Magnetic Compton profiles of disordered Fe 0.5 Ni 0.5 and ordered FeNi alloys
aa r X i v : . [ c ond - m a t . m t r l - s c i ] D ec Magnetic Compton profiles of disordered Fe . Ni . and ordered FeNi alloys D. Benea a , J. Min´ar b , H. Ebert c , L. Chioncel d,e a Faculty of Physics, Babes-Bolyai University, Kogalniceanustr 1, Ro-400084 Cluj-Napoca, Romania b University of West Bohemia, New Technologies - Research Center, Univerzitni 8, 306 14 Plzeˇn, Czech Republic c Department of Chemistry, University of Munich, Butenandstr. 5-13, D-81377 M¨unchen, Germany d Augsburg Center for Innovative Technologies, University of Augsburg, D-86135 Augsburg, Germany and e Theoretical Physics III, Center for Electronic Correlations and Magnetism,Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany (Dated: October 8, 2018)We study the magnetic Compton profile (MCP) of the disordered Fe . Ni . and of the orderedFeNi alloys and discuss the interplay between structural disorder and electronic correlations. TheCoherent Potential Approximation is employed to model the substitutional disorder within thesingle-site approximation, while local electronic correlations are captured with the Dynamical MeanField Theory. Comparison with the experimental data reveals the limitation of local spin-densityapproximation in low momentum region, where we show that including local but dynamic correla-tions the experimental spectra is excellently described. We further show that using local spin-densityapproximation no significant difference is seen between the MCP spectra of the disordered Fe . Ni . and a hypothetical, ordered FeNi alloy with a simple cubic unit cell. Only by including the electroniccorrelations, the spectra significantly separate, from the second Brillouin zone boundary down tozero momenta. The difference between the MCP spectra of ordered and disordered alloys is dis-cussed also in terms of the atomic-type decompositions. Finally based on the presented calculationswe predict the shape of the MCP profile for the ordered FeNi alloy along the [111] direction. I. INTRODUCTION
The dominant contribution, according to the non-relativistic limit of the X-ray scattering comes from theinteraction of the photon with the electron’s charge .The relativistic Compton amplitude does depend, how-ever, on the spin of the electron and on the polarizationof the X-rays . To obtain the scattering cross sectionthe Compton amplitude is squared and summed overall final states consistent with the energy conservation.Within the impulse approximation the magnetic scatter-ing cross section measures the spin moments throughthe integrated difference, R [ n ↑ ( ~p ) − n ↑ ( ~p )] d ~p , in themomentum distribution n ↑ ( ↓ ) ( ~p ) of spin up (down) elec-trons. The magnetic Compton scattering experimentscombined with theoretical calculations of the profile mayprovide also valuable information about the exchange andcorrelation effects in materials.The computed Magnetic Compton Profile (MCP) spec-tra for Ni and Fe have been previously re-ported in the literature. The analysis of spectra cov-ers the aspects of multiple scattering, core contribution,relativistic effects and electronic correlations. The com-parison with experimental measurements concerning theshape of the MCP spectra and the values of spin mo-ments were also discussed. Along the [111] directionprominent features of the MCP for Fe and Ni are (i)the negative polarization of s − and p − bands at low mo-mentum, (ii) dips in the MCP profiles near p z = 0 a.u. ,and the (iii) periodic features due to the Umklapp pro-cesses at momenta ~p = ~k ± n ~G , where ~G is the reciprocal-lattice vector and n ∈ Z . Generally, the theory overes-timates the MCP spectra near p z = 0 irrespective ofthe band-structure method used in the Density Func- tional Theory (DFT) calculations . This discrep-ancy have been attributed to the inadequate treatmentof the electron-electron correlations in the Local DensityApproximation (LDA) or its gradient corrected (GGA)type independent-particle-models for the exchange corre-lations of DFT. It was shown during the last decades thatDynamical Mean Field Theory (DMFT) , success-fully removes some of the observed inconsistencies inthe description of the ground state properties of 3d tran-sition metal elements. DMFT based calculations for theMCP profiles of Fe and Ni, showed indeed that thelow momentum discrepancies in the MCP are reduced,however high resolution measurements would be usefulto investigate specific features in the MCP profiles thatare still not well described.In this paper we report theoretical results on the MCPspectra for the disordered Fe . Ni . alloy using the ex-change correlation potential of LDA and the improvedLDA+DMFT method . We show that the discrepanciesbetween the LDA and the experimental spectra at lowmomentum are corrected including local dynamic corre-lations captured by DMFT. At the same time we identifythe correct magnitude of the Coulomb parameters on dif-ferent alloy components. To study the interplay of dis-order and correlation in momentum space, we compareMCP spectra of the Fe . Ni . alloy with the correspond-ing spectra of an ordered FeNi alloy with simple cubicsymmetry that has the same unit cell dimensions andchemical composition. We show that at the LDA levelthe total MCP spectra for both ordered and disorderedFeNi alloys are similar, while only within LDA+DMFTthe distinction between the two becomes apparent. Inparticular at low momenta the MCP spectra obtainedwithin the CPA calculation is reduced, while for the or-dered alloy the MCP spectra is slightly enhanced. Basedon the type decomposition of the MCP, we discuss dif-ferences and similarities between these spectra. As theMCP spectra of LDA+DMFT is found to be in agreementwith the experiment in the disordered case, and as we areunaware of any previous experimental measurements orcalculations of MCP spectra for the ordered FeNi alloywe thus, predict the MCP shape of FeNi along the [111]direction.In the following section (Sec. II) we provide anoverview of the LDA+DMFT computational procedureof the MCP profiles within the Korringa-Kohn-RostokerGreen’s function formalism for the disordered, Sec.II A,and ordered, Sec.II B, systems. The results are presentedin section Sec. III: the total MCP and type decomposi-tions are discussed in Sec. III A and Sec. III B for the dis-ordered respectively ordered alloys. The results for thedensity of states and magnetic moment calculations arepresented in Sec. III C, and finally the paper is concludedin Sec. IV. II. COMPUTATIONAL DETAILS
The most important ingredient in the analysis of theelectronic momentum density n ( ~p ) in disordered systemsis the impurity configuration averaged Green’s function.Such analysis was achieved for the first time by Mi-jnarends and Bansil within the muffin-tin frameworkof the Coherent Potential Approximation, (CPA) ,formulated using the multiple scattering theory, the so-called KKR-CPA method. Here we present re-sults using the spin-polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) method in the atomic sphereapproximation (ASA). The exchange-correlation poten-tials parameterized by Vosko, Wilk and Nusair wereused for the LSDA calculations. For integration overthe Brillouin zone the special points method has beenused . In addition to the LSDA calculations, a chargeand self-energy self-consistent LSDA+DMFT scheme forcorrelated systems based on the KKR approach has been used. The many-body effects are described bymeans of dynamical mean field theory (DMFT) andthe relativistic version of the so-called Spin-Polarized T-Matrix Fluctuation Exchange approximation impu- rity solver was used. The realistic multi-orbital inter-action has been parameterized by the average screenedCoulomb interaction U and the Hund exchange interac-tion J . Recent developments allow to compute the dy-namic electron-electron interaction matrix elements ex-actly . It was shown that the static limit of the screenedenergy dependent Coulomb interaction leads to a U pa-rameter in the energy range of 1 and 3 eV for all 3dtransition metals . As the J parameter is hardly af-fected by screening it can be calculated directly withinthe LSDA and is approximately the same for all 3d ele-ments, i.e J ≈ . a.u. and a BZ integration meshof 62 × ×
62 points was used.The computation of Compton profiles within the SPR-KKR formalism was worked out a decade ago .The Magnetic Compton Profile is given by the momen-tum distribution of valence electrons projected along thescattering vector p z . The spin projected momentum den-sity is expressed in terms of the Green’s function in themomentum representation, constructed from the real-space Green’s function, using the eigenfunctions of themomentum operator. The electron momentum densi-ties are usually calculated for the principal directions[001] , [110] , [111] using an rectangular grid of 200 pointsin each direction. The maximum value of the momentumin each direction is 8 a.u. . Here we present results onlyfor the [111] direction which allows us to compare withexperimental data for the disordered alloy. A. Magnetic Compton profiles for disorderedalloys, type decompositions
Given the eigenfunctions of the momentum operator,real-space integration is used to calculate G m s ( ~p, ~p, E ).This integration is performed over a unit cell, andsummed over the cells, as described in Ref. 46. In the mo-mentum representation the ensemble averaged Green’sfunction is given by: G m s ( ~p, ~p, E ) = 1Ω X q X A x q A h − X Λ ˜ M q A m s Λ m s + X ΛΛ ′ M q A m s Λ (cid:16) D q A τ q, qCP A ( E ) (cid:17) ΛΛ ′ M q A ∗ m s Λ ′ i (1)+ 1Ω X q X q ′ e − i~p ( ~R q − ~R q ′ ) X A = B x q A x q ′ B X ΛΛ ′ M q A m s Λ (cid:16) D q A τ nq,n ′ q ′ CP A ( ~p, E ) ˜ D q ′ B (cid:17) ΛΛ ′ M q ′ B ∗ m s Λ ′ . We denote by q ( q ′ ) the sites within the cells n ( n ′ ). With τ q, qCP A ( E ) we denote the site-diagonal andwith τ nq,n ′ q ′ CP A ( ~p, E ) the site-non-diagonal, parts of the scattering path operator. In addition τ q, qCP A ( E ) = R BZ τ CP A ( ~k, E ) d ~k . The type-projected scattering pathoperators D q A τ q, qCP A ( E ) and D q A τ nq,n ′ q ′ CP A ( ~p, E ) ˜ D q ′ B ap-pear as a consequence of the single-site approximationof the CPA, when computing the configuration average h τ nq,n ′ q ′ ΛΛ ′ i . Finally, M q,Am s Λ and ˜ M q,Am s Λ m s are the regu-lar and irregular Compton matrix elements for the alloycomponent A. The explicit form of these expressions waspresented in Refs. 46 and 47.In order to proceed with the decomposition of the MCPwe shall in the following analyze Eq. (1). A specific site q in the unit cell contains the components A(B), with theconcentrations x q A ( B ) . The site and component diagonalGreen function in momentum representation is: G A,Am s ( ~p, ~p, E ) = 1Ω X q x q A h − X Λ ˜ M q A m s Λ m s (2)+ X ΛΛ ′ M q A m s Λ (cid:16) D q A τ q, qCP A ( E ) (cid:17) ΛΛ ′ M q A ∗ m s Λ ′ i . The site-diagonal but component-non-diagonal ( A = B )Green function is obtained from the last term of Eq. (1): G A,B = Am s ( ~p, ~p, E ) = 1Ω X q x q A x q B · (3) X ΛΛ ′ M q A m s Λ (cid:16) D q A τ q, qCP A ( ~p, E ) ˜ D q B (cid:17) ΛΛ ′ M q B ∗ m s Λ ′ Accordingly, the spin resolved momentum densities inthe disordered system are obtained integrating the cor-responding Green’s functions: n A,B ; Xm s ( ~p ) = − π ℑ Z E F −∞ (cid:2) G A,B ; Xm s ( ~p, ~p, E ) (cid:3) dE (4)With X we denote the functional form in the band struc-ture calculation, X=LSDA(+DMFT) and m s = ↑ ( ↓ ).Using the expressions for the Green’s function the pureEq. (2) and the mixed Eq. (3) contributions in the mo-mentum density can be obtained. The double integral ofthe spin momentum density, projected onto the scatter-ing direction K , with ~p z || K , defines the magnetic Comp-ton profile (MCP): J A,B ; Xmag, K ( p z ) = Z Z [ n A,B ; X ↑ ( ~p ) − n A,B ; X ↓ ( ~p )] dp x dp y , (5) B. Site decomposition of the Magnetic ComptonProfile for ordered alloys
For systems with more atoms in the unit cell, the MCPspectra is usually decomposed into the site-projectedcontributions and the interference-like terms similar toRef. 46. The unit cell sites q, q ′ can be occupied by atomsof type A or B. The type- and site-diagonal Green’s func-tion has the form: G A,Am s ( ~p, ~p, E ) = 1Ω X q h X ΛΛ ′ M qm s Λ τ q,q ΛΛ ′ ( ~p, E ) M q ∗ m s Λ ′ − X Λ ˜ M qm s Λ m s i . (6) The summation over the sites ( q ) in Eq. (6) is restrictedto the sites occupied by the same type of atoms. Thesite-off-diagonal Green’s functions is: G A,B = Am s ( ~p, ~p, E ) = 1Ω X q X q ′ = q e − i~p · ( ~R q − ~R q ′ ) · X ΛΛ ′ M qm s Λ τ q,q ′ ( ~p, E ) ΛΛ ′ M q ′ ∗ m s Λ ′ (7)Eq. (7) contains the product of Compton matrix ele-ments M qAm s Λ , M q ′ B ∗ m s Λ ′ with the scattering path operatorweighted by the phase factor P q,q ′ e − i~p ( ~R q − ~R q ′ ) . In anal-ogy with elementary formulas for X-ray diffraction by anassembly of atoms, equations of type Eq. (7) can be in-terpreted as interference or structure factor functions forthe material. The momentum density and the magneticCompton profile are then computed using the formulasEq. (4) and Eq. (5). Accordingly, the magnetic Comptoninterference term is the MCP obtained using the Green’sfunction of Eq. (7). Note that the Compton interferencefunction can be an alloy-type non-diagonal or diagonaldepending on the occupation of the q ( q ′ )-sites. This in-terference term is an incoherent scattering contributionand the corresponding MCP signal shall have a weak am-plitude for all directions along p z . III. RESULTS
In Fig. 1 we depict the unit cell for the ordered FeNiand the disordered Fe . Ni . alloys in the simple cubicgeometry. The ordered sc-FeNi alloy consists of alternat-ing Fe and Ni layers with a unit cell containing two Feand two Ni atoms. In-plane atoms have neighbors of the FIG. 1. (Color on-line) Left: The simple cubic structure ofthe ordered FeNi. Fe/Ni red/blue spehres. Right: the “CPAeffective” atom (gray sphere) of composition Fe . Ni . . same type, while out-of-plane neighbors are of differenttypes. In this case, the calculation of the DMFT selfen-ergy is performed for each type separately, which allowsto use different Coulomb/exchange parameters.In the fcc-geometry, within the CPA the “effective”Fe . Ni . atom has the same neighbors in all directions.The DMFT (impurity) problem is still solved for eachcomponent Fe/Ni, in addition the CPA equation is im-posed for self-consistency. The charge self-consistencyinvolves the one “effective” atom unit cell. A. Total MCP and type decomposition spectra ofFe . Ni . alloy We have performed the LDA(+DMFT) calculations forthe Fe . Ni . alloy within CPA. Table I summarizes theresults for the spin and orbital magnetic moments. TheDMFT calculations were done for different values of thelocal Coulomb interactions for Fe and Ni. As best valuesfor the average Coulomb parameters we identified U F e =2 eV and U Ni = 3 eV . The average exchange parameterwas set to J = 0 . eV .The MCP spectra of Fe . Ni . alloy are presented inFig.2. With decorated dashed blue/red lines we presentraw LSDA/DMFT data. To allow the comparison withthe experimental spectra of Kakutani et al. the theo-retical profiles have been broadened using a broadeningparameter (full width at half maximum FWHM) equalwith the experimental momentum resolution for record-ing the spectra, which was ∆ p = 0 . a.u. . These areseen in Fig.2a) with solid blue/red lines.In the [111] direction one clearly observe the signif-icant discrepancy between theory and experiment for p z < . a.u. . The LDA+DMFT calculations capture thecorrect behavior at low momenta, similarly to the situa-tion in bulk Fe and Ni . Many of the specific featuresof the theoretical MCP can not be seen in the experimen-tal profile due to the relative limited resolution (∆ p ≈ . a.u. ). Within the first zone, p < p [111] F ≈ . a.u ,the theoretical spectra predict a first peak marked withA and situated at 0 . a.u . This peak is absent in experi-ment but its Umklapp is observed experimentally (peakC). Within the second zone, the theory predicts peaksmarked with B and C and outside the second zone the Dand E peaks are visible. Further Umklapp features canbe observed for larger momenta as shoulders at ∼ . a.u. , ∼ . a.u. etc. Because of the relatively large broadeningthe experimental spectra “melts” the peaks B, C and D,therefore the Umklapp of A into the second Brillouin zone(C) is slightly overestimated. Furthermore the highermomenta Umklapp shoulders in the experimental profilesare considerably smeared out.Based on Eq. (5), the total MCP along the [111] di-rection has been decomposed, as seen in Fig. 2b), intothe type-projected contributions. Both J F eF emag and J NiNimag spectra show a pronounced dip at p z = 0. At non-zero momenta we see that electronic correlations leadsto momentum density redistribution between differentBrillouin zones. The J F eF emag
DMFT spectra is situatedbelow the LSDA spectra for p z < p [111] F ≈ . a.u. andwithin the further Brillouin zones is above the LSDA.On the other hand the DMFT J NiNimag is situated belowthe LSDA spectra for the entire range of momenta. Theinset of Fig. 2b) shows the mixed MCP term, J F eNimag ob-tained from the formula Eq. (4) and the Green’s functionEq. (3). The mixed term shows no significant correlationeffects, its characteristic being the oscillatory structure.The type-resolved spectra has been scaled according momentum p z (a.u.) M C P p F F ExpLSDA ∆ p = 0.42 a.u.LSDA ∆ p = 0.00 a.u.DMFT ∆ p = 0.42 a.u.DMFT ∆ p = 0.00 a.u. [ 1 1 1 ]Fe Ni a)A B C D E momentum p z (a.u.) M C P Fe,Fe; DMFTFe,Fe; LSDANi,Ni; DMFTNi,Ni; LSDA p z (a.u.) -0.0100.01 M C P Fe,Ni; LSDAFe,Ni; DMFT mag, [111]
J (p z ) A,B; X b) FIG. 2. (Color on-line) a) Calculated total MCP of Fe . Ni . alloy along [111] direction. Blue solid line: LSDA(CPA); redsolid line: LSDA(CPA)+DMFT. The experimental spectra ofKakutani et al. (black circle). b) Type decomposition ofMCP profiles of Fe . Ni . alloy. LSDA/DMFT results arerepresented by dashed/solid lines. The inset shows the mixedterm. to the spin moment obtained by self-consistent calcula-tions, which is 1.57 µ B in the LSDA calculations and1.56 µ B in the LSDA+DMFT calculations with U F e = 2.0 eV and U Ni = 3.0 eV, respectively. Iron givesthe dominant contribution in MCP, as a consequence ofits large spin moment: 2.48 µ B LSDA and 2.46 µ B inLSDA+DMFT calculation, respectively. A significantlysmaller Ni spin moment is obtained 0.66 µ B (LSDA) and0.65 µ B (LSDA+DMFT) respectively. B. Total MCP spectra and the site decompositionsfor the ordered FeNi alloy
In Fig. 3a) we present the comparison between the to-tal MCP [111] profile of the ordered FeNi alloy obtainedusing the LSDA (black line) and the LSDA+DMFT (redline) methods. The computed spectra (no applied broad-ening) are normalized to the values of the magneticmoments obtained in LSDA/DMFT calculations respec-tively. momentum p z (a.u.) M C P LSDADMFT
FeNiFe Ni FeNiFe Ni [ 1 1 1 ] FeNi DMFTLSDA U Fe =2.0 U Ni =3.0 a) b)c) FIG. 3. (Color on-line) Calculated MCP of the ordered FeNialloy along [111] direction: panel a) the total MCP profilescomputed with LSDA and DMFT for the values of U F e/Ni =2 / eV and J = 0 . eV . Panels b) and c) the total MCPprofiles for the ordered and disordered FeNi alloys in LSDAand respectively in DMFT. Contrary to the disordered case, Fig. 2a) where corre-lation leads to a depleted spectra around the zero mo-menta, we predict that correlation effects enhance theMCP profile in the range up to p z ≈ . a.u. . For largermomenta p z > . a.u. , similarly to the disordered case,the DMFT corrections does not change much on theLSDA shape. We observe that at the LSDA level theMCP spectra can hardly distinguish between the orderedand disordered structures, Fig. 3b). Any broadening ap-plied to the spectra to account for the experimental res-olution would make the spectra identical. Including cor-relation effects Fig. 3c) the LSDA(CPA)+DMFT spec-tra separate starting from the maximum value down to zero momenta and match the experimental results. TheMCP spectra the red line Fig. 3a) is our prediction forthe shape of the MCP of ordered FeNi along the [111]direction.According to Eq. (6) the total spectra is further decom-posed into the MCP Ni Fig. 4a) and MCP Fe Fig. 4b)contributions. Note that Fe’s weight to the total spectrais about four times larger than that of Ni. The reasonwhy LSDA cannot distinguish between the ordered anddisordered structure Fig. 3b), become also apparent: theNi/Fe MCP-components for the ordered alloy under/overestimate the corresponding spectra of the disordered al-loy. The amount of under/over estimation nearly com-pensate each other producing a similar total spectra. Ni LSDAa) p F F A BC D
Ni bulkFe Ni FeNi Fe LSDAb) Fe Ni FeNi momentum p z (a.u.) M C P DMFT Ni Ni bulkFe Ni FeNi Fe DMFT Fe Ni FeNi
FIG. 4. (Color on-line) Type decomposition of MCP spectrafor the ordered FeNi alloy along [111] direction: panel a)/b)Ni respectively the Fe components computed within LSDA(upper part) and DMFT (lower part).
The DMFT calculations for the disordered alloy pro-duce slightly reduced spectra for Fe (red solid line) incomparison with the LSDA (red dashed line) as seen inFig. 2b) . For Ni a slightly stronger reduction takes place.For the ordered FeNi alloy, DMFT spectra of Fe/Ni areenhanced/diminished in the low momentum region, how-ever, the increase of the Fe’s MCP dominates the decreaseon the Ni side, and an overall enhanced spectra is ob-tained (Fig. 4a; lower part). Fig. 4a, contains also theresults of the Ni bulk MCP calculations. We mark thefirst essential peaks as in Fig. 2a). Obvious differencesare seen below p z < p F ≈ . a.u. (first two BZ). Forthe bulk-Ni the first two peaks (A and B) are shifted andcontained within the first BZ: the first (A) is positionedat about 0 . a.u. , the second (B) is in the vicinity of p F . The third, the fourth and the subsequent Umklapppeaks are similar for all three spectra. The intensity ofUmkalpps for Fe . Ni . are smeared out additionally be-cause of disorder effects. Electronic correlations enlargethe differences between the Ni-project spectra of the dis-ordered alloy with respect to the bulk and ordered FeNi.Additionally the Ni contribution for the disordered alloyis further smeared out. C. Density of states and magnetic data analysis
In this section we discuss the ground state properties(DOS and magnetic moments) for the FeNi alloys. InFig. 5, we show the total and atom resolved DOS forFe . Ni . (left column) and FeNi alloys (right column).The combined effect of correlation and disorder is mostremarkable for Fe’s DOS, as seen also in the MCP spectraFig. 2. There is a strong renormalization of the spectralfunction towards the Fermi level on the majority spinchannel (spin-up). For the minority spin channel theweight of DOS is suppressed. In both spin channels thespectra is broadened accordingly. -2-1012 Fe Ni LSDADMFT
FeNi
LSDADMFT-101
Fe Fe -9 -6 -3 0 3
E-E F (eV) -101 Ni -9 -6 -3 0 3 Ni FIG. 5. (Color on-line) The density of state results for thedisordered (left column) and ordered (right column) FeNi al-loy. The DMFT results were obtained the values U
F e = 2 eV ,and U Ni = 3 eV and J=0.9eV. Our calculations show that the majority spin chan-nel of Ni undergoes relatively small changes upon addi-tion of Fe. The minority states continue to remain occu-pied in the Fe . Ni . alloy, as in Ni pure. The spectralchanges are mainly limited to the weight reduction, inboth spin channels, which is more significant than thespectral weight transfer towards the Fermi level, seen forthe Ni majority spins.Comparing the cubic ordered with the fcc-disorderedalloy, one can easily recognize that the LSDA(CPA)-DOSis more broadened because of the imaginary part of thecomplex effective potential. In addition electronic cor-relations have a less dramatic effect in the case of theordered alloy, for the same values of the local Coulomband exchange parameters. Concerning the magnetic moments, the alloys have aferromagnetic ground state. The Fe magnetic moment isin the range about 2 . − . µ B while a value of about0 . µ B is obtained for Ni, depending on the strength ofthe local Coulomb parameters U F e/Ni . The Ni magneticmoment remains essentially at its value in bulk-fcc. Thefact that Fe in FeNi has a larger moment than in fccFe at the same lattice constant can be explained by asmaller Fe-Ni hybridization due to the more contracted3 d -orbitals of Ni. A larger hybridization tends to decreasethe magnetic moment on an atom by filling the minorityspin 3d orbitals, which are more extended. IV. CONCLUSIONS
The self-consistent spin polarized electronic structureand the Magnetic Compton profiles along the [111] di-rection have been computed for the disordered Fe . Ni . alloy. Disorder has been modeled using the CPA andthe electronic correlations were considered through amulti-orbital Hubbard model solved with the DMFT. Weshowed that the discrepancy at low momenta due to theinadequate treatment of electronic correlations in LSDAcan be corrected using DMFT. Note that DMFT has tobe “active” on both alloy components. We have checkedthat neglecting “electronic correlations” on one of com-ponents or using improper values of the Coulomb interac-tion parameter does not provide a good comparison withthe experimental spectra of Kakutani . Most notablythe LSDA(CPA)+DMFT with U F e = 2 eV , U Ni = 3 eV and J= 0 . eV resolves the discrepancy around p z = 0.Umklapp features of the total-MCP spectra can be iden-tified up to momenta p z < p F . Subject to electroniccorrelations, integrated spin-resolved momentum densityshow significant changes while integrated spin resolvedreal-space densities provide almost similar magnetic mo-ments. Due to the limited momentum resolution of theexperimental spectra (∆ p = 0 . a.u. ) no clear compari-son between theory and experiment can be performed forlarge p z momenta. High resolution measurements wouldbe useful to identify specific features of the computedMCP profile.To study further the interplay between disorder andelectronic correlations, in momentum space, we have per-formed calculations for the MCP spectra of the ferro-magnetic ordered FeNi alloy. The calculation has beenperformed in the supercell setup with a similar latticeparameter as the one of the disordered alloy. Our resultsshow that within the LSDA the MCP spectra for the or-dered and disordered alloys are similar. This additionalshortcoming of the LSDA can be explained by analyzingthe type decomposition of MCP. Beyond the LSDA, theMCP spectra of ordered and disordered alloys are dif-ferent, thus we predict within LSDA+DMFT the MCPshape of the FeNi alloy along the [111] direction.An interesting conclusion maybe drawn from the re-sults of the present work. Namely, MCP appears to bemore sensitive to changes in the strength of the elec-tronic correlations rather than in the different geomet-rical structures (different disorder realizations). In otherwords, electronic correlations affect the momentum dis-tribution more significantly than the chemical bonding induced by structural disorder. From an experimentalpoint of view high resolution measurements would bebeneficial to resolve further the theoretical features ofthe MCP profile at low momentum and the blurring ofthe Umklapp features for high p z . F e . Ni . DMFT U F e/Ni ; J F e/Ni
LSDA2 . / . . / . . / . . / . . / . . / . . / . . / . m s ( µ B ) 2.525 2.472 2.463 2.466 2.478 m l ( µ B ) 0.089 0.10 0.059 0.105 0.058Ni m s ( µ B ) 0.628 0.653 0.680 0.653 0.659 m l ( µ B ) 0.048 0.066 0.067 0.069 0.049 F eNi
DMFT U F e/Ni ; J F e/Ni
LSDA2 . / . . / . . / . . / . . / . . / . . / . . / . m s ( µ B ) 2.594 2.596 2.572 2.599 2.573 m l ( µ B ) 0.109 0.108 0.064 0.106 0.064Ni m s ( µ B ) 0.578 0.584 0.60 0.584 0.598 m l ( µ B ) 0.037 0.040 0.04 0.045 0.051TABLE I. Magnetic moments: spin and orbital components for the for the disordered Fe . Ni . and ordered FeNi alloy,computed with LSDA and for different values of U F e/Ni ; J F e/Ni using LSDA+DMFT.
V. ACKNOWLEDGMENTS
We are indebted to Prof. D. Vollhardt for very use-ful discussions and for reading our manuscript. Fi-nancial support of the Deutsche Forschungsgemein- schaft through FOR 1346, the CNCS - UEFISCDI(project number PN-II-RU-TE-2014-4-0009) and of theCOST Action MP1306 EUSpec are gratefully ac-knowledged. JM would like to thank CEDAMNFproject (CZ.02.1.01/0.0/0.0/15 003/0000358) funded bythe Ministry of Education, Youth and Sports of CzechRepublic. M. J. Cooper, Rep. Prog. Phys. , 415 (1985). P. M. Platzman and N. Tzoar,Phys. Rev. B , 3556 (1970). G. F. Chew and G. C. Wick, Phys. Rev. , 636 (1952). M. J. Cooper, E. Zukowski, S. P. Collins,D. N. Timms, F. Itoh, and H. Sakurai,Journal of Physics: Condensed Matter , L399 (1992). P. Carra, M. Fabrizio, G. Santoro, and B. T. Thole,Phys. Rev. B , R5994 (1996). S. W. Lovesey, Journal of Physics: Condensed Matter , L353 (1996). P. Rennert, G. Carl, and W. Hergert,physica status solidi (b) , 273 (1983). M. A. G. Dixon, J. A. Duffy, S. Gardelis, J. E. McCarthy,M. J. Cooper, S. B. Dugdale, T. Jarlborg, and D. N.Timms, J. Phys.: Condensed Matter , 2759 (1998). Y. Kubo and S. Asano, Phys. Rev. B , 4431 (1990). D. N. Timms, A. Brahmia, M. J. Cooper, S. P. Collins,S. Hamouda, D. Laundy, C. Kilbourne, and M.-C. S.Larger, J. Phys.: Condensed Matter , 3427 (1990). T. Baruah, R. R. Zope, and A. Kshirsagar, Phys. Rev. B , 16435 (2000). D. Benea, J. Min´ar, L. Chioncel, S. Mankovsky, andH. Ebert, Phys. Rev. B , 085109 (2012). L. Chioncel, D. Benea, H. Ebert, I. Di Marco, andJ. Min´ar, Phys. Rev. B , 094425 (2014). L. Chioncel, D. Benea, S. Mankovsky, H. Ebert, andJ. Min´ar, Phys. Rev. B , 184426 (2014). J. Minr, H. Ebert, and L. Chioncel,The European Physical Journal Special Topics , 24772498 (2017). S. P. Collins, M. J. Cooper, D. Timms, A. Brahmia,D. Laundy, and P. Kane, J. Phys.: Condensed Matter , 9009 (1989). Y. Tanaka, N. Sakai, Y. Kubo, and H. Kawata, Phys. Rev.Letters , 1537 (1993). W. Metzner and D. Vollhardt, Phys. Rev. Letters , 324(1989). A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg,Rev. Mod. Phys. , 13 (1996). G. Kotliar and D. Vollhardt, Phys. Today , 53 (2004). A. I. Lichtenstein, M. I. Katsnelson, and G. Kotliar, Phys.Rev. Letters , 067205 (2001). L. Chioncel, L. Vitos, I. A. Abrikosov, J. Koll´ar,M. I. Katsnelson, and A. I. Lichtenstein,Phys. Rev. B , 235106 (2003). J. Min´ar, L. Chioncel, A. Perlov, H. Ebert, M. I. Kat-snelson, and A. I. Lichtenstein, Phys. Rev. B , 045125(2005). A. Grechnev, I. D. Marco, M. I. Katsnelson, A. I. Lichten-stein, J. Wills, and O. Eriksson, Phys. Rev. B , 035107(2007). O. Granas, I. di Marco, P. Thunstr¨om, L. Nordstr¨om,O. Eriksson, T. Bj¨orkman, and J. Wills, Comp. Mat. Sci. , 295 (2012). G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko,O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. ,865 (2006). P. E. Mijnarends and A. Bansil, Phys. Rev. B , 2381(1976). P. E. Mijnarends and A. Bansil, Phys. Rev. B , 2912(1979). B. Velick´y, S. Kirkpatrick, and H. Ehrenreich, Phys. Rev. , 747 (1968). F. Yonezawa and K. Morigaki, Suppl. Prog. Theor. Phys. , 1 (1973). R. J. Elliott, J. A. Krumhansl, and P. L. Leath, Rev. Mod.Phys. , 465 (1974). J. M. Ziman,
Models of disorder (Cambridge UniversityPress, 1979). B. L. Gy¨orffy, Phys. Rev. B , 2382 (1972). D. D. Johnson, D. M. Nicholson, F. J. Pinski, B. L. Gy¨orffy,and G. M. Stocks, Phys. Rev. Lett. , 2088 (1986). Y. Wang, G. Stocks, W. Shelton, D. Nicholson, Z. Szotek,and W. Temmerman, Phys. Rev. Lett. , 2867 (1995). L. Vitos, I. A. Abrikosov, and B. Johansson,Phys. Rev. Lett. , 156401 (2001). H. Ebert, in
Electronic Structure and Physical Propertiesof Solids , Vol. 535, edited by H. Dreyss´e (Springer, Berlin, 2000) p. 191. H. Ebert, D. K¨odderitzsch, and J. Min´ar,Rep. Prog. Phys. , 096501 (2011). S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. ,1200 (1980). H. J. Monkhorst and J. D. Pack, Phys. Rev. B , 5188(1976). I. D. Marco, J. Min´ar, S. Chadov, M. I. Katsnelson,H. Ebert, and A. I. Lichtenstein, Phys. Rev. B , 115111(2009). J. Min´ar, J. Phys.: Condensed Matter , 253201 (2011). M. I. Katsnelson and A. I. Lichtenstein, Eur. Phys. J. B(2002). L. V. Pourovskii, M. I. Katsnelson, and A. I. Lichtenstein,Phys. Rev. B (2005). F. Aryasetiawan, M. Imada, A. Georges,G. Kotliar, S. Biermann, and A. I. Lichtenstein,Phys. Rev. B , 195104 (2004). D. Benea, S. Mankovsky, and H. Ebert, Phys. Rev. B ,094411 (2006). D. Benea, Ph.D. thesis, LMU M¨unchen (2004). Y. Kakutani, Y. Kubo, A. Koizumi, N. Sakai, B. L. Ahuja,and B. K. Sarma, J. Phys. Soc. Japan , 599 (2003). J. S. Faulkner, Prog. Mater. Sci. , 1 (1982). A. Gonis,