Stripe Liquid, Crystal, and Glass Phases of Doped Antiferromagnets

Abstract

A largely descriptive survey is given of the ordered phases of doped antiferromagnets, and of the long wavelength properties that can be derived from an order-parameter theory. In particular, we show that the competition between the long-range Coulomb repulsion and the strong short-distance tendency of doped holes to coalesce into regions of supressed antiferromagnetism leads to a variety of self-organized charge structures on intermediate length scales, of which "stripes" are the most common, both theoretically and experimentally. These structures lead to a rich assortment of novel electronic phases and crossover phenomena, as indicated in the title. We use the high temperature superconductors as the experimentally best-studied examples of doped antiferromagnets. Highly correlated materials have intermediate electron densities and are frequently doped Mott insulators, so that neither the kinetic energy nor the potential energy is totally dominant, and both must be treated on equal footing. The question arises, are there actual "interme-diate" low temperature phases of matter which interpolate between the high density "gas" phase (usually called a Fermi liquid) and the low density strongly insulating Wigner crystal phase? We have shown that, at least in the case of lightly-doped antiferromagnets, the tendency of the antiferromagnet to expel holes always 1-4 leads to phase separation which, when frustrated by the long-range piece of the Coulomb interaction, leads 5-7 to the formation of states which are inhomogeneous on intermediate length scales and (possibly) time scales. The most common self-organized structures which result from these competing interactions 7,8 are "stripes", by which, we generally mean d − 1 dimensional antiphase domain walls across which the antiferromagnetic order changes sign, and along which the doped holes are concentrated; 9 the term "stripe" is, of course, a reference to the important two-dimensional case relevant to the high temperature superconductors. Even if the discussion is confined to ordered phases of doped antiferromagnets we are left with an exceedingly complex problem. Many varieties of order have been observed in doped antiferromagnets, including spin and charge order and, of course, superconductivity. The spin and charge order can be commensurate or incom-mensurate, and both can be ideal or glassy. There are also various structural phases, such as the tetragonal and orthorhombic phases of La 1.86 Sr 0.14 CuO 4 , which may reflect important changes in the electronic state, as the structural order can couple to various forms of "electronic liquid crystalline" order. 10 Of course all of these types of order can compete or coexist in various ways. I. LANDAU THEORY OF COUPLED CDW AND SDW ORDER We begin by discussing density-wave order, and in particular the interplay between spin-density wave (SDW) and charge-density wave (CDW) order. This can be analyzed most simply by studying the Landau theory of coupled order parameters. 11 While it is possible to have various sorts of "spiral" spin phases, the only spin order in much of parameter space is collinear, so the discussion will be confined to this region. The resulting phase diagram is shown schematically in Fig. 1. Three features of the analysis, which is discussed in detail in Ref. 11, bear repeating: 1) In order for the SDW order parameter S q and the CDW order parameter ρ − Q to couple in the lowest possible order (third), it is necessary that the ordering vectors satisfy the relation Q = 2 q, or in other words, the wavelength of the SDW is twice that of the CDW; this gives precise meaning of the concept 5 of "topological doping" and implies that the charge is effectively concentrated along antiphase domain walls in the magnetic order. 2) It is possible to have a phase with CDW order, but no SDW order, whereas SDW order always implies CDW order. This is important to bear in mind when thinking about the experimentally determined phase diagram of the high temperature su-perconductors or any other doped antiferromagnet, since there are many good probes (such as NMR, µSR, and neutron scattering) that are sensitive to spin order or fluctuations, but fewer that are sensitive to charge order. Where incommensurate spin order is detected, we can directly infer the existence of charge order, but where no magnetic order is observed, there may or may not exist as yet undetected charge order. 3) Although Landau theory by its very character is relatively insensitive to the microscopic considerations conventionally referred to as the "mechanism" of ordering, an important classification of mechanisms follows directly from these considerations. If, upon lowering temperature, CDW order is encountered first and SDW order is either entirely absent or only appears at lower temperatures when the CDW order is already well developed, the density wave transition is "charge-driven", and we can infer that the SDW order is in some sense parasitic, i.e. driven by the interaction with the CDW. On the other hand, if both CDW and SDW order develop simultaneously, but with the CDW order turning on more slowly at the transition accord-1