Size, Dimensionality, and Strong Electron Correlation in Nanoscience

In electronic structure theory, electron-electron repulsion is normally considered only in an average (or mean field) sense, for example, in a single Hartree-Fock determinant. This is the simple molecular orbital model, which is often a good approximation for molecules. In infinite systems, this averaging treatment leads to delocalized electronic bands, an excellent description of bulk 3D sp(3) semiconductors. However, in reality electrons try to instantaneously avoid each other; their relative motion is correlated. Strong electron-electron repulsion and correlation create new collective states and cause new femtosecond kinetic processes. This is especially true in 1D and 2D systems. The quantum size effect, a single electron property, is widely known: the band gap increases with decreasing size. This Account focuses on the experimental consequences of strong correlation. We first describe π-π* excited states in carbon nanotubes (CNTs). To obtain the spectra of individual CNTs, we developed a white-light, right-angle resonant Rayleigh scattering method. Discrete exciton transitions dominate the optical absorption spectra of both semiconducting and metallic tubes. Excitons are bound neutral excited states in which the electron and hole tightly orbit each other due to their mutual Coulomb attraction. We then describe more generally the independent roles of size and dimensionality in nanoelectronic structure, using additional examples from graphene, trans-polyacetylene chains, transition metal dichalcogenides, organic/inorganic Pb iodide perovskites, quantum dots, and pentacene van der Waals crystals. In 1D and 2D chemical systems, the electronic band structure diagram can be a poor predictor of properties if explicit correlation is not considered. One- and two-dimensional systems show quantum confinement and especially strong correlation as compared with their 3D parent systems. The Coulomb interaction is enhanced because the electrons are on the surface. One- and two-dimensional systems can exhibit essentially molecular properties even though they are infinite in size. Zero-dimensional Qdots show quantum confinement and modest electron correlation. Correlation is weak in 3D bulk semiconductors. Strongly correlated electronic states can behave as if they have fractional charge and effectively separate the spin and charge of the electron. This is apparent in the "soliton" state of polyacetylene, the fractional charge quantum Hall state of graphene, and the Luttinger electrical conductivity of metallic CNTs.