Designing low-cost TaC virtual substrates for $Al_xGa_{1-x}N$ epitaxy
$Al_xGa_{1-x}N$ is a critical ultra-wide bandgap material for optoelectronics, but the deposition of thick, high quality epitaxial layers has been hindered by a lack of lattice-matched substrates. Here we identify the (111) face of transition metal carbides as a suitable class of materials for substrates lattice matched to (0001) $Al_xGa_{1-x}N$ and demonstrate the growth of thin film TaC which has an effective hexagonal lattice constant matched to $Al_{0.45}Ga_{0.55}N$. We explore growth conditions for sputtered TaC on sapphire substrates and investigate the effects of sputter power, layer thickness and incident plasma angle on film structure and in- and out-of-plane strain. We then show critical improvements to film quality by annealing films in a face-to-face configuration at 1600 $^\circ$C, which significantly reduces full width at half max (FWHM) of in- and out-of-plane diffraction peaks and results in a step-and-terrace surface morphology. This work presents a path toward electrically conductive, lattice matched, thermally compatible substrates for $Al_xGa_{1-x}N$ heteroepitaxy, a critical step for vertical devices and other power electronics applications.