Two competing soft modes and an unusual phase transition in the stuffed tridymite-type oxide<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>BaAl</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:math>

We investigated the structural phase transition of ${\mathrm{BaAl}}_{2}{\mathrm{O}}_{4}$, which has a network structure with corner-sharing ${\mathrm{AlO}}_{4}$ tetrahedra, via synchrotron x-ray thermal diffuse scattering measurements and first-principles calculations. ${\mathrm{BaAl}}_{2}{\mathrm{O}}_{4}$ shows the structural phase transition at ${T}_{\mathrm{C}}=451.4$ K from the $P{6}_{3}22$ parent crystal structure to the low-temperature superstructure with a cell volume of $2a\ifmmode\times\else\texttimes\fi{}2b\ifmmode\times\else\texttimes\fi{}c$. This phase transition is unusual, in which two energetically competing phonon modes at $M$ and $K$ points soften simultaneously. When approaching ${T}_{\mathrm{C}}$ from above, the $K$-point mode appears first. However, this $K$-point mode is overcome by the later-developed $M$-point mode. The thermal diffuse scattering intensities from both modes increase sharply at ${T}_{\mathrm{C}}$; therefore, both modes soften simultaneously. The first-principles calculations demonstrate that the $M$-point mode is electrostatically more preferable than the $K$-point mode and determines the eventual low-temperature structure, although these two modes are competing energetically. This competition is characteristic of ${\mathrm{BaAl}}_{2}{\mathrm{O}}_{4}$, which is ascribed to the structurally flexible network structure of this compound.