Formation of hyperfine fields in alloys A. K. Arzhnikov and L. V. Dobysheva Physical-Technical Institute, Ural Branch of Russian Academy of Sciences, Izhevsk, Russia ThisworkdealswiththeanalysisofexperimentaldataontheaveragemagnetizationofFe1−xMex 1 (Me=Sn,Si) disordered alloys, the average and local hyperfine fields (HFF) at the Fe nuclei. The 0 effect of the metalloid concentration on the HFF is studied with the help of the results of first- 0 principles calculations of ordered alloys. The disorder is taken into account by means of model 2 systems. The dependences obtained correspond to those experimentally observed. Experimental data on the ratio of the average HFF at Fe nuclei to the average magnetisation in alloys with n a sp-elements show that the ratio decreases proportionally with the metalloid concentration. This J change in the ratio is bound up with three factors. First, the contribution of the valence electron 4 polarizationbytheneighboringatoms,thatispositive(unlikethepolarization bytheownmagnetic moment), increases with thechange of thedisorder degree (increase of concentration). Second, the ] appearance of the impurities, i.e. metalloid atoms, in the nearest environment of Fe leads to the i c orbital momentincrease. And,finally,thechangeof thedisorderdegree,asin thefirstcase, results s in an increase in the orbital magnetic moment and its positive contribution to theHFF. The value - and thedegree of theinfluenceof these contributions to theHFFis discussed. l r t m 75.50.Bb, 71.15.Ap t. In the experimental physics of solids there are few disorderedalloysaretakenfromthefirst-principlescalcu- a methods for measuring local characteristics. Prominent lationsofthetranslationallyinvariantsystems,supposing m among them are nuclear techniques (e.g. Mossbauer the interactionbetween the clusters to be not significant - spectroscopy) determining the hyperfine magnetic fields because of small free length of electron in these alloys. d (HFF) at nuclei. It is widely believed that these tech- Nowadays, there is no possibility to conduct these cal- n o niquescanreflectsomechemicalandtopologicalfeatures culations for the disorderedsystems, moreover,it should c of the atomic surroundings of the excited nuclei. In- be noted that even calculations of the ordered alloys of- [ deed,innumerousstudiesthesespectraareinterpretedin ten do not give the required agreement with experiment terms of phenomenological models considering only the and revealonly the main features of the HFF and LMM 1 v nearest atomic environment (see e.g. [1-3]). There is behavior. 3 no doubt that such a description is often useful and effi- The calculations were performed by the full-potential 3 cient. Wethink,however,thattherearesomecaseswhen linearized augmented plane wave method (FLAPW) us- 0 sucha simplified approachis notjustified. Moreover,for ing the WIEN-97 program package [5]. The results are 1 the spectra of transition metals and alloys on their ba- presented in Table 1. 0 sis, the successful description within the framework of The systems were simulated on a BCC lattice which 1 such models is rather an exception than a rule, and ev- in the disordered alloys under consideration is retained 0 / erytimeadditionalargumentsareneededtojustifythese within a wide concentration range [1]. The lattice pa- t a limitations. For example, such an approach makes some rameters were chosen in accordance with the experi- m sense in the case of disordered systems due to strong lo- mental values for x=3.125 at.% and x=6.25 at.%. It - calizationofthed-electronsresponsibleforthe itinerant should be mentioned that even at small concentrations d magnetism of transition metals [4]. This is one of the the BCC lattice is somewhat distorted due to the re- n reasons why sometimes the spectra of disordered alloys pulsion/attraction by the Sn/Si atom of the surround- o are successfully described by the Jaccarino-Walker-type ing Fe atoms. As shown in our paper [6], the changes in c : models where the local magnetic moments (LMM) and magneticcharacteristicsbecauseofthisrelaxationarein- v HFFs are assumed to be proportional to the number of significant. Though the results in Table 1 were obtained i X metalloid atoms in the nearest environment. with allowance for this relaxation, we do not discuss it Inouropinion,eveninthiscasetheexperimentalspec- here. r a tra may provide more reliable and complete information 1. AVERAGE AND LOCAL MAGNETIC MO- without the use of phenomenologicalmodels and restric- MENTS. tiontothenearestenvironment. Oneshouldthenanalyze Fig.1 presents the experimental data and the calcu- the HFF peculiarities from the ”first-principles” calcula- latedaveragesofthe magneticmomentperone Fe atom. tions. This work is devoted to such an analysis of the The average magnetic moment M = P Mdi + Mint, i hyperfine magnetic fields, magnetization and local mag- whereMdi isthespinmagneticmomentoverthemuffin- netic moments and their interrelation in the disordered tin (MT) sphere of the i-th Fe atom (hereinafter we will Fe1−xSnx, Fe1−xSix alloys. Here the magnetic charac- refer to this value as local magnetic moment, LMM), teristicsofclusterswithagivenimpurityconfigurationin Mintisthespinmagneticmomentovertheunitcellwith- 1 2.38 third mechanism of the LMM reduction due to the dif- Fe-Si (exp.) 2.36 Fe-Si (Fcael-cS.-nin (teexrpp..)) ferenceintheimpurity-potentialscreeningbyd-electrons Fe-Sn (calc.-interp.) 2.34 (the difference in pushing out the impurity levels by the bandswithspinupanddown),whichwasrevealedinthis 2.32 workbyacomparisonofthenumberofd-electronswithin 2.3 the MT-sphere for different Fe-atom positions (there are 2.28 more d-electrons near the impurity, Table 1). M 2.26 C. The LMMs are concentration dependent. So the 2.24 LMM of the Fe atom closest to the Si atom is 2.181 2.22 µB at x=3.125 at.% (Fe31Si), and 2.262 µB at x=6.25 2.2 at.% (Fe15Si). This LMM increase with concentration 2.18 holds in general for all the non-equivalent positions (Ta- 2.16 ble 1). However, in spite of the LMM increase, the aver- 2.140 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 agemagneticmomentintheFe1−xSix systemsomewhat FIG.1. Theexperimentaldxa (tata.%)andthecalculatedaverages decreases 2.238 µB -¿ 2.230 µB (Fig.1). This is the re- sult of a higher probability of finding a Fe atom with a of the magnetic moment perone Fe atom neighboring impurity, which increases the number of Fe atoms having lower LMM values. outtheMT-spheres. ThemaincontributiontotheLMM 2. HYPERFINE MAGNETIC FIELDS AT NUCLEI. comesfromthed-electronsthatarealmostentirelyinside TheprogrampackageWIEN-97allowsonetocalculate the MT-sphere, whereas M is formed by the s- and p- the interaction between the nucleus magnetic moment int electrons and has a small negative value as compared to and the spin and orbital magnetic moments of the elec- the LMM. We do not take into account the contribution tron subsystem. The spin dipole contributions are small oftheorbitalmagneticmomentthatisabout0.045µ in ( 2 ÷ 3 kGs), they are suppressed due to the symmetry B ourcalculations. Inexperimentsthis valueis1.5?2times relations,andthemaincontributioncomesfromthespin larger and comprises 0.08 µ [7]. It is believed that polarization at the nucleus (Fermi-contact interaction) B the inclusion of the orbital polarizationin the exchange- and the orbital magnetic moment. The Fermi-contact correlationpotential[8]improvestheagreementbetween interaction may be divided into Hicore (the core-electron thecalculationsandtheexperimentalvalues,butherewe polarization) and Hival (the valence-electron polariza- didn’tusesuchapotentialsupposingthatthe calculated tion),andhence,theresultedfieldatthei-thsitecontains value can be multiplied by the factor close to two. The three terms: Hi = Hicore+Hival +Hiorb. For the core- main reason for the neglect of the orbital contribution electron polarization the simple relation Hicore =γsMdi lies inthe fact thatits variationswithconcentrationand issatisfied,whereγs does notdepend toahighaccuracy configuration of metalloid atoms are small as compared onneitherthemetalloidtypenoritsconcentration,andis toMevenwithallowanceoftheactualvalueandamounts determined only by the approximationfor the exchange- less than 1 % (see the last but one column in Table 1), correlation potential. Here we use the GGA approxima- whereas the Mdi variations range up to 15 %. Fig.1 dis- tion [11] which gives γs ≈ 123kGs/µB. In our opinion plays a rather good agreement of the magnetic moment just this simple expressionbetween the LMM and Hcore withtheexperimentaldata,thoughthetheoreticalvalues makes it possible to use phenomenological models ne- of the magnetic moment are somewhat higher than the glecting the effect of the atoms in the second, third, etc. experimentalones. Webelievethatthedisorder,whichis coordination spheres. Really, from Table 1 one can see not taken into account here, reduces the magnetization that the main distinctions in LMM at a certain concen- by 2-3 % [9]. Let us mention some peculiarities of the tration are connected with the presence or absence of magnetic moments formation. the metalloid atom in nearestenvironment. So, this also A.Themagneticmomentdoesnotdependonthemet- determines the variations in Hicore. Though the experi- alloid type and, as shown in Ref.10, is governed by the mentsshowedthattheproportionalitybetweenthemag- lattice parameter,a. Atequalconcentrations,the lattice netic moment and HFF is not as good as we found, and parameter of the Sn alloy is greater than that of the Si the proportionality coefficient essentially decreases with alloy,sothemagneticmomentinthefirstcaseisgreater. concentration. As we shall see later this is connected B. The LMM ofthe Fe atomclosestto the impurity is with the other two contributions to the HFF. of the smallest among others value (Table 1). As noted The proportionality Hiorb = γorbMiorb is fulfilled in a in Ref.10, this difference is defined by the competition somewhatworseway,butstillrathersatisfactorily. Note, between two mechanisms: the LMM reduction due to however,thatγorb is positiveandaboutfivetimes larger flattening of the d-band because of the s-d hybridization in magnitude than γs. Hence, if the changes of Miorb af- that is the strongest near the impurity, and the LMM fect the magnetic moment only slightly, the changes in increasedue to narrowingof the d-bandbecause of a de- Horbimay amountto 20 kGs evenatlow concentrations crease of the wave function overlap. There also exists (see Table 1). The nature of these changes in disordered alloysisdiscussedinmoredetailinRef.12. Herewemen- 2 6 355 Fe15Sn Fe31Sn 5 vertical bars show distaanncde iinn FFee3115SSnn 350 345 4 340 3 Hval (kGs/mu) 2 H (kGs) 333305 1 325 0 320 -1 315 HH10 ((FFee--SSnn)) H0 (Fe-Si) H1 (Fe-Si) -2 310 4 5 6 7 8 9 10 11 12 13 14 0 2 4 6 8 10 12 r (a.u.) x (at.%) FIG. 2. The polarizations of the valence electrons by a FIG.3. TheexperimentalH0andH1asafunctionofcon- Fe-atom magnetic moment as a function of distance centration -40 tion only the main features. The Horb increases along noon eim impupruitrieitsy iinn nneeaarreesstt eennvviirroonnmmeenntt i with the Morb increase with concentration. The Horb i i -35 takes the largest value at the atom closest to the metal- loid atom. The increase of Horb with allowance for the i -30 actual values of Morb, that are twice as large as the cal- i culatedone,comprises15÷20kGsevenatlowconcentra- Gs) tions ascomparedto thatinpureFe. Onthe strengthof Hval (k -25 thequalitativecharacterofthelaststatement,wecansay aboutthetendencyofthevariations. Finally,asshownin -20 [10],the orbitalcontributionincreasesalsowith disorder (that is, with concentration). -15 The Hval behaves in a more complicated way. This i is primarily associated with strong delocalization of the -10 0 1 2 3 4 5 6 7 s- and p-like electrons that interfere at sites with dif- x (at.%) FIG.4. The configurationally averaged negative contribu- ferent magnetic and charge properties, and therefore the Hval behavior cannot be in fact quantitatively pre- tion Hval i dicted. However, we succeeded in revealing some quali- tative regularities supported by experimental evidence. differences in the local magnetic moment that primarily First of all, we analyzed the valence contribution us- affect Hcore and Horb. Fig.3 presents the experimental ing the simple functional dependence of the magnetic H0 andH1 asafunctionofconcentration. Table1shows moment screening in the RKKY (Ruderman - Kittel that the difference between these quantities can be suc- - Kasuya - Yosida) theory, as it was done in Ref.9: cessfully explained by the LMM magnitude and hence Hival =A+BPjMjdcos(2kfr/T+φ)/r3. Suchasimpli- by the core-electronpolarization. However,if everything fiedtreatmentof Hval is hardlyjustified inour case,but were determined only by Hcore we should expect an in- i i we hope to havedetermined the main qualitative depen- crease in H0 and H1 with concentration in accordance dences. Solving the inverse task for the ordered cluster with the LMM increase,which is not the case,as for the ofsize200a.u. wereceivethemostprobablevaluesofA, Fe1−xSnx alloy the magnetic moment increases much B, T and φ in the alloys Fe15Sn and Fe31Sn. The cor- more quickly than H0 and H1, and there is no increase responding functions of Bcos(2k r/T +φ)/r3 are shown at all for Si. In reality the expected increase is com- f in Fig.2. Of special interest is the fact that for both pensated by the decrease in magnitude of the configura- concentrations the spin polarization of electrons is posi- tionallyaveragednegativecontributionHval. Fig.4gives tive for the I and II coordination spheres, which entirely theaveragedvaluesofHval ofadisorderedclusterofsize contradicttheresultsofasimilarprocessingoftheexper- 200 a.u. with a certain number of impurities in the first imentaldata. Thisisduetothefactthatduringprocess- coordination sphere. The averaging was performed in ing of the experimental data the difference between H0 assumption that the Fe atoms are distributed randomly (HFFatthenucleusoftheFeatomwithoutthemetalloid and polarize the conductivity electrons at distance r ac- atoms in its nearest environment) and H1 (with one im- cordingtothemodelfunctionforconcentrations6.25and purity atom in the nearest environment) was attributed 3.125 % (see Fig.2). The decrease of the averaged Hval to the changes in Hval, whereas in an alloy there are with concentration is due to the positive values of the 3 RKKY polarization at the distance of the first and sec- ondcoordinationspheres (see Fig.2). Two mainfeatures in the behavior of the Hval averagedover configurations can be noticed. First, the magnitude of H at the Fe val atom without impurities in nearest environment is less [1] E.P. Yelsukov. Phys. Met. Metallogr. 76 (1993) N5, 451. by 5-10 kGs than H at the Fe atom with one impu- val [2] N.N. Delyagin et al. Journal Eksp. i Teoret. Phys. 116 rity in the environment, and second, the magnitude of (1999) N 1(7), 130. [3] B. Rodmacq et al. Phys. Rev. B the averagedH decreases with concentrationfor both val 21 (1980) N 5, 1911. [4] A.K. Arzhnikov and L.V. Doby- configurations of the environment. sheva. JMMM 117 (1992) 87. [5] P. Blaha, K. Schwarz, Thus, the use of the ”first-principles” calculations and J. Luitz. WIEN97, A Full Potential Linearized Aug- makes it possible to explain the main peculiarities of mented Plane Wave Package for Calculating Crystal Prop- the HFF behavior in the low-concentration disordered erties (Karlheinz Schwarz, Techn. Universit”at Wien, Aus- alloys Fe1−xMex. The difference between H0 and H1 tria). (1999) ISBN 3- 9501031-0-4. [6] A.K. Arzhnikov and H0 −H1 ≈ −20 kGs consists of H0val −H1val ≈ 7÷10 L.V.Dobysheva.Phys.Rev.B62(2000)5324.[7]H.Akaiet kGs,H0core−H1core ≈−15÷−25kGsandH0orb−H1orb ≈ al.Progr.ofTheor.Phys.Suppl.101(1990)11.[8]J.Kunes −5÷−10kGs. The decreaseof the proportionalitycoef- andP.Novac.J.Phys.C11(1999)6301.[9]A.K.Arzhnikov, ficient between the HFF and the magnetic moment with L.V.Dobysheva,E.P. ElsukovandA.V.Zagainov. JETP83 concentration results from the decrease of magnitude of (1996) 623. [10] A.K. Arzhnikov, L.V. Dobysheva, and F. H andincreaseofmagnitudeofH thatareopposite Brouers. Phys. Sol. St. 42 (2000) 89. [11] J.P.Perdew et al. val orb to the H . Phys. Rev. Lett. 77 (1996) 3865. [12] A.K. Arzhnikov and core We believe that the general relations obtained here L.V.Dobysheva.IzvestiyaAkademiiNauk(tobepresented) willbe useful inprocessingthe experimentaldata onthe (in Russian). HFFs in disordered alloys of transition metals and non- magnetic impurities. The present work was supported by Russian Founda- tionforFundamentalResearches,GrantNo00-02-17355. 4 TABLE I. The results of the calculations: Configuration of impurities in the Fe-atom environment, [nm...] denotes the numberofmetalloidatomsinthefirst(n),second(m)etc. spheres. NumberofsuchFeatomsintheunitcell,NFe. Numberof d-electrons in the MT sphere, Nd. Magnetic moment, Md, in the MT sphere. Contribution of the core-electrons polarization to the HFF, Hcore. Contribution of the valence-electrons polarization to the HFF, Hval. Orbital magnetic moment, Morb. Orbital contribution to the HFF,Horb. [nm...], NFe Nd Md,µB Hcore,kGs Hval,kGs Morb, µB Horb,kGs Fe31Sn, [1000], 8 6.006 2.246 -277.50 -37.59 .0489 27.94 a=21.804a.u. [0003], 8 5.951 2.507 -310.35 -30.94 .0483 26.41 [0100], 6 5.964 2.432 -300.17 -34.84 .0482 26.62 [0020], 6 5.963 2.425 -299.79 -22.39 .0479 26.28 [0000], 2 5.987 2.391 -295.13 -34.34 .0441 23.66 [0000], 1 5.959 2.423 -299.68 -24.17 .0470 25.15 Fe15Sn, [1003], 8 6.114 2.442 -301.01 -20.25 .0548 29.77 2a=21.985a.u. [0200], 3 6.090 2.530 -312.64 -29.97 .0535 29.08 [0040], 3 6.084 2.542 -314.55 -2.46 .0550 29.38 [0000], 1 6.142 2.390 -294.33 -29.97 .0450 23.32 Fe31Si, [1000], 8 5.999 2.181 -269.44 -38.68 .0497 28.27 a=21.604 a.u. [0003], 8 5.967 2.413 -297.89 -35.15 .0471 25.55 [0100], 6 5.977 2.343 -288.51 -31.31 .0446 24.46 [0020], 6 5.965 2.385 -294.06 -16.92 .0468 25.32 [0000], 2 5.965 2.489 -308.13 -19.72 .0485 26.44 [0000], 1 5.969 2.389 -294.81 -13.56 .0451 24.28 Fe15Si, [1003], 8 6.130 2.262 -277.20 -29.11 .0594 32.38 2a=21.585a.u. [0200], 3 6.115 2.353 -289.52 -21.64 .0456 24.44 [0040], 3 6.094 2.432 -298.87 5.49 .0509 27.31 [0000], 1 6.094 2.536 -313.82 -13.89 .0496 26.83 5