Machine Learning Assisted Modeling of Amorphous TiO$_2$-Doped GeO$_2$ for Advanced LIGO Mirror Coatings

The mechanical loss angle of amorphous TiO$_2$-doped GeO$_2$ can be lower than 10$^{-4}$, making it a candidate for Laser Interferometer Gravitational-wave Observatory (LIGO) mirror coatings. Amorphous oxides have complex atomic structures that are influenced by various factors, including doping concentration, preparation, and thermal history, resulting in different mass densities and physical properties. Modeling at atomistic level enables capturing these effects by generating atomic structure models according to experimental conditions. In order to obtain reliable and physical amorphous models at an affordable cost, we develop classical and machine-learning potentials (MLP) to speed up simulations. First-principles calculations are used to train and validate MLP as well as validating structure models. To better reproduce properties such as elastic modulus, radial distribution function (RDF) and the variations in mass density of doped amorphous oxides, density functional theory (DFT) calculations are used to optimize the final models. We find that the mass densities of amorphous systems are correlated with the total void volume. The experimental mass density matches the models with the most symmetric potential energy wells under volume change. The elastic response of the metal-oxygen network is also studied. The 27\% TiO$_2$ doped GeO$_2$ system shows the least number of large atom-atom distance changes, while for 44\% TiO$_2$ doped GeO$_2$, a majority of Ti-O distances are significantly changed. In response to strains, the metal-oxygen network at low mass densities prefers to adjust bond angles, while at high mass densities, the adjustment is mainly done by changing atom-atom distance.