It is generally agreed that the key physics of the cuprate superconductors lies in a single CuO2 plane. The electronic system in these copper oxide layers are strongly correlated. Theoretical studies over the years have shown that, indeed such a two dimensional strongly interacting
electron system can give rise to many of the unusual properties observed in high-Tc super- conductors. Thus a number of theories proposed for the high-Tc superconductors have con- centrated mainly on the isolated copper oxide layers and the physics that emerges from the presence of more than one CuO2layer have received limited attention.
Nevertheless, interlayer coupling between different CuO2 layers in the cuprate supercon- ductors influences many of its physical properties (see Ref. [55] for a review). The crystal structure of these materials consists well separated blocks of n closely spaced CuO2 layers, where n can be 1, 2, 3 or 4 (see §1.1.1). The separation between the layers in a block or unit cell in multilayered compounds is very small (∼ 3.5 Å), whereas the blocks itself are sep- arated by large distances (∼ 6-8 Å). Therefore the coupling between different copper oxide layers within a unit cell (intracell) is generally distinguished from that between two adjacent blocks of the closely spaced CuO2layers (intercell coupling). The nature of intercell coupling in the cuprate superconductors and its role in the observed large c-axis anisotropies in various physical properties has been intriguing. For example, electronic band structure calculations predict that c-axis charge transport in the normal state should be metallic due to the finite hopping integral between layers[80]. However experimental evidences suggest that single particle hopping of electrons across the CuO2layers is highly incoherent. Several mechanism have been proposed to explain this phenomenon, such as, c-axis charge confinement due to a non-Fermi liquid ground state in the layers[146, 147, 148], renormalization of the interlayer hopping by inplane scattering[149] etc.
As regard to the CuO2layers in a single unit cell of multilayered cuprates, it is reasonable to expect coherent single particle motion across the layers because of their proximity. In fact, band structure calculations on bilayered superconductors, such as Bi2Sr2CaCu2O8+x(Bi2212), predict a splitting of the main electronic band crossing the Fermi level into a bondingε+(k) and an antibondingε−(k) band, with k representing a two dimensional wavevector in the first Brillouin zone for a single CuO2layer[150]. Such a splitting would originate from a coherent hopping electrons between the two CuO2layers in a unit cell. Further, calculations show that the splitting is k dependent and can be quantified as,
t⊥(k)= t⊥ 4
hcos(kxa)−cos(kya)i2
(1.9) with t⊥ being ∼ 0.1− 0.15 eV. Thus the splitting is maximum at (0, π) or (π,0) points and zero along the (π, π) direction. However, early ARPES study on Bi2212 reported absence of
any such bilayer band splitting[151]. But later ARPES studies with improved resolution on overdoped Bi2212 have clearly shown the presence of the band splitting as suggested by band theory[152,153]. More recently, finite bilayer band splittings in optimally doped Bi2212 and YBCO have been observed even in the nodal region[154,155,156].
The implications of these interlayer couplings on the physical properties of the cuprate superconductors are profound. The famous peak-dip-hump structure observed in the ARPES spectra of bilayer superconductors near the (π,0) point of the Fermi surface[76, 77, 157] is shown to be a consequence of the bilayer splitting (see [158] and references therein). In the magnetic excitation spectra away from half-filling, a commensurate (π, π) resonance peak in inelastic neutron scattering data is observed only in bilayer materials, such as, YBCO[159, 160] and Bi2212[161], whereas the the feature is absent in monolayer LSCO[162]. Fur- ther a gap in the spin spectrum observed in bilayered YBCO constitutes another distinguish- ing feature when compared with the monolayer compounds[163]. In the superconducting state, tunneling of Cooper pairs across the layers is believed to play a crucial role in high-Tc superconductors[26]. It has been proposed that interlayer pair tunneling provides the large condensation energy in high-Tccuprates leading to the high values of superconducting transi- tion temperature. In fact, the transition temperature, Tc of multilayered compounds are much higher than that of their single layered counterparts with the Tc of different members in ho- mologous series of compounds increasing linearly with the number, n of CuO2layers per unit cell (for n ≤ 3) (see§1.1.1). However, reliable theoretical work on multilayered systems are very few and hence the role played by the interplanar couplings in determining the physi- cal properties, especially in the superconducting state is largely left unexplored. This is the motivation for us to proceed with the current project on coupled bilayered systems.
Bilayer t-J model and variational Monte Carlo method
2.1 Introduction
In this chapter, we introduce the model for bilayer superconductors used in this study. Previ- ous results for this model in literature are discussed briefly. Next we describe the variational Monte Carlo method which we use for studying the model. The formalism and the algorithm for the method are described in details. For verification of our codes, we show comparison of our results with that of the published works.
The second part of the chapter (§2.4) briefly deals with a problem which is somewhat different from the main focus of the thesis. Here we have studied the effects of nonmagnetic impurities on the magnetic properties of the Hubbard model in a two dimensional lattice and a two-leg ladder using the Quantum Monte Carlo method. The results are interesting and agree with experiments on impurity doping in the cuprates.