Chemical controllability of charge states of nitrogen-related defects in ${\text{HfO}}_x{\text{N}}_y$: First-principles calculations

Relative stabilities of nitrogen-related defects in ${\text{HfO}}_x{\text{N}}_y$ have been extensively studied in terms of formation energies by using first-principles calculations. We have found that the two oxygen vacancies coupled with two substitutional nitrogen atoms at oxygen sites is the predominant defect in a low oxygen chemical potential $\left({\mu }_{\text{O}}\right)$ with doubly positive ${\left[{\left({\text{N}}_{\text{O}}\right)}_2{\left(V_{\text{O}}\right)}_2\right]}^{+2}$ and neutral ${\left[{\left({\text{N}}_{\text{O}}\right)}_2{\left(V_{\text{O}}\right)}_2\right]}^0$ forms for the $p$-type and $n$-type silicon substrates, respectively. On the other hand, the negatively charged substitutional nitrogen ${\left[{\left({\text{N}}_{\text{O}}\right)}_2\right]}^{-2}$ predominates in a higher ${\mu }_{\text{O}}$ condition. A neutral defect, ${\left[{\left({\text{N}}_{\text{O}}\right)}_2{V}_{\text{O}}\right]}^0$, is found to be stabilized in the midregion of ${\mu }_{\text{O}}$, which indicates that the oxygen partial pressure is an important factor to control fixed charges that cause malfunction in ${\text{HfO}}_x{\text{N}}_y$-based devices. An electron counting concept is shown to be valid to predict the stable forms of the defects.

Figure 1

(Color online) Formation energies of the complex defects with various charges as a function of oxygen chemical potential at the conditions of (a) ${\epsilon }_F=2.15\text{eV}$ and (b) ${\epsilon }_F=2.75\text{eV}$, which correspond to the Fermi energies in ${\text{HfO}}_2/\text{Si}$ junctions with the $p$ type, ${\epsilon }_F$ ($p$ type Si), and the $n$ type, ${\epsilon }_F$ ($n$ type Si), respectively (Ref. 29). The maximum and minimum values of ${\mu }_{\text{O}}$, respectively, correspond to an equilibrium between ${\text{HfO}}_2$ and an oxygen gas phase and to a precipitation condition of Hf metal in ${\text{HfO}}_2$ (Ref. 30).

Figure 2

(Color online) Relaxed structures for (a) the doubly negative charged two substitutional N at O sites ${\left[{\left({\text{N}}_{\text{O}}\right)}_2\right]}^{2-}$, (b) the neutral state of a $V_{\text{O}}$ coupled with two ${\text{N}}_{\text{O}}$’s ${\left[{\left({\text{N}}_{\text{O}}\right)}_2{V}_{\text{O}}\right]}^0$, and (c) doubly positive charged two $V_{\text{O}}$’s coupled with two ${\text{N}}_{\text{O}}$’s ${\left[{\left({\text{N}}_{\text{O}}\right)}_2{\left(V_{\text{O}}\right)}_2\right]}^{+2}$. Atomic positions and vacant sites are depicted by the balls. Bond lengths in angstroms are shown by the arrows and numbers.

Figure 3

(Color online) Formation energies of the complex defects with various charges as functions of the Fermi energy ${\epsilon }_F$ at (a) ${\mu }_{\text{O}}=-4.5\text{eV}$ and (b) ${\mu }_{\text{O}}=0\text{eV}$, respectively. The zero of ${\epsilon }_F$ is set to the valence band maximum. The slope of each line indicates the stable charge state of each defect at a condition of ${\epsilon }_F$. The charge states are also shown as numbers on each line. The positions of the transition level between different charge states are shown by the filled circles.

Figure 4

(Color online) Isosurface of the partial charge density [0.05 $\left(1/{\text{Å}}^3\right)$], corresponding to the doubly occupied Kohn–Sham state in the band gap for ${\left[{\left({\text{N}}_{\text{O}}\right)}_2{\left(V_{\text{O}}\right)}_2\right]}^0$. Here, the blue (light gray), red (small transparent), and green (dark gray) spherical or ellipsoidal particles denote Hf, O, and N atoms, respectively.