ELECTRONIC STRUCTURE CALCULATIONS AND MAGNETIC PROPERTIES D. G. PETTIFOR Department of Mathematics, Imperial College, London SW7, UK
Recent theoretical developments in the study of the electronic structure and magnetic properties of itinerant magnets are reviewed. The discussion fails naturally into two parts. The first part deals with the apparent success of the local spin density functional calculations in predicting ground state properties such as the magnetic moment or equilibrium atomic volume with the help of no adjustable parameters. The second part deals with the conditions for the existence of localized moments in the disordered paramagnetic phase and examines the energetics of different local moment configurations within a parametrized tightbinding framework. The relevance of these results to current theories of itinerant magnetism is stressed.
1. Introduction During the past few years numerous first principle calculations [15] have demonstrated that a simple band model of magnetism is capable of giving a good description of the ground state properties of itinerant magnets. This band model is based on the local spin density functional (LSDF) theory of Hohenberg, Kohn and Sham [6, 7] in which the selfconsistent field equations have the Hartreelike form but with exchange and correlation included explicitly in an effective local potential. These calculations with no adjustable parameters predict [1, 2] the equilibrium lattice constant to within 3% and the bulk modulus to within 30% for the ferromagnetic 3d transition metals and they reproduce [4] the sudden lattice expansion that occurs in the actinide series on going from Pu to Am resulting from the onset of localization of the 5f electrons. Further, the magnetic moment is given very accurately [13] and the enhanced spin susceptibility correctly predicts [2] that only iron, cobalt and nickel are possible candidates for ferromagnetism amongst the first 32 metallic elements. However, because the local spin density functional theory can be reduced within certain approximations to the familiar Stoner model [1, 5, 8], it is not surprising that it gives far too high a value of the Curie temperature Tc for the 3d ferromagnets [5]. As is well known [911], local moments persist in these itinerant magnets above To, whereas in Stoner theory the exchange splitting has collapsed in the paramagnetic state and with it any possibility of a local moment. Recently, two different approaches have been used for tackling this problem within a band theoretical framework. The first regards the paramagnetic state above T~ as a
disordered local moment state [1215]. By assuming that the moments may point either up or down the problem is analogous to a disordered binary alloy and may, therefore, be treated within the coherent potential approximation(CPA) [16]. Hasegawa [15] has worked through the thermodynamics of this model and shows very clearly why iron exhibits both itinerant and localized moment behaviour. The second approach regards the paramagnetic state above Tc as being characterised by the persistence of short range magnetic order in which the magnetization direction is assumed to change relatively slowly from atom to atom [7]. Because the band structure is determined locally on the scale of a few neighbours it will be only weakly affected by the magnetic transition so that a local exchange splitting and magnetic moment will continue to exist above Tc as observed experimentally[911 ]. In this paper electronic bandstructure calculations of both the ground state and the paramagnetic local moment state are reviewed. We begin in section 2 by presenting the LSDF ground state results and interpreting them within the Stoner framework. In section 3 we briefly discuss the AndersonFriedel [18] concept of a localized moment existing on a transition metal impurity in the light of recent ESDF calculations [19]. We then examine the conditions for the existence of a disordered local moment state [1215] and the behaviour of transverse spin density fluctuations in transition metals [2021]. 2. Ground state properties within LSDF theory. The spin density functional formalism is based on two theorems proved by Hohenberg and Kohn [6], which state that the ground state energy U of a
Journal of Magnetism and Magnetic Materials 1518 (1980) 847852 ©North Holland
847
848
D. G. Pettifor/ Electronic structure calculations
system is a functional of the electron density p(r) and that the functional U[p] is minimized by the correct ground state density. Kohn and Sham [6] wrote the energy functional as the sum of the usual Hartree energy UH[O] plus the contribution from exchange and correlation Uxc[P] which is neglected in the Hartree approximation i.e.
= u.[o] + Ux
[p].
(I)
By minimizing this energy subject to particle conservation, the ground state density can be written N
p(r) Y, Iq~,(r)]2,
(2)
iml
where the #i satisfy the following Hartreelike selfconsistent field equations
[  V 2 + Va(r ) + Vxc(r)] qJi(r) = e~#,(r), (3) where
Vx¢(,) =
(4)
Thus, the many body problem has been reduced to an effective single particle problem, which can be solved provided some approximation is made for the unknown exchange and correlation functional Uxc. Kohn and Sham [6] suggested the local density approximation, in which
u×~[p] = f,(,).xc(P(,))dr.
(5)
where exc(0 ) is the exchange and correlation energy per electron of a homogeneous electron liquid of density p. This approximation, although formally exact only in the limit of slow and weak spatial variations of the electron density, has been surprisingly successful in describing the properties of a number of inhomogeneous atomic, molecular, and solidstate systems [5]. Gunnarson and Lundqvist [7] have argued that this success is due to Uxc depending on the spherical average of the exchangecorrelation hole, so that distortions of the hole due to inhomogeneities integrate out. Further, the local density approximation satisfies the sum rule that the exchangecorrelation hole should contain one electron, thereby guaranteeing that the quasi particle is perfectly neutral [7]. We must stress, however, that the Lagrangian multipliers ei entering the selfconsistent field equations are not to be associated a priori with excitation energies as
Koopman's theorem does not apply to density functional theory [22]. We should also note that although the selfconsistent field equations have the independent particle form, this does not, of course, imply that the electrons are uncorrelated. For example, Gunnarson has shown that fluctuations in the number of electrons on a given atom in iron are heavily suppressed within the local density approximation, which is consistent with the sizeable value of the intra atomic Coulomb integral. The density functional formalism can be generalized to include spin polarization [7] with the consequence that the electrons feel a spin dependent local potential Vxc in eq. (3). The result of solving the spinpolarized selfconsistent field equations for the 3d transition metals is shown in fig. 1, where the equilibrium WignerSeitz radius, bulk modulus, magnetic moment and its pressure derivative are displayed [2]. We first note that the LSD exchangecorrelation potential passes an extremely sensitive test of magnetic stability by predicting correctly that only Fe, Co and Ni will be ferromagnetic amongst all the transition metals, unlike the earlier Slater and Xa exchange potentials which also included V and Pd in their list [22]. We see from fig. 1 that the LSDF magnetic moments [13] agree remarkably well with the experimental values. The differences between the measured and theoretical pressure derivatives of the moment are possibly due to the pressure dependence of the small orbital moment contribution which these calculations have neglected [2]. The LSDF values for the equilibrium WignerSeitz radius and bulk modulus [1, 2] show the same anomalous trend from Ni to Fe that is observed experimentally, namely the atomic volume increases and the bulk modulus decreases in contrast to the corresponding nonmagnetic 4d and 5d transition metals. (The absolute differences between theory and experiment for these ferromagnets are not atypical to those found for the nonmagnetic elements [2].) This anomalous behaviour is directly linked to the onset of magnetism as can be seen from fig. 1 where the nonspinpolarized LDF calculations show the expected 4d and 5d trend. This can be interpreted very simply within Stoner theory [9, 1, 2] to which the LSDF results can be reduced [1, 5, 8]. As the magnetism in the 3d transition metals arises primarily from the d electrons, we can neglect the sp electrons to a good
D. G. Pettifor/ Electronic structure calculations 1
•

,
I B ( M blot)
S(au)
T
V
which is the famous Stoner criterion. The resulting increase in pressure Pmag due to the onset of ferromagnetism can now be evaluated since LSDF results show that 1 is effectively volume independent [1] and the d band width may be assumed to vary inversely with volume to the fivethirds power [23]. It follows directly from equation (7) that
27  , ....'X.. ....• "....×. "'...... 26 ~.
"'X....... .
i i "'"'"'"x'"'"'"'" 2.~ _ _ Cr
M[n
Co V. . . . . . T
Nt
Fe
0 L__L____I Cr Mn
Fe
. @in~rn
0

20
o4~
_ _ ~ Co
Ni
\
10 I
(M b a r q)
03 t
02~
05
Cr
Mn
Fe
Co
Ni
Cr
Mn
Fe
CO
Ni
Fig. 1. The equilibrium WignerSeitz radius (au), bulk modulus (Mbar), magnetic moment (ixB) and its pressure derivative (Mbar1) for the magnetic 3d transition metals. The crosses, circles and squares are the experimental, spinpolarized and nonspin polarized results [2]. The triangles are estimated values [2] based on eq. (9).
approximation. (The importance of the sp contributions to certain magnetic phenomena, such as the hyperfine field, will be discussed by Professor Kanamori in his report to this conference.) As illustrated in fig. 2a and b the Stoner model characterizes the ferromagnetic state by split up and down spin bands. The amount of the exchange splitting A is given by = In,
(6)
where I is the Stoner parameter which LSDF theory gives a value of about I eV for the ferromagnetic 3d metals [13]. The change in energy on flipping the spins from the nonmagnetic to the ferromagnetic state will be
Utm =¼[m2/n(EF)]{l  In(EF) ),
(7)
where n(EF) is the density of states per spin at the Fermi energy. The first and second contributions represent the increase in kinetic energy and the decrease due to exchange energy respectively. Clearly the system is unstable to ferromagnetic ordering if Z,,(Er) > l,
(9)
3PmagV = ¼m2/n(EF),
r
®
1.5
849
(8)
where the right hand side is simply five times the increase in kinetic energy on going to the magnetic phase. Janak and Williams [2] have shown that this simple expression accounts for the increase in equilibrium volume on going to the magnetic state that the LSDF results display in fig. 1. They have used eq. (9) to estimate the corresponding change in atomic volume of Cr and Mn assuming the experimental value of m and the density of states for the bcc and fcc phases respectively. (This will be an overestimate of the magnetic pressure because the increase in kinetic energy will be lower in the observed antiferromagnetic phase.) The increase in atomic volume leads directly to a decrease in the equilibrium bulk modulus, because the valence s electrons are now no longer compressed to the same extent into the core region where they are repelled by orthogonality effects. Thus, the experimental trends of atomic volume and bulk modulus across the 3d series are well accounted for by the LSDF results. Recently, Skriver et al. [4] have obtained good agreement with experiment across the actinide series, the LSDF results reproducing accurately the sudden 30% expansion in volume that is observed in going from Pu to Am due to 5f localization. At this E~
E,
E ¸ Wclm
2
2 0
/_vvxl  ~ ,,k A/~,1
2 2
2 2 (a) (b) fc) Fig. 2. The ul~ and down spin density of states associated with a given atom in (a) the nonmagnetic, (b) the ferromagnetic and (c) the disordered local moment phases.
D. G. Pettifor/ Electronic structure calculations
850
conference they will be presenting similar results for the 3d transition metal monoxides. The density functional results of fig. 1 predict that the atomic volume of ferromagnetic iron would contract by 6% if the moment was totally destroyed. Shiga and Nakamura [9] had reached a similar conclusion within Stoner theory much earlier and they had, therefore, deduced that local moments must persist above the Curie temperature T¢ because the observed volume change was much smaller. This is consistent with the fact that Stoner theory predicts Curie temperatures for Fe, Co and Ni that are at least a factor of four too high [22]. A proper theory of itinerant magnetism must include the possibility of local spin fluctuations or moments above T c [12]. 3. Local moments
The conditions necessary for the presence or absence of local moments on transition metal impurities in nearlyfree electron hosts has been thoroughly examined within the FriedelAnderson model [18]. Very recent LSDF calculations of impurities in the noble metals Cu and Ag, which will be reported at this conference [19], bear out the basic concept of a virtual bound state that splits about the Fermi energy on the formation of a local moment. Within the tight binding framework the local splitting A on the impurity atom can be written A = e~d 
e~d = I ( N
t 
N J,) = I r a ,
(10)
where e~' t and N t' ~ are the atomic d level and number of d electrons, respectively, for up and down spins, m is the local moment and I is the LSD Stoner parameter. (In the nondegenerate Anderson model, I would be replaced by the intraatomic Coulomb integral.) Anderson [18] solved eq. (10) selfconsistently by evaluating N~ and N ] for a given splitting A assuming Lorentzian virtual bound states centred on e~ and e~. The virtual bound states arise from the mixing of the localized impurity d orbitals with the host nearly free electron band. However, as can be seen from fig. 1, of Zeller et al. [19] the situation in Cu is complicated by the presence of the d band which lies above the bottom of the nearly free electron sp band. The LSDF calculation finds only about 50% of the Mn
moment and 70% of the Cr moment associated with the virtual bound state, the remainder being due to states lying energetically in the vicinity of the C u d band. With a value of I of 0.8 eV for Cr and Mn [2] and with moments close to 3/zB [19], eq. (10) predicts an exchange splitting of about 2.4 eV, which is close to the computed splitting of the virtual bound states [19]. The LSDF calculations find that Ti and Ni impurities carry no moment in Cu. As pointed out by Edwards [24] these magnetic impurity calculations should provide another sensitive test of the SDF, especially in systems such as Cr in Ni where at present striking discrepancies exist between theory and experiment. The possibility of local moments existing above the magnetic ordering temperature has been studied by several authors [1215] by analogy with the disordered binary alloy system, in which the two components are characterized by atoms with their moments pointing either up or down, respectively. The disordered system is treated within the coherent potential approximation (CPA) [16], which regards atoms in the random alloy as impurities in a coherent medium. The local moment is found selfconsistently from eq. (10) by computing the CPA partial density of states. The results of these calculations can be understood within the following simple model illustrated in fig. 2. We have previously considered the ferromagnetic state in fig. 2b, where all the local moments are aligned up so that an upspin electron sees an exchange field  A / 2 at all sites and consequently has its energy lowered by this amount. However, in the disordered local moment state there are as many moments pointing up as down, so that the average exchange field seen by an up (or down) spin electron is zero, and therefore, the centre of gravity of the average up (or down) spin density of states remains unshifted. However, the average density of states will be broadened because the up spin electrons, for example, can redistribute themselves to spend more time on the more attractive up sites than down. This broadening is measured by the second moment of the density of states, which in tight binding theory leads [25] to the disordered local moment (dlm) density of states having the width Wdl m ~ (1 4 3 ( A / W ) 2 ) I / 2 w .
(11)
D. G. Pettifor/ Electronic structure calculations
Fig. 2c illustrates the local density of states of up and down spin electrons on an atom with its moment aligned up. The partial densities of states have been skewed so that the centres of gravity are at __+A/2, respectively [25]. The selfconsistency eq. (10) can now be solved to give the value of the disordered local moment, namely m = (1/V3)([~N(10
 N)] 2
2
(W/I) }
1/2 .
(12) Thus the dim state exists if
I/W
(13)
> [ 3 N ( 1 0  N ) ] I
which is to be compared with the Stoner criterion [eq. (8)] of I/W > 0.2 for a constant density of states. The magnetic energy of the dlm state can be obtained by adding up the band energies and subtracting off the exchange energy which has been double counted i.e. Udl m  
N) +¼Im2.
0(Wdlm  W ) N ( 1 0 
(14)
Fig. 3 shows the resulting regions of stability of the ferromagnetic (fm) and disordered local moment (dim) phases as a function of the renorrnalized exchange integral I/W and d band filling N. The values of N corresponding to the 3d metals are 03
I
02
Cr
o.1
1
5
Mn I
6
I
Fe I
7
I
Co
l 8
I
Ni I
I
g
10 N
Fig. 3. The regions of stability of the ferromagnetic and disordered local moment phases as a function of the renormalized exchange integral I / W a n d d b a n d filling N (see text for derails).
851
marked by fixing Ni with 0.6 holes. The fm and dim phases are stable for values of I/W above the critical curves ABC and DBE, respectively. In the region where both phases are stable, the fm and dim state have the lower energy in region FBE and ABF, respectively. These results are very similar to the CPA computations of Roth [14] and the Bethe lattice results of Liu [13]. Fig. 3 clearly illustrates why only iron amongst the 3d ferromagnets exhibits local moment behaviour [12, 15, 26]. We see that whereas the dlm stability curve lies below the fm stability curve in the vicinity of iron, the curves have crossed by the time cobalt is reached, so that neither cobalt which is only just ferromagnetic [2] nor nickel can sustain local moments in the completely disordered state. Thus Hasegawa's [15] very simple and elegant theory of magnetism in iron cannot be applied directly to cobalt and nickel, where the spin fluctuations do not behave like a set of welldefined local moments. Korenman and Prange [ 17] have proposed a band model of magnetism above T¢ in which the magnetization direction changes only relatively slowly from atom to atom, so that the local exchange splitting, and therefore, the local moment survive above the magnetic ordering temperature. They estimate [17, 21] that the maximum angle taken between neighbouring moments in iron and nickel above Tc is about 40 °. Recent electronic structure calculations have indeed shown that the moment in iron falls by only 6% for a 40 ° spiral spin wave [21] and remains unchanged in nickel until about 90 ° when it collapses catastrophically [25]. Unfortunately, however, the Korenman and Prange model can be criticized on thermodynamic grounds [26]. Finally, we briefly mention current research in two areas directly relevant to approximations made by the simple Hasegawa [15] model of iron, namely the Isinglike constraint of the moments being able to point only up or down and the neglect of any short range order above Tc. Hubbard [20] has recently looked at the transverse fluctuations of a local moment in ferromagnetic iron, and has found noticeable deviations from Heisenberg cos 0 behaviour (see his fig. 4). (This is partly due to lack of charge neutrality, see fig. 4 of ref. [15]). Heine et al. [21] found similar deviations in the energetics of spiral spin waves, which could not be attributed to
852
D. G. Pettifor / Electronic structure calculations
45*
90"
135"
e
180 t
I should like to thank Dr. D. M. Edwards for many helpful discussions.
w l References [1] O. K. Andersen, J. Madsen, U. K. Poulsen, O. Jepsen and J. Kollar, Physica 8688B (1977) 249. [2] V. L. Moruzzi, J. F. Janak and A. R. Williams, Calculated Electronic Properties of Metals (Pergamon Press, New York, 1978) and references therein.
007~1
[3] J. Callaway and C. S. Wang, Physica 91B (1977) 337. [4] H. L. Skriver, O. K. Andersen and B. Johansson, Phys. Rev. Lett. 41 (1978) 42. [5] O. Gunnarsson, J. Appl. Phys. 49 (1978) 1399 and refer
.
ences therein.
0,25
01C
energy
J
Fig. 4. The renormalized magnetic Umag/ W as a function of the angle 0 for N = 7.4 corresponding to iron. The curves are marked by different values of I~ W.
second nearest neighbour interactions for 0 > 30 °. We have extended the simple analytic model of the disordered local moment state illustrated in fig. 2c to consider the disordered alloy in which the two constituents are atoms with moments aligned at _+ 0 / 2 to the ferromagnetic axis. Fig. 4 shows the variation of magnetic energy with 0 for different values of I/W for the case of iron corresponding to N = 7.4. We see that the deviation from cos 0 behaviour reflects the fact that iron lies in the neighbourhood where the moment is equally stable in either the ferromagnetic or antiferromagnetic configuration, as has been postulated by Kaufman et al. [27] to account for the thermodynamic data of fcc iron alloys. We must be careful, however, before interpreting these results in terms of an effective nearest neighbour coupling, because the moment can be very sensitive to the particular local environment which has been averaged in the CPA calculations. For example, Miwa and Hamada [28] have included local environment effects in an extended CPA programme and have found that there can be a wide distribution of moments on the iron atoms in disordered bcc alloys.
[6] P. Hohenberg and W. Kohn, Phys. Rev. 136 0964) B864; W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) All33. [7] O. Gunnarsson and B. I. Lundqvist, Phys. Rev. BI3 (1976) 4274 and references therein. [8] S. H. Vosko, J. P. Perdew and A. H. MacDonald, Phys. Rev. Lett. 35 (1975) 1725. [9] M. Shiga and Y. Nakamura, J. Phys. Soc. Japan 26 (1969) 24; M. Shiga, AlP Conf. Proc. No. 18 (1974) 463. [10] H. A. Mook, J. W. Lynn and R. M. Nicklow, Phys. Rev. Lett. 30 (1973) 556. [11] D. E. Eastman, F. J. Himpsel and J. A. Knapp, Phys. Rev. Lett. 40 (1978) 1514, [12] W. E. Evenson, J. R. Sehrieffer and S. Q. Wang, J. Appl. Phys. 41 (1970) 1199; M. Cyrot, Phys. Rev. Lett. 25 (1970) 871. [13] S. H. Liu, Phys. Rev. BIT (1978) 3629. [14] L. M. Roth, Inst. of Physics Conf. Series No. 39, (London, 1978) 473. [15] H. Hasegawa, J. Phys. Soc. Japan 46 (1979) 1504. [16] P. Soven, Phys. Rev. 156 (1967) 809. [17] V. Korenman, J. L. Murray and R. E. Prange, Phys. Rev. BI6 (1977) 4032, 4048, 4058. [18] P. W. Anderson, Phys. Rev. 124 (1961) 41. [19] R. Zeller, R. Podloucky and P. H. Dederichs, paper given at this conference. [20] J. Hubbard, Phys. Rev. B19 (1979) 2626. [21] V. Heine, A. J. Holden, P. LinChung and M. V. You, to be published. [22] O. Gunnarsson, Electrons in Disordered Metals and at Metallic Surfaces, NATO Adv. Study Inst. Series (Plenum, New York, 1979) p. 1 and references therein. [23] V. Heine, Phys. Rev. 153 (1967) 673. [24] D. M. Edwards, Electrons in Disordered Metals and at Metallic Surfaces, NATO Adv. Study Inst. Series (Plenum, New York, 1979) p. 355 [25] D. G. Pettifor, Phys. Rev. Lett. 42 (1979) 846; Solid State Commun. 28 (1978) 621; and to be published. [26] D. M. Edwards, J. Magn. Magn. Mat. 1518 (1980) 262. [27] L. Kaufmann, E. Clougherty and R. J. Weiss, Acta Met. 11 (1963) 323. [28] N. Hamada and H. Miwa, Prog. Theor. Phys. 59 (1978) 1045; N. Hamada, J. Phys. Soc. Japan 46 (1979) 1759.