Deep electron and hole polarons and bipolarons in amorphous oxide

Amorphous (a)-${\mathrm{HfO}}_2$ is a prototype high dielectric constant insulator with wide technological applications. Using ab initio calculations we show that excess electrons and holes can trap in a-${\mathrm{HfO}}_2$ in energetically much deeper polaron states than in the crystalline monoclinic phase. The electrons and holes localize at precursor sites, such as elongated Hf-O bonds or undercoordinated Hf and O atoms, and the polaronic relaxation is amplified by the local disorder of amorphous network. Single electron polarons produce states in the gap at $\sim 2$ eV below the bottom of the conduction band with average trapping energies of 1.0 eV. Two electrons can form even deeper bipolaron states on the same site. Holes are typically localized on undercoordinated O ions with average trapping energies of 1.4 eV. These results advance our general understanding of charge trapping in amorphous oxides by demonstrating that deep polaron states are inherent and do not require any bond rupture to form precursor sites.

Figure 1

Typical spin density distributions and displacements (in Å) of Hf and O ions caused by localization of an extra electron (a) and a hole (b) in a-${\mathrm{HfO}}_2$. The arrows show the direction of displacements. Displacements less than 0.03 Å are not shown. The dashed arrows show the directions of ion displacements with respect to the neutral case. The distribution of spin density of the trapped electron (a) and hole (b) are shown in blue with different isovalue for clarity. The white spheres are Hf ions and the red spheres are O ions.

Figure 2

Distribution of the KS levels for the electron and hole trapping in a-${\mathrm{HfO}}_2$ structures with 96 atoms. The energy scale from 0.0 to 6.4 eV corresponds to the a-${\mathrm{HfO}}_2$ band gap as calculated using the PBE0-TC-LRC functional.