Lithium niobate $X$-cut, $Y$-cut, and $Z$-cut surfaces from ab initio theory

Density-functional theory calculations of the ${\text{LiNbO}}_3$ $\left(2\overline{1}\overline{1}0\right)$, $\left(1\overline{1}00\right)$, and (0001) surfaces, commonly referred to as $X$, $Y$, and $Z$ cuts, are presented. In case of the $Z$ cut, we find a pronounced dependence of the surface structure and stoichiometry on the direction of the ferroelectric polarization. In contrast, the influence of the chemical potentials of the surface constituents is limited. Rather electrostatics governs the surface stability. Different from the $Z$ cut, the stoichiometry of the $X$ cut and $Y$ cut is clearly dependent on the preparation conditions. The surface charge observed for the nominal nonpolar $Y$ cut is traced back to the formation of a strong surface dipole.

Figure 1

(Color online) Stacking sequence of the atoms in the ferroelectric phase $\left(T<T_C\right)$ of lithium niobate (a) and (c) and in the high-symmetry paraelectric phase $\left(T\ge T_C\right)$ (b) of LN. The displacement of the cations due to the ferroelectric phase transition and the two possible domain directions are indicated.

Figure 2

Conventional rhombohedral unit cell and conventional hexagonal unit cell of ferroelectric ${\text{LiNbO}}_3$. Li atoms are gray and Nb atoms are white. For the sake of clarity, the oxygen atoms are omitted.

Figure 3

(Color online) Relationships between the conventional hexagonal coordinate system and the Cartesian coordinate system used in this work (real-space representation). X and Y label the $x$ direction and $y$ direction, ${\mathit{t}}_{\mathbf{1}}$, ${\mathit{t}}_{\mathbf{2}}$, and ${\mathit{t}}_{\mathbf{3}}$ are the translational basis vectors of the hexagonal lattice. In round brackets the (real-space) vectors expressed in terms of the basis ${\mathit{t}}_{\mathbf{1}}$, ${\mathit{t}}_{\mathbf{2}}$, and ${\mathit{t}}_{\mathbf{3}}$, and in square brackets the corresponding Miller-Bravais indices indicating the crystallographic directions.

Figure 4

(Color online) Real-space representation of the $x$ axis, $y$ axis, $z$ axis and of the $X$ cut and $Y$ cut as defined and used in this work.

Figure 5

(Color online) Top view of the two supercells used to model the LN slabs. The orthogonal supercell is in the middle of the picture and the hexagonal supercell on the lower right corner.

Figure 6

(Color online) Stability range of $\Delta {\mu }_{\text{Li}}$, $\Delta {\mu }_{\text{Nb}}$, and $\Delta {\mu }_{\text{O}}$. Only within the region ABCD LN surfaces are formed.

Figure 7

(Color online) Phase diagram of the positive (upper part) and negative (lower part) LN (0001) surface at different growth conditions and in absence of foreign adsorbates. The unlabeled region corresponds to the Nb-terminated bulk cut. The shaded region indicates the thermodynamically range of the chemical potentials.

Figure 8

(Color online) Space-fill models of the stable positive (${\text{-Nb-O}}_3{\text{-Li}}_2$ terminated, left-hand side) and negative (O-Li- terminated, right-hand side) LN (0001) surface at growth conditions. White atoms are Nb, gray atoms Li, and small atoms O. The surface unit cell is outlined.

Figure 9

(Color online) Side view of the stable positive (${\text{-Nb-O}}_3{\text{-Li}}_2$ terminated) and negative (O-Li- terminated) LN (0001) surface. White atoms are Nb, gray atoms Li, and small atoms O.

Figure 10

(Color online) Partial density of states (DOS) of the last three LN trilayers of the slab used to simulate the stable positive (upper part) and negative (lower part) LN (0001) surface. The bulk DOS is reported for comparison. Neither the positive nor the negative termination drastically changes the fundamental band gap. The zero of the energy scale is the LN valence-band maximum.

Figure 11

Schematic energy-band diagram of the (a) negative, (c) positive, and (b) a nonpolar ${\text{LiNbO}}_3$ surface. ${\chi }_s$ and $\Delta \chi$, are the electronic affinity and its variation, $E_g$ is the band gap, $E_c$, $E_v$, and $E_{vac}$ are conduction band, valence band, and vacuum energies, $W$ is the ionization energy. Superscripts represent the quantities of positive and negative domains.

Figure 12

(Color online) Phase diagram of the LN $X$ cut at different growth conditions and in absence of foreign adsorbates. The shaded region indicates the thermodynamically range of the chemical potentials.

Figure 13

(Color online) Side view of the dominant (a) ${\text{-Li}}_{12}$ and (b) ${\text{-Li}}_6{\text{Nb}}_6{\text{-O}}_9$ terminations of the LN $X$ face. White atoms are Nb, gray atoms Li, and small atoms O.

Figure 14

(Color online) Phase diagram of the LN $Y$ cut at different growth conditions and in absence of foreign adsorbates. The unlabeled region represents the stoichiometric termination, i.e., the bulk cut plus a whole ${\text{Li}}_2{\text{Nb}}_2{\text{O}}_6$ layer. The shaded region indicates the thermodynamically allowed range of the chemical potentials.

Figure 15

(Color online) Side view of the stable (${\text{-Li}}_2{\text{-Nb}}_2{\text{-O}}_6$ terminated) LN $Y$ face at growth conditions. White atoms are Nb, gray atoms Li, and small atoms O.