O. Gunnarsson- Band model for magnetism of transition metals in the spin-density-functional formalism

8/3/2019 O. Gunnarsson- Band model for magnetism of transition metals in the spin-density-functional formalism

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1. Phys. F: Metal Phys., Vol. 6, No.4. 1976. Printed in Great Britain. 1976

Band model for magnetism of transition metals in the spin-density-functional formalism

o GunnarssontInstitute of Theoretical Physics. Fack, S-402 20 Goteborg 5, Sweden

Received 23 July 1975. in final form 24 November 1975

Abstract. The Hohenberg-Kohn-Sharn spin-density-functional (SDF} formalism is appliedto calculate magnetic properties of transition metals. A Stoner-like band model is derivedin the SDF formalism. The solutions of the SDr equations are assumed to have Blochcharacter and perturbation theory is used to show that the energy splitting betweenspin up and spin down states is approximately wave vector independent and proportionalto an energy-dependent Stoner parameter. This result makes it possible to obtain magneticproperties from the paramagnetic density of states and the Stoner parameter alone.

Results in the local-spin-density approximation for the Stoner parameters of V, Fe,Co, Ni, Pd, and Pt are presented. The relative stability of the para and ferromagneticstates is found to be correct for all the elements investigated and the total magnetizationcompares favourably with experiment. Values for the Curie temperature are systematicallytoo high. Experimental estimates of the Stoner parameter are compared with the calcu-lated values and the deviation is at most a few tenths of an eV.

1 . I nt ro duc ti onA major difficulty in the description of magnetism in transition metals stems fromthe d electrons being neither completely localized nor completely itinerant The bandtheories of magnetism, e.g., the Stoner model, stress the itineracy. They can explainthe low temperature properties (e.g., Herring 1966, Cracknell 1971, Freeman et al1975) of several transition metals in terms of an adjusted band structure thoughthe construction of a first principle potential required in the calculation is still amajor problem (Brinkman 1973).

The Hartree-Fock approximation is frequently used as a basis for band theories(Wohlfarth 1953. 1968). However, the importance of correlation for magnetic proper-ties was pointed out at an early stage by, e.g., Wigner (1934) and Wohlfarth (1953).Band theories can also be obtained in the Kohn-Sham scheme (Kohn and Sham1965) in its generalization to a spin-density-functional ( SDF ) formalism. This formalismallows the inclusion of correlation effects, while retaining the conceptual and compu-tational simplicity of the band picture. In principle, an improved description of corre-lation effects involves 'simply' an improved construction of the exchange-correlationfunction occurring in the theory.

Band theories can include a tendency to formation of localized moments, i.e.the propagating states may have site- and spin-dependent amplitudes, giving magnetict Present address: Institut fur Festkorperforschung der Kernforschungsanlage Julich, 517 Jiilich, Germany

587


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588 o Gunnarssonmoments varying in magnitude and direction from site to site (Friedel et al 1961 ,Lederer and Blandin 1966, Lederer 1966). This possibility is not taken into accountin this paper, where the electrons are assumed to have Bloch character. The purposeis (i) to derive a Stoner-like band model in the SDF formalism and (ii) to comparethe predictions of this model with experiment, using the local-spin-density (LSD)approximation for exchange and correlation. In the derivation, an expression forthe energy splitting of bands with different spins is calculated using perturbationtheory. The splitting is approximately wavevector independent and proportional toa generalized, energy-dependent Stoner parameter. Using the LSD approximation, thisparameter has been calculated for a number of transition metals (V, Fe, Co, Ni,Pd, Pt) and the results show systematic trends within and between the rows of theperiodic system. Neglect of the weak wavevector dependence of the band splittingallows magnetic properties to be expressed simply in terms of the density of statesand the Stoner parameter. By using densities of states from published band calcula-tions, the theory is found to give systematically correct predictions for the relativestability of the para- and ferromagnetic phases for all the elements studied. Thistest is significant, as the same analysis using the X C I . approximation (Slater 1974)predicts ferromagnetism for all these metals, except Pt, if recommended (Slater 1974)values of C I. are used. The difference is due to inclusion of correlation in the LSDapproximation. The results for the total magnetization are reasonable, but the calcu-lated Curie temperatures are systematically too high, as expected for a Stoner model.Comparison with experimental estimates of the Stoner parameter suggests that theerror in the calculated values is at most a few tenths of an eV.

The adequacy of the LSD approximation for the description of magnetism in transi-tion metals has independently been investigated by Madsen et al (1976). Their self-consistent band calculations using the atomic sphere approximation (Andersen 1973,1975) with neglect of hybridization were interpreted in terms of Stoner theory andcanonical d state density. Their approach is complementary, in that they focus onthe volume and structure dependence and hence on band theory.In 2 the SDF formalism for T = 1 = 0 is presented. Following the derivation ofa Stoner-like theory and stability criteria, formulae for the magnetic moment andthe Curie temperature are obtained. In ~ 3 results for the Stoner parameter for V,Fe, Co, Ni, Pd, and Pt are presented. The relative stabilities of the para- and ferro-magnetic phases are compared and results for the total magnetization and Curietemperature are given. In ~ 4 it is shown that charge fluctuations are substantiallysuppressed in the LSD approximation, the trends in the calculated Stoner parametersare explained, the calculated values are compared with experimental estimates ofthe Stoner parameter and the high results for the Curie temperature are discussed.

2. Derivation of a Stoner-like model in the SDF formalismThe main features of the Stoner model (see Stoner 1939 and references therein)are (i) the itinerant description of the electrons, originally assumed to be in a parabolicband, and (ii) the introduction of an exchange energy, which is proportional to themagnetization squared-the constant of proportionality being called the Stoner para-meter. This model has been used in discussing many magnetic properties, includingthe first semiquantitative discussion of a transition metal band structure by Slater(1936) for Ni, which was correctly predicted to be ferromagnetic. The Stoner model


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Band m odel for m agnetism of transition m etals 589has been derived in the Hartree-Fock approximation by, e.g., Wohlfarth ( 19 53 , 1 96 8) ,who also gave an explicit formula for the Stoner parameter including some correla-tion.

In the present derivation of a Stoner model the main approximations are theuse of a Bloch representation for all states and the LSD approximation, which isintroduced to calculate Stoner parameters. This approach neglects effects of localized,fluctuating moments, which may be present in the system. The main problem inthe derivation is to determine the energy splitting of bands with different spins. Thesplitting is due to a lowering (raising) of the exchange-correlation potential, andthus a lowering (raising) of the energy of majority (minority) electrons, when thesystem is spin polarized. Using first order perturbation theory, the band splittingis obtained as the expectation value of potential splitting, which is related to thespin polarization. It is shown in appendix 1 that the spin polarization is essentiallydue to spin-flips of d electrons at the Fermi surface. Neglecting other contributionsan explicit expression for the spin polarization is obtained (equation 2.9). In appendix2 the nonspherical part of the spin polarization is found to be of little importancefor the properties studied, and neglecting it a formula for the potential splittingis derived. The calculation of the expectation value of this potential is simplifiedby the small weight in the d band of angular quantum numbers different from twoand a simple expression for the band splitting is obtained (equations 2.14 and 2.15)and compared with the results of a spin polarized band calculation.

2.1 . The SD F formalismThe SDF formalism for T = 0 has been discussed in several papers (Kohn and Sham1965, Stoddart and March 1971, von Barth and Hedin 1972, Gunnarsson e t al 1 " 9 7 2 ,Rajagopal and Callaway 1973, Gunnarsson and Lundqvist 1976). In the latter paperthe extension to T = 1 = 0 was treated and we give the basic equations for this case.The equations are in a particularly simple form for a ferromagnetic metal, for whichthe spin-density everywhere is parallel or antiparallel to some fixed direction, thesame all over space. The spin-density at the temperature T is given by

(2 .1 )where 5 = +, - is a spin index,f{Ei.s) = {I + exp[(Ei.s - . u ) / k T ] } -1is the usual Fermi-Dirac function and 1 1 is the chemical potential. The functions t / J i.5(r)are the solutionsof a Schrodinger-like equation

(172 {72 . ( ) 2 f p(r') d3, .x c ( )) .t, () - ,I, ()-2niV +vsr +e Ir-,I r +vs r 'i'i,sr -ti,s'i'i,sr (2.2)

where vj_r) is some external, spin-dependent potential and p(r) = p+(r) + p_(r) is thetotal density. Equations (2.1 and 2.2) are in principle exact. In practice approximationsfor the exchange-correlation potentials are necessary, and the local-spin-density (LSD)approximation is used in the numerical calculations. This approximation is exact in thelimit of spatially slow and weak density variations and has been found to have a broadrange of applicability (Tong and Sham 1966, Lang 1973, Smith et al 1973, Lang andWilliams 1975, Janak et al 1975, Gunnarsson and Johansson 1976). The equation (2.2)is not invariant under rotations in spin space if the LSD approximation is used. However,this is of minor importance for a ferromagnetic system, which has a preferred direction


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590 0 Gunnarssondefined by the magnetic moment. In the LSD approximation the exchange-correlationpotentials v ~ c are related to the exchange-correlation part f X C of the free energy perelectron of a homogeneous electron liquid. For applications in the temperaturerange considered here, however, data for T = 0 can be used (Gunnarsson andLundqvist 1976). Then one gets

v~C (r) ~ ap~r) {p(r)Exc [p + (r), P _ (r)] J (2.3)where EXC is the exchange-correlation energy per electron of a homogeneous electronliquid. Numerical results for these potentials are given in the literature (von Barthand Hedin 1972, Gunnarsson and Lundqvist 1976). Potentials from the latter paperare used here and are given as an interpolation formula in terms of the densityparameter rs (4nr:a5/3 = lip) and the fractional spin-polarization (= mlp(m = p+ - p_ )

X C ( r) x ( f l + 1 b( )v :! r s'" = 1 1 f1 - " 3 1 (" . (2.4)Here J ,t = -2/(mxr s), C( =(4/91t)113,3 = 1 + 0'0545r5In(1 + 114/r s),b = 1 - 0'036r5 - 1 36 r J (l + lO r5) and y =0.297.For the magnetic properties the most important parameter is b , which for smallspin-polarizations determines the splitting of the potentials. The b used here andthe result of von Barth and Hedin (1972), both with correlation included, are plottedin figure 1. In the limit of small spin-polarizations the potentials (2.4) can be comparedwith the x~ potential (Slater 1974) and the Slater potential (Slater 1951a), whichwould then correspond to b equal to !r i and !, respectively. These two values forb are plotted in figure 1 using ri=0'70, a value typical of those recommended for

12-,-'"

BH . . . _ - - - - - --------

Figure 1. The function b(r, ) describing the spin splitting of the exchange-correlation poten-tials (equation (2.4)). The full curve shows the b(r, l used here and the broken curvethe result of von Barth and Hedin (BH) (l972). The chain curves give the b( rs l derivedfrom the X:< and Slater potentials. Correlation effects are included in the former twocurves, which give considerably smaller b(r, ) .


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Band model for magnetism of transition metals 591transition metals (Slater 1974). If only exchange is included in the potential (2.3) theso called Kohn-Sham-Gaspar potential is obtained (Kohn and Sham 1965). Thispotential is represented by '].=iand corresponds to b(rs) = = 1. Figure 1 shows thatcorrelation reduces 5 substantially.

It has been argued that some correlation can be included by using a ex valuelarger than ~ as the X']. potential then is closer to the potential (2.4) in the paramag-netic limit. However, figure 1 shows that to simulate the correlation effects on thespin-dependence of the potential (2.4) an']. value smaller than f is required.

The interpolation formula (2.4) was constructed to fit the calculated electron liquiddata in the rs range 1-5. For the paramagnetic phase the interpolation formula (2.4)gives a somewhat more attractive potential for small rs than the result of an rsexpansion (Carr and Maradudin 1964). This is not important in the present paper,where we are just interested in the spin-dependence. In a full band calculation onemight, however, use a more accurate paramagnetic potential (Hedin and Lundqvist1971) and obtain the spin-dependence from equation (2.4).

2.2 . Band splitting and spin polarizationFirst the band splitting is related to the spin polarization. The band energies andwavefunctions of the paramagnetic state are assumed to be known. The effects ofintroducing a spin polarization are calculated using perturbation theory. As in theoriginal Stoner theory, the net magnetization per atom

(2.5)

is used as an expansion parameter, and effects of first order in t1 m are considered.The exchange-correlation potentials v~ c in equation (2.4) can be expanded as

(2.6)where v(l\ is determined from the spatial distribution of the spin polarization.Equation (2.6) gives the main contribution to the band splitting, which to firstorder is given by the expectation value of equation (2.6)

In addition, the spin polarization causes a shift of the electrostatic potential, whichcontains a small first order term in t1m. To lowest order in t1m, however, this shiftis spin-independent and does not contribute to equation (2.7).The next step is to find the spin polarization m(r) , from which v(i) and &,," can becalculated. In the remaining part of this subsection it will be shown that the sphericalaverage of the spin polarization is

(m(r) = (~m/41!)~(r,Ed (2.8)The function ,(r,E) is the solution of the radial Schrodinger equation for the energyE and the angular quantum number t, normalized to the Wigner-Seitz sphere.


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592 o GunnarssonThe spin polarization is caused by two effects: ( i) Electrons in states close to

the Fermi surface flip their spins due to the changes in the chemical potentials forspin up and spin down electrons when the system is spin polarized. (ii) Spin polariza-tion causes modifications of the potentials, giving changes in the wavefunctionsof all states. These changes are different for spin up and spin down states givinga spin polarization. In appendix 1 it is argued that the first effect dominates. Accord-ing to equation (2.1) it gives the spin polarization

m ( r ) =L I t l t k n { r W U(ek n + ) - ! ( E k n - ) ]kl l (2.9)to lowest order in ~m.For the wavefunctions t I t , , " it IS convenient to use a one-centre expansion, (e.g.Seitz 1940)

(2.10)

where the vectors R give the positions of the lattice sites, N is the number of sites and

< 1 > k n { r )=L c l m ( k n ) 4> , (r , E k n ) Y i m ( r )1m

inside the Wigner-Seitz sphere=0 elsewhere. (2.11)

Here and in the following the Wigner-Seitz cell is approximated by a sphere. Thefunctions Y i m ( r ) are spherical harmonics and c ' m ( k n ) are expansion coefficients. Fora muffin-tin potential the wavefunction can always be written as in equation (2.11).

Only states close to the Fermi level contribute to the spin polarization (2.9),and we will now show that these states have essentially d character. In the lowestsix unhybridized valence bands the number of electrons per atom with d and spcharacter are five and one, respectively. Further, the width of the sp band is a factortwo to three times larger than the d bandwidth. Assuming that the density of statesis proportional to the number of electrons and inversely proportional to the band-width, one finds that the d electrons contribute about 90--95 per cent to the densityof states within the d band. These simple arguments are supported by band calcula-tions. For instance, for Cu with a band structure similar to the FCC transition metals,it has been reported that within the d band energy range the d character is alwaysmore than 90 per cent and for most states more than 95 per cent (Wood 1967).It is therefore a good approximation to assume that only 1 =2 terms contributeto the spin polarization. This simplifies the calculations considerably.

In appendix 2 the non-spherical parts of the spin polarization are found to contrib-ute little to the properties considered (in, for instance, the stability criteria (2.16and 2.17) below it gives a contribution which is less than four per cent of the leadingone). Therefore, only the spherical average


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Band model for magnetism of transition metals 593where the energy argument E F in 4 > 2 ( 1 ' , E ) is correct to first order in L\m. As4>2(r ,EF) is normalized to the Wigner-Seitz sphere the sum in equation (2.12) has asimple relation to the net magnetization L\m, and one obtains formula (2.8).

2.3 . S toner param eterFrom the spin polarization (2.8) the exchange-correlation potentials in equation (2.4,6)can easily be calculated asv~ (r) - v~ (r) = ilm v ~f) (r) = 6 1n ,u'(rs} 6 (r s) L \m [

~(r 'E~t~(r 'EF) . (2.15)This is the final formula for the energy splitting. According to the definition (2.7)this quantity depends a priori on both the wavevector k and the energy E, whileour final expression depends on k only implicitly through the energy. Besides theexplicit formula for I (E) , the k independence is the main result of the analysis. Itfollows from the physically well motivated assumptions that (i) the states under con-sideration are mainly of d character and that (ii) the non-spherical part of the spinpolarization gives a small contribution to the band splitting.

The general form as well as the magnitude of the results in equations (2.14 and2.15) was tested by a comparison with a full spin polarized band calculation forFe of Wakoh and Yamashita (1966). As this calculation was done using the X ' Y . .method with 'Y..= 0.5. we have in this particular case calculated the Stoner parameterwith a b value which is consistent with the X'Y..method, i.e. with b = ! : J . . . The wavefunc-tions q)2( r ,EF) and q)2( r ,E) were obtained by solving the radial Schrodinger equationwith the potential being an average of the majority and minority spin potentialscalculated by Wakoh and Yamashita. The results for the band splitting as a functionof the energy is shown in figure 2. The results calculated from equations (2.14 and2.15) are compared with the values obtained by Wakoh and Yamashita for stateswith predominantly d character at various symmetry points. The figure shows thatthe predicted k independence is essentially fulfilled and that the present results alsoagree well in magnitude with the band calculation, the deviation being typically 5-10per cent. The agreement is a test of all the approximations except the use of aspherical exchange-correlation potential, as Wakoh and Yamashita, too, neglectednon-spherical components of the potential.

The energy dependence is a consequence of the shape of the wavefunctions (seeequation (2.15)). The spin polarization is caused by spin flips of electrons at the


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594 o Gunnarsson20 .... ------------ .. .....

'"l

OO_~-- .....--~--"---~--~-05 -03 -01 00

Figure 2. The spin splitting of the energy bands as a function of the energy. The trianglesshow the splittings for various symmetry points in the Brillouin zone obtained in afull band calculation (Wakoh and Yamashita 1966).Our results calculated from equations(2.14 and 2.15)are given by the curve. The figure shows that the band splitting is energy-dependent but has only a weak k dependence.

Fermi energy, which is near the top of the d band for the metals to the right ofthe 3d series, e.g., iron, and the corresponding wavefunctions are rather contractedtowards the nuclei (Wood 1960). Thus the spin polarization is concentrated to theinner parts of the atoms and the difference between the potentials for spin up andspin down electrons is consequently largest there. This difference is effectively pickedup by the contracted wavefunctions at the top of the band, while states at the bottomwith more extended wavefunctions are less influenced.

The discussion so far has essentially been within the LSD approximation, butthe arguments are believed to be of more general validity. Assumption (i) aboutthe d character is essentially independent of the approximation for the exchange-cor-relation potential, while this approximation of course has to be specified to discussassumption (ii) about a spherical potential. For potentials with a smooth non localdependence on the density there should however be a tendency to average out thenon-spherical part of the density. For such potentials assumption (ii) might thereforebe even better than in the LSD approximation. The only change in the derivationabove would then be a replacement of the exchange-correlation potential (2.13), whichwould give a new expression for the Stoner parameter.

2.4 . M agnetic propertie sThe magnetic ground-state properties can be derived from equations (2.1, 2.2, 2.14and 2.15). The essential k independence of the band splitting enables these propertiesto be related to the density of states, which contains the necessary information aboutthe band structure, and the Stoner parameter, which gives the required informationabout the wavefunctions. Below we give some results, which are needed for the calcu-lations. The Curie temperature is the temperature at which (Stoner 1936)

(2.16)

where N(e) is the density of states per atom and spin. At T = 0 this equation gives the


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Band model for magnetism of transition metals 595Stoner criteria for ferromagnetism (Stoner 1936)

I(E F ) N (E F ) > 1 .The susceptibility is given by:

(2.17)

(2.18)where X O is the susceptibility for non-interacting electrons. To calculate themagnetization at T = 0, we introduce E (n ) , the energy to which the band has to befilled to accumulate n electrons per atom and spin.r " dn'E{n) = Jo N[E(n') ]Ifthe spin up and spin down bands are filled with n + = (n + t1.m)j2 and n : = (n - t1.m)j2 ,respectively, the chemical potentials become

(2.19)

Il =E{n: r ) = + = 11.m I (EF )to lowest order in Sm. Equilibrium requires

tu: 1 ( C E C E ) 1o(t1.m) ="2 cn+ - on_ = 2 :{Il+ - Il-) = 0(2.20)

(2.21)

where E is the total energy of the system. Using equations (2.19 and 2.20) thecondition (2.21) is rewritten as (Lomer 1967)

1 n ~ dn'I (EF ) = t1.m1 N[E(n') ] = = F (t 1.m ) (2.22)

This gives the magnetization as a function of the Stoner parameter. If there are morethan one solution, the condition 02 E/o(t1.m)2 > 0 can be used to show that at least onesolution is metastable.02 E 1 0 i 1 of(t1.m)0{t1.m)2 = 2: c (t1.m ) {t1.m [F (~m) - I ( E d ] J = "![F {~m) - I ( E F ) ] + ~m o (~m) (2 .23 )

At equilibrium the first term is zero according to equation (2.22). This new resultshows that the state is metastable if F(t1.m) has a negative derivative.

3. ResultsCalculations are made on V. Fe, Co. Ni, Pd, and Pt, i.e, the transition metals whichare ferromagnetic or almost ferromagnetic under normal conditions. The relativestability of the para- and ferromagnetic phases is calculated. For the ferromagneticmetals the Curie temperature and the total magnetization are given.

The key quantity in the calculation of the magnetic properties are the density ofstates for the paramagnetic phase and the Stoner parameter. In a consistent treatmentthese quantities should be computed from a band calculation using the exchange-correlation potential (2.4). As such band calculations are not available, band struc-tures obtained with other potentials are used. This introduces uncertainties in theMpln6A-~H


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596 o GunnarssonTable 1. The Stoner parameter I ( EF ) (equation (2.15)) in the LS D approximation, the X ((method and with the Slater potential. We have used the recommended values for (((Slater 1974) for V-Ni and extrapolated to the values 070 and 069 for Pd and Pt.respectively. The Slater potential corresponds to IX = 10. We give a range of values forthe quantity I ( EF )N(E F ) obtained by using values of N(EF ) from various band calculations.The theory predicts ferromagnetism if I ( EF )N(EF) > 1 and P(F) shows that the metal isparamagnetic (ferromagnetic) in nature at T =O.

LS D X", Slater potentialMetal l ( EF ) I (EF1N(EF) I(E F ) I(E f)N (E F ) I( Er) l ( EF )N(EF)

V (P) 080 0'8--09" 101 10-12 1 0 4 1 1 0 4 - 1 '7 "Fe.sec (F) 0'92 15-17b 114 1'8-2-2b 160 26-30b

Co (F) 0'99 1'6-18< 121 19-2-2' 171 2'7-l]'Ni (F) ],0] 2,] d ]23 26d 174 l6dPd (P) 070 08' 086 10' '23 l:5'Pt (P) 063 05' 078 07' 113 10'

(a) Yasui e t a l (1970) and Papaconstantopoulos et al (1972)(b) Mattheiss (1965). Snow and Waber (1969) and Connolly (1970)(c) Wakoh and Yamashita (1970) and Ishida (1972)(d) Hodges et al (1966 )(e) Andersen (1970)

results, since the densities of states differ appreciably between different band calculationsdue to different potentials, methods of calculation and numerical accuracy. Themain conclusions obtained in the following seem, however, to be independent ofthe band calculations chosen.

To calculate the Stoner parameter one needs to known the electron density andthe radial d wavefunction. The choice for these quantities is described in appendix3. The results for the Stoner parameter obtained with the LSD approximation, theX~ method and the Slater potential are compared in table 1. It shows that theX~ and Slater potentials give larger Stoner parameters than the LSD approximation.This is due to the improper treatment of correlation in the former two approxima-tions.

Fe

IIi(FJIIIj,jIIIIIIIIII

Figure 3. The magnetic moment of iron in Bohr magnetons as a function of the Stonerparameter I (equation (2.22)). The curves are obtained for different densities of states(Snow and Waber (SW) 1969, Mattheiss (M) 1965, Connolly (C) 1970). The arrow showsthe experimental moment and the broken line our calculated value for the Stoner par-ameter.


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Band model for magnetism of transition metals 597

T

05 10I r eV)

Figure 4. The Curie temperature of iron as a function of the Stoner parameter (equation(2.16)) for the same densities of states as in figure 3. The arrow marks the experimentalCurie temperature.

Some paramagnetic band calculations have been collected to check the stabilitycriterion (2.17). Table 1 shows the range of values for the quantity I (E F )N (EF ) withN (E F ) obtained from different band calculations. We notice that for all the metalsstudied the LSD approximation gives the correct result for the relative stability ofthe para- and ferromagnetic states.

The X : 1 and Slater potentials give a stronger tendency to ferromagnetism thanthe LSD approximation, the Slater potential predicting all the metals studied, exceptpossibJy Pt, to be ferromagnets. The same seems to be true in the X : 1 method, too.In the X'Y. approximation, however. V and Pd are close to the phase transition andthe uncertainty in the band calculations makes it difficult to definitely assess thestates of these two metals in the X'Y. scheme. Thus Cronklin et al (1972) have reportedV to be paramagnetic in the X :l approximation.To calculate the magnetic moment at T =0 equation (2.22) has been used. Thequantity F(~m) for iron is plotted in figure 3 for various calculations of the densityof states. The calculated magnetic moments fall in the range 2'1-2,7 Bohr magnetonscompared with the experimental result 22. For both Co and Ni the majority spind band is found to be full. This is correct for Ni, as is well known, and probablyalso for Co (Wohlfarth 1970). The actual values of the magnetic moment for thelatter two metals are of less interest in this context as they essentially just reflectthe number of unfilled d states in the paramagnetic phase.The Curie temperature is calculated from equation (2.16). The value of

( _ J X a Z ( E ) N(E) d E ) - I- f OEfor iron as a function of the temperature is shown in figure 4. The Curie temperaturesobtained for the ferromagnetic metals considered are shown in table 2. The results aresystematically much larger than the correct ones.4. DiscussionThe description of charge fluctuations in transition metals has attracted considerableinterest and Hartree-Fock band theories have been criticized for allowing large


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598 o GunnarssonTable 2. The Curie temperature (KJ in the LSD approximation compared with experiments.Metal LSD ExperimentFeCoN i

44()()--'62003300-48002900

10401400631

fluctuations in the number of d electrons on a particular atom. For this [casonthe Hubbard model (Hubbard 1963) was proposed, which due to its relative simplicityallows a treatment beyond the Hartree-Fock approximation. In this model theimportance of the fluctuations is determined by the Coulomb repulsion U, and foran infinite U there would be no fluctuations at all. Estimates of U (Herring 1966,Watson 1973, Cox et al 1974) indicate that U is not very large for transition metals andthat the fluctuations are reduced but not negligible. A measure of the fluctuationsIS

~ = ~ f d3r f d3r'(p 2 ( r , O F ) . As b has roughly the same spatial dependence for allthe metals, varying between 08-09 over the region of interest (cf figure 1), the othertwo factors are the important ones for the trends. Plots of (r,/r)2 and < l > 4 ( r , E d againstthe scaled radius r / r w s for some typical metals are given in figure 5.

Figure 5a illustrates the decreasing trend of (I ' J r ) 2 along the 3rd row series V~Niand along the VIn group series Ni, Pd, and Pt, which reflects the increasing densityalong these series. A high electron density means that a certain spin polarizationgives a smaller fractional spin polarization , and thus a smaller difference betweenthe majority and minority spin potentials (see equation (2.4)). This is the main sourceof the rs dependence in equation (4.3) and gives a factor r; . In addition the quantityJi in the exchange-correlation potential contributes a factor r; 1. The wavefunctionsgradually contract along the series V-Ni, a'S is indicated by the results for V andNi in figure 5b. The contraction is partially due to a more binding potential atthe end of the series, connected with the incomplete screening of the increased nuclear


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Band model fo r m agne tism o f transition m etals

"0

", .';-. PtI ' \ I ,., '\, '.

N : " . \: I .. . \:; " \"-', . . . \, '= !'

I: , " \I , ,. \I ,10 ) I : / ,0,\ Ib)I : ,0,I I ':_~/ :'/ . : c :. . . . . . - _ _ V _~ .. .

00 05 10 00 1 -0r I r W 5 r I r W 5

Figure 5. la) The quantity (r,!r)" (appearing in equation (4.3) as a function of rfrws,where rand I'ws are the distance from the nucleus and the Wigner-Seitz radius, respect-ively. The bulk densities are obtained by superposing the atomic densities. (b ) The wave-function $(r. Ef) = rcP2(r. Ed to the fourth power. The figure shows the trends for the3rd row and the VIII group series.

599

charge (cf. Slater's rules (Slater 1951b and Coulson 1961)). A second reason is theenergy-dependence of the wavefunctions. These gradually contract as their energiesare increased (Wood 1960). Therefore the successive filling of the d band alongthe 3rd row series gives a further contraction of $(r, EF ) ' In the series Ni, Pd, andPt, on the other hand. the increasing number of nodes makes the wavefunction expand.The integral (4.3) can be considered as a weighted average of (rs/r)2 b(rs)$2(r ,EF)with the normalized weighting factor $2(r, E d. If the wavefunction $(r ,EF) isconcentrated within some region of space, the main contribution to this averagecomes from a region where $(r, E F ) tends to make the average equantity large. Thistends to give a large Stoner parameter and reflects the importance of exchange ifthe wavefunctions are essentially confined to a small region of space.

The two factors (r5/1/ and $4(r, E F ) work in the opposite direction in the seriesV. F e. Co. and Ni. The numerical results and the figure show, however, that thechange in the wavefunction is the more important one. In the series Ni, Pd, andPt both factors tend to decrease the Stoner parameter in accordance with table 1.

Finally. figure 6 shows the integrand of equation (4.3) for Ni. Due to the factor< 1 > 4 - ( 1 ' , lOt) most of the contribution to the Stoner parameter comes from the high

10r Ir W 5

Figure 6. The integrand of equation (4.3) as a function of rirws for Ni. The figure illus-trates the importance of the high-density region in the inner parts of the d shell.


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600 0 Gunnarssondensity region. with r s typically in the range 0,3-1,0. The r s values correspondingto the average density are considerably larger, being about 1,2-1,6.

In * 3 magnetic properties calculated in the LSD approximation were comparedwith experiments. The opposite route will now be followed and estimates of theStoner parameter extracted from experiments will be compared with calculated values.

For Fe Gold et al (1971) have calculated de Haas-van Alphen data assuminga constant band splitting. They obtained fairly good agreement with experimentsif they used 099eV (compared to 092eV calculated in the LSD approximation) forthe Stoner parameter, the value for which they found the correct net magnetization.

Wakoh and Yamashita (1970) found in a calculation on Co that a spin splittingof 12eV between the spin up and spin down densities of states gave good agreementwith photoemission data for the ferromagnetic phase. This number corresponds tothe value 076 eV (LSD 099 eV) for the Stoner parameter.

For Ni, Zornberg (1970) has adjusted a parametrized band structure to fit alarge amount of experimental data, including Fermi surface data, optical data andthe total magnetization. He obtained a Stoner parameter in the range 0,75-1,0 eV(LSD 101eV). Mook et a t (1969) have used a band calculation to describe the intensityvariation in neutron scattering against spin waves, obtaining a Stoner parameterof 10eV.For Pd and Pt estimated values for the susceptibility enhancement (Andersen1970) and formula (2.18) have been used to estimate the Stoner parameters. The

values 074 ( L SD 070) and 086 ( L SD 0'63) eV were obtained for Pd and Pt, respectively.Finally, Asano and Yamashita (1973) have made self-consistent x~ calculationsfor the ferro- and antiferromagnetic 3rd row metals. For each metal they adjustedthe band splitting until they obtained the correct magnetic moment. With ~=08a Stoner parameter in the range 0,7-0,8 eV was required. They also remarked thatthis value would increase by about 005 eV if they used ~ = j- , a value which givesparamagnetic bands closer to those used here. These estimates indicate that the calcu-lated values for the Stoner parameter are fairly accurate.The X ~ approximation gives Stoner parameters which are substantially larger(20-60 per cent) than the estimates above (except for Pt). The enhanced tendencyto ferromagnetism has also been observed in band calculations by, e.g., Wakoh andYamashita (1966) and DeCicco and Kitz (1967).

The calculated Curie temperatures T c are much too high (table 2) as is expectedfor a Stoner-like model (e.g., Friedel et a t 1961, Mott 1964). Figure 4 shows thattoo high a result for T; is obtained for all the band structures considered, and thata moderate change in I ( EF ) is not sufficient to give the correct value for 4. Theinadequacy of the Stoner model at high temperatures is also indicated by experiments(Fadley and Wohlfarth 1972) and in particular, there is evidence that for the paramag-netic phase there exist localized moments with lifetimes long enough to be observed.Effects of the fluctuations of these moments, for instance an increased entropy, havenot been included in the present treatment. Band theories can, however, describethe formation of localized moments (Friedel et al 1961, Lederer and Blandin 1966,Lederer 1966).The Curie temperature should then be mainly determined by the inter-action between the localized spins and not by the Stoner parameter (Feiedel et al1961, Slater 1974). In a straightforward application of the SDF formalism this hasto be described through the exchange-correlation functional. An alternative approachof obtaining the Curie temperature, which demands less from the functional, wouldbe to calculate the excitation energies of the system, using the same formalism as


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Band model for magnetism of transition metals 60 1for the ground state. The free energy, for instance, is then obtained by summingover all excited states according to the formula F =- kT In[I" exp( - E"/kT)] . Thissum includes states which differ by not having the same orientation of the localizedmoments. The free energy would then contain effects of the spin fluctuations, e.g.,it would give a high entropy.

In summary, it has been found that the LSD approximation gives a reasonabledescription of the zero-temperature magnetic properties. Comparison with estimatesfor the Stoner parameter extracted from experiments indicates that the errors inthe calculated values should be at most a few tenths of an eV. The results obtainedsuggest that the LSD approximation should be useful for the calculations on transitionmetal systems at T = O .

AcknowledgmentsIt is a pleasure to thank B I Lundqvist for many valuable discussions and suggestions.R 0 Jones and J Harris are acknowledged for critical reading of the manuscript.The work has been done with support from the Swedish Natural Science ResearchCouncil.

Appendix 1In a spin-polarized system the potentials for spin up and spin down electrons aredifferent, and the wavefunctions are therefore spin-dependent. This effect (called (ii)in 2.2) gives rise to a spin polarization. In the following this spin polarizationwill be shown to be small compared with that caused by spin-flips (effect (i)).

Perturbation theory is used and the spin polarization is calculated to lowest orderin the net magnetization Sm. For the wavefunctions of the paramagnetic state therepresentation in equations (2.10 and 2.11) is used. Orthonormality requiresL c[m(kn) Clm ( kn ') ( kn lk n') / = 6n"

1m(Al.1)

where(knlkn')l = = f : ' " r2 dr l(r,e"n) l(r,e" nl

To lowest order in 1 1 m the spin polarization due to distortions of the wavefunctions is_ O~~ kn'lvi1)lkn> * )m(r) = p+ (r) - p_ {r) = t1.m L L !/I,,",(r)!/Ikir) + c.c. (Al.2)

lin n' i= I! fkn - Ekll'All terms in equation (Al.2) with n ' occupied cancel. Appendix 2 shows that it is areasonable approximation to assume v(,f){r) to be spherically symmetrical and utilizingthis approximation one gets

occ uno" (


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60 2 0 GunnarssonThe orthonormality condition (A1.1) provides a relation between the coefficientsc1m(kn). It can be used to replace all the coefficients with a certain I value inequation (A 1 .3 ). As the bands considered are predominantly of d character, it isconvenient to remove all the terms with 1 = 2.----

occ lIl1O"" ( * (k ') kn 'lv (f)lkn)1 * )m ( r ) = ~ m IL I e 'm in e l m kn ) I /! k n (r ) l / lk n ( r ) + c.c.k n n ' 1+ 2 E k n - Ekn' (Al.4)m

where 2 4 > 2 ' For brevity the arguments of thetwo functions c P l ' r, E"n' and r, Ek", respectively are omitted. The sum over RandR ' (cf equation (2.10)) disappears because the density in one single cell is consideredand because functions centred in different cells do not overlap.

The potential v ( ~ ) (r ) is large in the regions of space where the d wavefunctionsare large. Therefore the integral 1 is neglected. Using EFfor the energy arguments it is assumedthat

(A 1.6)where (EFlv(~)IEF>I=2=-l(EF) is the expectation value of v('h for the wavefunctionc P 2(1', EF). Numerical calculation shows that there are appreciable cancellationsbetween the two terms in (kn'lv(l)lkn),. For E'm and Ekn' in the energy range of thelowest six valence bands and for l s 2 we find the absolute value of the left handside of equation (A1.6) to be overestimated by about 50-100 per cent. As the expec-tation value of the spin polarization will be calculated, the spin polarization in thed wavefunction region is particularly important. Therefore the last term in Mr isassumed to dominate and the approximation

(A 1 .7 )is made. Numerical estimate shows that this approximation, too. means that thecontribution of the wavefunction distortion to the Stoner parameter is overestimatedby about 50-100 per cent. The energy denominator in equation (A1.5) is the differencein energy between an unoccupied and an occupied state and is always nonzero.It is approximated by the d bandwidth W This might be an underestimate of equation(A1.5). To get an upper limit for the sums over the coefficients clm(kn) the resultsof a band calculation (Wood 1967 ) is used. For the d-like bands of Cu it has beenfound that

L hm (kn)12 ;? : 0 9m (A 1 .8 )


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Band model for magnetism of transition metals 603Using Schwartz inequality one obtains

IIt n ( k n ') c 1 m ( k n ) 1 2 ::( I I C l m ( k n 'W Ic 1 m ( k n W s;1*2 1*2 I",,~

001 if both nand n' are d-likebands

m m m(A 1 .9 )

01 if either n or n ' is ad-likebandFirst the contribution to equation (A1.S) from the six lowest valence bands isconsidered. It is assumed that the number of nonzero terms in the k sum equal

Nn~ccn~~occ, where l1~cC and 1J~~occ are the numbers of occupied and unoccupied statesper atom and spin in the bands nand n', respectively. This overestimate gives

2~m< m(r) ~ --I{E)A,2(r E ) ( 0 ' 1 1 1 ' ) c - : nUIlD,': + 01 n'.lccnunocc + 0.01 nOCC nUnOCC)--- 4n W f 'f' 2 ' ~ u, p 'r u d dUsing the calculated values for I ( EF ) and typical numbers for n ~cc, n ~~ c and W forvarious transition metals equation (Al.lO) is found to contribute about 5-10 percent of the total spin polarization.

In the estimate of (A1.5) only the six lowest valence bands were included. Inthis way effects of different hybridization for majority and minority spin electronswere taken into account. In addition, the functions c P 2 ( r , E ) tend to contract (expand)for the majority (minority) spin electrons. To include this effect fully one wouldalso have to include higher bands in equation (A1.5). Instead, an explicit calculationon the function c P 2 ( r , E ) for both spin states in iron has been performed. It is foundthat the distortion of the radial wavefunction contributes only a few (1-3) per centto the band splitting.

In. e.g .. iron there is in the interstitial region probably a very weak spin polariza-tion which has opposite sign to the net magnetization (Shull 1967). This negativemagnetization is supposed to be due to the above mentioned contraction of themajority spin states towards the nuclei (DeCicco and Kitz 1967). It is, however,not important for the splitting of the d bands as the d electrons are mostly in theregion of the very much larger, positive spin polarization.

It is concluded that the spin polarization due to the deformation of the wavefunc-tions l / I k n is small and could be neglected in this calculation.

(A1 .10 )

Appendix 2In the derivation in ~ 2. the non-spherical part of the spin polarization was neglected.This approximation is discussed here. assuming cubic symmetry of the metal.

Quite generally the spin-density can be written as

m(r) ~


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604 0 Gunnarssonwhere

(A2.3)is a cubic harmonic introduced by von der Lage and Bethe (1947). In appendix1 we showed that the spin-density is essentially due to spin-flips of d electrons atthe Fermi surface. The spherical and non-spherical parts of the spin polarizationtherefore have the same r dependence, described by 4 > ~ (r , EF ) ' We can thus assumef ( r ) to be just a number f. This number can be determined experimentally from,e.g., neutron diffraction experiments.

U sing the result (A2.2) for the spin polarization in equation (2.13 and 2.7) weget the ratio between the spherical and non-spherical contribution to the band split-ting of a state (kn)

r 'm =f L c ~ m ( k n ) c z m ( k n ) J c t n Y im ( r ) Y zm ( f ) K 4,1 (f). (A2A)mm

Using the explicit form for K4, I r) in (A2.3) we obtainr k n =U/)21 (3 ~ I c z m ( k n ) j 2 a m + 5 Re C z , z ( k n ) Ct-2( k n ) )

where a m = 1 if m = 0, - jif Im l = 1 and * if Im l = 2. If a state contains fractions04 + X k n and 06 - X k n of functions with Eg symmetry (3z2 - r2, X2 - y 2 ) and T2 gsymmetry (x y, yz, zx), respectivelyr.; = (5flj21)xk" (A2.6)

(A2.5)

Estimates from neutron scattering data (Shull 1967) gives that f : : s : 02 for Fe,Co, and Ni. For pure r, and Tzg states this gives r : : s : 013 ( E g ) and r : : S : -0,09 (T2 g ) .The effects of the nonspherical spin polarization may therefore be of importancewhen calculating for instance the Fermi surface. In this paper, however, only integralproperties of the band splitting are considered and for these there are large cancella-tions between states with positive and negative r. For instance, to derive the stabilitycriteria in ~ 2.4 the magnetization .lm due to an external magnetic field is considered.From equation (2.1) one gets

(A2.7)IfH is a weak field the Fermi-Dirac function can be expanded

11m =L [- 1 1 m I (EF)( l + (5/v 2 1 ) f x k n ) - 2Jl.BH] C f ~ E k n )k n EHowever, the average of xkll over the Fermi surface is v 2 T fl5 and

11m =(- 11m I (F ) (l + f2) - 2Jl.sH) I _ L x a - z ) N ( E ) ds (A2.9)A non-zero magnetization can be obtained in the limit H = 0 if the Stoner parametersatisfy

(A2.10)


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Band model for magnetism of transition meta ls 605This is the criteria (2.16) except for the factor (1 + f 2) . Thus in this case the non-spher-ical part of the spin-polarization effectively increases the Stoner parameter with atmost four per cent.

Appendix 3In our calculation of I ( EF ) in equation (2.15) we have to know the electron densityp(r) and the radial d wavefunction 2(r, E d . For the electron density we have usedthe spherical average of superposed atomic densities. The atomic densities calculatedby Herman and Skillman (1963) have been used. An electron density obtained froma full selfconsistent band calculation would give only a small change in the resultfor I (EF) ' For most elements results for the wavefunction are not available and wehave therefore estimated it in various ways. For V and Fe we have solved the radialSchrodinger equation with potentials from band calculations by Yasui et a t (1970 ,1973), respectively. For Co we have used the renormalized atom result of Hodgese t a t (L972). Finally. for Ni, Pd, and Pt we have utilized the proximity of the Fermilevel to the top of the d band and the fairly strong binding of the electrons, whichimplies that the wavefunctions resemble the atomic ones. As the wavefunction atthe top of the d band is approximately zero at the Wigner-Seitz radius, we havesmoothly changed the atomic wavefunctions to satisfy this condition. This modifica-tion of the wavefunction changes I ( EF ) by at most ten per cent, which could be con-sidered as an upper limit for the uncertainty in I ( EF ) due to errors in the wavefunction.

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606 o GunnarssonHubbard J 1963 Proc . R. S oc . A 276238-57Ishida S 1972 J . Phys. S O (, J IJ P "1 1 33 369-79Janak J F. Moruzzi V L and Williams A R 1975 Phy , ReT. B 12 1257-61Kohn Wand Sham L J 1965 P hys. Re t. 14 0 A 1133-8von der Lage F C and Bethe H A 1947 Ph ys. Rev. 71 612-22Lang N D 1973 S olid S ta te P hysic s eds H Ehrenreich. F Seitz and D Turnball 28 225-300Lang N D and Williams A R 1975 Phys. Rer. L ett. 34 531-4Lederer P 1966 P hil. M a4 . 14 1143-56Lederer P and Blandin A 1966 P hil . . ' 1 1 " 4 . 14 363-H ILomer W M 1967 Theory of Maqnetism in Transition M etals. International School of Physics Enrico

Fermi (New York: Academic Press) pp 1-21Madsen 1. Andersen 0 K. Poulsen U K and Jepsen 01976 to be publishedMattheiss L F 1965 P hvs. Rev. 13 9 A1893-904Mook H A. Nicklow R M. Thompson E D and Wilkinson M K 1969 J. App l . Phys. 40 1450-1Mott N F 1964 Adr. Phys. 13 325-422Papaconstantopoulos D A, Anderson J R and McCaffrey J W 1972 P hys. Rev. B 5 1214-21Rajagopa1 A K and Callaway J 1973 P hys. Rev. B 7 1912-9Seitz F 1940 The Modern Theory of S olids (New York: McGraw-Hili) p 331Shull C G 1967 M agnetic and Inelastic S cattering of N eutrons by M eta ls. M eta llu rg ic al S oc ie ty C on fe re nc es

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O. Gunnarsson- Band model for magnetism of transition metals in the spin-density-functional formalism