PHYSICAL REVIE%" B VOLUME 48, NUMBER 22 Model for c-axis transport in high-T, cuprates 1 DECEMBER 1993-II A. G. Rojo' and K. Levin James Franck Institute, The University of Chicago, 66/0 South Ellis Avenue, Chicago, Illinois 60687 (Received 30 August 1993) We present a model for the c-axis resistivity p in the cuprates which incorporates interplanar disorder. Higher-order perturbative calculations demonstrate that this special disorder stabilizes a low Tme-tallic state and in the dynamical limit leads to temperature-dependent slopes dp /dT which are negative (positive) for low (high) hole concentrations. Predictions are presented for correlations which associate a nonlinear planar resistivity with the magnitude of negative dp /dT. Understanding the nature of c-axis transport in the copper oxides has fundamental consequences for theo- ries of the normal as well as superconducting state. It has been argued that the "semiconductinglike" tempera- ture dependences which are &equently observed indicate a failure of Fermi liquid theory: localization in the c di- rection alone is inconsistent with the scaling theory of disordered systems. Thus it is claimed that there can be no two-dimensional metallic (i.e. , Fermi liquid) state. The situation is made even more complex by the presence of strong Coulomb efFects (which underlie the Mott insu- lating phase) and by the particular nature of the disorder which is predominantly interplanar or "ofI'-diagonal. " In this paper we examine these issues microscopically within the context of a theory, which treats incoherent c-axis conduction as in a highly disordered but never- theless metallic state. Here the word "incoherent" des- ignates a situation in which there is finite conductivity although the relevant Bloch waves are not defined over several lattice constants. Static as well as dynamic in- I terplanar disorder is included and systematic expansions in the (static) scattering matrix element are shown to be fully consistent with a metallic ground state. In this way there is no contradiction with the general concepts of lo- calization theory, which have been raised as objections to a more conventional treatment of the cuprates. However, just as in highly disordered metals a negative resistivity slope dp/dT can arise at sufficiently high temperatures T from phonon or other boson assisted hopping processes. A second and important goal of this paper is to make direct contact with existing experimental trends as well as present experimental tests of our picture. An interest- ing consequence of our model is the correlation between nonlinearity in p g and the magnitude of the increase with decreasing T of p, . We show here that these efFects become more pronounced with decreasing carrier concen- tration. Elsewhere this deviation &om linearity has been associated with "spin gap" effects. We consider the following model of an anisotropic three-dimensional disordered system. H = ) W„n„+ t~~ ) (c~t c„+ + H. c. ) + ((t~ + V„)ct c„+i + H. c. ) + Hab X)fA where the ct and c's refer to creation and annihilation operators, m is a plane index, w is a two-dimensional co- ordinate within each plane, and R and V are random variables which introduce "diagonal" and "ofF-diagonal" disorder, respectively. In (1), H b contains in-plane cor- relation effects, that in particular give rise to a linear inverse lifetime: x & — — ImZ g — — 2vrAkT, where A 0. 2— 0.4, as determined &om ac conductivity measurements. This term gives rise to the linear (in temperature) ab resistivity in the case of decoupled planes. We will be interested in the regime of high anisotropy in which the ratio of the interplane to intraplane hopping matrix elements, t~/t~~ && 1. We treat Coulomb efFects only insofar as they are incorporated into renormaliza- tions of tI~ and t~ . These renormalizations restrict the hopping so that the closer the system is to the insulating state the smaller are both tlt and t~ . Our model contains two limiting cases which have been discussed in the literature. In the limit V = 0, the c-axis conductivity yields p g oc p for both a Bloch wave de- scription (t~ ) T) and for the Giaver tunneling or per- turbative limit (t~ & T). The opposite case in which 7. b ~ oo, t~ = 0, and V g 0 has been discussed in Ref. 6 where it is shown within a Boltzmann-Born approxima- tion that static disorder leads to a lifetime in the ab plane whereas it provides the hopping mechanism for transport in the c direction. Thus p g oc p, . A combination of these two cases, such as we advocate here, is discussed in Ref. 7, but the full diagrammatic resummation (and therefore a demonstration of self consistency) is not ad- dressed, nor are the detailed experimental consequences explored. We consider separately the two regimes of static and dynamic ofI' diagonal disorder. In the latter case, V in (1) is replaced by a dynamical bosonic variable V2(b + bt ) which may be viewed as phonon or spin fluctua- tion assisted hopping. In addition the Hamiltonian in- cludes a term that reflects the decoupled dynamics of these bosonic variables. In the static case we consider uncorrelated disorder such that (V ) = 0, and analo- gously for diagonal disorder Wx ~. We argue that the static and dynamic limits are qualitatively similar, with 0163-1829/93/48{22)/16861(4)/$06. 00 48 16 861 1993 The American Physical Society