Quasiparticle bands and superconductivity for the multiple-layer and

PHYSICAL REVIEW B

VOLUME 57, NUMBER 18

1 MAY 1998-II

Quasiparticle bands and superconductivity for the multiple-layer and three-dimensional superlattice t-J models Wei-Guo Yin Department of Physics, Nanjing University, Nanjing 210093, China

Chang-De Gong CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China and National Key Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China ~Received 16 May 1997; revised manuscript received 17 October 1997! We present a propagator-matrix approach to calculate the one-hole spectrum for the multiple-layer and three-dimensional ~3D! superlattice t-J models using the self-consistent Born approximation and then obtain the superconducting transition temperature T c within the antiferromagnetic van Hove scenario. The flat region around k i 5( p ,0) in each band of the quasiparticle is observed for a trilayer and a 3D superlattice with a bilayer per unit cell, while strong many-body effects give rise to a large reduction of the band splitting. It is found that for a trilayer the outer planes are slightly less doped than the inner ones and for a 3D bilayer cuprate one can observe little dispersion along k z , in agreement with recent experimental data. Our calculations for rather different lattice structures and boundary conditions indicate that a large density of states leading to the boost of T c mainly results from the in-plane antiferromagnetic correlations rather than from the band effects. @S0163-1829~98!04405-1#

I. INTRODUCTION

The dispersion of the hole quasiparticles in the normal state of the two-dimensional ~2D! CuO 2 planes is of crucial importance in the explanation of many anomalous properties of the high-temperature superconductors ~HTSC’s!. The small bandwidth, only ;0.3 eV, shown by recent angleresolved photoemission ~ARPES! data1 for Y-Ba-Cu-O and Bi2 Sr2 CaCu2 O8 ~Bi2212!, both near optimal doping, suggests that strong correlations are crucial for a proper description of carriers in the cuprates. However, even for the simplest Hamiltonian including strong electron correlations, the 2D t-J model which is considered to contain much of the low-energy physics of HTSCs,2 there has been no exact result. An essential first step in the solution of this problem is to understand the motion of a hole in the quantum antiferromagnetic ~AF! Heisenberg model.3 This calculation is expected to be able to provide some important clues for fundamental issues of HTSC’s such as the form of the Fermi surface at moderate doping. For example, the flat band naturally obtained from these small concentration calculations survives near the Fermi surface observed in ARPES.4 An intuitive picture is that as long as the AF correlation length is robust, the holes behave as if moving in a nearly perfect AF background. This idea is supported by recent quantum Monte Carlo simulations which show that nearest- and next-tonearest-neighbor spins tend to be antiparallel even with hole densities as large as 0.25.5 Furthermore, using the dispersion of a single hole to approximate the observed quasiparticle dispersion at finite doping, Dagotto, Nazarenko, and Moreo ~DNM! illustrated even in a conventional BCS-like theory that the resulting enhancement of the Van Hove singularities in the quasiparticle ~QP! density of states can lead to a boost in T c and predict that the optimal doping occurs when the 0163-1829/98/57~18!/11743~9!/$15.00

57

Fermi level reaches the flatness of the QP dispersion.6,7 Consequently, extensive studies have focused on the single-hole problem within the 2D t-J model.8–14 Excellent agreement is found between analytic results for the Green function obtained by the self-consistent Born approximation ~SCBA! introduced by Schmitt-Rink, Varma, and Ruckenstein3 and Kane, Lee, and Read,13 and the Green function obtained by an exact diagonalization of small systems for a wide range of parameter values of J/t.14 Using SCBA, one can clearly recognize that the motion of the hole always leaves behind a ‘‘string’’ of overturned spins which can then be healed by the quantum fluctuations associated with the transverse exchange interactions, thus leading to the coherent motion. The well-defined quasiparticle peak in the low-energy side of the hole spectral function is corresponding to this Bloch-wave-type solution. Thereafter it was successfully applied to the 2D t-t 8 -J model to compare its predictions with the photoemission spectra of Sr 2 CuO 2 Cl 2 , 15 and the case of bilayers and 3D antiferromagnets.16,17 Since all the systems previously studied are simple hypercubic lattices ~in d dimension!, the well-defined low-energy single peak structure of the hole spectral function makes it easy to tell the quasiparticle eigenstates in the corresponding d dimensional momentum k space ~the bilayer is analogous to a 3D simple lattice with k z 50,p ). Coming to the case of more realistic complex lattices such as any n.2 layers ~e.g., Hg-1223! or a 3D lattice with more than one layer per unit cell ~e.g., YBCO and Bi-2212!, there has been no theory for the dispersion of quasiparticles. The role of multiple layers in HTSC’s continues to be one of the most intriguing puzzles.18 It is experimentally established that in the layered cuprates the T c increase with increasing n while it peaks at n53 and then decreases as shown in the HgBa 2 Ca n21 Cu n O 2n121 d series.19 Some authors phenom11 743

© 1998 The American Physical Society

WEI-GUO YIN AND CHANG-DE GONG

11 744

enologically explained the dependence of the T c on n assuming a heavily inhomogeneously hole distribution between the outer and inner planes.20 However, this is controversial to the recent 63Cu nuclear magnetic resonance ~NMR! spectra.21,22 On very general grounds, one expects that the new quasiparticle operators in these complex lattices are the combinations of bare hole operators in different layers, i.e., the n CuO 2 layers in a unit cell should hybridize to produce n quasiparticle bands. Thus there are in general n quasiparticle peaks in the low-energy side of the hole spectral functions. Unfortunately, the strong coupling of the hole and magnon makes it very difficult to find these quasiparticle operators and tell the quasiparticle eigenstates in these complex lattices. Thus any effort is nontrivial and interesting to explore a theoretical framework to generalize the SCBA approach to a variety of lattice structures with different boundary conditions, in particular, the open boundary condition in the z direction for a multiple layer and the periodical one for a superlattice. In this paper such a generalization is first formulated using the hole propagator-matrix approach. We investigate this explicitly for the trilayer system which has drawn particular interest because of its highest T c 5133 K and for the 3D bilayer cuprates due to the most extensively studied copper oxide compounds YBCO and Bi-2212. Despite the rather different lattice structures ~also with apparently different boundary conditions! taken into account, the quasiparticle band always manifests a flat region around ( p ,0), thus leading to the boost of T c within the van Hove scenario.16,17 Our results suggest that a large density of states that enhances T c results from the antiferromagnetic correlations in the t-J model rather than from the band effects. The paper is organized as follows. Section II gives an outline of the hole propagator-matrix approach of the calculation of the quasiparticle dispersions for a 3D t-J model with multiple layers per unit cell. Section III contains a detailed description of the SCBA for trilayer and 3D bilayer cuprates. Numerical results for quasiparticle dispersion, spectral functions, and the calculation of T c are also given in this section. We summarize our results in Sec. IV. An appendix gives details of the diagonalization of the Heisenberg model for complex lattices considered here in the linear spinwave approximation.

FIG. 1. A simplified model of layered cuprates. The CuO 2 layers are represented by lines. m 51,2, . . . ,n is the layer index in the mth unit cell.

and the in-plane sites index i. m 51,2, . . . ,n. For a pure multiple layer, m51 and the open boundary condition in the z direction is used. To simplify the notation, on the B sublattice we perform a rotation of the spins by 180° about the S x axis3: 7 S6 r →S r ,

(

rr 8 s

c ~ t rr 8 ˜

J rr 8 Sr •Sr 8 , ( rr 8

˜ c rs→ ˜ c r2 s .

~2!

t rr 8 5t d i 8 ,i1aˆ d m 8 , m d m 8 ,m 1 d i 8 ,i @ t'1 d m 8 , m 61 d m 8 ,m 1t'2 ~ d m 8 ,n d m ,1d m 8 ,m21 1 d m 8 ,1d m ,n d m 8 ,m11 !# ,

As shown in Fig. 1, we consider a single-band t-J model for the 3D superlattice with n coupled planes per unit cell. These layers, which are to simulate the copper oxide layers, are all placed parallel to the x-y plane, each described by a 2D t-J model. Let the distance between two nearestneighbor ~NN! layers inside the same unit cell be c 1 ; that between two NN layers between cells c 2 . The structure is periodical in the z direction with periodicity c5(n21)c 1 1c 2 . The Hamiltonian of the system can be written as H5

S zr →2S zr ,

This canonical transformation changes the Ne´el configuration into a ferromagnetic state with all spins up and removes the further necessity to distinguish between sublattices. In Eq. ~1!, the nearest-neighbor in-plane interactions t,J, the between-plane one inside the same unit cell t'1 ,J'1 , and NN intercell interactions t'2 ,J'2 are taken into account only when they are accurate enough to reproduce the realistic spin properties of YBa2 Cu3 O72 d for a bilayer system. Then the interactions in the 3D superlattice system are given by

II. THEORETICAL FRAMEWORK

† ˜ r s c r 8 s 1H.c. ! 1

57

~1!

in the standard notations. The site coordinate → r 5r→m 1r→m → 1r i are specified by the layer index m in the mth unit cell,

J rr 8 5J d i 8 ,i1aˆ d m 8 , m d m 8 ,m 1 d i 8 ,i @ J'1 d m 8 , m 61 d m 8 ,m 1J'2 ~ d m 8 ,n d m ,1d m 8 ,m21 1 d m 8 ,1d m ,n d m 8 ,m11 !# , ~3! where i,i1aˆ are the NN sites in a layer. Therefore the model can be rewritten as H5

( ~ t rr 8˜c rr s 8

z

G

2S zr S r 8 .

† rs

˜ c r 8 2 s 1H.c.! 1

F

1

1 2 2 J rr 8 ~ S 1 ( r S r 8 1S r S r 8 ! 2 rr

8

~4!

Following an approach proposed by Schmitt-Rink et al.,3 we define hard-core boson operators b r , such that S †r 5b r , S 2 r

57

QUASIPARTICLE BANDS AND SUPERCONDUCTIVITY . . .

5b†r , S zr 5 21 2b †r b r , and the fermion operators for spinless holes~holons!3 so that h †r 5 ˜ c r↑ and ˜ c r↓ 5h †r b r . The localfermion Hilbert space is thereby mapped onto the product space u hole& ^ u boson& , with n hr 50,1 and n br 50,1, where @ h r ,b r 8 # 50. Besides hard-core restriction to bosons, there is another constraint ~C2!: At a given site there cannot be both a hole and a boson. The Hamiltonian then has the form H5H t 1H J within linear-spin-wave theory,3,16 with H t5

† t rr 8 h †r h r 8 ~ b r 1b r 8 ! 1H.c., ( rr

8

1 † † H J5 J ~ b † b 1b r 8 b r 8 1b †r b r 8 1b r b r 8 ! . 2 rr 8 rr 8 r r

(

h km5

b qm5

1

mi AN (

1

mi AN (

b re

,

where the holon-magnon coupling M m n a (k,q),F m n a (k,q) are defined as 1

AN

* an# . @ t m n ~ k2q ! v q am 1t m n ~ k ! u 2q ~11!

F m*n a ~ k2q,2q ! 5M n ma ~ k,q ! .

1 † † † H J5 @ J ~ 0 !~ b q† m b q m 1b 2q n b 2q n ! 1J m n ~ q ! b q m b 2q n 4 q, m n m n

(

~7!

where t m n ~ k ! 54t g k d m n 1t'1 e 6ik z c 1 d m , n 61 1t'2 ~ e ik z c 2 d m ,1d n ,n 1e 2ik z c 2 d m ,n d n ,1! , J m n ~ q ! 54J g k d m n 1J'1 e 6ik z c 1 d m , n 61 1J'2 ~ e ik z c 2 d m ,1d n ,n 1e 2ik z c 2 d m ,n d n ,1! ,

~8!

where g k 5(coskx1cosky)/2,t n*m (k)5t m n (k)5t n m (2k), * J n m (q)5J m n (q)5J n m (2q). The AF spin background may be well treated within linear-spin-wave theory. In the 3D momentum space, we diagonalize H J 5 ( q a v q a b q† a b q a using a Bogoliubov-like transformation23 ~see the Appendix!

H t5

(

M mma ~ k,q ! h k†m h k2q, m b q a

1

(

kq am

kq a , m Þ n

h k†m h k2q n @ M m n a ~ k,q ! b q a

m 51,2, . . . ,n, ~9!

~13!

In Eq. ~13!, it is difficult to find the QP eigenoperators mixed with different-layer bare hole operators because of the strong coupling between the holes and magnons. Let us introduce the zero-temperature propagator matrix for holes, G m n ~ k,t ! 52i ^ t h k m ~ t ! h †k n ~ 0 ! & ,

m , n 51,2, . . . ,n, ~14!

where t is the time-ordering operator, and ^ & represents an average over the ground state. The Green’s functions satisfy matrix Dyson equations: G m n ~ k, v ! 5G m0 n ~ k, v ! 1

G m0 l ~ k, v ! S l r ~ k, v ! G rn ~ k, v ! , ( lr

~15!

G m0 n (k, v )’s

where the are unperturbed Green’s functions. Within SCBA which includes only ‘‘noncrossing’’ diagrams the self-energies S m n ~ k, v ! 5

(

ql r a

* a ~ k,q ! M m l a ~ k,q ! M nr

3G l r ~ k2q, v 2 v q a ! ,

n

~12!

Then the Hamiltonian Eq. ~10! has the following form in linear-spin-wave theory:3,16

† 1F m n a ~ k,q ! b 2q a # 1H.c.

1H.c.,

1J m*n ~ q ! b 2q n b q m # ,

functions

* an# , @ t m n ~ k2q ! u q am 1t m n ~ k ! v 2q

AN 1

~10!

M m*n a ~ 2k,2q ! 5M m n a ~ k,q ! ,

~6!

† ( h k†m h k2q n @ t m n~ k2q ! b q m 1t m n~ k ! b 2q n# 2 AN kq, m n

† ( ~ u q am b q a 1 v q am b 2q a!, a 51

† 1F m n a ~ k,q ! b 2q a # 1H.c.% ,

Using t n*m (k)5t m n (k)5t n m (2k),u q*am 5u 2q am , v q*am 5 v 2q am , we hold the following relations:

1

b qm5

( a(51 $ h k†m h k2q n @ M m n a~ k,q ! b q a

F m n a ~ k,q ! 5

where k i 5(k x ,k y ), k x 5(2 p /a)l 1 ,l 1 50,1, . . . ,L 1 21, k y 5(2 p /a)l 2 ,l 2 50,1, . . . ,L 2 21; k z 5(2 p /c)l 3 ,l 3 50,1, . . . ,L 3 21. a(c) is the x-y plane (z direction! lattice constant, c5(n21)c 1 1c 2 . N5L 1 L 2 L 3 is the number of the unit cell. Thus H t5

n

1 H t5 2 kq, m n

~5!

h r e 2i @ k i •r i 1k z ~ r m 1r m !# ,

2i @ q i •r i 1q z ~ r m 1r m !#

where b a† ( b a ) are the a th mode magnon operators with the spin-wave energy v q a . Using the relation J n*m (q)5J n m (2q), we have u q*am 5u 2q am , v q*am 5 v 2q am :

M m n a ~ k,q ! 5

We take the Fourier transformation

11 745

~16!

can be found self-consistently together with Eq. ~15!. Using Eq. ~12!, we have

WEI-GUO YIN AND CHANG-DE GONG

11 746

57

TABLE I. The hole-magnon coupling functions M m n a (k,q) defined in Eq. ~11! for three coupled t-J planes. t 8 is defined in Eq. ~18!.

AN

(mna)

4t

S m n ~ k, v ! 5S n m ~ 2k, v ! ,

4t

g k2q u q111 g k v q11 ( v q121u q11)•t 8 /4t g k2q u q121 g k v q12 g k2q u q311 g k v q31 (u q121 v q11)•t 8 /4t v q31•t 8 /4t

~111!,~331! ~121!,~321! ~221! ~113!,2(333) ~211!,~231! ~213!,2(233)

III. RESULTS

S D v q3 2J

G 13~ k, v ! 5

5 ~ 12 g q ! 2 1

a' @~ 61 a' ! 2

2 A~ 61 a' ! 2 232~ 12 g 2q !# ,

S D v q2 2J

2

5 ~ 12 g q ! 2 1

a' @~ 61 a' ! 2

1 A~ 61 a' ! 2 232~ 12 g 2q !# ,

F

1 1 @ G 0 ~ k, v ! 21 2S 22~ k, v !# 2 @ G 011~ k, v ! 21 2 D 22

G

G 22~ k, v ! 5

1 @ G 0 ~ k, v ! 21 2S 11~ k, v ! 2S 13~ k, v !# , D 11 1 •S ~ k, v ! , D 12

~20!

D5 @ G 011~ k, v ! 21 2S 11~ k, v ! 2S 13~ k, v !#@ G 022 ~ k, v ! 21 ~18!

The spin part of the Hamiltonian is diagonalized using a Bogoliubov-like transformation expressed in the Appendix. The three spin-wave modes split into one acoustic and two optical modes, 2

G

where

J 11~ q ! 5J 22~ q ! 5J 33~ q ! 54J g q ,

v q1 2J

F

1 1 @ G 0 ~ k, v ! 21 2S 22~ k, v !# 1 @ G 011~ k, v ! 21 2 D 22

G 12~ k, v ! 5

t 21~ k ! 5t 23~ k ! 5t'1 [t 8 ,

~19!

2S 11~ k, v ! 1S 13~ k, v !# 21 ,

t 11~ k ! 5t 22~ k ! 5t 33~ k ! 54t g k ,

J 21~ q ! 5J 23~ q ! 5J'1 [J 8 .

5 ~ 11 a' ! 2 2 g 2q ,

2S 11~ k, v ! 1S 13~ k, v !# 21 ,

A. Trilayer system

Using the propagator-matrix approach, we explicitly calculate the quasiparticle bands for slightly doped three coupled t-J planes, i.e., m51,n53,c52c 1 1c 2 ,t'2 5J'2 50. The open boundary condition in the z direction is used, thus k z 5q z [0 in our scheme. Interactions in Eq. ~8! are given by

2

where a' 5J 8 /4J. Note that all the transformation coefficients are real numbers. Because the two outer planes are equivalent, the following relations hold: S 11(k, v ) 5S 33(k, v ), S 12(k, v )5S 23(k, v ). Therefore, the independent Green’s functions are G 11~ k, v ! 5

In this section, we explicitly calculate the Green’s functions for the trilayer system which has drawn particular interest because of its highest T c 5133 K and for the 3D bilayer cuprates due to the most extensively studied copper oxide compounds Y123 and Bi-2212.

M m n a ~ k,q !

g k2q u q211 g k v q21 ( v q221u q21)•t 8 /4t g k2q u q221 g k u q22 u q31•t 8 /4t (u q221 v q21)•t 8 /4t 0

~112!,~332! ~122!,~322! ~222! ~123!,2~323! ~212!,~232! the others

G m n ~ k, v ! 5G n m ~ 2k, v ! . ~17!

We stress that the problem for pure n-coupled t-J planes ~i.e., m[1) corresponds to the case of t'2 5J'2 50. In this case, the periodic boundary condition in the z direction is not applicable and as a result we set k z 5q z [0 in the present scheme.

S D

AN

(mna)

M m n a ~ k,q !

2S 22~ k, v !# 22 S 12~ k, v ! 2 . 0 (k, v )5( v 1i h ) 21 ,for h →0 1 , and the other unperG mm turbed Green’s functions vanish. The coupling functions M m n a (k,q) are defined in Table I. Let us study the one-hole spectrum using realistic values of the parameters J50.4, J 8 /J5(t 8 /t) 2 50.16 suggested by band-structure calculations and some experiments ~we will refer to all quantities in units of t from now on!. The iteration steps are carried out on the 3316316 sites, and in the v mesh which was set 1000 points from 25 to 5. We chose an artificial broadening h 50.01 for the d functions. The quasiparticle dispersion relation E k1 ,E k2 have been determined through the relations D50 and E k3 5S 11(k,E k3 )2S 13(k,E k3 ), as shown in Fig. 2. The main features of this solution reproduce those in the ARPES data.24 Each band is a ‘‘flat band’’ around the wave vectors

57

QUASIPARTICLE BANDS AND SUPERCONDUCTIVITY . . .

11 747

TABLE II. The fit coefficients for the one-hole dispersions in Eq. ~22!. The parameters that we use are J50.4, J 8 /J5(t 8 /t) 2 50.16.

m

x m( 1 )

x m( 2 )

x m( 3 )

x8

1 2 3

22.067 22.106 22.062

0.0692 0.0715 0.0706

0.0480 0.0468 0.0462

0.0157 0.0157 0

the inner plane and the outer ones may take the form of « 8k 54x 8 g k and the excellent fits of the one-hole spectra on the whole momentum space correspond to 1 E k1 5 ~ « k1 1« k2 ! 1 A~ « k1 2« k2 ! 2 /412« k8 2 , 2 FIG. 2. Dispersion relations E k a of the quasiparticle bands along symmetry lines in the Brillouin Zone on a 3316316 sites: J 50.4, J 8 /J5(t 8 /t) 2 50.16. The best fits are for a 51 ~solid line!, 2 ~dashed line!, 3 ~dotted line!.

k5( p ,0) and k5(0,p ) which has a width of about 2J. The minimum energy is at „p /2,p /2…, and the maximum at (0,0) and ( p , p ), similar to those observed in a single AF layer. In Fig. 3, we show spectral functions A m n (k, v )5 21/p Im G m n (k, v ) at the high symmetry points. According to Eq. ~20!, it is appropriate to define the even function A 6 (k, v )5A 11(k, v )6A 13(k, v ). One sees that there are two slight shifted Lorentzian-like peaks at the low-energy side of A 1 (k, v ) and A 22(k, v ); the maximal distance between them is at k5(0,0) and ( p , p ). The positions of the two peaks in A 1 (k, v ) are the same as those in A 22(k, v ) while the spectral weights are different, which implies the existence of QP eigenoperators mixed with h q2 and h q1 5(h q1 1h q3 )/ A2. However, we observe a single peak in each spectrum along the AF Brillouin zone boundary (0,p )→( p ,0) line. These results indicate that the effective interlayer hopping between

FIG. 3. SCBA spectral functions A(k, v ) at high symmetry points for the same values of parameters used in Fig. 2. We define the even function A 6 (k, v )5A 11(k, v )6A 13(k, v ).

1 E k2 5 ~ « k1 1« k2 ! 2 A~ « k1 2« k2 ! 2 /412« 8k 2 , 2 E k3 5« k3 ,

~21!

where « k m 5x ~m1 ! 14x ~m2 ! j k 14x ~m3 ! g 2k ,

m 51,2,3;

~22!

here j k 5coskxcosky , and the fit parameters are defined in Table II. The main effective contribution to E k arises from the in-plane hole and the between-plane hopping processes to first- and second-nearest neighbors on the same sublattice, to avoid distorting the AF background.4 The effective interlayer hopping strength x 8 is one order of magnitude smaller than the bare one t 8 which supports the suggestion of a greatly reduced interlayer hopping originating from nontrivial many-body effects25 and is consistent with recent photoemission experiments data.27 In addition, the effective transverse matrix element, « k9 5(« k1 2« k3 )/2, directly between the two outer layers is much smaller and can be neglected. The distinguished energy shift between the E k2 and E k1 (E k3 ) causes the particular evolution of the Fermi surface ~FS! as the SCBA hole dispersion is populated in the rigidband picture. At very low hole density it starts with hole pockets around ( p /2,p /2) in the E k2 band at first. The pockets are longer along the (0,p )2( p ,0) direction than along the main diagonal in the Brillouin zone and enclose smaller pockets in E k1,3 band. As these pockets grow in size, the Fermi level in the E k2 band eventually hits the saddle points, while the hole pockets remain in the E k1,3 ones. When the Fermi level in the E k1,3 band reaches the flatness, the FS in the E k2 band changes its topology, becoming a large FS. The small effective interlayer hopping leads to a weak mixing of the bare holes in different layers. Therefore E k2 basically reflects the hole behavior in the inner plane and E k1 (E k3 ) reflects the symmetry ~antisymmetry! states energy in the two outer planes. So we conclude that the outer planes are slightly less doped than the inner ones. We stress that our results come from the simplest t-J model and will be changed by the introduction of other hole-hopping process and the modification of the spin couplings in the system.

WEI-GUO YIN AND CHANG-DE GONG

11 748

57

in our microscopic calculations, which agrees well with the phenomenological estimate.20 Figure 4 shows that the system does not have a single critical temperature, in particular, the inner layer has a higher T c than the outer ones in the underdoped and optimally doped regions. This controversy comes from our above rough evaluation. Since the outer planes are less doped than the inner ones, the spin-wave excitations are less heavily softened in the outer layers than those in the inner ones. Therefore, the effective NN contact attractive interaction }J eff , as we assumed in our above calculations, is more reduced in the inner layer which may lead to the same transition temperature in three planes. B. 3D bilayer

For a superlattice with more than one plane per unit cell, the periodic boundary condition in the z direction is applicable. We calculate the quasiparticle bands for slightly doped 3D bilayer t-J planes, i.e., n52,c5c 1 1c 2 . Interactions in Eq. ~8! are given by t 11~ k ! 5t 22~ k ! 54t g k , ik z c 1 t 21~ k ! 5t * 1t'2 e 2ik z c 2 5t 8k , 12~ k ! 5t'1 e

J 11~ q ! 5J 22~ q ! 54J g q , FIG. 4. The dependence of T c on the chemical potential for a trilayer. We chose the energy unit t50.4 eV. The solid line stands for the holes doped in the inner plane, and the dashed line for that in the outer ones.

Using the trilayer SCBA hole dispersion Eq. ~21! and the hole-hole NN attraction mediated by AF fluctuations, following DNM, we have studied superconductivity.6,16 we employ a value V50.6J for the in-plane NN interaction, while for the interaction in the z direction V' 50.6J 8 will be used. Solving numerically the gap equation on large clusters within the BCS formalism, we observed that the symmetry of the superconducting order parameter is d x 2 2y 2 for all the hole densities where superconductivity exists. Notice that the strict separation of the transition temperatures implies that T c for d x 2 2y 2 wave pairing is independent of V' . 28 This is in agreement with the recent NMR spectra which questions the pure interlayer spin-pairing picture.21 The dependence of T c on the chemical potential is shown in Fig. 4. The shape for the inner layer band is very similar to that for the outer layers because of nearly the same coefficients of x (2) ,x (3) in Eq. ~21!,7 while the particular evolution of the FS leads to the shift between them. The optimal doping occurs when the chemical potential reaches the flatness of each QP band with T max c 5135 K ~we chose the energy unit t50.4 eV!. The corresponding value of hole density in each band is about 0.35

iq z c 1 J 21~ q ! 5J * 1J'2 e 2iq z c 2 5J 8q . 12~ q ! 5J'1 e

The spin-wave spectrum v q a is given by23

S D v qa 2J

2

5 ~ 11 a' ! 2 2 ~ g q 6 a' u g' u ! 2 ,

AN 4t

~111!, ~221! * ~121!, ~211! *

M m n a ~ k,q !

g k2q u q111 g k v q11 8 u q111t 8k v q12)/4t (t k2q

~24!

where a' 5J 80 /4J, g' 5J 8q /J 80 . The eigenmode labels a 51 (2) corresponding to the sign 1 (2) in Eq. ~24! denote the acoustic ~optical! mode. The coupling functions M m n a (k,q) are given in Table III. Noting that M 111~ k1Q,q1Q ! 5M 112~ k,q ! , M 221~ k1Q,q1Q ! 52M 222~ k,q ! , M 121~ k1Q,q1Q ! 5M 122~ k,q ! , M 211~ k1Q,q1Q ! 52M 212~ k,q ! ,

~25!

where Q5( p , p ,0). In addition to the general relation S 12(2k, v )5S 21(k, v ), using Eq. ~15! and ~25! the following relations hold: S 11~ k1Q, v ! 5S 11~ k, v ! ,

S 12~ k1Q, v ! 52S 12~ k, v ! ,

S 11~ k, v ! 5S 22~ k, v ! ,

TABLE III. The hole-magnon coupling functions M m n a (k,q) defined in Eq. ~11! for a 3D bilayer antiferromaget. t 8k is defined in Eq. ~23!. (mna)

~23!

(mna)

AN 4t

~112!,2(222) * ~122!,2(212) *

M m n a ~ k,q !

g k2q u q211 g k v q21 8 u q211t 8k v q22)/4t (t k2q

~26!

QUASIPARTICLE BANDS AND SUPERCONDUCTIVITY . . .

57

FIG. 5. SCBA bonding band spectrum E k1 and antibonding band spectrum E k2 of one hole in an 3D bilayer AF along symmetry lines in the Brillouin Zone on a (83832)38 sites: J50.4, J'1 /J5(t'1 /t) 2 50.16, J'2 /J5(t'2 /t) 2 50.04. The best fits are for k z 50 ~solid line!, p /2 ~dashed line!, p ~dotted line!.

FIG. 6. SCBA spectral functions A 11(k, v ) at high symmetry points for the same values of parameters used in Fig. 5.

is consistent with the recent photoemission experiments data.26,27 The resulting excellent fits of the one-hole spectra on the whole momentum space correspond to « k1 522.05110.304j k 10.196g 2k

due to the equivalence of the two layers. Therefore, the independent Green’s functions are

F

G 11~ k, v !

G F

1 5 D G 12~ k, v !

G 011~ k, v ! 21 2S 11~ k, v ! S 12~ k, v !

G

,

~27!

where D5 @ G 011(k, v ) 21 2S 11(k, v ) # 2 2S 12(k, v )S 21(k, v ). 0 (k, v )5( v 1i h ) 21 , for h →0 1 , and the other unperG mm turbed Green’s functions vanish. Coming to the case of the 3D bilayer system, we also study the one-hole spectrum using the realistic values of parameters J50.4, J'1 /J5(t'1 /t) 2 50.16, J'2 /J5(t'2 /t) 2 50.04. The iteration steps are carried out on the (838 32)38 sites, and in the v mesh which was set 1000 points from 25 to 5. We chose an artificial broadening h 50.01 for the d functions. On very general grounds, one expects that the two CuO 2 layers in a unit cell should hybridize to produce a bonding ~1! and an antibonding band (2). These quasiparticle dispersion relations E k6 have been determined through the relations D50, i.e., E k6 5S 11~ k,E k1 ! 6 AS 12~ k,E k1 ! S 21~ k,E k1 !

~28!

as shown in Fig. 5. According to Eq. ~26!, E k1Q,1 5E k,2 . We would like to point out that we can observe little dispersion along k z , indicating a greatly reduced interlayer hopping originating from nontrivial many-body effects25 which

11 749

64 g k At 21 1t 22 12t 1 t 1 cosk z ,

~29!

where t 1 50.016 denotes the effective between-plane hopping processes to NN on the same sublattice inside the same unit cell, and t 2 50.01 , the intercell one. In Fig. 6, we show spectral functions A m n (k, v )5 21/p Im G m n (k, v ) at the high symmetry points. One sees that there are two slightly shifted Lorentzian-like peaks at the low-energy side of A 11(k, v ) corresponding to bonding and antibonding bands, respectively. The maximal distance between them is at (0,0,0) and ( p , p ,0). These maxima coalesce in each spectrum along the AF Brillouin zone boundary (0,p )→( p ,0) line in agreement with Eqs. ~26! and ~29!. If we consider that the spectral weights at (0,0,0) and ( p , p ,0) are much smaller than those around the bottom of the quasiparticle as shown in Table IV, the experimental measurements cannot resolve the dispersions along the k z TABLE IV. Z(k)5 @ 12( ] / ]v )S(k, v ) u v 5E k # 21 of the QP for a 3D bilayer antiferromagnet. kz 0 p /2 p

Z(0,0,k z )

Z( p , p ,k z )

Z( p /2,p /2,k z )

Z( p ,0,k z )

0.106 0.122 0.146

0.217 0.201 0.179

0.364 0.349 0.336

0.371 0.362 0.348

11 750

WEI-GUO YIN AND CHANG-DE GONG

direction. The overall shape of the spectral functions is also similar to that for a pure bilayer.16 Thus the predictions obtained from a pure bilayer remain nearly unchanged when the intercell couplings are taken into account. Our explicit calculations support the suggestion that it is at least qualitatively enough to focus on the quasiparticle behavior in a pure bilayer for most researchers interests. The main functions of the intercell couplings are to develop three-dimensional ordering, including the AF Ne´el order and the superconductivity state when the already well-correlated bilayers couple together along the c axis even at nonzero temperature.

57

~A2!

A5OB,

so that we can diagonalize the considered Hamiltonian as ~A3!

H5B † DB,

where D is a diagonal matrix. Let us introduce the normalization matrices N by N5 @ A,A † # 5 @ B,B † # ,

IV. SUMMARY

To understand the number of layers dependent on the quasiparticle bands in high T c cuprates, we developed a propagator-matrix approach on the basis of the selfconsistent Born approximation to the effective hole-magnon Hamiltonian to calculate the quasiparticle spectrum for a slightly doped pure multiple-layered t-J model and the 3D superlattice system with n planes per unit cell. Then the antiferromagnetic van Hove model is used to obtain T c . The explicit results for the trilayer system and 3D bilayer show that the main effective contribution to the quasiparticle dispersion relations E k arises from hole hopping processes on the same sublattice, to avoid distorting the antiferromagnetic background. Despite the rather different lattice structures ~also with apparently different boundary conditions! taken into account, strong many-body effects in the t-J model cause a flat band around k i 5( p ,0) in each band and give rise to a great reduction of band splitting. Our results suggest that a large density of states leading to the boost of T c results from the antiferromagnetic correlations rather than from the band effects. For the realistic parameter, it is found that for a trilayer the outer planes is slightly less doped than the inner one. The calculations for a 3D bilayer superlattice explicitly prove that there is little dispersion along k z , supporting the suggestion that the physics of cuprates focus on the already well-coupled bilayer and the intercell coupling can be neglected for most researchers interests.

~A4!

with the relation N2 5I but NÞI for the boson operators. I is the identity matrix. The equation of motion in the representation of $ B n % is given by @ B,H # 5NDB5 @ O21 A,H # 5O21 NUOB. Hence the following relation holds: ND5O21 KO,

~A5!

where K5NU. Thus the Bogoliubov-type transformation matrix O diagonalizes the motion matrix K. In addition, O should satisfy certain normalization conditions, i.e., the following relation holds: ~A6!

N5ONO† .

The eigenenergy matrix E with elements E aa 8 5 v a d aa 8 can be expressed as E5DN2 5D.

~A7!

For the case of an AF Heisenberg model, we usually choose

ACKNOWLEDGMENTS

We are grateful to Y. J. Wang, M. R. Li, and J. X. Li for their helpful discussions. This research was supported by the National Natural Science Foundation of China and the Doctoral Training Program Foundation of the Chinese State Education Commission. APPENDIX:

BOGOLIUBOV-TYPE TRANSFORMATION FOR AF HEISENBERG MODEL

In linear spin-wave theory, we need a general procedure of a Bogoliubov-type transformation to diagonalize the quadratic Hamiltonian H consisting of a set of boson operators $ A n % , where A n is either a creation or annihilation operator one. Formally this model is that †

H5A UA,

~A1!

in matrix notation. Our aim is to find a Bogoliubov-type transformation,

A 2 ~ q ! 5A †1 ~ 2q ! , . . . ,A n ~ q ! 5A †n21 ~ 2q ! ,

~A8!

then an additional relation for further simplification holds: O5O1 5E1 OE1 ,

~A9!

where N5

F

1 21 

G F G ,

1 21

E1 5

0

1

1

0



0

1

1

0

.

57

QUASIPARTICLE BANDS AND SUPERCONDUCTIVITY . . .

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