawan, and A. I. Lichtenstein, J. Phys.: 68 P. Taylor, Phys. Rev. 131, 1995 1963 . Condens. Matter 9, 767 1997 . 69 M. Kaganov and A. Semenenko, Sov. Phys. JETP 23, 419 1966 .