Solid State Communications, Vol. 78, No. 4, pp. 307309, 1991. Printed in Great Britain.
00381098/91 $3.00 + .00 Pergamon Press plc
HOLE PAIRING IN THE RVB STATE Shi Daning and Shi Wei Department of Mathematics, Physics and Mechanics, Nanjing Aeronautical Institute, P.R. China and Li Zhengzhong Department of Physics, Nanjing University, P.R. China
(Received 31 May 1990, in revised form 20 December 1990 by L. Hedin) The RVB model has been restudied in the slavefermion theory. The superconducting state can be explained by fermionic hole pairing. This gives a more reasonable transition temperature than in earlier work.
1. INTRODUCTION ANDERSON has first suggested the resonatingvalencebond (RVB) which he proposed in 1973 as a theoretical description of highTo superconductivity [1]. Before long Baskaran, Zou and Anderson have used the slaveboson (SB) method to formulate the RVB theory for the nearly halffilled Hubbard Hamiltonian [2]. In the framework of the SB theory, the charge carriers, namely the holes, are bosons and t h e spin carriers, namely the spins, are fermions. Recently, Read and Chakraborty [3] indicated that in order to obtain the lowest energy of a hole in the RVB state, the situation is just the opposite: the holes obey Fermi rather than Bose statistics and the spins obey Bose statistics. Therefore, the RVB model has been restudied in the framework of the slavefermion theory [4]. The other problem is how the holes in the RVB state can give a highT, superconductor? The original answer of Anderson was the BoseEinstein condensation of the holes in the RVB vacuum, but this condensation in the two dimensional CuO2 plane is a controversial subject. In order to clarify this paradox, Anderson and coworkers [5] have pointed out that due to a Josephsonlike coupling between the CuO2 layers which are in RVB states, the holes in each layer might be pairing. Superconduction in highTc materials is thus due to (eke_k) condensation of the holes in one layer. We must notice that it is a bosonic pair in the SB theory. So, the bosonic pair orderparameter equations are different from those of BCS theory. In this letter, we develop the slavefermion (SF) mean field theory for the RVB model. As a complement of [4], the transition temperature of the RVB state as a function of the doping parameter is given numerically. If the Josephson pair mechanism is domi
nant, we would expect that the condensation of the fermionic hole pairs gives rise to highTo superconductivity. The fermionic pair orderparameter equations are similar to those of the BCS theory. The transition temperature which will come out in Section 3 is more reasonable than that of Anderson. 2. THE SF MEAN FIELD THEORY FOR RVB MODEL Our starting point is analogous to that of BZA [2] who used the SB theory for a twodimensional singleband Hubbard model. In the largeU limit the Hubbard model takes the form [2, 4]: H =
t ~
+ eiej+ Sj,  J ~ bq+ b,j. S,.o
(i,j)o
(1)
(i,j)
The spin and hole sectors are included in the Hamiltonian [1]. However, in contrast to the BZA work, the spin operators Si,+ , Sjo obey Bose rather than Fermi statistics and the hole operators ei, e7 obey Fermi statistics. Let us first consider the spin sector and do a mean field approximation for the hole sector. The wellknown tJ model is then obtained: /'/spin
=
 
t6 E
S~+ Sjo  J E b+ bij
(i,j)o
(2)
(ij )
where 6 is the doping concentration and bi~ is defined by: b;;
=
1 (sit qz

s,;
).
(3)
We transform the Hamiltonian (2) into the Bloch representation and make the HartreeFock factoriz
307
H O L E P A I R I N G IN T H E RVB STATE
308 ation, which gives
Vol. 78, No. 4
0.00
H = ~ (ek  11)S~Sk~  JA ~ (f(k)S~ S~ + h.c.) ka
k
+ NJ(P 2 + A2)
P
(4)
where the order parameters A, P are defined as A = x/~(b~j), P = (S~+Sjo) and
ek = f(k)
(2/~ + PJ)(cos kx + cos ky)
(5)
i(sin ks + sin ky).
=
0.25
(6)
The Hamiltonian (4) can be diagonalized by the Bogoliubov transformation and the quasiparticle energy be obtained:
Ek = [(ek  #)2 + j2A2f2(k)]~/2"
(7)
The order parameters A and P, and the bosonic chemical potential # can be calculated selfconsistently from:
2 J 
1~ f2(k)
1  6 =
coth(Ek/2T) Ek
~
coth
1
P =
2N ~ (cos kx +
(8)
cos
ky)(E k



1
(9)
11)
coth (Ek/2T)
Ek
0.50
! .00
"N \ \
\
0.50
!
\
\
O. 0 5
0.I0
Fig. 2. The order parameter P vs the doping concentration and t/J = 10. (dashed line). P is drawn in Fig. 2. In our numerical calculation, we choose t/J = 10. We note that T R in the SF theory is higher than in the SB scheme [2]. This would confirm that the fermionic holes and bosonic spins favor the RVB state. We also point out that T R is not the transition temperature of the superconducting state, and in the following section we study the hole pairing in RVB state. 3. H O L E P A I R I N G IN T H E RVB STATE
(lO) where N is the number of sites. We consider the transition temperature of the RVBstateT R.AtT= T R,A = 0, a n d i n t h e 5 = 0 case, it is clear the P = 0 and ~k = 0. So, we have calculated T Ranalytically from equations (8)(10) and got TR(5 = 0) = 0.91 J, 11 = J. Away from the halffilled band, we have solved equations (8)(10) numerically and show T R as a function of 5 in Fig. 1
,
O. O0
In this section, we turn to the hole sector which is responsible for the superconducting state. Anderson initially suggested that the mechanism of the highTc superconductor is the BoseEinstein condensation of the bosonic holes. However, the holes are charged fermions in the SF theory. Hence, in order to get the superconducting state the possible physical mechanism is hole pairing as far as we know. However, the exact nature of this mechanism must be clarified. Within the framework of the RVB theory, one approach is to consider the Josephsonlike coupling between adjacent CuO2 layers. Let us discuss two layers a, b. According to [5], the dominant interaction between layers a, b is due to singlet pair hopping. Thus, layer b can be taken as the reservoir of the hole pairs in layer a and vice versa. From this suggestion and the Hamiltonian (1) the hole sector inplane effective Hamiltonian can be written as:
Hhole = 0.0o
0.0°
0.,,
Fig. 1. The transition temperature of the RVB (dashed line) and the highTc superconductor (solid line) vs the dopin2g concentration, t/J = 10. 1, 2, 3, correspond to (t/t,b) = 10, 20, 30.
~ Fke~ek

k A ~ / e~+k e +
k
+
h.c.)
(11)
k
Fk is the hole band energy relative to the fermionic chemical potential: Fk =
4tP(cos kx +
cos
ky)
(12)
A is the vertex describing the pair hopping process.
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HOLE P A I R I N G IN THE RVB STATE
Anderson [5] has suggested that A should be considered as a constant:
A =
(13)
JIt2ab
where tab is the interlayer electron hopping matrix element and J l approximately describes the spin response function. Diagonalizing the Hamiltonian (11) by the Bogoliubov transformation gives the quasiparticle energy: Ek =
(Fk "1 ~2)1/2
309
Fig. 1. One is that the transition temperature of the highTo superconductor increases nonlinearly with increasing doping concentration, which is different from the result in [5] but in agreement with some experimental results [6]. The other is that there exists a maximum of T~ as 6 increases to 6c. The maximum and 6c depend on t/tab which depends on the high T,. material. 4. CONCLUSION
(14)
In this communication, within the framework of the slavefermion mean field theory we have studied l = A ~ th(ff, u/2T)_~ . (15) BCStype hole pairing vs Josephsonlike coupling. The transition temperature of the RVB state and the k 2G highTo superconducting state are given. Especially, a While solving this equation, we should bear two nonlinear dependence of the transition temperature points in mind. First, the inplane hole bandenergy on the doping concentration has been obtained. We which is larger than that of the spins by more or less 1 have also found that the transition temperature is not order is about 4tP which increases with increasing hole only dependent on the doping concentration, but also concentration. Secondly, since the spin is involved as on the ratio of inlayer and interlayer electron hopping a virtual excitation when a hole pair is tunneling, the matrix elements. These results are in better agreement cutoff energy in equation (15) must be the spin band with experiment than those in [5]. energy 2t6 + PJ. If these two points are taken into account, we can solve equation (15) and get the superREFERENCES conducting transition temperature T~ approximately: where the order parameter ,~ is determined by
Tc =
1.13(2t6 + P J ) e x p
4
t IPI • (16)
From [5] we estimate (tit,b) z > 10. Thus, the Tc variation with 6 can be computed using equation (10). The result is shown in Fig. 1 (solid line) for different choices of (t/tab) z, the curves 1, 2, 3 corresponding to (t/t~b)2= 10, 20, 30 respectively. (To simplify the numerical calculation we assume that P takes the values in Fig. 2.) There are two interesting results in
1. 2. 3. 4. 5. 6.
P.W. Anderson, Science 235, 1196 (1987). G. Baskaran, Z. Zou & P.W. Anderson, Solid State Commun. 63, 973 (1987); Z. Zou & P.W. Anderson, Phys. Rev. B37, 3786 (1988). N. Read & B. Chakraborty, Phys. Rev. 1340, 7193 (1989). K. Flensberg, P. Hedegard & M.B. Pederson, Phys. Rev. 1340, 850 (1989). J.M. Wheatley, T.C. Hsu & P.W. Anderson, Phys. Rev. B37, 5897 (1988). M.W. Shater et al., Phys. Rev. B36, 4047 (1987).