  The Hubbard model is one of the primary models for understanding the
essential many-body physics in condensed matter systems such as Mott insulators
and cuprate high-Tc superconductors. Recent advances in atomically precise
fabrication in silicon using scanning tunneling microscopy (STM) have made
possible atom-by-atom fabrication of single and few-dopant quantum dots and
atomic-scale control of tunneling in dopant-based devices. However, the complex
fabrication requirements of multi-component devices have meant that emulating
two-dimensional (2D) Fermi-Hubbard physics using these systems has not been
demonstrated. Here, we overcome these challenges by integrating the latest
developments in atomic fabrication and demonstrate the analog quantum
simulation of a 2D extended Fermi-Hubbard Hamiltonian using STM-fabricated 3x3
arrays of single/few-dopant quantum dots. We demonstrate low-temperature
quantum transport and tuning of the electron ensemble using in-plane gates as
efficient probes to characterize the many-body properties, such as charge
addition, tunnel coupling, and the impact of disorder within the array. By
controlling the array lattice constants with sub-nm precision, we demonstrate
tuning of the hopping amplitude and long-range interactions and observe the
finite-size analogue of a transition from Mott insulating to metallic behavior
in the array. By increasing the measurement temperature, we simulate the effect
of thermally activated hopping and Hubbard band formation in transport
spectroscopy. We compare the analog quantum simulations with numerically
simulated results to help understand the energy spectrum and resonant tunneling
within the array. The results demonstrated in this study serve as a launching
point for a new class of engineered artificial lattices to simulate the
extended Fermi-Hubbard model of strongly correlated materials.
