Electronic band structure, phonon dispersion, and magnetic triple- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>q</mml:mi> </mml:math> state in GdGaI

We theoretically investigate the physical properties of the magnetic van der Waals material GdGaI. Using first-principles calculations, we compute the phonon dispersion of GdGaI and show no imaginary phonons, suggesting that phonon-driven phase transitions are unlikely to occur in GdGaI. Our band calculation reveals that the electronic bands near the Fermi energy are composed of Gd 5d and Ga 4p orbitals. We construct a tight-binding model that incorporates the Gd 5d and Ga 4p orbitals to investigate the magnetic structure. We introduce Kondo coupling between electrons in Gd 5d orbitals and localized spins in Gd 4f orbitals and present the modified band structure when localized spins form a magnetic order characterized by three q vectors that connect the valence and conduction bands. We discuss the origin of the spin order based on the Ruderman-Kittel-Kasuya-Yosida mechanism and suggest that Coulomb interactions acting on electrons near the Fermi level can contribute to the ordering of localized spins.