Diffusion of acceptor dopants in monoclinic $\beta$-Ga$_2$O$_3$

$\beta$-Ga$_2$O$_3$ is a leading ultra-wide band gap semiconductor, but its performance depends on precise control over dopant incorporation and stability. In this work, we use first-principles calculations to systematically assess the diffusion behavior of eight potential deep-level substitutional acceptors (Au, Ca, Co, Cu, Fe, Mg, Mn, and Ni) in $\beta$-Ga$_2$O$_3$. We consider two key diffusion mechanisms: (i) interstitial diffusion under non-equilibrium conditions relevant to ion implantation, and (ii) trap-limited diffusion (TLD) under near-equilibrium thermal annealing conditions. Our results reveal a strong diffusion anisotropy along the b and c axes, with dopant behavior governed by competition between diffusion and incorporation (or dissociation) activation energies. Under interstitial diffusion, Ca$^{2+}_{\text{i}}$ and Mg$^{2+}_{\text{i}}$ show the most favorable combination of low migration and incorporation barriers, making them promising candidates for efficient doping along the b and c axes, respectively. In contrast, Au$^{+}_{\text{i}}$ diffuses readily, but exhibits an incorporation barrier that exceeds 5 eV, rendering it ineffective as a dopant. From a thermal stability perspective, Co$^{2+}_{\text{i}}$ shows poor activation but high diffusion barriers, which may suppress undesirable migration at elevated temperatures. Under trap-limited diffusion, the dissociation of dopant-host complexes controls mobility. Mg$^{2+}_{\text{i}}$ again emerges as a leading candidate, exhibiting the lowest dissociation barriers along both axes, whereas Co$^{2+}_{\text{i}}$ and Fe$^{2+}_{\text{i}}$ display the highest barriers, suggesting improved dopant retention under thermal stress. Our findings guide dopant selection by balancing activation and thermal stability, essential for robust semi-insulating substrates.