Probing a Molecular Hubbard Cluster System with Non-Contact Atomic Force Microscopy

Electronic correlations drive many important phenomena in materials that cannot be described by a single-electron picture. The Hubbard model captures a remarkable range of this behaviour by introducing an on-site interaction – the “Hubbard U” – that competes with the kinetic energy of the electrons, or delocalization through wavefunction overlap – described by the hopping parameter t. While Hubbard models are not generically analytically or even exactly solvable, model systems such as clusters often are and can lend insights into more complex extended structures, while “quantum simulators” – controllable quantum ensembles with features of these models – may provide solutions for larger systems and a mapping of parameter space. The success of such models in describing correlated electron phenomena like metal-insulator transitions, charge density waves, magnetically ordered states, and superconductivity¹, is typically assessed by comparing to macroscopic quantities. However, as a model that captures the local character of electron-electron interactions, local tools provide a unique perspective. Noncontact atomic force microscopy (ncAFM) offers a particularly intriguing view through the ability to probe charge states and electrostatic interactions locally.
We used clusters of PTCDA molecules adsorbed on NaCl bilayer films on silver as an experimental prototype of a 4-site Hubbard model in a regime where U>>t. PTCDA clusters have weak in-plane hybridization (t small), and isolated molecules have a Hubbard U in gas phase of ~3eV, reduced to ~1.4eV by screening of the nearby silver². The large electron affinity of PTCDA leads to electron transfer from the silver substrate, so that isolated molecules carry a charge of -1. Thus, these clusters represent a Hubbard model system where localization is expected, connected to a reservoir of charge. We used STM, STS, ncAFM and electrostatic force spectroscopy to probe the structures, charge states^3,4, and charging energies^5,6 of two different geometries of 4-molecule clusters as well as isolated PTCDA^- molecules. Equilibrium charge distributions of asymmetric clusters show charge segregation, while symmetric clusters show uniform charge. A 4-site extended Hubbard model was mapped onto the experimental results to identify the necessary interactions required to describe the ground state and charge excitations of the system corresponding to the jumps seen in Δf(V) spectra. Once above a threshold where U/t drives localization, intersite energy differences and repulsion compete with hopping, driving the observed charge segregation and unexpected correlated behaviour. Such molecular model cluster systems, probed by scanning probe microscopy, can provide insight into correlated behaviour and verification data for extended Hubbard models.

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