A two-band model for intermediate valence compounds

t

Solid State Communications, Vol.36, pp.19—24. Pergamon Press Ltd. 1980. Printed in Great Britain.

A TWO—BAND MODEL FOR INTERMEDIATE VALENCE COMPOUNDS* J.R.Iglesias—Sicardi and I .Aveline Instituto de F~sica Universidade Federal do Rio Grande do Sul 90000 Porto Alegre, RS, Brasil (Received

30

November 1979 by R.C.C.Leite)

The change of valence as a function of pressure in intermediate valence rare—earth compounds is computed in a two hybridized band model: a conduction band and a 4f—band, being V the hybridization parameter. A Coulomb interaction G between 4f and conduction electrons is also included. For different values of G/V we obtain first or second order transitions from a semiconductor to a metallic state.

Resistivity and optical experiences on SmS, SmSe and SmTe show that these compounds present a pressure—induced semiconductor to metal transition due to

The first

term, Hc~ is the energy

of the conduction s—band.

a change of valence of the Samarium The transition is isostructural

Hc

(NaCL — NaCi) and it is accompanied by a large volume decrease. The Sm valence, deduced from lattice constant and MBssbauer isomer shift measurements, changes from 2 at normal pressure to an intermediate value of about 2.7 at the

band, assumed as a

~

=

t~ a~

0 akO

k,0

—

—

(2)

—

—

)

where a+ (a

creates

(annihilates) an

electron of wave vector k and spin tk is the corresponding energy. —

metallic phase . It is generally admitted that this transition involves the delocalization of a fraction of the six 4f—electrons of Sm to the conduction 2~we have band. calculated the valence change of the Sm In a previous work~ atoms in Sm 1_~Y~Salloys, describing

0

and

The 4f—electrons are described by a Hubbard—Hamiltonjan Hf

~ ~

=

IE r-

n.

L°

I

n.

+

10

2

10

n. ~1 i—Oj

(3)

the system by two bands: a zero width band of energy E0 (the 4f—level of Sm) interact between them finite and a conduction band. via The a4f—electrons intra—atomic Coulomb repulsion U, which splits the f—band in two sub—bands at energies H 0 and E0+IJ. The f— and

with

conduction bands hybridize, being V the hybridization parameter. Within this model it is possible to obtain a continuous valence transition from a non—magnetic to a non—magnetic or magnetic ground—state varying the ratio U/V. In order to obtain an abrupt transition we include now a Coulomb repulsion G between f— and conduction

operators for an f—electron at the site i with spin 0, and U the Coulomb integral between f—states at the same atomic site. The mixing term, Hmjx~ is:

1 b. 10 b 10

~ n~ +

being b~0(b~0)the

H.

~ ~

electrons~~~,preserving the finite value ofU. The Hamiltonian of the system is:

+

V*e

r

creation (annihilatiox~

~.

+

Ve L —ik.R.

ak

——1

—

Hc

+

Hf

+

Hmjx

+

Hc

*) Work supported in part by ~Pq

b.

+

—

1 b~0 akaj —

The last term, HG. is H

(4)

(1)

the intra—atomic

Coulomb interaction between f and conduction

and FINEP. 19

20

MODEL FOR INTER~EDIATEVALENCE CO~0UNDS

electrons:


>


=

H

G

e

N

—1-

——

ItT ki’ ,c’

+

~

a~o~ n~

—

(6)

>

+

s1~

S

i(k—k’) .R.

Vol. 36, No. I
(12b) S

Replacing eqs.of (10) in eqs. get the densities states of the(7)

we

—

hybridized f— and s— bands, and integrating self—consistently p~(z) and Our goal is to compute, at T = 0, the two bands densities of states which are defined by:

0 5(z)

Nilk

p

p~(z)

—

=

—

Im G~k(z)

(7a)

=

—

In G~k(z)

(7b)

up to a Fermi level determined by the total number of electrons, we can calculate the occupation of each band. To apply the model to Sm monochalcogenides, we take the state with two 4f—electrons as corresponding to the 6 configuration of Sm and the state 4f with one electron to the 4f5 configuration. If E 3 is the baricenter of the conduction band and W its half~idth, for PfSZ)

LEB

where G~k(z) are Green’s the Fourier transforms of and the G~k(z) retarded functions as defined by Zubarev~4~, and z is the complex energy E + ~ (~~O) The equations of motion are decoupled in Hubbard I approximation for the Coulomb

(E0

1 the effect of the In have this two hybridization is small 2) and. we 4f—electrons at E0 + u( case, the system behaves as a semiconductor with a gap given the difference the bottom of the byconduction band between EB — W —

c

HG

~

[f n.

itTO’


itT

>

+

10


U)1 1W

>>

and E 0

interaction U, as in ref. (2), and in a mean field approximation for the Coulomb interaction G. Expression (6) is then approximated by:

+

+

U.

Decreasing the energy separation

(EB E0)/W the Fermi level comes into the conduction band giving a metallic state. For the numerical calculations we assume a density of states for the unhybridized s—band,

n~0~]

of

(8) p(E)

=

~

-

the form:

)]

1E _EB\2fl ~

(13)

where

i(k—k’) .R. n.10

=

~,

—1

e

~.

+ aktTak,tT — —

(9)

with this expression, eqs. (10) are computed, as indicate in ref. 2, using the following values of the parameters: V02 0.4

The Green’s functions are

tT (z)

=

G

G~k(z) —

=

_!_

~

2ir

Z

—

+

~

~

Ek

~

then obtained:

G)

(lOa)

G~k~Z)~

(lob)

—

where

NV2[z_Eo_G_IJ(l_
C

>p_J —

(11)

In fig. 1 the number of electrons in the f—band is plotted as a function of (EB —E0)/W (for a total number of 2 electrons) 3. The decrease in theof cases flf is G/V smooth = 0,1,2, for and G/V = 0 or 1, and, for bigger values of which G/V aone to final corresponds has state a discontinuous oftoabout the one high—pressure transition f—electron, phase of Sm monochalcogenides. To explain the intermediate value of the valence, one must add to n~ a fraction of the s—electrons which lies below the gap. These electrons are, because of the hybridization, of character f and do not contribute to the conduction (2) ~ that way, we calculate

(z—E—G~zm>)(z—E 0—U—G) with

=



+



(l2a)

process the valence of S~ and the resulti are shown on figure 2 for three different values of C/V. On fig. 3 and 4, the densities of

Vol. 36, No. I

MODEL FOR INTER~DIATEVALENCE COMPOUNDS

2.0

-

3.0

~.

1.0

N

1.0

0.8

I’

0.6

I

0.4

0.2

0.0

(EB~EO)/w

Figure I: The number of f—electrons as a. function of (EB

—

E

0)IW for

G/V=O, I, 2 and 3.

2.4

2.2

-

2.4

-

2.2

-

2.4

-

2.2

-

h\~Tr3.0

LU

0

z >

2.0

Figure

2:

0.8

0.6

0.4 0.2 (EB_EO)/W

The valence, number

of

calculated

conduction

as a function

0.0

of

(E3

as the

s—electrons —

E0) /W for

G/V= 1, 2 and 3.

states are shown for two representative values of G/V, respectively, G/V 1 and 3, and for three values of (EB —E0)/W: 0.6, 0.4 and 0.2. On fig. 3 one observes a continuous transition from semiconducting

to metallic state. On fig. 4, for G/V3, the density of states practically does not vary in cases a) and b) . After the transition the metallic state(compare is similar to that previously obtained fig.

21

22

MODEL FOR INTER~DIATEVALENCE COMPOUNDS

a)

p(E)

Vol. 36, No. I

b)

p(E)

~LO

~1.O V

V

2.0-

1.0

2.0-

-

I /

1.0

“S _____________________________

/~~)_~If’

_________________ ______________

-0.5

-_____________ //

,--

‘-—----

/

II I

~J

SS

S _____________________________

E0.0

0.5

_____________

I

1.0

1.5 (E-E~tw

-1.0

I

/

-0.5

______________________

I

0.0

0.5

1.0

(E-E

8)/W

c)

p(E)

2.0

-

1.0

-

/

/

,

—

—

/

S I I

I

I II

1

/ /

/

/

0.0

Figure

-0.5

0.0

0.5

1.0

(E-EB)/W

3: Plot of the f—density of states (full line) and the s—density of states (dashed line) corresponding to the case G/Vl for a) (EB —E0)/W —0.6, b) 0.4 and c) position

0.2. EF indicates of

the Fermi

3c and 4c) . The occurrence of only one gap in fig.+ 4a) and 4b)arises , near from the energy U + G, having two

E

electrons into the f—band. In all the other cases there are two gaps: near

gives a qualitative explanation of the properties of Sm monochalcogenides. We

of

+

G

the

level.

and E0

+

U

+

G. The baricenter

the conduction band is at E8 + G. The calculation presented here

Vol. 36, No. I

MODEL FOR INTERIDIATE VALENCE COMPOUNDS

p(E)

a)

23

b)

pE)

~3.0

3~3.0

2.0

2.0

-

1.0

1.0

-

,

I I I

I

S..

/ / /

“

I

1.0

I

/

Ill

I

1.5

2.0

_______

(E-E

9)/W

/

~

I

I\

___________________________________

0.0

EF

1.0

1.5

2.0 (E~EB)/W

c)

p(E)

2.0

-

1.0

-

-5—’

, ,

/ // /

I -0.5

/

/

0.0

I

-‘1~

‘-

\I~ L

L 0.5 EF

S.

i 1.0

1.5

2.0 (E-E8)/W

Figure 4: Same as figure 3 for G/V have considered non—magnetic solutions, which is the case of SmS, SnSe and SmTe. For the chosen value of U/W = 0.4, we think this is a valid assumption. In ref. 2, the appearance of magnetic solutions was estimated at U/W=O.5; however, calculations now in course, indicate that greater values of C increase the stability range of those solutions, As in the 3~,the Ramirez, and s—f Falicov Coulomb repulsion Kimball C, treated model~ in nean field approximation, renornalizes self—consistently the band positions,

S.

/

/

...J........ L_.—I-— 0.0 EF 0.5

/ /

“

driving a continuous

or abrupt

3.

transition to a metallic state, which explains the difference between SnTe and SnSe on one side and SmS on the other. Finally, we remark from fig. 1, that the occurrence of an intermediate valence (Or a fractional number of f—electrons) does not necessarily coincide with the semiconductor to metal transition. In some cases, as it is shown by the density of states plotted in fig. 3a, there phase. is a seniconducting intermediate valence This can be useful to describe the properties of TmSe, which seems to present this kind of behaviour~6~.

24

MODEL FOR INTER~DIATE VALENCE COMPOUNDS

Acknowledgement

—

We gratefully acknowledge

Vol. 36, No. I

H.Nazareno.

useful discussions with Drs. A.A.Comes and REFERENCES 1)

2)

3)

For a complete review on intermediate valence see: J.M.Robinson, Physics Reports 51, 1 (1979). I.Aveline ai~ J.R.Iglesias—Sicardi, Journal of Low Temperature Physics 35, 433 (1979) , R.Ramirez, L.M.Falicov and J.C.Kimball, Physical Review B 2, 3383 (1970) and R.Ramirez and L.M.Falicov,

Physical Review B 3, 2425 (1971). D.N.Zubarev, Soviet Physics Uspekhi3, 320 (1960) 5) I.Aveline and J.R.Iglesias—Sicardi, Proceedings of the International Conference of Magnetism 79 (MUnich). To be published in Journal of

4)

Magnetism

6) O.Pe~a, Ph.D.

and Magnetic

Thesis.

Materials.

Unpublished.