Magnetic transition state approach to ferromagnetism of metals: Ni

Journal of Magnetism and Magnetic Materials 36 (1983) 125-130 North-Holland Publishing Company

125

MAGNETIC T R A N S I T I O N STATE A P P R O A C H TO F E R R O M A G N E T I S M OF M E T A L S : Ni

A.I. LIECHTENSTEIN, V.A. GUBANOV Institute for Chemistry, Sverdlovsk GSP- 145, USSR

M.I. KATSNELSON and V.I. ANISIMOV Institute for Metal Physics, Sverdlovsk, USSR Received 9 June 1982; received in revised form 13 September 1982

The generalization of Slater's magnetic transition state approach for the case of continuous energy distribution of d-states in metals is suggested. The method developed allows the first-principle calculations of local spin excitation energies. The connection of these magnetic excitations to the spin-wave spectrum and Curie temperature T~ for an arbitrary type of magnetic interaction is established. Using the X a - S W cluster method with k-dependent boundary conditions the numerical calculations of magnetic excitation energies and T~ for Ni have been carried out in very reasonable agreement with experimental values available.

I. Introduction

The recent development of spin-fluctuation theory [ 1-7] has lead to significant progress in understanding of the transition metal magnetism. Ground-state properties of ferromagnetic metals are described reasonably well by the band theory, whereas the thermodynamical properties are defined by the collective spin excitations which are the rotations of local magnetic moments. In refs. [1-7] such phenomena are considered in terms of the Hubbard Hamiltonian, ignoring the interaction of d-electrons on different sites, s-d hybridization and, usually, the degeneracy of d-states. A first-principle calculation of the collective spin excitation spectrum could be of much interest here. In ref. [8] ,the self-consistent band-structure calculations of ferromagnetic and hypothetical antiferromagnetic iron have been made and the difference of total energies of these configurations has been obtained. However, the inverse of half of the spins in the systems is surely a very strong perturbation and it is not clear in which way this energy difference is related to the spectrum of low-lying spin excitations near the ferromagnetic ground state. "Carious spin excitations in metal

iron corresponding to different spin arrangements were considered in ref. [9] based on the tight-binding model. The results explain well the high-temperature magnetism of iron but the introduction of Stoner intra-atomic exchange parameter in ref. [9] make this approach rather close to the Hubbard model. It should be noted also that the spin dynamic quantum effects are neglected in any approach using static local exchange fields [1]. First principle calculations of local spin excitation energies can be performed when one uses the magnetic transition state (MTS) approach introduced by Slater [10], In this method a complicated evaluation of the total energy differences is replaced by the calculations of one-electron energies for corresponding so called "transition states". This approach appeared to be surprisingly successful in magnetic excitation studies of the systems with well localized magnetic moments [ 11]. But this scheme should be generalized for the case of MTS in the continuous spectrum of d-metals. Then the problem of elucidating thermodynamic properties, the Curie temperature T~ in particular, in terms of such energy differences should be solved also. Here some quantum-statistical model should be used but as seen below its particular

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A.L Liechtenstein et al. / Magnetic transition state approach for Ni

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form is not very essential for this purpose. Besides, the calculation of local spin excitation energies is interesting in itself. In this paper we present the generalization of MTS method for the case of delocalized electronic states in metals. Using Ni metal as an example, we show the possibilities of such an approach. The energies of magnetic excitations and T¢ for Ni are calculated by the X ~ - S W method, being in a good agreement with the experimental estimations. 2. Magnetic transition state in a continuous spectrum

The main value which is calculated in the MTS scheme [10] is the energy AE of the local magnetic moment flip when the spin polarization of surrounding is fixed. F o r the case of magnetic insulators when the Heisenberg model is rather appropriate AE is connected with exchange parameters and T~ in quite a simple way [10,11]. This is not the case for d-metals. First of all one should obtain the exact expression for AE which is independent of the type of magnetic interactions. Let us consider the spin flip on some site in a ferromagnetic crystal. In this case the variation of transverse spin density components in Fourier representation is of the form: 8(Sq+ ) = 2 S 0,

8(S~)=2S

0,

where SO= (S*) is the value of the spin moment for the ground state. The corresponding magnetic field which induces such an inversion equals:

h;

= [X+-(q, 0 ) ] - ' a < S ; > ,

(1)

where q belongs to the first Brillioun zone. Taking into account the formula by Mori and Kawasaki for the spin wave frequency w(q) [12]:

hw (q) = 2 NSo/X + - (q, 0 ),

(3)

we obtain:

AE = ~

Y'.hw(q).

(4)

q

The expression (4) can be considered as some generalization for an arbitrary ferromagnetic system of the known formula for AE in the Heisenberg model [10,11]. It has a simple physical interpretation: AE equals the average spin wave energy multiplied by the extreme spin deviation value. Expression (4) can be used as a basis in the direct comparison of the calculated AE with the experimental parameters. However, the spin wave spectrum in metals is well defined only in the rather small region of q-space due to the existence of Stoner continuum [13]. Therefore we shall discuss here the connection of AE with the T~ values. The relation between AE and Tc can be established in the framework of a quantum-statistical model only. Nevertheless, the ratio T J A E is practically independent of the model parameters used. This allows us to believe that it has a rather general character. In ref. [3] the simple expression for Tc in the framework of the Hubbard model has been obtained in the static approximation. In terms of X + -(q, 0) it is as follows: Tc = ~2sg - ~ art / I - l +~_,X+-(q, 0) ] -' ,

(5)

q

h ; = [X-+(q, 0)]-'$
where I is the Hubbard parameter. Taking into account that T~ << I we can neglect I - 1 in eq. (5). Expression (5) corresponds to the Tyablicov approximation [14] in the Heisenberg model. The mean-field approximation differs from (5) by the replacing of [ZqX + -(q, 0)]-1 for Zq[X + -(q, 0)]-1, which gives (see eq. (2)):

aE = 1 E (hga+ hgS)

T~ = A E / 6 k B.

q

= 4Sg }--~[X + - ( q , 0)] --1, q

(2)

It can be shown that this value is the rigorous upper limit for T~. The most important deficiency

A,I. Liechtenstein et al. / Magnetic transition state approach for Ni

of eq. (5) is the neglect of the quantum nature of spin as a consequence of the static approximation [1]. The quantum corrections probably may be taken into account by the substitution of S2 for ( S 2 ) in the expression for Tc as has been proposed [6,7]. Therefore we suggest the following formula for the Curie temperature:

AE ( S z) T~ = 6k B S~

(6)

In the Heisenberg model ( S 2) = S(S + 1) so eq. (6) transforms to the usual mean-field formula for magnetic insulators (see ref. [11]). In the case of metallic systems with essentially delocalized electronic states the calculation procedure for AE needs some modifications compared • I to the magnetxc insulators case [11]. In the scope of the X~ method for a discrete spectrum the Slater's transition-state approach gives for the excitation energy [10,11,15]:

AE = E ( nVio- n],, ),i°,

(7)

i,o

where n~dI is the occupation number of the ith level with spin projection o in the final (F) and initial (I) states, c°, - the one-electron energies in 0 the transition-states with occupation numbers nio I F ~(nio+ ,,Io). For the spin flip excitation the variation of the occupation number in the magnetic nondegenerate states li) is: n,V, - n i ti = - 1 and nV~ - n~1 = 1. Therefore the energy of spin flip is:

127

corresponding to the magnetic (unpaired) electronic states. As a first approximation we shall neglect the weak polarization of s and p electrons and consider d electrons only. It should be mentioned that in the case of "itinerant" magnetics localized magnetic moments do not really exist, but we can consider the energy AE connected with the inversion of the local exchange field at the one of atoms [4]• Then only the partial density of states of the excited atom is essentially changed• Similar conclusion has been made by the authors [9] from the calculations of total energy difference for the ground ferromagnetic state and for the state with the one reversed spin moment on iron. So we suggest, that the generalization of eq. (8) can be taken as follows:

AE=f~ii~,N°~(,)dc-.f~ii~,N°~(,)d,,

(9)

where N°o(c) is the partial density of d-states at a particular site in the MTS considered./~1 t and/~z determine the energy range of magnetic states for spin-up electrons,/~1 + and/~2 + are the corresponding values for spin-down electrons. As is obvious:

f iN°o(,)d

= ¢., >, = <., >,

(lO)

=

ae = E'(,°~ -,°~).

(8)

i

Here i runs over the magnetic states connected with the site whose spin undergoes an inversion• For the magnetic insulators the sum (8) includes the small number of d-states [11], because of their localization on the excited atom. In metals we should consider a continuous spectrum and spin flip at the given site of the crystal lattice affects many one-electron states in the finite energy range. Here the sum on the discrete levels in eq. (8) should be replaced b y the integral with the weights corresponding to the densities of the proper states in the energy interval

where (no) is the number of d-electrons with spin projection o in the ferromagnetic ground state at the site which undergoes the spin flip• 3. The calculation of T~ for nickel As the spin flip at a particular site breaks down the translational symmetry of the system, the conventional band theory methods are not applicable in magnetic transition state calculation. The cluster approaches which describe well the local excitations in insulators [11], require a consideration of very large groups of atoms in order to describe itinerant d-states in metals. Therefore for calculation of AE in nickel we have used an intermediate approach - the X~-SW method with k-depending boundary conditions [16] which reproduces to a good approximation the structure of delocalized

A.L Liechtenstein et al. / Magnetic transition state approach for Ni

128

states making use of the small clusters. The nearest neighborhood Nil3 cluster in fcc lattice of nickel was chosen for the calculation. The crystal potential was constructed as the superposition of atomic potentials from the atoms of eleven coordination spheres. For the ferromagnetic ground state we used the atomic configuration: 3d'~63d~°4sl'4 which was obtained for nickel in refs. [ 17,18] from the self-consistent band-structure calculations. The use of k-dependent boundary conditions, as is shown in ref. [16], results in the continuous energy spectrum and the density of electronic states. In fig. 1 the partial density for the central atom d-states obtained from the calculations for the ground state of Nil3 cluster is presented (light line). A comparison of these resuits with that of band-structure calculations [18,19] shows that, though some of the fine structure details of the electron states distribution are

/~ I N~(E)

iI

t0t

not reproduced perfectly, the width of 3d-band and the energy distances between the main peaks are in a good agreement with the data [18,19] (see ref. [20]). In order to find AE it is necessary to calculate N°o(~). The magnetic transition state here is defined as the state halfway between the initial ferromagnetic state and the final state with. the inverse spin of central atom (fig. 2). The MTS occupation numbers are the mean values between n(F) and n(I). Therefore the central atom potential in the MTS calculation is initially spin-restricted whereas the surrounding is spin-polarized. So for a simple nonself-consistent transition state calculation we •have used the central atom 3d'~33d~'34s14 configuration when constructing the cluster MTS spindensity, The response of a "spin-restricted" atom on the polarization of the surroundings permits the calculation of the upper limit on the magnetic excitation energy connected with such spin flip in the direct way. The results of the MTS calculations for the partial of the central atom Nd°o(~) isdensity shownofin d-states fig. 1. Compared to theNiground state, both positions and intensities of the main peaks are varied; the spin splitting decrease is

,A ~

,i i~

!EF

Fig. 1. Partial density of central atom d-states in Nil3 cluster for ground state (light line) and magnetic transition state (heavy line). The range of magnetic state in both cases are indicated also.

Fig. 2. Cluster Nil3 from bcc nickel in magnetic transition state.

129

A.L Liechtenstein et al. / Magnetic transition state approach for Ni

particularly noticeable. The use of eq. (9) leads to the following value of the magnetic energy excitation (in mRy): AE= E ° - E ° = (-313.9)-

MTS ones results in: A E ' = E~ - E t = ( - 3 1 1 . 8 ) -

(-339.6)

= 27.8 mRy.

(321.8)-- 7.9 mRy.

tor

In order to illustrate the physical meaning of AE' the model of rigid band spin-splitting may be used:

;== 3<(,,,-.~ 2 >

Nd ~ ( , ) = Nd t (c + Apo,).

An accurate estimation of the "quantum" fac-

So

Then eq. (9) applied to the ground ferromagnetic state lead to the simple formula:

where 2S 0 = (n ¢ > - (n ~ > in formula (6) for T~ is difficult because of the two particle averaging ((n ¢ - n ~ )2) which is not reduced to one-particle values. For the metal Ni case it can be evaluated using the concentration of the 3d-holes: nh/10 = 0.14 as a small parameter. Then the probability of the states with more than two holes is quite low for the ground state and, consequently, 2(n h ~nh, > = nh -- 2S0 is the number of nonmagnetic holes (see ref. [7] also). Therefore using the following identity relations:

a E ' = 2S0apo,.

(12)

The mean value of the polarization band splitting for the ground state of the Nil3 cluster in our calculations equals ApoI = 43.6 mRy. Then from eq. (12) A E ' = 26.2 mRy can be found in a good agreement with the calculated magnitude of the integral value (9) for the ground state. Spin splitting Apo~ is obviously due to intra-atomi¢ Hund exchange and determines the Stoner parameters I s = Apol/2S 0. Our calculations results in the following value:

<(F/ T -- /'/ ,L )2> = ( ( r / h 1' -- nh,L )2>

I s = zaE'/4So2 = 77 mRy. -- ( ( n h T

and

+nh,L)2>--4
),

ignoring the charge density fluctuations ~ n2), one obtains:

( < ( r t h 1, 4- n h + ) 2 > ~ < n 2 >

~= 3

nh(n

h --

2) + 4S o

4sg

3.0.

(ll)

It is worthwhile to note that the factor i S 0 + l)/S0 which has been introduced in ref. [21] as an analog to the Heisenberg model equals 4.3 and results in the strong overestimation of the Curie temperature. Using the value of AE obtained we find from eq. (6) T¢ = 620 K which is in a surprisingly good agreement with the experimental value T¢ = 631 K. It worth noting that the factor ~ is quite sensitive to the variation of the occupation numbers. For example for n h = 1.42, So = 0.28 (see ref. [18]) the ~" value is 2.84 and T~ equals 590 K. The calculation of AE' value following eq. (9), but with the ground state characteristics instead of

In ref. [22] I s --73 mRy has been obtained from the band structure calculation. Thus, AE' defines the energy of Stoner excitations, but A E is the energy of collective spin excitations, connected with inter-site spin interactions only. The "Heisenberg" exchange parameter I H can be introduced according to the following formula: IH =

zIE 4S2z

= 1.8 mRy

where z = 12 is coordination number. The decrease of AE in comparison with AE' is determined by the variation of peak intensities rather than by the decrease of the spin splitting (the later is approximately 1.5, on the other hand A E ' / A E = 3.5). Therefore we believe that a good agreement of the results of ref. [21] with the experiment is accidental because T¢ has been estimated through the atomic level spin-splitting in the transition state.

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A.I. Liechtenstein et a L / Magnetic transition state approach for Ni

It is interesting to note that our calculations show quite different roles of t2g and eg d-states in the ferromagnetism of nickel. The former gives the negative contribution to AE and the latter gives the positive one. Thus, the approach developed in the present paper allows the study, in a direct way, of both Stoner (ground state) and spin collective excitations (MTS) without using any model assumptions, and can be applied to the analysis of magnetic interactions in other transition metals.

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