Destruction of antiferromagnetism by vacancies for the Hubbard and Anderson lattice models

Solid State Communications, Vol. 70, No. 1, pp. 93-96, 1989. Printed in Great Britain.

0038-1098/89 $3.00 + .00 Pergamon Press plc

DESTRUCTION OF A N T I F E R R O M A G N E T I S M BY VACANCIES FOR THE HUBBARD AND A N D E R S O N LATTICE MODELS C. Bastide and C. Lacroix Laboratoire Louis N6el, C.N.R.S., 166 X 38042 Grenoble-Cedex, France

(Received 26 September 1988 by P. Burlet) We study the itinerant antiferromagnetism of fermions with hard-core repulsion like in the weak hopping Anderson lattice or the strong U Hubbard Hamiltonian. It is found that the critical temperature Ts decreases very quickly when vacancies are injected. The effect of the dynamic three-sites magnetic interaction is studied. WE STUDY the depression of the N6el temperature for fermions with hard-core repulsion when the electronic density deviates from half-filling. Such strong local repulsions are believed to play an important role both in the heavy fermions compounds and in the high-T,, cuprate materials. The strong-U single band Hubbard model [1,2] and the weak hopping Anderson lattice model [3, 4] can both be described by an effective Hamiltonian involving only one kind of particles with infinite on-site repulsion and nearest-neighbour two and three-sites second-order interactions:

than one; n' = 12 - n l for 1 < n < 3. Note also that those effective particles have a charge + e for 1 < n < 2insteadof-e. For the Hubbard model, the parameters are different:

t2 = O, g = 1/4 = - t 2 / U ,

We will keep only the magnetic terms of the second order interactions, so we will study the effective Hamiltonian:

H:fr' : Herr = ~i;~ tlbi~bj~ + + ~ t2bt~+ bl~ + 1 tjla

I~/S/Sj+ 2 ..

"

2 g ' bt,+ bt, nj - ~ I ' Sit Sj qt~ ijr

where the couples of sites 0" and fl are nearest neighbours. The operators bt, are defined by bt, = (1 - a~-,at_,)at, so that double occupancy is forbidden for the effective particles described by ai, (fermions operators). Moreover nt = Zobt,+ bt~ and St and Su are expressed with the bt, as usual: (Sf,St

,S{)

=

~ tlbi~bj _ ~1 .__ ~, ISi. Sj _ -~ I,Si .

(3)

gntnj

(1)

St =

g' = I'/4 = - t 2 / 2 U .

1

+

( b ~ b t + , b ¼ b , T , ~ t r b t , bt,,)

We will estimate the N~el temperature for antiferromagnetism by the difference of energy at zero temperature between the non-magnetic state and the one with a staggered magnetization m. The value of this antiferromagnetic magnetization will be found by minimizing the energy (He~r) with respect to m. We consider a lattice with no frustration; square or simple-cubic. We approximate the energy coming from the two last terms of (3) by: 1 ~ I
(4)

which is supposed to be the main contribution to the

1

Su = ( S f , S,[, Si~) = (b~ bt+, b~ bn, ~ ~ ab + b,,). (2) For the Anderson lattice, the general values for the parameters of Hat are given in [4]. The magnetic interaction I is negative (antiferromagnetic) or equal to zero (n > 2), while the three-sites magnetic term I' can be either positive or negative depending on the number of electrons and on the ratio Eo/V. The number of effective particles n' in the effective Hamiltonian (1) is shown on Fig. 1 and is always smaller 93

(n')

0

I

2

:3

(n)

Fig. 1. Number and charge of the effective particles as a function of the number of electrons per site in the Anderson lattice.

94

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D E S T R U C T I O N OF A N T I F E R R O M A G N E T I S M

m-dependent part, and where nearest neighbour magnetic correlations are neglected. This approximation which gives the mean-field result in the localized limit is hoped to be valid at least in three dimensions. Anyway, this approximation favors the long-range ordered magnetic state. We must describe the motion of the particles by a method which gives the good variation of the kinetic energy as a function of the staggered magnetization. Some works are related to this question [5-10], but we use here a very simple description: The Hubbard's 1 approximation [11] contains the main point necessary to our purpose which is the reduction of the bandwidth with increasing magnetization. The equation of motion for the Green's function Gim ~ = ( ( b ~ , b+.,.)) is in this approximation: e)G,.,~ =

( l - n, . ) + ( l - ni_~) ~ t,,Gtm~.

(5)

I

We impose a given magnetization m = Z~ a~(,b~+.bi,~) where a = + 1 is an index for the two sublattices; a = + 1 is the sublattice with spins up. Note that we have not included the effect of the interactions I and I ' in the equation of motion (5) because we just want to calculate (bi+~bj,,) (and (bia+ bt,) for the kinetic magnetization (S~)) which is governed by the first term of H¢~-the hopping amplitude t~-which is at least an order of magnitude larger than I and I'. These second order interactions nevertheless come into play indirectly through the imposed magnetization m. The equation (5) is easily solved introducing two propagators (intra- and inter-sublattice): G~

=

~ G,,j~ exp [-ik(R~, - R:,)] ict

=

(n, 1 -'~

+ as

;)o co2 -

E~

hopping must verify:

Fefr=

( 1 .2 m)( 2 t 1--5-+ .

t

(8)

which gives the same reductive factor (7). The particle density is easily calculated from (6) where f is the Fermi-Dirac function: 1 ~ f door(co) - 7[ 1 Im G~2

n'

=

~/

n' 2 -- n'

=

1 ~ -~ . f ( E , ) ,

f(x)

=

(ep~,- ~,)

(9)

+ 1)-

I

(lO) The sum over the k-vectors in (9) initially restricted to the Brillouin zone of one sublattice, has been extended to the whole Brillouin zone of the full lattice, because G~_ e = G~~ with Q = (Tr, zt, zt). The magnetization can also be reobtained from (6), m,

=

1~

a f dmf(m) - nl i m G k 0 : "

(11)

This description is not exactly self-consistent since this calculated magnetization m, is not exactly equal to m: m,/m

=

=

# / ( 2 - n')

(1 -- 6)/(1 + 6)

(6 = 1 -- n').

(12)

However, when n' ~ 1, the two magnetizations are nearly equal, and as we will only need to consider the nearly localised regime 6 = l-n' ~ 1, the calculation will remain sound. The kinetic energy (first term of (3)) is easily obtained from the inter-sublattice Green's function: K =

K0

1

(2_#)2

(13)

(6) Gi-~~

=

~ G, ,j, exp [-ik(R,_, - Rj,)] i_ a

where Y~. means the summation is restricted to the a-sublattice and where n' is the number of effective particles per site in the effective Hamiltonian (1), and: E,

=

~1/( 1 - ~ - ) 2 - - ~m2 - - e,

(7)

The reductive factor (7) for the bandwidth is easily understood: The Hubbard's I approximation means the effective hopping of a particle with spin a to site i is reduced by the factor (1 - ni_,) which is alternatively 1-n'/2-m/2 and 1 - # / 2 + m / 2 when the particle moves through the lattice. So the mean effective

where K0 is the kinetic energy of the non-magnetic state. This reduction of the kinetic energy is in fact not sufficient; when the staggered magnetization is maximum, i.e., equal to n', the particles are not allowed to move since if they do, some particles would be on the wrong sublattice and the magnetization not maximum. This is why a maximum antiferromagnetic moment is impossible as soon as there are some vacancies. However, if this idea was correctly taken into account, one should obtain a reductive factor (7) or (13) which goes to zero when m = n'. As this is not verified, our approximation underestimates the reduction of the kinetic energy, at least for a large moment m, but it is likely that this assumption holds true for all values of m. So, this approximate reductive factor (13) favors the antiferromagnetic state (as the mean-field treatment (4) did). Similarly, the kinetic

Vol. 70, No. 1

DESTRUCTION OF ANTIFERROMAGNETISM

95

magnetization is derived: -zI/8

1

~

~rn

1

f(Ek) cos [k(Ri -- R,)]

I

2 N

E-,.In

(14)

and then the contribution of I' to the energy per site: -- m:

n'

4

1

-- z27n' + N

f(Ek)

(15)

where z is the number of nearest neighbours. The tight-binding form for gk is useful in the derivation of (15). One can show that the expression (15) changes of sign at n' = 2/3. A positive I ' favors antiferromagnetism for n' > 2/3 and favors ferromagnetism for n' < 2/3 (and reversely for a negative I'). This is due to the sign of the next-to-nearest neighbour hopping (bi+~blo ) which depends upon the electronic density. When 6 ,~ 1, the energy (15) can be written:

z(z - 1)6 I'm 2

(16)

and the total energy per site is:

~/ E =

K0

m2 1

z

(1 + 6) 2 + g ( I -

4(z -

1)6I')m 2. (17)

By minimizing E with respect to m, we find the critical value 6, for the number of vacancy at which antiferromagnetism disappears:

I/4 6,

-

2ltl-

(z -

1)I"

(18)

We recall that both in the Anderson lattice and in the Hubbard model, I is negative. A negative I' like in the Hubbard model reduces 6,. But in the validity domain of the perturbation expansion in t which lead to (1), the second order interactions I an I ' are far smaller than t, and the effect o f / ' is very small and can be neglected. The magnetization as a function of 6 is: rn =

~/1 - 64 62

t2

(19)

and the difference of energy between the antiferromagnetic and non-magnetic state is: AE -

8zt2 (6 -+- 1 ) 2 I 8-M

(3) Fig. 2. Destruction of the antiferromagnetic phase as a function of the number of vacancies 6 = 1 - n. The staggered magnetization m goes from 1 to zero. energy loss in the kinetic energy of the vacancies when increasing the antiferromagnetic magnetization. 6, is a small value; for example in the Hubbard model where

I = 21' = - 4 f l / U 1

6, =

2U --

jtl

(21) +

2(z -

1)

This quick destruction of the antiferromagnetic phase is due to the kinetic energy of the vacancies which is larger in the non-magnetic state than in the magnetic one. Yokoyama and Shiba [12] as well as Kaxiras and Manousakis [17] have obtained with numerical calculations a similar destruction with however a larger 6, than [21]. The three sites magnetic interaction increases this destruction further when I' < 0 (as in the Hubbard model) and this is in agreement with the work of Inui, Doniach and Gabay [13] who have found with a different method an even stronger effect from this three sites interaction. For the Anderson lattice, using the parameters calculated in [4], antiferromagnetism is then found only in the vicinity of n = 1 [14, 15] and leaves the superconductivity appear for n > 1 [16]. REFERENCES

(20) 1.

These two quantities are shown in Fig. 2. The critical temperature TN is expected to have the same shape as AE and one will notice the upward curvature of AE(6) and the difference of shape with re(f) 2. The total energy gain AE is smaller than rn2 because of the

2. 3. 4.

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DESTRUCTION OF ANTIFERROMAGNETISM L.N. Bulaevskii & D.I. Khomskii, Soy. Phys. JETP 25, 1067 (1967). W.F. Brinkman & T.M. Rice, Phys. Rev. B2, ! 324 (1970). R. Joynt, Phys. Rev. B37, 7979 (1988). B.I. Shraiman & E.D. Siggia, Phys. Rev. Lett. 61, 467 (1988). S. Schmitt-Rink, C.M. Varma & A.E. Ruckenstein, Phys. Rev. Lett. 60, 2793 (1988). Y. Takahashi, Z. Phys. BT1, 425 (1988). J. Hubbard, Proc. Roy. Soc. A276, 238 (1963). H. Yokoyama & H. Shiba, J. Phys. Soc. Jap. 56, 3570 (1987). M. Inui, S. Doniach & M. Gabay, Phys. Rev. B38, 6631 (1988).

14.

15.

16.

17.

Vol. 70, No. 1

C. Bastide, C. Lacroix & A. da Rosa Sim6es, International Conference on Magnetism, July 1988, Paris, to appear in Le Journal de Physique ( Colloques). We have not considered the ferromagnetism, but the possibility of a transition from antiferromagnetism to ferromagnetism when increasing 6 should be studied. A. da Rosa Sim6es, C. Lacroix & C. Bastide, Sixth International Conference on Crystal-Field Effects and Heavy Fermion Physics, July 1988, Frankfurt, to appear in J. Magn. Magn. Mat. 76-77 (1988). E. Kaxiras & E. Manousakis, Phys. Rev. B37, 656 (1988).