568 Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 568 570 North-Holland, Amsterdam
M O D E L STUDY O F A N T I F E R R O M A G N E T I S M IN HIGH-T,, S U P E R C O N D U C T I N G O X I D E S Andrzej M. OLES * and Jan Z A A N E N
M a x - P l a n c k - l n s t i t u t f o r Festki3rperforschung, D - 7 0 0 0 Stuttgart 80, Fed. Rep. G e r m a n )
The antiferromagnetic ground state in high-T,, superconducting oxides (HTSO) is studied with a two-band model Hamiltonian by using a Hartree-Fock approximation and the Gutzwiller ansatz. It is found that the magnetic moments of Cu atoms and the antiferromagnetic gap agree well with the available experimental data for HTSO.
1. Introduction
The understanding of antiferromagnetic (AF) order in high-T c superconducting oxides (HTSO) is one of the important questions in the search for the microscopic mechanism of superconductivity. The AF phase of these compounds is interesting in itself. It is highly anisotropic with large exchange constant within CuO 2 planes J = 1300 K [1] and weak magnetic coupling between the planes. It is therefore important to understand the magnetic properties of a CuO 2 plane. In doped systems (La 2 ~Sr.~CuO 4 and YBazCu306+ x, x > 0) anti- ferromagnetism disappears quickly with the in- creasing number of holes N within CuO 2 planes. In the localized picture one may understand that as a consequence of frustration caused by ferro- magnetic interactions induced by holes [2]. The observed magnetic moments of 0.4-0.6#B per Cu atom agree then also with what is expected for a two-dimensional Heisenberg magnet where quan- tum fluctuations are significant.
In spite of this qualitative success of the local- ized picture, the evidence increases that the
itinerant description of HTSO is more appropriate.
The Cu(3d) holes are localized only if both U (Coulomb interaction) and D (the energy cost of a charge fluctuation d 9 --~ d 10 + p-hole) are larger than the p - d hybridization V [3]. The analysis of electron spectroscopy data [4], as well as a direct calculation of these parameters within the local spin density formalism [5] give that U / V - - - 5, but
D ~ V < 1. Therefore, the charge degrees of freedom are important in HTSO. Below we discuss the
* Permanent address: Institute of Physics, Jagellonian Univer- sity, PL-30059 Krak6w, Poland.
results obtained by an itinerant description of the AF phase of HTSO.
2. Model Hamiltonian and antiferromagnetic ground state
We use the following two-band model Hamilto- nian for HTSO
- - ( . + + H - d Z d i o d i o + { p Z a m o a .... i f m o + + U Y ' n , T ~, + V E ( & a ° , o h.c.) + n, imo t (1)
with D = {p - %. It describes holes in Cu(3d,2 ,.e) and O(2p,~,~) orbitals within a CuO 2 plane and has three parameters: D ~ V, U / V and the number of holes per unit cell N [3].
The A F phase is studied in the H a r t r e e - F o c k approximation (HFA) by solving a Hamiltonian H o in place of H, where the interaction term proportional to U is replaced by an alternating field rio = -Tv e x p ( i Q - R , ) with a wave vector Q
= ( , r / a , v / a ) characteristic of the two-sublattice
A F structure observed in HTSO; the signs corre- spond to o = 1" and $. From the diagonalization of H o one obtains an A F band structure and the energy minimization gives v = Urn, where m is a magnetic moment per one Cu atom related to the + ~ [ n d -- local hole densities by n , o = ( d , , d , o ) = ~ +
m exp(iQ- R,)].
A.M. Oleg, J. Zaanen / Anttferromagnetism of high- T c superconductors" 569 a n d O a t o m s in a similar w a y to that used for the
periodic A n d e r s o n model [7,8]. The reduction of kinetic energy m a y be described by introducing the following effective H a m i l t o n i a n
/ateff = Z (Ed --
~io)di+dio +
EpE
a+~amo
io
mo
+ ~_, V,o(n,o, nd, d)(d+oamo + h.c.),
(2)i m O
where
Vio(nio, n a, d)= q~/Z(nio, rid, d)V,
q,o(nio, n d, d)
1
- n i o ( 1 - n , o ) { [ ( 1 - n d + d ) ( n i o - d ) ]
'/2
+ n , o - ,
(3)
/L~, are alternating potentials which fulfill /.ti~ = /zj_, for i a n d j belonging to different sublattices, n d is the average hole density per Cu site and d is the average n u m b e r of Cu atoms occupied by two holes. The derived renormalization factor
q,o(n~o,
n d, d ) agrees with that of the A n d e r s o n lattice [8] a n d reduces to the well-known q , o = [ ( 1 - n d )/
(1 -n,o)] l/2
at U = o0 [7]. The variational p a r a m - eters { ~ , t , ~,~, d ) are found by minimizing the energy E o =(Heft) + Ud+ E~oyio(d~odio ).
O u r ef-fective H a m i l t o n i a n (2) interpolates between the b a n d and localized limit and reproduces the exact g r o u n d state for a finite cluster consisting of one
Cu
a t o m s u r r o u n d e d by four O a t o m s and filled by two holes.3. Results and discussion
A t N = 1 one finds an A F g r o u n d state for U :> 0 which is a consequence of the perfect nest- ing instability of our half-filled band. Since the results of an mean-field type theory are indepen- dent of the actual dimensionality, we have sim- plified our numerical analysis b y considering the A F phase o f an alternating C u - O chain in one dimension (1D). T h e n we find an analytic f o r m u l a for the A F gap in the a n t i b o n d i n g b a n d
where
DnF = D - ½Un d
is the energy difference0 , 8 l i z
m
/ /
/
0.6. . . .
,,,/
/0.4
/
/ 0.2 / 0 / ' (o) 0.2 0./, 0.6 0.8 1.0 n d 2.0 , , ,,,
V'O" 1.
/,;,'"
1.0
.//
0.5 /J
(b)
0 ~ ' ' ' O 0.2 0.4 0.6 0.8 m Fig. 1. M a g n e t i c m o m e n t m as a f u n c t i o n of n,~ (a) a n d A F g a p A / V as a f u n c t i o n o f rn (b) in H F A ( d a s h e d ) a n d G A (full lines) for N = 1, U / V = 5 a n d V = 1.8 eV. D o t t e d linesi n d i c a t e the v a l u e s of n d a n d A / V for 0.4 < m < 0.6.
between the one-hole levels in H F A . In 2 D the gap is anisotropic but its m i n i m u m in the (1, 1) direction is given also by eq. (4). If
DRy ~ oo,
A ~ U m
and only then A F phase of the C u O 2 plane reduces to that of the H u b b a r d model.Instead of
Dnv,
it is more convenient to use n a as a p a r a m e t e r being directly measured by p h o t o - emission. The resulting rn and A are shown in fig. 1. Taking the experimental magnetic m o m e n t of 0 . 4 < m < 0 . 6 , we find n a = 0 . 6 a n d A = V . The G A is a simple way to include electron correla- tions. Therefore, n d values areenhanced,
as also f o u n d elsewhere for n o n m a g n e t i c states [5]. The A F gap A isreduced and, in spite of the simplifi-
cations of o u r model, agrees with the experimental value of 2.0 eV [9].570 A.M. OleoS, J. Zaanen / Antiferromagnetism of hlgh- T superconductors long-range order and suggests that the
spin bags
formed by excess holes are important [10]. Indeed, numerical simulation of finite two-dimensional lattice shows that such defects form and condense into domain walls which destroy the AF order
[111.
We would like to thank P. Fazekas and O. Gunnarsson for valuable discussions. The finan- cial support of the Polish Research Project CPBP 01.03. is acknowledged.
References
[2] A. Aharony et al., Phys. Rev. Lett. 60 (1988) 1330. [31 J. Zaanen and A.M. Ole.~, Phys. Rev. B 37 (1988) 9423. [4] Z. Shen et al., Phys. Rev. B 36 (1987) 8414.
[5] J. Zaanen et al., Physica C 153-155 (1988) 1636; M. Schluter, J. Hybertsen and N.E. Christensen, ibid. 153 155 (1988) 1217~
[6] A.M. ()leg, J. Zaanen and P. Fulde, Physica B 148 (1987) 26(/.
[7] T.M. Rice and K. Ueda, Phys. Rev. B 34 (1986) 6420. [8] V.Z. Vulovid and E. Abrahams, Phys. Rev. B 36 (1087)
2614.
[9] J.M. Ginder et al., Phys. Rev. B 37 (1988) 7506.
[10] J.R. Schrieffer, X.G. Wen and S.C. Zhang, Phys. Rev. Lett. 60 (1988) 944.
[11] J. Zaanen and O. Gunnarsson, to be published.
