Electronic structure of the Anderson lattice with finite Coulomb interaction

Solid State Commumcations, Vol. 68, No. 5, pp. 477-481, 1988. Printed in Great Britain.

0038-1098/88 $3.00 + .00 Pergamon Press plc

E L E C T R O N I C S T R U C T U R E OF THE A N D E R S O N LATTICE W I T H FINITE COULOMB INTERACTION L.G. Brunet*~t, Rejane M. Ribeiro-Teixeira* and J.R. Iglesias* *Instituto de Fisica, Universidade Federal do Rio Grande do Sul, P.B. 15051-91.500 Porto Alegre, RS, Brasil ~Departamento de Fislca, Universidade Federal de Santa Maria, 97.119 Santa Maria, RS, Brasil

(Received 13 April 1988 by E.F. Bertaut) An approximate diagrammatic technique is applied to the periodic Anderson Hamiltonian with finite Coulomb interaction. A Kondo peak at the Fermi level is obtamed, it is spht by the hybridization gap. We investigate the dependence of the gap and of the density of states as functions of the parameters of the Hamiltoman: the hybridization, the Coulomb repulsion and the energy of the f-level. IN A RECENT publication [1] it was described as an approximate diagrammatic method to treat the periodic Anderson Hamiltonian. Within this scheme the hybridization between f and conduction electrons is included in the exact solution of the atomic part of the Hamiltonian and the hopping terms are treated through an approximate Dyson equation [2]. In [1] the case of infinite Coulomb repulsion, U, was considered. An hybridization gap and a Kondo peak were obtained near the Fermi level and the system could be either metalhc or insulating depending on the strength of the hybridization and the depth of the f-level [1]. Here we are interested in the details of the electromc structure, including the behaviour of the system as a function of U, the Coulomb repulsion. It is well known [3] that for an isolated impurity the width of the Kondo peak (and the Kondo temperature) are proportional to: -1 exp { ~ } ,

A~

(1)

where the atomic Hamiltonlan is

I41 = E y~ f~+f, + EB Y~ G c,o t~

+ V ~ (f+c,, + h.c.) + Un{rn ¼

J

=

E F

--

V2 E + U"

being f , + (c,~) + the field operator that creates an f-(c-) electron of spin a at the site i. E(EB) is the energy of thef-(c-) level, V is the hybridization between f- and conduction electrons and U the Coulomb repulsion between f-electrons. The band term H' is:

H'

= ~ tvc,+~cj,,

(5)

t ¢j

where t,j is the hopping matrix element between the i- and j-sites. The f-orbitals are supposed with no overlap. Also the Coulomb repulsion between c-c and f - c electrons is neglected. The eigenstates and energies of H i are presented in Table 1, and the atomic Green functions are calculated by their spectral representation [6] at T = 0 K.

(2)

In the case of an Anderson Lattice the coherent hybridization generates a gap at the Fermi level [1, 5]. A peak appears on both sides of the gap and the width depends on the parameters of the Hamiltonian. In this work, the Hamiltonian reads:

H = y~ (Hi) + H'

(4)

tT

where J is the coupling between conduction and localized spins and Q(EF) is the density of states at the Fermi level. According to the Schrieffer-Wolff transformation [4] we can expect that V2 E F -- E

tr

(3)

t

477

' ~ A , B~o,,T= o =

[- <0lAin>

L <01Bin> ] - ~ - ~ - ~ o Z ~-~-)j

(6)

where E0 and 10) are the energy and state vector of the ground state of the system for a given particle number. In a Kondo lattice the mean number of electrons is two. This does not mean that the atomic Green function should be calculated just from the ground state of two particles. The band term adds the possibility of atoms with different occupation numbers. So, as in [1], we assume a collection of N atoms being N,,, of them in the m-particle state, and we define the

478

ELECTRONIC STRUCTURE OF THE ANDERSON LATTICE N

E

z~4-E,o E9 - E2

2

21 3,4

E~I -E 2

I

E12 - E I o E4 -E 1 = E6 -E 2 EB E13 -E 6 = E16 -E14 EIO -E 2

2,3

0,1 3,4 1,2

E2 -E 1 E12 - E l l

0,1 3

EIO -E 4

2

(A)

= ~ Pm(m]Alm)

3

m

Fig. 1. Electronic energy levels of the isolated atom, withEB = 0, E = - 1 . 5 , V = 0.4, U = 4.

such an average is also given by the imaginary part of the Green function associated to A. If we consider Pit = P u and e3r = e3+, the system is non-magnetic and the averages of four different observables are necessary for the complete specification of the PIn'S. The values of the P,,'s are determined by the set of equations: (t/f)

2Pl cos 2 (k + 2P2[(a]0) 2 + (b]0) 2]

=

relative weight of each ground state: P"-

N~ N

+2(1 + sen 20)P3 + 2P4 (7)

(na)

(n¢)

go = ~ Pmgo.m

(H)

(8)

m

where go,,. is the Green funcUon calculated through equation (6). So, with equation (8) we can determine the complete Green function through a Dyson equation, obtained from an approximate diagrammatic method [2]: go + g0WG

(9)

where W is the one electron hopping matrix connectlng different sites. Now, we focus our attenhon in the following atomic Green functions: goI!~(o)) =

((f~,f+))~

(10.a)

go.,,(o)) = ((c., c+)),,, -

(lO.b)

g~/~(o)) =

((c., f+))~o

(10.c)

goJio(o)) =

[g~,{o(co)]+

(iO.d)

The explicit expressions of these Green functions are presented in the appendix. The electronic energy levels, gwen by the poles of the Green functions, are plotted in Fig. 1 (for E8 = 0,

=

(12.a)

2Pl sen 2 q5 + 2Pz[(a,o) 2 + (c,0)z] +2(1 + cos: O)P3 + 2P4

The mean atomic Green function go, which averages over all possible number of particles, can then be introduced:

G =

(11)

m

m

E12 -E 9

-3

E = --1.5, V = 0.4, U = 4). The corresponding number of particles is indicated on the right. Note that the same pole can be found in different atomic Green functions. It is important to emphasize that the electronic levels are one-particle levels and the residue at each pole is a fractionary number, as in Hubbard-I approximation [7]. In order to evaluate go one needs the values of the Pin's. They are obtained with the assumption that the average of any observable calculated through the Green function for the total Hamiltonian is equal to the average over the atomic ground states, that is:

m

0

-2

L~ - S ~

Vol. 68, No. 5

(12.b)

- 2 P i sen ~b cos ~b + 2P2[a]oqo + aloblo]

=

--2P3 sen 0 cos 0

(12.c)

= 2PIE2 + P2Elo + 2P3E]2 + P4EI6

(12.d)

together with the normalization condition:

E ~=l m

(12.e)

where ( H ) is the average energy value and the coefficmnts a,0, b~0, c,0 are defined in Table 1. When one computes the weights from the atomic expressions the approximation is equivalent to Hubbard-I decouphng scheme [2]. The P,,'s values are obtained self-consistently from equations (12) and from the mean values (n/), (n,) and (n,/) derived from the Green functions that, after Fourier transforming read:

Gf](O)) = gUT(o)) + (g°/(o)))2 8(k) g~o'(o)-----~~ g~' '((o) -- e(k) (13.a) 1

G,c(o)) =

~g'o'-'(o))

G,r(o) )

g'?(o))

_

- ~(k) !

g~' (o)) ~ g~'-'(o)) - ~(k)

(13.b) ( i 3.c)

Vol. 68, No. 5

ELECTRONIC

STRUCTURE

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479

Table 1. Eigenstates [01> 102> 103 > 104> 105>

= [0> ---- cos = cos = sin = sin

#)fr~ 10> ~f~+ 10> ~bfT+ IO> ~bf~+ 10)

--+ -

sin sin cos cos

~bc~-I0> ~bc{ 10> ~bc~ IO> ~bc{ 10>

Energies

Number of electrons

E, = 0

0

E2 = ½ E - [¼ E 2 + V2] ''2

1

E3 = E2

1

E4 = ½ E + [¼ E 2 + V2] '/2

1

E5 = E4

1

10~> = frc?lO>

2

107> = f~+ C~-10>

E 6 = E7 = Es = E

2

{A + c{ [0>}

2

W,~> = 5 109> ~-- a9{A+c~ - -- Z+c~-}10> q- b9fT+f~+ 10> -4- C9¢?C:10> 10,o> = ato{fT+c~- -- fl+c~ - }[0> + blt~f?+fl + 10> + Clo¢~ C~-I0> 101, > = a,, {fr+c{ -- f~+ c~ }10> +blt/T+fl+]0> + c,,c~-c{[O>

10'2> = sin Ofr+Z+c~-IO> + cos Oft+ c~- c~{ 10>

E 9 = 2 X/-S--Q cos 8,/3

2

E,o = 2 ~

cos (8,/3 + 120 °)

2

E,, = 2 x / ~ - Q cos (0,/3 + 240 °)

2

E,a = ½ {(3E + U) - IE + U[ (1 +

3

(e + u) ~] J 10,3> = sin 0fT+f~+c~-I0>

El3 = El2

+ cos 0 f : c?- c[ r0> 1014> = - c o s

OfT+f~+c:10>

E,4 = ½ { ( 3 E + U) + [ E + U ' ( I

+ sin OfT+ c~- c{ 10>

4V 2

+

~1/2~

( e Y b ) ~] J 10,5 > =

- cos OfT+Z+ < Io> + sin 0fT + c~- c~-10>

E,5 = El4

11]/16> = A+Ac: c; 10>

El6 + 2E + U

With

tgc~ = V{½ E + [¼ E 2 + V211/2} - I a, = {2 + 4 V 2[E, 2 + (E, - 2 E -

2V

U)-2]} '/2,b, = E, - 2 E -

2V = - ~ , a,,

ua"c'

01 = arc cos R / - Q 3/2, R = ~ [ 2 ( E 3 + E~) - 3(E2Ev + EE 2) + 1 8 V - ' ( 2 E Q = -~[12V 2 + E 2 + E~-

EEul, Eu = 2E + U, tgO = V{ I ( E +

The average energy value is:

= E + U + 2V

U)

i = 9, 10, !1

Ev)],

[ ( 1+

1 + (E + U)2.] J J

obtain: (13.d)

where is calculated through a similar formalism, that is, defining:

A{f(fo) =

<(nfTf~;f~ + )),

(14.a)

A~(o)) =

<>.

(14.b)

We can extend equations (8) and (9) for A :y and we

A::(~o) = Aio:(O)) +

A~f(co) g~)f(co)

g'o' (o))

e(k)

~ g'o'-' (~o--)-- e(k)' (15)

the exphcit expressions for Ag are shown in the appendix. The integral of the imaginary part o f A:: up to the Fermi level give us the m e a n value
480

E L E C T R O N I C S T R U C T U R E OF T H E A N D E R S O N L A T T I C E

N

f

.....

f

I

d

| |

I

I

!

I I

I

r' i

/ / !

I

Vol. 68, No. 5

I

\ \\ \\ \

I

I/I

\

\

\ \

\ ,

J~i it

I

-24 -21 =1.

- 5

'I.

0

I

Z2 24 ~/W

Fig. 2. Density off-(full hne) and conduction-(dashed line) states for the symmetric case (I2EI = U). The system is insulating with n, = nI -- 1. The values of the parameters are U = 4, E = - 2 , EF = 0, V = 0.4.

band, we consider a model density of states [8] given by:

e°(to) =

~

1 -- ~

,

Icol ~< W

(16)

I~ol > W being W the half width of the conduction band. We study first the density of states for a particular set of values of the parameters and how it changes as a function of E, V and U. Figure 2 shows the density o f states for a typical K o n d o c a s e : E = - 2 , V = 0.4, U = 4, W = l. The f-level is well below the conduction band and the Ferm~ level is exactly at zero, inside the gap. A structure with three peaks, each one separated by a gap, reside the conduction band is obtained [1]. This is a consequence of the electronic structure o f the isolated a t o m that is preserved in this approach. It is also clear from Fig. 2 that the system is insulating with n, = nI = 1 [5, 9-11]. Such a symmetric density of states occurs whenever I2E] = U and does not depend on the hybridization value. Decreasing the hybridization just makes sharper the peaks around the central gap. If we change to a region where U > IEI the system is metallic with the Fermi level greater than zero (inside the third sub-band of Fig. 2). Besides, the density of states is asymmetric with n, > n I. A feature we want to study is the variation of the central gap, Ex, as a function of the Hamiltonian parameters U, V and E. The gap dependence with the hybridization V is shown in Fig. 3a. It decreases for decreasing hybridization and is non zero even for V ~ 0.1. Figure 3b shows the gap dependence on E.

The it is and gap

gap is a monotonically increasing function of E, minimum in the symmetric situation (12El = U) increases with the asymmetry (U > I2EI). The dependence on U is shown in Fig. 3c. We have tried to fit the variation of the gap as a function of V, E and U. A simple relation between the energy of the gap and the hybridization V could be: Eg ",,

exp {-- const./xf-V}.

(17)

This result obviously differs from that o f the one impurity case. It should be emphasized that this relation was obtained through numerical calculations. No simple explanation is available for this type of dependence. Concluding, we have studied the electronic structure of the Anderson lattice at zero temperature as function of E, the depth of the f-level, U, the Coulomb repulsion and V, the hybridization. The approximation utilized enables us to obtain a structure of K o n d o (or Abrikosov-Suhl) peaks near the Fermi level, as well as hybridization gaps [1]. In the symmetric case the system is insulating with n, = nI = 1, and ]t is metallic when [2E[ < U. The gap is an exponential function of the hybridization V [equation (17)], it is a monotonically increasing function of E (Fig. 3b) and a monotomcally decreasing function of U (Fig. 3c). Finite temperature calculations are m progress. - We would like to thank Prof. M.E. Foglio and A.S. Rosa SimSes for helpful discussions. This work has been partially supported by the Brazilian agencies CNPq, F I N E P and CAPES. Acknowledgements

Vol. 68, N o 5

ELECTRONIC

STRUCTURE

OF THE ANDERSON

LATTICE

481

APPENDIX We present here, as an example, the Green function for spin direction for the f-electrons calculated in the ground states o f 0, l, 2, 3 and 4 particles:

Eg 2

0gg

=

c°s2 q~ + sin2 E~ co-E,

(A.I)

~2

4

6

8

/

1 EQ

I

-12

(3/2) sin 2 (k

tO - - E 2

co -- E6 + E2

(b, cos ~ - a, sin q~)2

+ ,=9 ~

2gg =

(A.2)

co-E,+E2

(bin cos (~ - at0 sin ~b)2

co -- E m + E2

J

0 I -20 -16

cos 2 q~

lglof = - - +

v/w

I

-08

I

-04

(a,o cos ~b + bm sin ~)2 co - Elo + E5

+

(b) 00

E/W

(Cl0 COS 0 - - al0 s i n 0) 2

q

ca - Ej2 + Ej0 -~

Eg l

(alo cos ~b + blo sm ~b)2 60 - - El4 +

(3/2) sin 2 0

3gg =

cos z 0

+

co -- E~2 + E m

i

I

2

4

I

6. 0/W

I

I

I

8

lO

12

,=9/~ 14

F~g. 3. T h e central gap width, Eg, dependence on the H a m i l t o n i a n parameters: a) V (U = 4, E - - 2 ) ; b) E (U = 4, V = 0.4); c) U (V = 0.4, E = - 0 . 5 ) . REFERENCES 1. 2. 3. 4. 5. 6. 7. 8 9. 10. 11.

A.S.R. Sim6es, J.R. Iglesias, A. Rojo & B.R. Alascio, J. Phys. C21, 1941 (1988). E.V. Anda, J. Phys. C14, L1037 (1981). F o r review articles see: G. Czycholl, Physics Reports 143, No. 5 227 (1986); and N.B. Brandt & V.V. M o s h c h a l k o v , Adv. Phys. 33, 373 (1984). J.R. Schrieffer & P.A. Wolff, Phys. Rev. 149, 491 (1966). C. Lacrolx & M. Cyrot, Phys. Rev. B20, 1969 (1979). D . N . Z u b a r e v , Soc. Phys. Usp. 3, 320 (1960). J. H u b b a r d , Proc. R. Soc. A276, 238 (1963). A.S.R. Sim6es, J.R. lglesias & E.V. Anda, Phys. Rev. B29, 3085 (1984). M. Lavagna, C. Lacrolx & M. Cyrot, J. Phys. F 12, 745 (1982). C. Lacrolx, J. Magn. Mat. 60, 145 (1986). R. Jullien, J.N. Fields & S. Doniach, Phys. Rev. Lett. 38, 1500 (1977); Phys. Rev. BI6, 4889 (1978).

'gg

CO - - El2 -4-

cos 2 0

=

co - El6 + E~2

(c, cos 0 - a, sin 0) z

+ (c)

(A.3)

Em

co - - E~6 +

(A.4)

E, sin 2 0

+ Et2

co - - Et6 +

(a.5) Et4

We also present the G r e e n functions defined by equation (14.a) calculated on the ground states of 2, 3 and 4 particles:

2M=

b,o cos ~b(C,o cos ~b - a,o sin ~b) CO - - E m +

E2

+ blo sin ~b(alo cos ~ - blo sm 4') to -- E . + E,o + a,o sin O(a,o sin 0 - cm cos 0) co - El: + E 6 -+ at0 cos O(alo cos 0 + el0 sin 0) co - El4 + Eio

3M=

(3/2) sin 2 0 co -- Et2 + E 6 +

+

(A.6)

cos 2 0 co - - El6

-']'- El2

a, sin O(a, sin 0 -- c, cos 0)

,=9 ~ co - El2 + E, cos 2 0 sin 2 0 + co - - El6 + E~2 co -- Er6 + Er4

(A.7) (A.8)