<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:msub> <mml:mtext>-Ga</mml:mtext> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi mathvariant="normal">O</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mrow>

We present a comprehensive investigation of reconstructions on <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:mi>β</a:mi> <a:msub> <a:mtext>-Ga</a:mtext> <a:mn>2</a:mn> </a:msub> <a:msub> <a:mi mathvariant="normal">O</a:mi> <a:mn>3</a:mn> </a:msub> <a:mrow> <a:mo>(</a:mo> <a:mn>001</a:mn> <a:mo>)</a:mo> </a:mrow> </a:mrow> </a:math> combining first-principles calculations with experimental observations. Using atomistic thermodynamics and replica-exchange grand-canonical molecular dynamics simulations, we explore the configurational space of possible reconstructions under varying chemical potentials of oxygen and gallium. Our calculations reveal several stable surface reconstructions, most notably a previously unreported <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mn>1</c:mn> <c:mo>×</c:mo> <c:mn>2</c:mn> </c:mrow> </c:math> reconstruction consisting of paired <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:msub> <d:mi>GaO</d:mi> <d:mn>4</d:mn> </d:msub> </d:math> tetrahedra that exhibits remarkable stability across a wide range of experimental growth conditions. In this reconstruction, two Ga atoms share one oxygen bond and are separated by a distance of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:mn>2.64</e:mn> <e:mspace width="0.16em"/> <e:mi>Å</e:mi> </e:mrow> </e:math> along the [010] direction. High-angle annular dark-field scanning transmission electron microscopy imaging of homoepitaxially grown (001) layers is consistent with the predicted structure. Additional investigations of possible indium substitution at the surface sites, which can occur during indium-mediated metal-exchange catalysis via molecular beam epitaxial growth, reveal a cooperative effect in In incorporation, with distinct stability regions for In-substituted structures under O-rich conditions. Our findings provide an understanding for controlling surface properties during epitaxial growth of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mrow> <g:mi>β</g:mi> <g:msub> <g:mtext>-Ga</g:mtext> <g:mn>2</g:mn> </g:msub> <g:msub> <g:mi mathvariant="normal">O</g:mi> <g:mn>3</g:mn> </g:msub> <g:mrow> <g:mo>(</g:mo> <g:mn>001</g:mn> <g:mo>)</g:mo> </g:mrow> </g:mrow> </g:math> .