$\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$ is a promising material for next-generation power electronics because of its ultrawide band gap and high critical breakdown voltage. However, realizing its full potential requires 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 $\beta \text{−}{\mathrm{Ga}}_2{\mathrm{O}}_3$. We consider two key diffusion mechanisms: (i) interstitial diffusion under nonequilibrium 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, ${\mathrm{Ca}}_{\text{i}}^{2+}$ and ${\mathrm{Mg}}_{\text{i}}^{2+}$ 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, ${\mathrm{Au}}_{\text{i}}^{+}$ diffuses readily, but exhibits an incorporation barrier that exceeds 5 eV, rendering it ineffective as a dopant. From a thermal stability perspective, ${\mathrm{Co}}_{\text{i}}^{2+}$ 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. ${\mathrm{Mg}}_{\text{i}}^{2+}$ again emerges as a leading candidate, exhibiting the lowest dissociation barriers along both axes, whereas ${\mathrm{Co}}_{\text{i}}^{2+}$ and ${\mathrm{Fe}}_{\text{i}}^{2+}$ 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.

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• Defects
• Diffusion
• Semiconductors
• Wide band gap systems
• Density functional theory
• First-principles calculations