. ~ Solid State Communications, Vol.53,No.10, pp.827-830, 1985. 0038-1098/85 $3.00 + .00 Printed in Great Britain. Pergamon Press Ltd.

SHORT RANGE ORDER EFFECTS AND THE FALICOV-KIMBALL MODEL

Gast6n Martfnez

Departmento de Ffsica, Universidad de Tarapac~, Arica, Chile

and

Jaime Rgssler*

Facultad de Ciencias B~sicas y Farmac~uticas, Universidad de Chile, Casilla 653, Santiago, Chile

and

Miguel Kiwi *t

Facultad de Fisica, Universidad Cat61ica de Chile, Casilla II4-D, Santiago, Chile

(Received 15 November 1984 by H. Suhl)

The model of Falicov and Kimball is investigated incorporating short range order effects. This modifies both the electronic density of states and the configurational entropy, relative to the fully disordered case. The way this affects the conditions under which phase transitions occur is evaluated and discussed.

] . I ntroduction

The model of Falicov and Kimball I (FK), has been extensively used to obtain an under- standing of a wide variety of phenomena such as first- and second-order phase transitions, 2 the electronic and magnetic properties of the rare- earths 3-5 and of excitonic phases. 3

Here we incorporate short range order ef- fects, and the consequent modification of the configuration entropy, in the formulation and

H = [li>£o<il + [ li>tij<J I - ~'

i <i,j> i

solution of the FK model. These effects were treated qualitatively by Sakurai and Schlottmann,Swho suggested the formation of a short range ordered superstructure as a mechanism to stabilize a valence of 2/3 of Sm in SmS. Also Lazo et al 7 gave some attention to these effects in previous work.

In this paper we first formulate the prob- lem writing an appropiate Hamiltonian. Next, we outline the techniques of R~ssler and Lazo 8 and of Kikuchi, 9 which allow to handle the ef- fects due to short-range-order on the electronic density of states and to evaluate the configu- rational entropy, respectively. Finally, the scheme used to obtain numerical results, as well as the results themselves are presented, its implications are discussed and some relevant conclusions are drawn.

* Supported in part by CONICYT under Project # 1098

+ Supported in part by the Organization of American

827

II. Model Hamiltonian and its Helmholtz Free Energy

Falicov and Kimball (FK) introduced I their Hamiltonian as a simple theoretical tool to study metal-insulator phase transitions. Sub- sequently, Ramlrez, Falicov and Kimball dis- cussed it in detail 2 using the mean-field a~- proximation. Later on RSssler and Ramfrez, ~ and also Trias, Ramirez and Kiwi I0 reformulated the FK model as an alloy problem and obtained ad- ditional information of its features. We follow the notation of the latter work I0 and write H as

li>G<ij, c2.1)

where li> denotes a localized Wannier state on lattice site ~i, G>O describes the attractive interaction between an electron and a hole at

site Ri; actually, the prime in the last summa- tion in (2.1) implies that it is limited to

ionized sites only. The hopping parameter t ij= -t connects two first neighbor atoms denoted by <i,j>. In this way an equivalence is estab- lished between the fn+l rare-earth atoms and say

tile A-type species of ~ binary alloy, and be- tween the fully ionized 4f n atomic states and species B. (For example, in the fluctuating valence compound SmS we identify the Sm ++ (4f 8) and the Sm 4-~+ (4f 5) with species A and B, respectively).

Using the notation {RI, R 2 .... R N } = {R.} to denote the set of lattlce sites B with B-#ype atoms, we can write the total Helmholtz free energy as

and by the DIB of the U. of Chile.

States (OAS).

828 SHORT RANGE ORDER EFFECTS AND THE FALICOV-KIMBALL MODEL Vol. 53, No. I0

1 {~j })3 --F = - 2kBT I in [i + exp~ (~ - ~ N

where c {Rj} are the eigenvalues of

H{R.}]~ > = ¢ ]~> (2.3) 3

and ~ is the chemical potential. Smazn and Sconf are the magnetic and configuratio~al entropies, respectively and ]Ef[ is the ionization energy.

In principle we have to minimize Fto t with respect to {Rj}. However, thls is a very dif- ficult task. Thus, we limit ourselves to in- elude nearest neighbor correlations, which allows us to write the total Helmholtz free energy as

+ NB(E f + B) - T(Smag n + Sconf) , (2.2)

figurations. The probability of each one of them was evaluated self-consistently. 12 In terms of these probabilities, and using an ap- proximation due to Kikuchi, 9 Sconf is computed for different values of x and 7. This method yields exact results for 7 = 0, -1 and +I.

Next, the total Helmholtz free energy Fto t is minimized, subject to the conservation of charge constraint

i T D(c; x, G, y)

x 2 ~ c - ~ + 1 de, (2.6)

s B

F(T, x, G, 3") = x(~ - A] - x kBT in q - TSconf (x , ~()

I gT e- ~ (~ _ N~ 2kBT dgD(£; x, G, ~() in [i +

E B

where D(e; x, G, ~) is the density of electron states per atom and spin direction, for fixed values of the concentration x= NB/(N A + NB) ' of the electron-hole interaction G and of the Cowley short range orderllparameter y, defined as

PAB . (2.5)

X = 1 - --

Here PAB is the probability of finding an A next to a B type ion and 19 7 >- mim[(l-~/x, x/(l-x)] . The ratio q = (2JB+I)/(2J A + I) de- termines the ionic spin entropy and k = min [ [Efl + c(k)] is the energy gap between local- ized f- and itinerant - electron states. The free energy of Eq (2.4) is the same as the one treated in Ref. 10, except that now short range order effects are also included through the

parameter 7- For simplicity we consider a simple cubic

lattice. The density of electron states D(e; x, G, 7) is obtained using the coherent locator approximation of RUssler and Lazo. 8

This is a self-consistent single-site procedure, which incorporates short range order effects and has the following characteristics: i) It is independent of dimensionality, in con- trast with other procedures II which also include short range order effects; 2) Only one complex parameter fully determines the Green function; 3) It reproduces exactly the coherent potential approximation results (CPA) in the case of a random alloy (7 = 0); and 4) It yields exact results for the electronic density of states in the fully segregated (7 = +I) and binary superstructure (y = -I) cases, as well as for the pure crystal in the

appropiate limit. The configurational entropy S(x,y) has been

evaluated considering an 8 atom cubic cluster; this cluster can adopt 28 = 256 different con-

(2.4)

where the factor of two accounts for the spin degeneracy. The results obtained on the basis of the scheme outlined above are presented in the following section.

III Results and Conclusions

The results reported below were obtained through the implementation of the following scheme: 1) The electronic density of states D(E) is evaluated for fixed values of the concentration x and of the short range order parameter y; 2) The chemical potential ~(T) is computed from the conservation of charge requirement (2.6); 3) The Helmholtz free energy F is evaluated by means of Eqs. (2.2) and (2.4); 4) Equilibrium properties are determined minimiz- ing F(T; x, G, ¥) with respect to the relevant parameters.

Several plots of the density of states D(E) for G = 21.4/70 and for various values of the parameters y and x are displayed in Fig. 1. The energy E is measured (from now on) in units of W = c T - c B = 12t = i. Where the density is split into two sub-bands the lower one contains a fraction x of the total number of accesible states; however, due to spin degeneracy it is only half-full. The strong influence of short range order is quite apparent.

In Fig. 2 we display the stability curve in the x-7 plane, for the same G = 21.4/70, indicating a clear trend towards the stabili- zation of spatial configurations with 7 ~ O~ which becomes stronger as the temperature in- creases. Thus, at high temperatures (i.e. T > 0.05), when a large density of ions is generated, they tend to distribute themselves at random in space. Only at low temperatures and therefore for relatively small x, does a tendency to segregation of ions of equal valence become apparent. However, even then it is highly unlikely, because of the large

Vol. 53, No. 10 SHORT RANGE ORDER EFFECTS AND THE FALICOV-KIMBALL MODEL 829

D .E6). (°) (b)

0.2-

0-(el EF t=o.o (d) EF __ ~-=o.z, 0.4-~ 0.2-

0 , ~ -2.0 EF 0.0 1.5 EF 09 1.5

E/W Fig. 1 Density of states vs. energy for various values of the short range order parameter y. All plots, expect (f) are for x = 0.5.

o" 1.0

0.81

0.6

0.4

0.2-

0.0

T=O.O02

~ T 0'005 .01

2 =0.07

0'.2 04 0.6 0.8 X

Fig. 2 Stability curve in the x - y plane.

Coulomb energy implied; to treat this diffi- culty properly the inter-atomic interaction energy has to be incorporated into our analysis. This leads to a renormalized value of the electron-hole interaction G (to be determined self-consistently) which tends to reinforce the stability of the ¥ m 0

configurations. Details of this renormali- zation procedure will be reported elsewhere. 12

While for G = 21.4/70 the behavior of n c = x vs___. T is continous)for larger values of G, for example G = 30/70, we obtain a discontinous plot for the number of conduction electrons per site n c = x as a function of the ionization energy of the f-level. In Fig. 3 we plot n c v_~s. A=mi~IEfl + ~(k)] for various values of the temperature T. Since it has been convincingly argued 3 that the position of the f-level is linearly related to the pressure p, we can also think of Fig. 3 as an x vs___.p plot. It is noticed that the sharp discontinuity at T = 0 gives way first to a two step transition, as pressure is increased, at T = 0.005 and finally to smooth curves for T ~ 0.01. The above mentioned two step process recalls the T +~ ~ ~ a' transition 4 in Ce and the

B +-+ B' ÷+ M one 13 in Sml_ x Yx S. These two step transitions were also obtained theo- retically 14 by Khomskii and Kocharjan.

X 1.0

05

o o o o o o Q

i / T000si i 0.12 0.1 0.08 A/W

Fig. 3 Average number of conduction electrons per atom as function of the position of the f- level, for T = 0.005 and 0.0] (in units of W=I). The dotted line corresponds to T = 0.

]n summary, we have obtained a solution of the FK model which includes short range order effects. We find a weak tendency towards segregation of equally ionized atoms at very low temperatures; however, this tendency is largely supressed when inter-atomic Coulomb interactions 15 are included in the Hamiltonian. On the other hand, for large G values and very low temperatures we find abrupt increases in x as p grows; at higher temperatures the x V@Sop plots become smoother. All discontinuities dissapear for T > 0.01.

References

i. Falicov L.M. & Kimbal J.C.,Physica) Review Letters 22, 997 (1969).

2. Ramfrez R., Falicov L.M. & Kimbal J.C., Physical Review B2, 3383 (1970).

3. Ramfrez R. & Falicov L.M.,Physical Review B3, 2425 (1971).

4. Kiwi M. & Ramfrez R., Physical Review B6, 3700 (1972)-

5. RSssler J. & Ramfrez R., Journal of Physics C9, 3747 (1976).

6. Sakurai A & Schlottmann P., Solid State Communications 27, 991 (1978).

SHORT RANGE ORDER EFFECTS AND THE FALICOV-KIMBALL MODEL Vol. 53, No. I0 830

7. Lazo E., RSssler J. & Moraga L., Physica Status Solidi (b) 108 , 341 (1981)

8. RSssler J. & Lazo E., Journal of Physics C14, 3499 (1981).

9. Kikuchi R., Physical Review 81, 988 (1951). i0. Trias A., Ramfrez R., & Kiwi M., Physical

Review B19, 5877 (1979). ii. RSssler J. Martfnez G. & Kiwi M., Physical

Review B21, 5211 (1980).

12. Martfnez G., RSssler J. & Kiwi M. (to be published).

13. Jayaraman A., Dernier P. & Longinotti, Physical Review BII, 2783 (1975).

14. Khomskii D.I. & Kocharjan A.N., Solid State Communications 18, 985 (1976).

15. Fletcher G. & Mahanti S.D., Solid State Communications 48, 773 (1983).