Quasiparticle and excitonic effects in the optical response of ${\mathrm{KNbO}}_3$

The cubic, tetragonal, and orthorhombic phase of potassium niobate (${\mathrm{KNbO}}_3$) are studied based on density-functional theory. Starting from the relaxed atomic geometries, we analyze the influence of self-energy corrections on the electronic band structure within the $GW$ approximation. We find that quasiparticle shifts widen the direct (indirect) band gap by 1.21 (1.44), 1.58 (1.55), and 1.67 (1.64) eV for the cubic, tetragonal, and orthorhombic phase, respectively. By solving the Bethe-Salpeter equation, we obtain the linear dielectric function with excitonic and local-field effects, which turn out to be essential for good agreement with experimental data. From our results, we extract an exciton binding energy of 0.6, 0.5, and 0.5 eV for the cubic, tetragonal, and orthorhombic phase, respectively. Furthermore, we investigate the nonlinear second-harmonic generation (SHG) both theoretically and experimentally. The frequency-dependent second-order polarization tensor of orthorhombic ${\mathrm{KNbO}}_3$ is measured for incoming photon energies between 1.2 and 1.6 eV. In addition, calculations within the independent-(quasi)particle approximation are performed for the tetragonal and orthorhombic phase. The novel experimental data are in excellent agreement with the quasiparticle calculations and resolve persistent discrepancies between earlier experimental measurements and ab initio results reported in the literature.

Figure 1

Different phases of ${\mathrm{KNbO}}_3$: (a) cubic, (b) tetragonal, and (c) orthorhombic (left: conventional orthorhombic unit cell, right: primitive rhombic unit cell) with lattice parameters, angles, and space groups in Hermann-Mauguin notation. The deformations are actually very small and are exaggerated here for clarity.

Figure 2

Band structure in the Brillouin zone corresponding to the primitive unit cell of the orthorhombic phase calculated with the PBE functional (black solid lines) and with $\mathrm{ev}G{W}_0$ quasiparticle shifts (dashed blue lines).

Figure 3

Real (solid lines) and imaginary parts (dotted lines) of the linear dielectric function of orthorhombic ${\mathrm{KNbO}}_3$ calculated with different methods (top to bottom: DFT, DFT with scissors shift, $\mathrm{ev}G{W}_0$, BSE with scissors shift, $\mathrm{ev}G{W}_0$-BSE). The experimental data (gray lines) are taken from [7].

Figure 4

Real (solid red lines) and imaginary (dotted red lines) parts of the dielectric function for cubic (top), tetragonal (middle), and orthorhombic (bottom) ${\mathrm{KNbO}}_3$ calculated from the BSE with a scissors shift.

Figure 5

Components of the imaginary part of the dielectric function for the (top to bottom) cubic, paraelectric tetragonal, ferroelectric tetragonal, paraelectric orthorhombic, and ferroelectric orthorhombic phase calculated within DFT.

Figure 6

Band-decomposed charge density (yellow) of the three highest valence bands (left) and the three lowest conduction bands (right) of ${\mathrm{KNbO}}_3$ for the cubic phase (a) and tetragonal phase in the paraelectric (b) and ferroelectric (c) configuration. Potassium, niobium, and oxygen atoms are indicated by large purple, medium green, and small red balls, respectively.

Figure 7

Nonlinear susceptibility of orthorhombic ${\mathrm{KNbO}}_3$ calculated using DFT with a scissors shift (blue lines). The red marks indicate the values measured in this work.

Figure 8

Nonvanishing components of the nonlinear susceptibility for tetragonal ${\mathrm{KNbO}}_3$ calculated using DFT without (black lines) and with a scissors shift (blue lines), compared to theoretical values from a bond-charge model [34] (green crosses) and from LDA with a scissors shift [36] (blue crosses).

Figure 9

Nonvanishing components of the nonlinear susceptibility for orthorhombic ${\mathrm{KNbO}}_3$ calculated using DFT without (black lines) and with a scissors shift (blue lines) as well as measurements performed in this work (red lines). The green crosses and brackets indicate the mean value and the spread of the experimental data from [30, 31, 32, 33, 34, 35].