Electronic and Optical Properties of Sodium Niobate : A Density Functional Theory Study

In recent years, much effort has been devoted to replace the most commonly used piezoelectric ceramic lead zirconate titanate Pb[ZrxTi1−x]O3 (PZT) with a suitable lead-free alternative for memory or piezoelectric applications. One possible alternative to PZT is sodium niobate as it exhibits electrical and mechanical properties that make it an interesting material for technological applications. +e high-temperature simple cubic perovskite structure undergoes a series of structural phase transitions with decreasing temperature. However, particularly the phases at room temperature and below are not yet fully characterised and understood. Here, we perform density functional theory calculations for the possible phases at room temperature and below and report on the structural, electronic, and optical properties of the different phases in comparison to experimental findings.


Introduction
e most widely used piezoelectric ceramic to date is lead zirconate titanate Pb[Zr x Ti 1−x ]O 3 (PZT), where the composition x is used to tailor specific properties for memory or piezoelectric devices.However, due to the toxicity of these lead-containing devices, much effort has been devoted in recent years to find suitable lead-free alternatives to PZT.One promising alternative materials system is the solid solution sodium potassium niobate (Na,K)NbO 3 [1,2].While the structural and electronic properties of the one end member potassium niobate (KNbO 3 ) are relatively well known, this is much less the case for the ferroelectric (FE) perovskite sodium niobate (NaNbO 3 ).
Like many other perovskites, NaNbO 3 exhibits a large range of structural phase transitions, accompanied by changes in the ferroelectric behaviour.A first comprehensive discussion of the different structural phase transitions in NaNbO 3 was reported by Megaw [3].According to Megaw [3], the high-temperature phase of NaNbO 3 is paraelectric (PE) and crystallises in the simple cubic perovskite structure (Pm3m), before it undergoes a phase transition to a PE tetragonal T 2 phase (P4/mbm) at a transition temperature T c1 � 913 K. Next, there appear three distinct phase transitions into orthorhombic phases: to the PE T 1 phase (Cmcm) at T c2 � 848 K, to the PE S phase (Pnmm) at T c3 � 793 K, and to the antiferroelectric (AFE) R phase (Pmnm) at T c4 � 753 K, respectively.e orthorhombic AFE R phase (Pmnm) undergoes a phase transition into the orthorhombic AFE P phase (Pbcm) at T c5 � 633 K, which is the commonly assumed crystal structure at room temperature and stays stable over a wide temperature range down to T c6 � 173 K. Below T c6 , NaNbO 3 crystallises in the rhombohedral FE N phase (R3c).
However, while the structure of the high-temperature crystalline phases of NaNbO 3 is commonly agreed on, there is still an ongoing discussion about the crystalline phases at room temperature and below.Darlington et al. [4] and Cheon et al. [5] reported on a possible admixture of a monoclinic phase (Pm) into the room-temperature orthorhombic AFE P phase (Pbcm) based on X-ray diffraction and neutron powder diffraction measurements.ere are also reports about a room-temperature phase transition into a FE phase (P2 1 ma) induced by an applied electric field [6,7], by nanoparticle growth [8] or by growth as a strained thin film [9], respectively.A full list of recent experimental data can be found in [2,[10][11][12].
While there is quite a lot information available on the structural phase transitions from experiment, this is much less the case for theoretical investigations.Diéguez et al. [13] reported a first-principle study of epitaxial strain in perovskites, including KNbO 3 and NaNbO 3 , while Li et al. [14] reported density functional theory (DFT) calculations for epitaxially strained KNbO 3 /NaNbO 3 superlattices, thereby including the unstrained simple cubic perovskite phase (Pm3m) as well.Finally, Machado et al. [15] reported on the relative phase stability and lattice dynamics of NaNbO 3 from first principles.A rigorous assessment of the performance of different exchange-correlation functionals within DFT calculations and applied to the possible crystalline phases of NaNbO 3 at room temperature and below is still missing to date.
e focus of the present work is on the reported possible crystalline phases of NaNbO 3 at room temperature and below, especially on the coexistence of the rhombohedral FE N phase (R3c) with the monoclinic AFE P phase (Pm) and the orthorhombic AFE P phase (Pbcm).Since the delicate interplay of structural and electronic properties determines properties like the spontaneous polarisation, an improved description would help demystify the crystalline phases of this material at room temperature and below and to tailor it better for technical applications.Here, we present results of DFT calculations for the structural, electronic, and optical properties of the crystalline phases of NaNbO 3 at room temperature and below, with a special emphasis on the performance of different flavours of the generalised gradient approximation (GGA) to the unknown exchange-correlation potential.e results include calculations based on the conventional PBE parametrisation of Perdew et al. [16], the AM05 parametrisation [17], and the PBE parametrisation revised for solids (PBEsol) [18].In addition, we also perform benchmark calculations for the simple cubic PE perovskite phase (Pm3m) utilising the hybrid functional PBE0 [19], where a quarter of the exchange potential is replaced by Hartree-Fock exact-exchange to better account for electronic correlation effects [20].In accordance with similar investigations for the other end member KNbO 3 [21] of the solid solution sodium potassium niobate (Na,K)NbO 3 , we find that the improved GGA approximations of AM05 and PBEsol perform better for the structural, electronic, and optical properties compared to the conventional PBE approximation.e results will be beneficial for future theoretical works concerning strain influences from underlying substrates or calculations of the spontaneous polarisation.
is paper is organised as follows.Section 2 introduces the necessary theoretical background and details of the calculations.Section 3 is devoted to the discussion of the obtained structural properties in comparison to available experimental data, the electronic properties, and finally, the optical properties.e final section provides a summary of the presented results and their main conclusion.
Structural relaxations have been performed within a scalar-relativistic approximation with a plane wave energy cutoff of 500 eV.Γ-centred k point meshes have been used to sample the Brillouin zone and amounted to 6 × 6 × 6 for the simple cubic perovskite Pm3m phase, 6 × 2 × 6 for the monoclinic Pm phase, 6 × 6 × 2 for the orthorhombic Pbcm phase, and 6 × 6 × 2 for the rhombohedral R3c phase, respectively.
To evaluate the performance of different exchangecorrelation functionals, the structural and electronic properties have been calculated employing the GGA in the conventional PBE parametrisation of Perdew et al. [16], the AM05 parametrisation [17], and the PBE parametrisation revised for solids (PBEsol) [18].Both AM05 and PBEsol have been developed to increase accuracy in structural properties for crystalline solids [18,26].For the smallest unit cell of the simple cubic perovskite Pm3m phase of NaNbO 3 , the results have additionally been benchmarked against hybrid functional calculations using the PBE0 functional [19] to better account for electronic correlation effects [20].
e obtained relaxed ground state structures served as a starting point for subsequent calculations of the electronic band structures and the real and imaginary parts of the dielectric functions.ereby, the imaginary part of the dielectric tensor (in VASP) is determined by a summation over empty states using ε (2)  αβ (ω) � where c and v denote the conduction and valence band states, respectively, and u ck is the cell periodic part of the orbitals at k.In order to ensure converged results, the number of empty bands in the calculations has been increased by a factor of three.e real part of the dielectric tensor is obtained via a Kramers-Kronig transformation: where P denotes the principal value of the integral.Details of the method can be found in [27].e real and imaginary parts of the dielectric functions for the noncubic phases have been obtained by diagonalising the dielectric tensors for every energy point and averaging over the resulting main diagonal values, respectively.

Crystalline Phases at Room Temperature and
Below. e present work focuses on the following crystalline phases at room temperature and below: the rhombohedral FE N phase (R3c, SG 161, Z � 6), the monoclinic AFE P phase (Pm, SG 6, Z � 8), and the orthorhombic AFE P phase (Pbcm, SG 57, Z � 8).In addition, benchmark calculations have been 2 Advances in Materials Science and Engineering carried out for the high-temperature simple cubic perovskite phase (Pm3m, SG 221, Z � 1).All four phases have been initially set up using experimental data reported by Jiang et al. [28] for the Pm3m phase, by Cheon et al. [5] for the Pm phase, by Johnston et al. [6] for the Pbcm phase, and by Darlington and Megaw [4] for the R3c phase, respectively.For all four phases, we performed a full geometry optimisation for several unit-cell volumes centred around the experimentally reported ones.e geometries have been fully optimised employing three different GGA functionals (PBE, PBEsol, and AM05), until the forces on each atom were smaller than 0.001 eV Å−1 .In addition, for the simple cubic perovskite Pm3m phase, we also employed the PBE0 hybrid functional.Together with the plane wave energy cutoff and the k point meshes reported in Section 2.1, this ensured well-converged structural and electronic properties.Exemplary, the PBEsol relaxed structures for the four different phases are shown in Figure 1, and its CIF files can be found in the Supplemental Material (available here).e volume dependence of the total energies for the four different phases gives access to the bulk modulus B 0 , defined as where E tot is the total energy and V 0 is the equilibrium bulk volume.For cubic crystals, the bulk modulus can also be expressed in terms of the elastic moduli C 11 and C 12 [29][30][31]: Typically, the total energies are fitted to the Murnaghan equation of state [33]: giving access to the bulk modulus B 0 and its pressure derivative B ′ 0 as well as the ground state unit-cell volume V 0 , respectively.
e structural properties for the simple cubic perovskite Pm3m phase of NaNbO 3 (SG 221, Z � 1) have been calculated for a range of unit-cell volumes around the initial experimental unit-cell volume [28] using different exchange-correlation functionals.e total energy curves shown in Figure 2(d) are obtained by a cubic-spline fit to the theoretical data and are rescaled to zero energy corresponding to the lowest energy for each functional.
From the total energy curves, the ground state lattice constants and bulk moduli B 0 have been obtained and are compared to available experimental data in Table 1.
It can be seen from Figure 2(d) that the PBE functional overestimates the unit-cell volume, and that both PBEsol and AM05 yield improved and nearly indistinguishable structural properties for the simple cubic perovskite Pm3m phase of NaNbO 3 .
e hybrid functional PBE0 slightly overbinds, resulting in a too short lattice constant and a too large bulk modulus, similar to other investigations on oxide semiconductors [29][30][31].As has been shown recently for oxide semiconductors, this might improve with a self-consistent determination of the amount of Hartree-Fock exactexchange mixed into the hybrid functional [20].However, as already mentioned by Machado et al. [15], it is not correct to directly compare the high-temperature measurements of the simple cubic perovskite Pm3m phase with the zerotemperature DFT calculations.
eir GGA calculations yielded a lattice constant a � 3.9516 Å and a bulk modulus B 0 � 193.02GPa, in favourable agreement with our results.
Based on the slightly different ground state properties, for each of the exchange-correlation functionals, the electronic band structures have been calculated and are shown in Figure 3.
e most obvious result is that all three GGA functionals yield very similar electronic band structures. is is also reflected by the indirect (direct) Kohn-Sham energy gaps (Table 1) that amount to 1.652 eV (2.404 eV), 1.639 eV (2.386 eV), and 1.642 eV (2.383 eV) for the PBE, the PBEsol, e hybrid functional PBE0 yields a wider indirect (direct) Kohn-Sham gap of 3.756 eV (4.567 eV), which can also be seen in Figure 3(d), and a slightly broader valence-band bandwidth.e experimental bandgap has been measured by Li et al. and amounted to 3.29 eV.It is indirect as well, in agreement with our calculations.As to be expected, our GGA calculations underestimate the bandgap, while the indirect PBE0 Kohn-Sham gap of 3.756 eV is slightly larger than the experimental value.
Based on the obtained relaxed ground state structures for the simple cubic perovskite Pm3m phase, we calculated the optical properties.e real (orange) and imaginary (green) parts of the dielectric function are shown in Figure 4 for the di erent exchange-correlation functionals.Similarly to the electronic band structures shown in Figure 3, the three GGA functionals yield nearly indistinguishable dielectric functions.e onsets in the imaginary parts re ect the similar bandgaps already seen in the electronic band structures, and the heights and widths of the di erent peaks re ect the similarity in regions of nearly parallel bands in the band structures where large transition matrix elements give rise to stronger features in the imaginary part of the dielectric functions.e most striking di erence in the PBE0 calculated dielectric functions is the shifted onset in the imaginary part, re ecting the larger bandgap obtained for the electronic properties.However, since the overall regions of nearly parallel bands in the band structure of Figure 3(d) are similar to the band structures of the plain GGA calculations, the peak structure in the imaginary part of the dielectric function and the widths of the peaks remain similar to the plain GGA calculations, only the heights are reduced.e PBE0-calculated dielectric functions agree best with available experimental results [35,36], mostly in the onset of the imaginary part of the dielectric function (showing best agreement for the bandgap) and the low-energy onset of the real part of the dielectric function.

Monoclinic Pm Phase.
e total energy curves for the monoclinic Pm phase are shown in Figure 2(c), calculated employing di erent GGA functionals.It can be seen that the Exchange-correlation functionals include the conventional PBE functional of Perdew et al. [16], the PBEsol functional revised for solids [18], the AM05 functional [17], and the PBE0 hybrid functional [19].Zero energy has been rescaled corresponding to the lowest energy for each functional.e vertical black dashed lines correspond to the experimental ground state volumes.[34] Given are the lattice parameter a, equilibrium unit-cell volume V 0 , the bulk modulus B 0 and its pressure derivative B ′ 0 , and the direct E dir KS (Γ-Γ) and indirect E ind KS (M-Γ) Kohn-Sham energy gaps, respectively, calculated with di erent exchange-correlation functionals.2. e electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state volume is shown in Figure 5(c).
e direct Kohn-Sham energy gap of 2.290 eV is comparable to the direct Kohn-Sham energy gap of 2.386 eV of the simple cubic perovskite Pm3m phase, but much larger than its indirect Kohn-Sham energy gap of 1.639 eV.However, the valence-band bandwidths are very similar between the monoclinic Pm phase and the simple cubic perovskite Pm3m phase, respectively.An additional PBE0 hybrid functional calculation based on the PBEsol ground state structure yielded an increased direct Kohn-Sham energy gap of 4.453 eV.
e real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure 6(c) for the monoclinic Pm phase.Broad ranges of nearly parallel bands in the electronic band structure (Figure 5(c)) give rise to a very broad absorption peak centred around 5 eV.A second broad peak appears to be centred around 8.5 eV.

Orthorhombic Pbcm Phase.
e total energy curves for the orthorhombic Pbcm phase are shown in Figure 2(b), calculated employing di erent GGA functionals.It can be seen that the di erence between the AM05 and PBEsol functional is again larger compared to the simple cubic perovskite Pm3m phase and similar to the monoclinic Pm phase.e conventional PBE functional again overestimates the unit-cell volume.Both improved approximations to the GGA functional, AM05 and PBEsol, slightly overestimate the unit-cell volume as well, with the PBEsol functional performing better.e obtained ground state structural properties are given in Table 3.
e electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state structure is shown in Figure 5(b).e direct Kohn-Sham energy gap of 2.298 eV is again comparable to the direct Kohn-Sham energy gap of 2.386 eV of the simple cubic perovskite Pm3m phase, but much larger than its indirect Figure 3: NaNbO 3 electronic band structures calculated for the simple cubic perovskite Pm3m phase employing di erent exchangecorrelation potentials: (a) the conventional PBE functional of Perdew et al. [16], (b) the PBEsol functional revised for solids [18], (c) the AM05 functional [17], and (d) the PBE0 hybrid functional [19].Zero energy has been rescaled to the valence band maximum (at the M point).

Advances in Materials Science and Engineering
Kohn-Sham energy gap of 1.639 eV.However, the valenceband bandwidth is again very similar to the simple cubic perovskite Pm3m phase and the monoclinic Pm phase, respectively.An additional PBE0 hybrid functional calculation based on the PBEsol ground state structure yielded an increased direct Kohn-Sham energy gap of 4.461 eV, very close to the PBE0 Kohn-Sham energy gap of 4.453 eV of the monoclinic Pm phase.It is, however, much larger than the experimental gap of 3.45 eV, determined by Li et al. [34] using a Tauc plot and assuming an indirect bandgap.e real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure 6(b) for the orthorhombic Pbcm phase.Again, similar to the monoclinic Pm phase, broad ranges of nearly parallel bands in the electronic band structure (Figure 5(b)) give rise to a very broad absorption peak centred around 5 eV.A second broad peak appears to be centred around 8.5 eV.It should be noted that the dielectric functions for the orthorhombic Pbcm phase and the monoclinic Pm phase are nearly indistinguishable.It is understandable if one assumes that the only di erence in the structural properties (di erent orientations of the oxygen octahedra) has only little in uence on the electronic and subsequently the optical properties, respectively.
e total energy curves for the rhombohedral R3c phase are shown in Figure 2(a), calculated employing di erent GGA functionals.Similar to the orthorhombic Pbcm phase, the conventional PBE functional overestimates the unit-cell volume the most, whereas the AM05 and PBEsol functionals again only slightly overestimate, with the PBEsol functional performing best for the structural properties.e obtained ground state structural properties are given in Table 4.
e electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state volume is shown in Figure 5(a).Due to the smaller amount of atoms in the unit cell, Z 6 for the rhombohedral R3c phase compared to Z 8 for the orthorhombic Pbcm and the monoclinic Pm phases, there are fewer bands in the electronic band structure.
e direct Kohn-Sham energy gap of 2.660 eV is slightly larger than the direct Kohn-Sham energy gap of 2.386 eV of the simple cubic perovskite Pm3m phase and much larger than its indirect Kohn-Sham energy gap of 1.639 eV.It is also larger than the direct Kohn-Sham gaps of 2.298 eV and 2.290 eV for the orthorhombic Pbcm and the monoclinic Pm phases, respectively.e valenceband bandwidth, however, is again very similar to the simple cubic perovskite Pm3m, the orthorhombic Pbcm, and the monoclinic Pm phases, respectively.An additional PBE0 hybrid functional calculation based on the PBEsol ground state structure yielded an increased direct Kohn-Sham energy gap of 4.840 eV, larger than the PBE0 Kohn-Sham gaps of 4.461 eV and 4.453 eV for the orthorhombic Pbcm and the monoclinic Pm phases, respectively.e real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure 6(a) for the rhombohedral R3c phase.Broad ranges of nearly parallel bands in the electronic band structure (Figure 5(a)) give rise to slightly narrower but still very broad absorption peaks compared to the orthorhombic Pbcm and the monoclinic Pm phases, now centred at slightly smaller energies.

Conclusions
In summary, we presented a detailed DFT investigation on the structural properties of the crystalline phases of NaNbO 3 Figure 4: NaNbO 3 dielectric functions calculated using di erent exchange-correlation potentials.e real (orange) and imaginary (green) parts of the dielectric functions for (a) the conventional PBE functional of Perdew et al. [16], (b) the PBEsol functional revised for solids [18], (c) the AM05 functional [17], and (d) the PBE0 hybrid functional [19].Given are the lattice parameters a, b, and c, the equilibrium unit-cell volume V 0 , the bulk modulus B 0 and its pressure derivative B ′ 0 , and the direct (Γ-Γ) Kohn-Sham energy gaps E dir KS , respectively, calculated with di erent exchange-correlation functionals.e last row gives the energy di erence ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3m phase.6 Advances in Materials Science and Engineering at room temperature and below, end member in the solid solution sodium potassium niobate (Na,K)NbO 3 , and a promising lead-free alternative to PZT. e calculations assessed the performance of di erent GGA exchangecorrelation functionals, namely, the conventional PBE parametrisation, the AM05 parametrisation, and the PBEsol parametrisation.For the simple cubic perovskite Pm3m structures, additional calculations also employed the PBE0 hybrid functional.It could be shown that the improved GGA functionals AM05 and PBEsol perform better for the structural properties compared to the conventional PBE  Given are the lattice parameters a, b, and c, the equilibrium unit-cell volume V 0 , the bulk modulus B 0 and its pressure derivative B ′ 0 , and the direct (Γ-Γ) Kohn-Sham energy gaps E dir KS , respectively, calculated with di erent exchange-correlation functionals.e last row gives the energy di erence ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3m phase.Given are the lattice parameters a and c, the equilibrium unit-cell volume V 0 , the bulk modulus B 0 and its pressure derivative B ′ 0 , and the direct (Γ-Γ) Kohn-Sham energy gaps E dir KS , respectively, calculated with di erent exchange-correlation functionals.e last row gives the energy di erence ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3m phase.
Advances in Materials Science and Engineering approximation. is is in line with similar investigations for the other end member KNbO 3 [21] of the solid solution sodium potassium niobate (Na,K)NbO 3 .Based on the PBEsol ground state geometries, the electronic band structures and the optical dielectric functions have been calculated and discussed with respect to available experimental data, with the PBEsol functional performing best.
is can serve as a basis for future theoretical works, calculating the spontaneous polarisations in the different phases of NaNbO 3 at room temperature and below or to investigate the influence of strain effects on the structural, electronic, and optical properties, respectively.

Figure 2 :
Figure 2: NaNbO 3 total energy curves calculated with di erent exchange-correlation functionals for (a) the rhombohedral R3c phase (SG 161), (b) the orthorhombic Pbcm phase (SG 57), (c) the monoclinic Pm phase (SG 6), and (d) the simple cubic perovskite Pm3m phase (SG 221).Exchange-correlation functionals include the conventional PBE functional of Perdew et al.[16], the PBEsol functional revised for solids[18], the AM05 functional[17], and the PBE0 hybrid functional[19].Zero energy has been rescaled corresponding to the lowest energy for each functional.e vertical black dashed lines correspond to the experimental ground state volumes.

Figure 5 : 2 Figure 6 :
Figure5: NaNbO 3 electronic band structures for the crystalline phases at room temperature and below calculated using the PBEsol functional revised for solids[18].e electronic band structures for (a) the rhombohedral R3c phase (SG 161), (b) the orthorhombic Pbcm phase (SG 57), and (c) the monoclinic Pm phase (SG 6) are shown.Zero energy has been rescaled to the valence band maximum.

Table 1 :
Structural properties of the simple cubic perovskite Pm3m phase of NaNbO 3 in comparison to experimental data.

Table 2 :
Structural properties of the monoclinic Pm phase of NaNbO 3 .

Table 3 :
Structural properties of the orthorhombic Pbcm phase of NaNbO 3 .

Table 4 :
Structural properties of the rhombohedral R3c phase of NaNbO 3 .