Relationship between ferroelectric polarization and stoichiometry of ${\mathrm{HfO}}_2$ surfaces

We used density functional theory to assess the stability of nonpolar tetragonal $(P{4}_2/nmc) \mathrm{Hf}{\mathrm{O}}_2\left(110\right)$ reconstructed surfaces and the effect of polarization on the stability of the corresponding surfaces of polar orthorhombic $\left(Pca{2}_1\right) \mathrm{Hf}{\mathrm{O}}_2\left(001\right)$. The models consisted of nine-formula-unit-thick slabs with two-formula-unit surface unit cells. We determined an oxygen-terminated surface reconstruction to be the most stable surface for the nonpolar ${\mathrm{HfO}}_2$ slab, with one oxygen atom per formula unit on both sides of the slab (1.0-O/1.0-O). For the same surface composition, the ferroelectric displacements in the polar orthorhombic slab result in band bending that leads to the migration of charge carriers to the surface (with sign opposite to that of the surface polarization charge) which metallizes the surface to eliminate or reduce the net out-of-plane dipole. Ionic passivation also is effective at screening the polarization charge, which therefore alleviates band bending, leading to stabilization. This is achieved via a nonstoichiometric surface reconstruction, in which the most stable positively polarized side is oxygen-terminated with 1.5 oxygen atoms per formula unit, while the negatively polarized side has one oxygen atom per formula unit (P+:1.5-O/P−:1.0-O). This work establishes a link between the stability of the surface reconstruction and the ferroelectric polarization in ${\mathrm{HfO}}_2$, which is important for the technological need to control ferroelectric performance at the nanoscale.

Figure 1

(a) Structure of the conventional two-formula-unit nonpolar tetragonal $\left(P{4}_2/nmc\right) {\mathrm{HfO}}_2$ phase. (b) four-formula-unit supercell of the tetragonal phase shown in (a), reoriented such that the new $a, b$, and $c$ axes are now along the $\left[\overline{1}10\right]$, [001], and [110] directions of the conventional tetragonal vectors, respectively. (c) four-formula-unit supercell of the polar orthorhombic $\left(Pca{2}_1\right) {\mathrm{HfO}}_2$ phase. The polarization vector lies along the $c$ axis [001] direction, as indicated by the overlaid arrow. For all structures, Hf and O atoms are shown as green (large) and red (small) spheres, respectively. The fainter atoms are farther away from the viewer.

Figure 2

Relaxed structures for compositionally (a) symmetric and (b) asymmetric nonpolar tetragonal ${\mathrm{HfO}}_2$ supercell slabs. Profile views are shown only for the most stable compositionally symmetric and asymmetric slabs among all compositions studied. Top and bottom views of the slabs are shown for the most stable configuration for a given composition. The composition of the outermost layers in terms of atoms per surface unit cell is labeled for the top and bottom layer above each structure. The fainter atoms are farther away from the viewer. Outermost atoms are circled within a 1 × 1 lateral unit cell that contains two formula units (purple-dashed box).

Figure 3

Relaxed structures for most stable compositionally (a) symmetric and (b) asymmetric orthorhombic ${\mathrm{HfO}}_2$ supercell slabs. Profile views are shown only for the most stable compositionally symmetric and asymmetric slabs among all compositions studied. Top and bottom views of the slabs are shown for the most stable configuration for a given composition. The composition of the outermost layers in terms of atoms per surface unit cell is labeled for the top (P+) and bottom (P−) layer above each structure. To disambiguate the nomenclature for the compositionally asymmetric slabs, we use P+ and P− to refer to the composition of the positively and negatively polarized surfaces, respectively. The fainter atoms are farther away from the viewer. Outermost atoms are circled within a 1 × 1 lateral unit cell that contains two formula units (purple-dashed box).

Figure 4

Plot of surface energy as a function of (a) temperature from 100 to 1100 K and (b) pressure from ${10}^{-12}$ to ${10}^2$ bar for compositionally symmetric and asymmetric tetragonal $\mathrm{Hf}{\mathrm{O}}_2\left(110\right)$ slabs.

Figure 5

Plot of surface energy as a function of (a) temperature from 100 to 1100 K and (b) pressure from ${10}^{-12}$ to ${10}^2$ bar for compositionally symmetric and asymmetric orthorhombic $\mathrm{Hf}{\mathrm{O}}_2\left(001\right)$ slabs. To disambiguate the nomenclature for the compositionally asymmetric slabs, we use P+ and P− to refer to the composition of the positively and negatively polarized surfaces, respectively.

Figure 6

Layer-by-layer Bader charge deviation for (a) nonpolar tetragonal and (b) polar orthorhombic ${\mathrm{HfO}}_2$ slabs for three different compositions. The values correspond to the average Bader charge deviation per atom for each Hf and O half-layer relative to their respective bulk phase. Note that the left- and right-hand sides of the plots correspond respectively to the bottom and the top of the slabs. In the surface nomenclature, the composition of the top surface is given first. Figures 2 and 3 show the corresponding atomic structures of the compositions indicated.

Figure 7

Plane-averaged electrostatic potential comprised of the ionic and Hartree potentials (blue) and the $z$-averaged (5.09 Å window) potential (red), calculated along the surface normal for the tetragonal(110) surface with (a) symmetric 1.0-O/1.0-O composition and orthorhombic(001) surfaces with (b) symmetric 1.0-O/1.0-O, (c) symmetric 1.5-O/1.5-O, and (d) asymmetric P+:1.5-O/P−:1.0-O compositions. The potentials are referenced to the Fermi level. The horizontal dashed lines mark the positions of the vacuum level for each surface. The difference in the work functions (ΔΦ) is the difference in the vacuum potentials of the two surfaces multiplied by a unit of elementary charge $e=1$. Note that the left- and right-hand sides of the plots respectively correspond to the bottom and the top of the slabs. In the surface nomenclature, the composition of the top surface is given first. For the corresponding structures, see Figs. 2 and 3.

Figure 8

Layer-by-layer projected densities of states (pDOS) for the tetragonal $\mathrm{Hf}{\mathrm{O}}_2\left(110\right)$ surface with (a) symmetric 1.0-O/1.0-O composition and orthorhombic (001) surfaces with (b) symmetric 1.0-O/1.0-O, (c) symmetric 1.5-O/1.5-O, and (d) asymmetric P+:1.5-O/P−:1.0-O compositions. The electronic energies reference to the valence-band edge or the Fermi level (dashed vertical lines mark Energy $=0\mathrm{eV}$). The top O layer of the polar orthorhombic slab corresponds to the P+ surface and the bottom O layer corresponds to the P− surface. Hf half-layer spin up/down: green/light green; O half-layer spin up/down: red/pink. The values are shifted so that the pDOS of the top and bottom layers correspond respectively to the top-most and bottom-most curves. Figures 2 and 3 show the corresponding atomic structures.