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Finite concentrations of localized moments in metals
ANNALS
OF PHYSICS
Finite
95, 267-280 (1975)
Concentrations
of Localized
Moments
in Metals
J. C. STODDART Department of Physics, The University, Shefield, SlO 2TN, England Received November 4, 1974
In earlier work, density functional methods have been used to study the magnetism of pure metals and the problem of local moment formation on a single magnetic impurity. In the present paper these methods are reviewed and applied to the case of an arbitrary concentration of impurity atoms. It is shown that the Hartree-Fock treatment of the system and a local environment effect can be reproduced very easily in the density functional framework. More importantly, the present formulation allows one some insight into the effects of correlation. Although correlation density functionals are difficult to obtain, it is shown that a class of models of the many impurity system can be solved exactly.
1. INTRODUCTION In a previous paper (Stoddart and March [l], referred to as SM), a theory based entirely on the spin densities p+ and p- in a magnetic metal has been formulated. Ref. [l] was concerned solely with periodic metallic magnets, whereas in a second paper the spin-density description was applied to localized moments in metals (Stoddart, March, and Wiid [2], referred to as SMW). The approach adopted by SM and SMW was taken up by von Barth and Hedin [3] and by Rajagopal and Callaway [4]. These workers give a somewhat more fundamental discussion of the use of the spin densities p+ and p- to characterize the magnetic system. The basic problem one encounters, of course, in the spin-density description is the lack of knowledge of the energy functionals. Some attempt to get round this has been made by von Barth and Hedin [3] for the special case of the uniform gas, though unfortunately this is not the interesting limit in magnetic metals, the tight-binding methods being much more appropriate in, say, the d bands of ferromagnetic nickel. Nevertheless, neglecting correlation effects, one can write down the usual generalization of the Dirac-Slater exchange potential to deal with the magnetized case. Such exchange potentials have been used in band structure calculations on Ni (see, for example, [5]). 267
Copyright All rights
6 1975 by Academic Press, Inc. of reproduction in any form reserved.
268
J. C. STODDART
The present paper is concerned with the problem of the type of magnetic impurities in metals for which the Anderson Hamiltonian affords a useful starting point. While the Anderson model of a single magnetic impurity was discussed by SMW, the case of a finite concentration of magnetic impurities, which can interact together, is the main problem tackled here. Some work on finite concentration has been carried out by Kim [6], using standard many-body theory. However, he imposes a Hartree-Fock condition on the interaction term. It is shown first (Section 5) that Kim’s results can be regained from the spin-density formalism. Furthermore, some insight can be gained into the likely effect of relaxing the Hartree-Fock constraint on the interaction, i.e., the effect of correlation. 2. THE SPIN POLARIZED
CASE
Following von Barth and Hedin [3], we can write the total ground-state energy for the spin polarized system as
where T, is the kinetic energy functional for a system of noninteracting Fermions, EcouL is the classical Coulomb energy, Eze is the exchange and correlation contribution, and aae is the spin-dependent external potential. Minimization of E[pJ with respect to pNswill give an Euler equation involving w,, , provided that these functionals are known. We consider the general case when w,is spin independent, and is in fact just the static potential V(r) due to the ionic lattice. A procedure, introduced in SM, is very useful in formulating Anderson type models, and will be used later. We describe it here for the exact functional (2.1). We assume that we have a set of Wannier functions UJ corresponding to some periodic potential (e.g., V(r)), and expand the density matrix, pars(r)= C n$w *(r - tJ o(r - tj),
(2.2)
where the ti label ionic sites. If N, is the total number of electrons, then we have C [ dr p,,(r) = c c n:t = Nl .
Substitution n$! gives,
of (2.2) into (2.1), and minimization
aT, + g$+p ana? 13 23
(2.3)
of E with respect to the numbers
r w*(r - ti) V(r) w(r - tj) + hd,Jij
= 0,
(2.4)
LOCALIZED
MOMENTS
IN
269
METALS
where Er is the total electron-electron interaction energy. By comparison with the third term in (2.4) it was shown in SM that we could interpret the second term as,
34 = __ WB23
s
dr o*(r
- tJ VyB(r) w(r - ti),
where W-9
is the contribution from the electron interaction to the (spin-dependent) potential V,, in which the electrons move, i.e.,
one-body
v,dr) = V(r) + P(r).
(2.7)
In the next section we show how these equations can be generalized to describe a system of many magnetic impurities in a nonmagnetic host metal. For example, much is known experimentally about systems like PdNi, RhNi, and CuNi over a range of Ni impurity concentrations and it is to these types of systems that the present formulation will apply.
3. THE MANY-IMPURITY
MODEL
IN DENSITY
FUNCTIONAL
THEORY
We consider a system of many mutually interacting impurities, assumed to be identical, in a host metal that has a conduction band defined by a given one-body potential V(r). This conduction band is described by wavefunctions &(r) (or Wannier functions w(r)) with corresponding eigenvalue c(k). Since the Coulomb interaction between the conduction electrons of the host metal may be strong if it is a transition metal, then some extra potential V,(r) must be introduced to describe this interaction. This potential will obviously be density dependent, and later we choose this dependence appropriate to a Stoner model of &function interactions. The electron interactions are obviously very complicated in the system that we are trying to describe. As in SMW, the problem is simplified by separating parts of the one-body potential that describe particular effects. These parts of the total one-body potential are assumed to be density independent, but they have special properties, which are expressed through their matrix elements, as we show later. We have that the t,‘s label ionic sites in the system and introduce a potential V(r - t& localized round a particular site ti , which defines an impurity at this site. The impurity wavefunction defined by V is &(r - tJ and corresponding
270
J. C. STODDART
impurity level Ed (assumed to be a single level, as usual in the Anderson type models). In any model for this system we have interactions between the conduction electrons and the impurity (a) electrons, and a part of the one-body potential V,(r - ti) describes this effect. One can think of V, as being localized round an impurity site. It is assumed to be density independent. Lastly, we may include a term to represent the direct interaction between the impurities in the system, which obviously will be important at high concentrations, and important, as explained by Kim[6], to describe environmental effects in transition metal alloys. This interaction can be introduced through a density independent potential V?(r), which may be regarded as localized between the two impurities at td and tj . These potentials are made to describe the particular physical effects noted, through properties of their matrix elements, as we show later. We may now write the total (ground-state) energy for the system in the form (see SM and SMW),
E = Tsb+l + TAP-1+ Wb, , p-1,
(3.1)
where EIT = 1 dr WI@+(r)
+ p-(r)> + 1 VB(r)(p+(r)
+ p-(r))
+ Ct j dr W - fd)CO+(r)+ p-(r)) +
C z
J
h
K(r
-
t&p+(r)
+
p-(r)>
i- i,mc 1h ~?(r)@+(r) + p-(r)> + E&+,p-1.
(3.2)
In (3.2) impurity sites are labeled by i and m, and EI represents the electronelectron interactions that are not included in the tist terms. If precise contact is to be made with the Anderson type model of this system, then EI would represent the interaction energy between electrons in the impurity states rjd . However, it can be left quite general for the moment. The one-body potential that we obtain by minimizing the energy (3.1) with respect to the diagonal parts p+ , p- of the density matrix then has the general form, v,(r)
= v(r) + C V(r - t,) + V,(r) + C Vdr - ti) + C CYr)
*
z
+ v&)-
i,m (3.3)
LOCALIZED
MOMENTS
IN
271
METALS
Here u,(r) is obtained by functional variation of EI , i.e., (3.4)
and is dependent upon the spin densities in general. We may vary the model that we have just set up by changing v, . The first two terms on the right-hand side of (3.3) define eigenfunctions I& (or Wannier function w(r)) and +d, respectively. As discussed above, the physical meaning of the contribution to V,, will be expressed through their matrix elements. The only matrix elements that we take to be nonzero are
W9 t4&9 sdrIcrk*(r) - tJVr- ti)q&(r) sdrAt*@
(3.5) (3.6)
and vki
=
J
dr
#k*(r)
v,(r
II,, = l dr &*(r
-
ti1
$d(r
- tJ V?(r) &(r
-
h>
- tj)
(3.7) (3.8)
where 17ii = 0. The one-body potential (3.3) defines spin-dependent eigenfunctions &Jr) through the Schrodinger equation, which are then used in the usual way to construct the spin densities, f&)
=
C
GXr> 9Ldr).
?dOCC)
(3.9)
In analogy with SMW, we expand the & (and so the pa(r terms of the complete set of functions #k (or w(r)) and C#Q. This expansion obviously takes the general form, PO(r) = C n$*(r j.1
- tJ w(r - ti)
+ 2 b&&*(r j.m
- tm) w(r - tj) + c.c.
(3.10)
272
J.
C.
STODDART
where i, m vary over impurity sites only andj, 1 vary over all ionic sites. Obviously then the interaction energy EI can be regarded as a function of the expansion parameters n*, mu, and Ia. According to the theory of SM and Section 2, we can then interpret the derivatives of EI as follows: -aJ-3 = anyj
s
dr w*(r - tJ u,(r) w(r - tj)
(3.11)
a& = __ s dr &*(r abkj
- t,J u,(r) w(r - tJ
(3.12)
- a4
- t,) q,(r) &(r
(3.13)
alki
= s dr &*(r
- tJ.
Thus, through (3.5)-(3.8) and (3.1 l)-(3.13), provided that we know the interaction functional E, , we can completely specify the one-body spin-dependent potential for the model of magnetic imputities. In the next section we use these equations, through the one-body Green’s function, to set up the Euler equations for the magnetic state of the system.
4. THE ONE-BODY GREEN’S FUNCTION
We work with the one-body Green’s function G, defined in terms of the one-body eigenfunctions & by (4.1) The spin densities are given directly in terms of G, by,
Obviously, we may expand G, , in a way similar to p0 in (3.10), in terms of the complete set of functions w and 4 as follows:
CArroE+) = $31 c gj”,@+) w*(r- 6)4m - 6) + I.m c kXE+)w*(r- tj) &(r, - t,J + C.C. + C &LdE+) A*@- G>vWO- L>. i.m
(4.3)
LOCALIZED
MOMENTS
IN
213
METALS
The equation that G, satisfies is given in terms of the one-body potential (3.3) by [ -$V:
- E] G,(rr,E+)
= - V,(r) G,(rr,E+)
- 6(r - r,,).
(4.4)
We require, of course, equations for the unknown expansion coefficients go, ka, and of (4.3) into (4.4), multiplying both sides by w(r - to) w*(ro - tn), - tf), $,(r - tf) o*(q - t,), +,(r - tf) cbd*(r,, - t,.), in turn and integrating, we obtain directly, using (3.1 l)-(3.13), the following set of equations for the coefficients in (4.3): ho. Substitution w(r - t,) &*(r
-
-
(4.6) -
(4.7)
-4.
(4.8)
In these equations, Tnj is defined by - $V:w(r
and
- tj) + v(r) w(r - tj) = C T,+(r n
P
V,, = J dr w*(r
- t,> Vdr - t3 $dr - t3,
and we have assumed that
s
&*(r
- tn)
- tJ o(r - ti) dr = 0
(4.9)
274
J. C. STODDART
for all i andj. It is also important to note that the indices i, m, r, andflabel impurity sites in (4.5)-(4.8), other indices covering the entire range. In (4.5) and (4.6), V,(jc) is defined by V,(jc) = J dr w*(r - t,) &(r) w(r - t,)
(4.10)
where V,(r) is the potential that describes the interaction between the lost metal conduction band electrons. This potential should be spin dependent, and to be explicit we choose a Stoner model to represent these electron interactions, in which VB is density dependent. It is easily seen from (4.5) and (4.6) that we can incorporate the effect of V,cic> in aE@zjo, and the Stoner model amounts to assuming that the no dependence of EI is given by (4.11)
- -iv c (no)” CT=* where n, = nyj , which is independent of host metal site j. Thus, we put
-8J-G = - JhJj, an;, in (4.5) and (4.6) and drop the term involving V,(jc). This choice of a Stoner model corresponds essentially to that of Kim (1970) in the Hartree-Fock model. Equations (4.5)-(4.8) determine the expansion coefficients in terms of nq b”, and P. The final step in the density functional procedure is to write down the Euler equations that determine the ground-state values of V, b”, and 10, and so the magnetic state through (3.10). From (3.10), (4.2), and (4.3), we find these equations directly as
Z;i = - &
J:
m
dE (h;,(E+)
- h;,(E-)).
(4.15)
The formal problem is thus well defined, although we require knowledge of the density-dependent part of V, discussed in Section 3, and of course the electronelectron interaction function EI to make the model quantative. We show in the next section that the Anderson model in Hartree-Fock theory for the many impurity model, as discussed by Kim 161,follows directly by a simple choice of EI . Although lack of precise knowledge of the effect of correlation effects on the functional EI makes it difficult to go beyond Hartree-Fock theory, it is possible to examine general features of models of this type.
LOCALIZED
5.
MOMENTS IN METALS
THE HARTREE-FOCK
275
APPROXIMATION
A particular model in this present framework is specified by giving the interaction functional E, , either as a functional of the spin densitiesp+ and p- , or as a function of the parameters PP,IV, and I”. In the Anderson model, Er describes the interaction of the electrons when they are in the impurity states of the system and so corresponds to an EI that depends only on the parameter P. The Hartree-Fock limit of the Anderson model was discussedin SMW for the single impurity case. A straightforward generalization of this result for many impurities gives E,HF= I c l;l,; , z (impurity
(5.1)
sites)
where the parameter I measuresthe strength of the Coulomb repulsion between electrons of opposite spin in the sameimpurity state. Thus, in (4.5)-(4.8) we have (5.2)
With (5.2) and (5.3) in (4.5)-(4.8) we have the HF theory of the many impurity Anderson model. The equivalence with the work of Kim [6] in HF theory is easily shown. In HF theory, we have just one parameter to determine self-consistently through (4.12), i.e., the Green’s function parameters g”, k”, and h” depend only on the 1; . The solution of (4.5)-(4.8) with the matrix elements of the potential V, given above easily can be shown to be I&” where,
595/95/2-4
Vi2F,,“(E+) - 2: (E,) + ic -’
(5.4)
276
J. C. STODDART
and Eb”(E+) = G@+) (5.7) and Vi E Vka given by (3.7), and assumed to be k independent. The solution (5.4) can then be substituted into (4.12) with m = i to determine self-consistently the impurity state occupation number lZ; . It is easily verified that (at T = 0) this is the same result as that obtained by Kim [6]. The function cf contains the “environment” effect in transition metal alloys discussed extensively by Kim. This function leads to a local modification in the electronic structure at a certain impurity site due to the presence of other impurities nearby. Through (5.5) it is easily seen that Cl depends, through the summation over impurity sites j, upon the impurity structure near to the ith site. This dependence, together with V,2F0u(E), can provide the explanation of local moment formation at a certain impurity site upon the presence of other impurities near this site. The simplest approach to the problem is to assume that (5.8) is approximately independent of E (and equal to its value when E = p) and also independent of the spin. Then the Euler equation (4.12) becomes (for m = i) Im Ai Ii0 = + S1:, dE (E _ cd - Zl;O - Re AJ2 + (Im Ai) (5.9) = ;
1
cot-l
Ed
i- ilip”+ Re Ai - p Im Ai
If we define ImAi n,qE) = -!n- (E - cd - Zl;” - Re Ai) + (Im Ai)2 as, in the present approximation scheme, the impurity density of states for the ith impurity, then the condition for a localized moment on the ith impurity is given by the Hartree-Fock condition, ml”‘(p)
> 1,
(5.11)
where rp(p)
= n;(p)
at
Ii0 = 0.
Obviously, the environment effect enters through the dependence of n(“Vj~) on A, and a qualitative explanation can be given of the effects observed in transition metal alloys that seem to depend sensitively upon local environment.
LOCALIZED
6. MODELS THAT
277
MOMENTS IN METALS
INCLUDE CORRELATION
EFFECTS
We have shown that the usual Hartree-Fock results follow from the simple interaction function (5.1). It is difficult at the present time to go beyond HartreeFock theory. Knowledge of correlation functionals has so far been restricted to the case of the uniform electron gas, and extensions of this via gradient expansions that have little bearing in the present case. In this section, we want to discuss some general features that would hold for any model specified by an interaction function EI (no, b”, I”).
The basic equations of the theory are given by (4.5)-(4.8) in which we assume a general EI that can depend on the parameters PP, b”, and 20. The effect of the dependence on both b” and no can be seen quite easily. Firstly, if we allow E, to depend on b” then the terms involving aEI/8b” can be incorporated exactly by the replacement in these equations of VCi by, (6.1)
Thus, the dependence of EI on b” just alters the coupling of the band states to the impurity states, in effect making this coupling spin dependent and density dependent. Secondly, if E, depends upon no in a more general way than in the Stoner model discussed earlier then it is easily seen that the terms that depend on 8EI/&zu can be incorporated exactly by the replacement of Tcj (given by (5.9)) by (6.2)
Using Tcj = $ c e-ik’(tr-fj)c(k) k
and (3.11) we have
If we can approximate,
s dr
$k*@)
f-44
#qW
-
fW
d kg
65)
then the no dependence of EI just causes the replacement, 44 ---f 44 + f”(k).
66)
278
J. C. STODDART
Thus, the effect is to make the band states in the host metal spin dependent and also density dependent. These two effects may well turn out to be significant. However, at present we know of no experimental evidence to guide us on this point. We shall focus all attention, therefore, in what follows, on the effect of the dependence of EI on the parameters I& . We shall make, again in the absence of other information, the further restrictions to models in which E, is dependent only on the diagonal parameters 1: = lia. We expect, in general, an interaction function of the form,
where i runs over the impurity sites. Thus, in the simplified model discussed in Section 5, in which 4, given by (5.8) is independent of energy, we have the Euler equation that determines the local moment on impurity site i given by (Im 4
+ (Im d,)’ &
Ii” = $ .i-L FEE,“)”
where Eio = l d + 2
1
+ Re Ai
(6.9)
Thus, as in (5.9), we can write (6.8) as, E
+ d
aEI(4+,
,i-> ario
+ Re Ai = (Im Ai) cot(+‘).
It can be seen easily that the only change from the HF theory of Section 5, in particular, Eq. (5.9), is the replacement,
in the equation that determines the local moment. An important point to emphasize is that due to this close connection of better models, specified by or ) to the HF theory, then many of the conclusions reached in the HF model are still true. In particular, it is easy to see through (6.10) that the same qualitative description of the effect of the local environment on the formation of a local moment at site i, through the function di still holds true, even though a rather different condition for the moment formation would be obtained. Secondly, it is easy to show that upon investigation of the susceptibilities, the same conclusion reached by Kim [6] in HF theory still holds. The impurity susceptibility X, can be shown to diverge
LOCALIZED
MOMENTS IN METALS
279
at the same time as the host metal susceptibility, and thus, the condition for the occurrence of a local moment in the alloy is the same as the condition for the ferromagnetism of the entire system. This conclusion can be seen easily by a generalization of the method of Kim [6] in HF theory. The replacement (6.11) in Kim’s method obviously doesnot affect any of the arguments. The above result is solely a result of the impurity-band coupling and the conduction-band interactions and would hold with any E]. Finally, we would like to stress the possible importance of Eq. (6.10) in the investigation of the correlation effects in alloys. Obviously the environment factor d, is difficult to handle, unlesssomesort of average “environment” can be assumed. The correlation effects are poorly understood and the present formulation of the alloy problem offers a rather direct handle on them. The correlation energy function enters very simple into Eq. (6.10) which essentially describesthe local moment liU. At the very least, measurementsof local moments in alloys could through (6.10) investigate the validity of HF theory.
7. SUMMARY In this work, a description of transition metal alloys, based upon the density functional theory of the ground state of a spin polarized system has been set up. The particular model being used dependedupon specifying an interaction function EI , which dependsupon parameters 10,b”, and no, which give the electron density in the alloy system through (5.10). It is emphasized that derivatives of E, with respect to these parameters give through (3.1l)-(3.13) the matrix elements of the one-body spin-dependent potential between the expansion functions. The most important parameters are the Z,‘i, and it is shown that the system can be solved exactly once the dependence of E, on Ii” is known. Thus, Anderson and similar models are included in the general formulation, although exact contact has proved difficult. It is shown that the Hartree-Fock theory of Kim [6], which can explain many featues of transition metal alloys qualitatively, can be obtained with a very simple Hartree-Fock functional, which corresponds in this approximation to the Anderson model. Although models superior to the HF case are difficult to formulate at the moment, the density functional picture of these transition metal alloys allows certain predictions to be made. Simple changes in the HF equations allow us to seesome of the effect of introducing electron correlations into the system. Qualitatively, the effect of impurity environment on local moment formation is the same as in HF theory. It is felt that the present formulation of the alloy problem in the language of
280
J.
C. STODDART
density functional theory allows one additional physical insight into a rather difficult problem. Through local moment measurement, it allows one also a method of investigating the correlation problem in these alloy systems. ACKNOWLEDGMENT The author would like to thank Professor N. H. March for a critical reading of an earlier draft of this work and for many useful comments. REFERENCES 1. J. C. STODDARTAND N. H. MARCH, Ann. P/qvs. (N. Y.) 64 (1970), 174. 2. J. C. STODDART,N. H. MARCH, AND D. WIID, Intern&. J. Quantum Chem. 5 (1971),745. 3. U. VON BARTH AND L. HEDIN, J. Phys. Ser. C 5 (1972), 1629. 4. A. K. RAJACXIPAL AND J. CALLAWAY, Phys. Rev. B7 (1973),1912. 5. J. LANGLINAIS AND J. CALLAWAY, Phys. Rev. BJ (1972),124. 6. D. J. KIM, Phys. Rev. Bl (1970),3725.
OF PHYSICS
Finite
95, 267-280 (1975)
Concentrations
of Localized
Moments
in Metals
J. C. STODDART Department of Physics, The University, Shefield, SlO 2TN, England Received November 4, 1974
In earlier work, density functional methods have been used to study the magnetism of pure metals and the problem of local moment formation on a single magnetic impurity. In the present paper these methods are reviewed and applied to the case of an arbitrary concentration of impurity atoms. It is shown that the Hartree-Fock treatment of the system and a local environment effect can be reproduced very easily in the density functional framework. More importantly, the present formulation allows one some insight into the effects of correlation. Although correlation density functionals are difficult to obtain, it is shown that a class of models of the many impurity system can be solved exactly.
1. INTRODUCTION In a previous paper (Stoddart and March [l], referred to as SM), a theory based entirely on the spin densities p+ and p- in a magnetic metal has been formulated. Ref. [l] was concerned solely with periodic metallic magnets, whereas in a second paper the spin-density description was applied to localized moments in metals (Stoddart, March, and Wiid [2], referred to as SMW). The approach adopted by SM and SMW was taken up by von Barth and Hedin [3] and by Rajagopal and Callaway [4]. These workers give a somewhat more fundamental discussion of the use of the spin densities p+ and p- to characterize the magnetic system. The basic problem one encounters, of course, in the spin-density description is the lack of knowledge of the energy functionals. Some attempt to get round this has been made by von Barth and Hedin [3] for the special case of the uniform gas, though unfortunately this is not the interesting limit in magnetic metals, the tight-binding methods being much more appropriate in, say, the d bands of ferromagnetic nickel. Nevertheless, neglecting correlation effects, one can write down the usual generalization of the Dirac-Slater exchange potential to deal with the magnetized case. Such exchange potentials have been used in band structure calculations on Ni (see, for example, [5]). 267
Copyright All rights
6 1975 by Academic Press, Inc. of reproduction in any form reserved.
268
J. C. STODDART
The present paper is concerned with the problem of the type of magnetic impurities in metals for which the Anderson Hamiltonian affords a useful starting point. While the Anderson model of a single magnetic impurity was discussed by SMW, the case of a finite concentration of magnetic impurities, which can interact together, is the main problem tackled here. Some work on finite concentration has been carried out by Kim [6], using standard many-body theory. However, he imposes a Hartree-Fock condition on the interaction term. It is shown first (Section 5) that Kim’s results can be regained from the spin-density formalism. Furthermore, some insight can be gained into the likely effect of relaxing the Hartree-Fock constraint on the interaction, i.e., the effect of correlation. 2. THE SPIN POLARIZED
CASE
Following von Barth and Hedin [3], we can write the total ground-state energy for the spin polarized system as
where T, is the kinetic energy functional for a system of noninteracting Fermions, EcouL is the classical Coulomb energy, Eze is the exchange and correlation contribution, and aae is the spin-dependent external potential. Minimization of E[pJ with respect to pNswill give an Euler equation involving w,, , provided that these functionals are known. We consider the general case when w,is spin independent, and is in fact just the static potential V(r) due to the ionic lattice. A procedure, introduced in SM, is very useful in formulating Anderson type models, and will be used later. We describe it here for the exact functional (2.1). We assume that we have a set of Wannier functions UJ corresponding to some periodic potential (e.g., V(r)), and expand the density matrix, pars(r)= C n$w *(r - tJ o(r - tj),
(2.2)
where the ti label ionic sites. If N, is the total number of electrons, then we have C [ dr p,,(r) = c c n:t = Nl .
Substitution n$! gives,
of (2.2) into (2.1), and minimization
aT, + g$+p ana? 13 23
(2.3)
of E with respect to the numbers
r w*(r - ti) V(r) w(r - tj) + hd,Jij
= 0,
(2.4)
LOCALIZED
MOMENTS
IN
269
METALS
where Er is the total electron-electron interaction energy. By comparison with the third term in (2.4) it was shown in SM that we could interpret the second term as,
34 = __ WB23
s
dr o*(r
- tJ VyB(r) w(r - ti),
where W-9
is the contribution from the electron interaction to the (spin-dependent) potential V,, in which the electrons move, i.e.,
one-body
v,dr) = V(r) + P(r).
(2.7)
In the next section we show how these equations can be generalized to describe a system of many magnetic impurities in a nonmagnetic host metal. For example, much is known experimentally about systems like PdNi, RhNi, and CuNi over a range of Ni impurity concentrations and it is to these types of systems that the present formulation will apply.
3. THE MANY-IMPURITY
MODEL
IN DENSITY
FUNCTIONAL
THEORY
We consider a system of many mutually interacting impurities, assumed to be identical, in a host metal that has a conduction band defined by a given one-body potential V(r). This conduction band is described by wavefunctions &(r) (or Wannier functions w(r)) with corresponding eigenvalue c(k). Since the Coulomb interaction between the conduction electrons of the host metal may be strong if it is a transition metal, then some extra potential V,(r) must be introduced to describe this interaction. This potential will obviously be density dependent, and later we choose this dependence appropriate to a Stoner model of &function interactions. The electron interactions are obviously very complicated in the system that we are trying to describe. As in SMW, the problem is simplified by separating parts of the one-body potential that describe particular effects. These parts of the total one-body potential are assumed to be density independent, but they have special properties, which are expressed through their matrix elements, as we show later. We have that the t,‘s label ionic sites in the system and introduce a potential V(r - t& localized round a particular site ti , which defines an impurity at this site. The impurity wavefunction defined by V is &(r - tJ and corresponding
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J. C. STODDART
impurity level Ed (assumed to be a single level, as usual in the Anderson type models). In any model for this system we have interactions between the conduction electrons and the impurity (a) electrons, and a part of the one-body potential V,(r - ti) describes this effect. One can think of V, as being localized round an impurity site. It is assumed to be density independent. Lastly, we may include a term to represent the direct interaction between the impurities in the system, which obviously will be important at high concentrations, and important, as explained by Kim[6], to describe environmental effects in transition metal alloys. This interaction can be introduced through a density independent potential V?(r), which may be regarded as localized between the two impurities at td and tj . These potentials are made to describe the particular physical effects noted, through properties of their matrix elements, as we show later. We may now write the total (ground-state) energy for the system in the form (see SM and SMW),
E = Tsb+l + TAP-1+ Wb, , p-1,
(3.1)
where EIT = 1 dr WI@+(r)
+ p-(r)> + 1 VB(r)(p+(r)
+ p-(r))
+ Ct j dr W - fd)CO+(r)+ p-(r)) +
C z
J
h
K(r
-
t&p+(r)
+
p-(r)>
i- i,mc 1h ~?(r)@+(r) + p-(r)> + E&+,p-1.
(3.2)
In (3.2) impurity sites are labeled by i and m, and EI represents the electronelectron interactions that are not included in the tist terms. If precise contact is to be made with the Anderson type model of this system, then EI would represent the interaction energy between electrons in the impurity states rjd . However, it can be left quite general for the moment. The one-body potential that we obtain by minimizing the energy (3.1) with respect to the diagonal parts p+ , p- of the density matrix then has the general form, v,(r)
= v(r) + C V(r - t,) + V,(r) + C Vdr - ti) + C CYr)
*
z
+ v&)-
i,m (3.3)
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MOMENTS
IN
271
METALS
Here u,(r) is obtained by functional variation of EI , i.e., (3.4)
and is dependent upon the spin densities in general. We may vary the model that we have just set up by changing v, . The first two terms on the right-hand side of (3.3) define eigenfunctions I& (or Wannier function w(r)) and +d, respectively. As discussed above, the physical meaning of the contribution to V,, will be expressed through their matrix elements. The only matrix elements that we take to be nonzero are
W9 t4&9 sdrIcrk*(r) - tJVr- ti)q&(r) sdrAt*@
(3.5) (3.6)
and vki
=
J
dr
#k*(r)
v,(r
II,, = l dr &*(r
-
ti1
$d(r
- tJ V?(r) &(r
-
h>
- tj)
(3.7) (3.8)
where 17ii = 0. The one-body potential (3.3) defines spin-dependent eigenfunctions &Jr) through the Schrodinger equation, which are then used in the usual way to construct the spin densities, f&)
=
C
GXr> 9Ldr).
?dOCC)
(3.9)
In analogy with SMW, we expand the & (and so the pa(r terms of the complete set of functions #k (or w(r)) and C#Q. This expansion obviously takes the general form, PO(r) = C n$*(r j.1
- tJ w(r - ti)
+ 2 b&&*(r j.m
- tm) w(r - tj) + c.c.
(3.10)
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J.
C.
STODDART
where i, m vary over impurity sites only andj, 1 vary over all ionic sites. Obviously then the interaction energy EI can be regarded as a function of the expansion parameters n*, mu, and Ia. According to the theory of SM and Section 2, we can then interpret the derivatives of EI as follows: -aJ-3 = anyj
s
dr w*(r - tJ u,(r) w(r - tj)
(3.11)
a& = __ s dr &*(r abkj
- t,J u,(r) w(r - tJ
(3.12)
- a4
- t,) q,(r) &(r
(3.13)
alki
= s dr &*(r
- tJ.
Thus, through (3.5)-(3.8) and (3.1 l)-(3.13), provided that we know the interaction functional E, , we can completely specify the one-body spin-dependent potential for the model of magnetic imputities. In the next section we use these equations, through the one-body Green’s function, to set up the Euler equations for the magnetic state of the system.
4. THE ONE-BODY GREEN’S FUNCTION
We work with the one-body Green’s function G, defined in terms of the one-body eigenfunctions & by (4.1) The spin densities are given directly in terms of G, by,
Obviously, we may expand G, , in a way similar to p0 in (3.10), in terms of the complete set of functions w and 4 as follows:
CArroE+) = $31 c gj”,@+) w*(r- 6)4m - 6) + I.m c kXE+)w*(r- tj) &(r, - t,J + C.C. + C &LdE+) A*@- G>vWO- L>. i.m
(4.3)
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METALS
The equation that G, satisfies is given in terms of the one-body potential (3.3) by [ -$V:
- E] G,(rr,E+)
= - V,(r) G,(rr,E+)
- 6(r - r,,).
(4.4)
We require, of course, equations for the unknown expansion coefficients go, ka, and of (4.3) into (4.4), multiplying both sides by w(r - to) w*(ro - tn), - tf), $,(r - tf) o*(q - t,), +,(r - tf) cbd*(r,, - t,.), in turn and integrating, we obtain directly, using (3.1 l)-(3.13), the following set of equations for the coefficients in (4.3): ho. Substitution w(r - t,) &*(r
-
-
(4.6) -
(4.7)
-4.
(4.8)
In these equations, Tnj is defined by - $V:w(r
and
- tj) + v(r) w(r - tj) = C T,+(r n
P
V,, = J dr w*(r
- t,> Vdr - t3 $dr - t3,
and we have assumed that
s
&*(r
- tn)
- tJ o(r - ti) dr = 0
(4.9)
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J. C. STODDART
for all i andj. It is also important to note that the indices i, m, r, andflabel impurity sites in (4.5)-(4.8), other indices covering the entire range. In (4.5) and (4.6), V,(jc) is defined by V,(jc) = J dr w*(r - t,) &(r) w(r - t,)
(4.10)
where V,(r) is the potential that describes the interaction between the lost metal conduction band electrons. This potential should be spin dependent, and to be explicit we choose a Stoner model to represent these electron interactions, in which VB is density dependent. It is easily seen from (4.5) and (4.6) that we can incorporate the effect of V,cic> in aE@zjo, and the Stoner model amounts to assuming that the no dependence of EI is given by (4.11)
- -iv c (no)” CT=* where n, = nyj , which is independent of host metal site j. Thus, we put
-8J-G = - JhJj, an;, in (4.5) and (4.6) and drop the term involving V,(jc). This choice of a Stoner model corresponds essentially to that of Kim (1970) in the Hartree-Fock model. Equations (4.5)-(4.8) determine the expansion coefficients in terms of nq b”, and P. The final step in the density functional procedure is to write down the Euler equations that determine the ground-state values of V, b”, and 10, and so the magnetic state through (3.10). From (3.10), (4.2), and (4.3), we find these equations directly as
Z;i = - &
J:
m
dE (h;,(E+)
- h;,(E-)).
(4.15)
The formal problem is thus well defined, although we require knowledge of the density-dependent part of V, discussed in Section 3, and of course the electronelectron interaction function EI to make the model quantative. We show in the next section that the Anderson model in Hartree-Fock theory for the many impurity model, as discussed by Kim 161,follows directly by a simple choice of EI . Although lack of precise knowledge of the effect of correlation effects on the functional EI makes it difficult to go beyond Hartree-Fock theory, it is possible to examine general features of models of this type.
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5.
MOMENTS IN METALS
THE HARTREE-FOCK
275
APPROXIMATION
A particular model in this present framework is specified by giving the interaction functional E, , either as a functional of the spin densitiesp+ and p- , or as a function of the parameters PP,IV, and I”. In the Anderson model, Er describes the interaction of the electrons when they are in the impurity states of the system and so corresponds to an EI that depends only on the parameter P. The Hartree-Fock limit of the Anderson model was discussedin SMW for the single impurity case. A straightforward generalization of this result for many impurities gives E,HF= I c l;l,; , z (impurity
(5.1)
sites)
where the parameter I measuresthe strength of the Coulomb repulsion between electrons of opposite spin in the sameimpurity state. Thus, in (4.5)-(4.8) we have (5.2)
With (5.2) and (5.3) in (4.5)-(4.8) we have the HF theory of the many impurity Anderson model. The equivalence with the work of Kim [6] in HF theory is easily shown. In HF theory, we have just one parameter to determine self-consistently through (4.12), i.e., the Green’s function parameters g”, k”, and h” depend only on the 1; . The solution of (4.5)-(4.8) with the matrix elements of the potential V, given above easily can be shown to be I&” where,
595/95/2-4
Vi2F,,“(E+) - 2: (E,) + ic -’
(5.4)
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J. C. STODDART
and Eb”(E+) = G@+) (5.7) and Vi E Vka given by (3.7), and assumed to be k independent. The solution (5.4) can then be substituted into (4.12) with m = i to determine self-consistently the impurity state occupation number lZ; . It is easily verified that (at T = 0) this is the same result as that obtained by Kim [6]. The function cf contains the “environment” effect in transition metal alloys discussed extensively by Kim. This function leads to a local modification in the electronic structure at a certain impurity site due to the presence of other impurities nearby. Through (5.5) it is easily seen that Cl depends, through the summation over impurity sites j, upon the impurity structure near to the ith site. This dependence, together with V,2F0u(E), can provide the explanation of local moment formation at a certain impurity site upon the presence of other impurities near this site. The simplest approach to the problem is to assume that (5.8) is approximately independent of E (and equal to its value when E = p) and also independent of the spin. Then the Euler equation (4.12) becomes (for m = i) Im Ai Ii0 = + S1:, dE (E _ cd - Zl;O - Re AJ2 + (Im Ai) (5.9) = ;
1
cot-l
Ed
i- ilip”+ Re Ai - p Im Ai
If we define ImAi n,qE) = -!n- (E - cd - Zl;” - Re Ai) + (Im Ai)2 as, in the present approximation scheme, the impurity density of states for the ith impurity, then the condition for a localized moment on the ith impurity is given by the Hartree-Fock condition, ml”‘(p)
> 1,
(5.11)
where rp(p)
= n;(p)
at
Ii0 = 0.
Obviously, the environment effect enters through the dependence of n(“Vj~) on A, and a qualitative explanation can be given of the effects observed in transition metal alloys that seem to depend sensitively upon local environment.
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6. MODELS THAT
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MOMENTS IN METALS
INCLUDE CORRELATION
EFFECTS
We have shown that the usual Hartree-Fock results follow from the simple interaction function (5.1). It is difficult at the present time to go beyond HartreeFock theory. Knowledge of correlation functionals has so far been restricted to the case of the uniform electron gas, and extensions of this via gradient expansions that have little bearing in the present case. In this section, we want to discuss some general features that would hold for any model specified by an interaction function EI (no, b”, I”).
The basic equations of the theory are given by (4.5)-(4.8) in which we assume a general EI that can depend on the parameters PP, b”, and 20. The effect of the dependence on both b” and no can be seen quite easily. Firstly, if we allow E, to depend on b” then the terms involving aEI/8b” can be incorporated exactly by the replacement in these equations of VCi by, (6.1)
Thus, the dependence of EI on b” just alters the coupling of the band states to the impurity states, in effect making this coupling spin dependent and density dependent. Secondly, if E, depends upon no in a more general way than in the Stoner model discussed earlier then it is easily seen that the terms that depend on 8EI/&zu can be incorporated exactly by the replacement of Tcj (given by (5.9)) by (6.2)
Using Tcj = $ c e-ik’(tr-fj)c(k) k
and (3.11) we have
If we can approximate,
s dr
$k*@)
f-44
#qW
-
fW
d kg
65)
then the no dependence of EI just causes the replacement, 44 ---f 44 + f”(k).
66)
278
J. C. STODDART
Thus, the effect is to make the band states in the host metal spin dependent and also density dependent. These two effects may well turn out to be significant. However, at present we know of no experimental evidence to guide us on this point. We shall focus all attention, therefore, in what follows, on the effect of the dependence of EI on the parameters I& . We shall make, again in the absence of other information, the further restrictions to models in which E, is dependent only on the diagonal parameters 1: = lia. We expect, in general, an interaction function of the form,
where i runs over the impurity sites. Thus, in the simplified model discussed in Section 5, in which 4, given by (5.8) is independent of energy, we have the Euler equation that determines the local moment on impurity site i given by (Im 4
+ (Im d,)’ &
Ii” = $ .i-L FEE,“)”
where Eio = l d + 2
1
+ Re Ai
(6.9)
Thus, as in (5.9), we can write (6.8) as, E
+ d
aEI(4+,
,i-> ario
+ Re Ai = (Im Ai) cot(+‘).
It can be seen easily that the only change from the HF theory of Section 5, in particular, Eq. (5.9), is the replacement,
in the equation that determines the local moment. An important point to emphasize is that due to this close connection of better models, specified by or ) to the HF theory, then many of the conclusions reached in the HF model are still true. In particular, it is easy to see through (6.10) that the same qualitative description of the effect of the local environment on the formation of a local moment at site i, through the function di still holds true, even though a rather different condition for the moment formation would be obtained. Secondly, it is easy to show that upon investigation of the susceptibilities, the same conclusion reached by Kim [6] in HF theory still holds. The impurity susceptibility X, can be shown to diverge
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MOMENTS IN METALS
279
at the same time as the host metal susceptibility, and thus, the condition for the occurrence of a local moment in the alloy is the same as the condition for the ferromagnetism of the entire system. This conclusion can be seen easily by a generalization of the method of Kim [6] in HF theory. The replacement (6.11) in Kim’s method obviously doesnot affect any of the arguments. The above result is solely a result of the impurity-band coupling and the conduction-band interactions and would hold with any E]. Finally, we would like to stress the possible importance of Eq. (6.10) in the investigation of the correlation effects in alloys. Obviously the environment factor d, is difficult to handle, unlesssomesort of average “environment” can be assumed. The correlation effects are poorly understood and the present formulation of the alloy problem offers a rather direct handle on them. The correlation energy function enters very simple into Eq. (6.10) which essentially describesthe local moment liU. At the very least, measurementsof local moments in alloys could through (6.10) investigate the validity of HF theory.
7. SUMMARY In this work, a description of transition metal alloys, based upon the density functional theory of the ground state of a spin polarized system has been set up. The particular model being used dependedupon specifying an interaction function EI , which dependsupon parameters 10,b”, and no, which give the electron density in the alloy system through (5.10). It is emphasized that derivatives of E, with respect to these parameters give through (3.1l)-(3.13) the matrix elements of the one-body spin-dependent potential between the expansion functions. The most important parameters are the Z,‘i, and it is shown that the system can be solved exactly once the dependence of E, on Ii” is known. Thus, Anderson and similar models are included in the general formulation, although exact contact has proved difficult. It is shown that the Hartree-Fock theory of Kim [6], which can explain many featues of transition metal alloys qualitatively, can be obtained with a very simple Hartree-Fock functional, which corresponds in this approximation to the Anderson model. Although models superior to the HF case are difficult to formulate at the moment, the density functional picture of these transition metal alloys allows certain predictions to be made. Simple changes in the HF equations allow us to seesome of the effect of introducing electron correlations into the system. Qualitatively, the effect of impurity environment on local moment formation is the same as in HF theory. It is felt that the present formulation of the alloy problem in the language of
280
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C. STODDART
density functional theory allows one additional physical insight into a rather difficult problem. Through local moment measurement, it allows one also a method of investigating the correlation problem in these alloy systems. ACKNOWLEDGMENT The author would like to thank Professor N. H. March for a critical reading of an earlier draft of this work and for many useful comments. REFERENCES 1. J. C. STODDARTAND N. H. MARCH, Ann. P/qvs. (N. Y.) 64 (1970), 174. 2. J. C. STODDART,N. H. MARCH, AND D. WIID, Intern&. J. Quantum Chem. 5 (1971),745. 3. U. VON BARTH AND L. HEDIN, J. Phys. Ser. C 5 (1972), 1629. 4. A. K. RAJACXIPAL AND J. CALLAWAY, Phys. Rev. B7 (1973),1912. 5. J. LANGLINAIS AND J. CALLAWAY, Phys. Rev. BJ (1972),124. 6. D. J. KIM, Phys. Rev. Bl (1970),3725.