Strongly Correlated Electronic Materials

The Simplest model of a periodic solid is a periodic array of valence orbitals embedded in a matrix of atomic cores

If correlations between different orbitals may be neglected, solving the problem of one of the orbitals is often equivalent to solving the whole system

These orbitals hybridize to form a valence band

If we may ignore nonlocal Coulombic correlations, these states are filled by electrons starting from the bottom of the band. This simple one-electron picture ignores both intersite and even some intrasite correlations between the electrons. Nevertheless, this approach has been very successful at describing many of the properties of periodic solids. However, electronic correlations are responsible for some of the most fascinating properties of materials.

Superconductivity: For example, in conventional superconductors, such as lead, the phonons mediate an attraction between the electrons. As illustrated below, first one electron moves quickly through the lattice, causing the positively charged lattice ions to distort towards its path. Due the the different time scales of the lattice and the electrons, once the lattice is fully distorted, the first electron is far away (more than 1000 Angstroms). Thus a second electron can be attracted to the region of positive charge along the path of the first electron, without feeling its Coulomb repulsion. At sufficiently low temperatures this attractive interaction causes the electrons to simultaneously pair into bosons and condense into a superconducting state.

Magnetism: Electronic correlations are also responsible for both the formation of magnetic moments and their ordering in magnetic materials. To understand the former, note that higher angular momentum states are precluded from sampling the region with the highest ionic potential by the angular momentum barrier. As a result, higher angular momentum states are more strongly screened, and are therefore promoted in energy as shown in the potential energy sketch below. In elemental Ce, for example, the 4f, 5d and 6s states all participate in the valence shell. However, since the principle quantum number n determines the size of the orbital, the 4f orbitals are far smaller, and hence more correlated than the 5d or 6s orbitals. These correlations move the doubly occupied (singlet) 4f state to higher energy. Thus, 4f electrons tend to not be paired into singlets and so form a magnetic moment.

Adjacent moments can become antiferromagnetically correlated through superexchange. If two orbitals have a hybridization overlap t, then moments on adjacent sites tend to have opposite spin so that they can gain hybridization energy.

High Temperature Superconductivity: The cuprate high temperature superconductors are basically doped antiferromagnets, or nearly antiferromagnetic metals. We can gain some idea of how holes in an antiferromagnet pair by considering a classical spin system pictured below. Holes on adjacent sites cost less exchange energy than do separated holes, since they break fewer antiferromagnetic bonds. This mediates a pairing potential between the holes.

Heavy Fermion Behavior: Electronic correlations are also responsible for Heavy Fermion behavior in which certain materials act as conventional metals with a strongly enhanced electronic mass (which can exceed 1000 electron masses). This behavior is observed in materials with a periodic array of moments embedded in a metallic matrix, and results from the competition between the antiferromagnetic exchange energy which tends to localize the conduction electrons into singlets and the kinetic energy of these states at the Fermi surface

Correlation and Disorder in Organic Nanostructures: Electronic correlations and disorder become more important in lower dimensional systems, including the nearly two dimensional cuprates and one-dimensional nanostructures (see this paper for an introduction). They strongly effect charge transport in organic nanostrucutres, including DNA, molecules, and carbon Nanotubes. This is rapidly emerging and wide open field. In DNA, the most puzzling discovery is that electron-transfer (ET) occurs over very large distances, despite the fact that DNA is an insulator with a very-large HOMO-LUMO gap. Many conflicting theories have been proposed to explain this. One possibility is that polarons form due to the weak hybridization between base pairs and strong coupling to acoustic phonon modes. Nanotubes are fascinating because the reduced dimensionality enhances the effects of both disorder and correlations. In one dimension, disorder always leads to localization and correlations to Luttinger liquid behavior; however, their combined effect is largely unknown. The numerical tools we have developed are ideal to study models of both DNA and nanotubes. We recently obtained funding from the NSF (ITR/NANO) and from the DOE (NSET through ORNL) to study the effect of disorder and correlations in these lower dimensional systems.

Methodology: One feature that all of these systems share is that they cannot be fully explained by traditional techniques based upon perturbation theory. In most cases, perturbation theory does not converge since there is no small parameter (i.e. the bandwidth W is roughly the same as the screened electron-electron correlation energy U). However, in lower dimensional systems such as nanotubes, perturbative approaches fail whenever U is finite (even if U << W).

Numerical simulations are used when perturbative approaches fail. Here, we typically (but not always) abandon perturbation theory, and attempt to solve the problem by enumerating the states of the system. However, this may be difficult since the number of systems states grows exponentially with the system size. Consider the simple spin-½ system shown, where each spin can be either up or down. The number of system states grows like exp[N ln(2)], where N is the number of spins in the system. For a reasonably large number of spins N, it is impossible to enumerate all of these states. Some simplification has to occur.

In Quantum Monte Carlo (QMC) techniques, the simplification is accomplished by only generating spin configurations that have a large contribution to the Boltzman weight. This is called importance sampling. (For an introduction to QMC methods, see my computational physics links.) In numerical renormalization group techniques, the simplification is accomplished by integrating out less important (usually high energy) states. In our recently introduced Dynamical Cluster Approximation, we accomplish this simplification by parsing the problem into long and short length scales. Correlations and fluctuations on short length scales are treated explicitly, while those only longer length scales are treated at the mean field level. These longer length scales may be integrated out thereby mapping the problem onto a self-consistently embedded cluster. We have developed a variety of numerical methods needed to solve this cluster problem. For example, we can reduce the complexity even further by simulating the cluster problem with QMC methods.

We have used the DCA formalism to study models of the high Tc cuprate superconductors and heavy Fermion systems (for a very general discussion of some of this research, see this link). However, this formalism is ideal for the study of lower dimensional nanoscale systems, since it allows us to treat both the correlations within the molecule as well as the coupling to the environment (the substrate). Furthermore, when combined with conventional Density Functional Theory techniques, the DCA will allow us to study correlation effects with ab initio calculations in realistic (not model) systems.