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

Authors: Daniel Fritsch
Journal: Advances in Materials Science and Engineering
 (2018)

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## Abstract

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. The 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.


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## Body

## 1. Introduction

The most widely used piezoelectric ceramic to date is lead zirconate titanate Pb[ZrxTi1−x]O3 (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)NbO3 [1, 2]. While the structural and electronic properties of the one end member potassium niobate (KNbO3) are relatively well known, this is much less the case for the ferroelectric (FE) perovskite sodium niobate (NaNbO3).Like many other perovskites, NaNbO3 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 NaNbO3 was reported by Megaw [3]. According to Megaw [3], the high-temperature phase of NaNbO3 is paraelectric (PE) and crystallises in the simple cubic perovskite structure (Pm3¯m), before it undergoes a phase transition to a PE tetragonal T2 phase (P4/mbm) at a transition temperature Tc1 = 913 K. Next, there appear three distinct phase transitions into orthorhombic phases: to the PE T1 phase (Cmcm) at Tc2 = 848 K, to the PE S phase (Pnmm) at Tc3 = 793 K, and to the antiferroelectric (AFE) R phase (Pmnm) at Tc4 = 753 K, respectively. The orthorhombic AFE R phase (Pmnm) undergoes a phase transition into the orthorhombic AFE P phase (Pbcm) at Tc5 = 633 K, which is the commonly assumed crystal structure at room temperature and stays stable over a wide temperature range down to Tc6 = 173 K. Below Tc6, NaNbO3 crystallises in the rhombohedral FE N phase (R3c).However, while the structure of the high-temperature crystalline phases of NaNbO3 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. There are also reports about a room-temperature phase transition into a FE phase (P21ma) 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, 10–12].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 KNbO3 and NaNbO3, while Li et al. [14] reported density functional theory (DFT) calculations for epitaxially strained KNbO3/NaNbO3 superlattices, thereby including the unstrained simple cubic perovskite phase (Pm3¯m) as well. Finally, Machado et al. [15] reported on the relative phase stability and lattice dynamics of NaNbO3 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 NaNbO3 at room temperature and below is still missing to date.The focus of the present work is on the reported possible crystalline phases of NaNbO3 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 NaNbO3 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. The 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 (Pm3¯m) 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 KNbO3 [21] of the solid solution sodium potassium niobate (Na,K)NbO3, 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. The results will be beneficial for future theoretical works concerning strain influences from underlying substrates or calculations of the spontaneous polarisation.This paper is organised as follows. Section2 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. The final section provides a summary of the presented results and their main conclusion.


## 2. Materials and Methods

### 2.1. Computational Details

The results of the present work have been obtained by DFT calculations employing the Vienna ab initio simulation package (VASP 5) [22–24] together with the projector-augmented wave (PAW) formalism [25]. For the latter, standard PAW potentials supplied with VASP were used, providing 9 valence electrons for Na atoms (2s22p63s1), 13 valence electrons for Nb atoms (4s24p65s14d4), and 6 valence electrons for O atoms (2s22p4), respectively.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 Pm3¯m 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 exchange-correlation 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 Pm3¯m phase of NaNbO3, the results have additionally been benchmarked against hybrid functional calculations using the PBE0 functional [19] to better account for electronic correlation effects [20].The 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. Thereby, the imaginary part of the dielectric tensor (in VASP) is determined by a summation over empty states using(1)εαβ2ω=4π2e2Ωlimq→01q2∑c,v,k2wkδϵck−ϵvk−ω×uck+eαquvkuck+eβquvk∗,where c and v denote the conduction and valence band states, respectively, and uck 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. The real part of the dielectric tensor is obtained via a Kramers–Kronig transformation:(2)εαβ1ω=1+2πP∫0∞εαβ2ω′ω′ω′2−ω2+iηdω′,where P denotes the principal value of the integral. Details of the method can be found in [27]. The 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.


### 2.2. Crystalline Phases at Room Temperature and Below

The present work focuses on the following crystalline phases at room temperature and below: the rhombohedral FEN 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 carried out for the high-temperature simple cubic perovskite phase (Pm3¯m, SG 221, Z = 1). All four phases have been initially set up using experimental data reported by Jiang et al. [28] for the Pm3¯m 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. The 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 eVÅ−1. In addition, for the simple cubic perovskite Pm3¯m 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). The volume dependence of the total energies for the four different phases gives access to the bulk modulus B0, defined as(3)B0=V0∂2Etot∂V2V=V0,where Etot is the total energy and V0 is the equilibrium bulk volume. For cubic crystals, the bulk modulus can also be expressed in terms of the elastic moduli C11 and C12 [29–31]:(4)B0=13C11+2C12.Figure 1
Crystalline phases of NaNbO3 at room temperature and below in comparison to the cubic perovskite phase (Pm3¯m, SG 221) (d). The rhombohedral R3c phase (SG 161) (a), the orthorhombic Pbcm phase (SG 57) (b), and the monoclinic Pm phase (SG 6) (c) are shown. Pictures have been generated using VESTA [32].
(a)
(b)
(c)
(d)Typically, the total energies are fitted to the Murnaghan equation of state [33]:(5)EV=EV0+B0VB0′V0/VB0′B0′−1+1−B0V0B0′−1,giving access to the bulk modulus B0 and its pressure derivative B0′ as well as the ground state unit-cell volume V0, respectively.


## 2.1. Computational Details

The results of the present work have been obtained by DFT calculations employing the Vienna ab initio simulation package (VASP 5) [22–24] together with the projector-augmented wave (PAW) formalism [25]. For the latter, standard PAW potentials supplied with VASP were used, providing 9 valence electrons for Na atoms (2s22p63s1), 13 valence electrons for Nb atoms (4s24p65s14d4), and 6 valence electrons for O atoms (2s22p4), respectively.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 Pm3¯m 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 exchange-correlation 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 Pm3¯m phase of NaNbO3, the results have additionally been benchmarked against hybrid functional calculations using the PBE0 functional [19] to better account for electronic correlation effects [20].The 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. Thereby, the imaginary part of the dielectric tensor (in VASP) is determined by a summation over empty states using(1)εαβ2ω=4π2e2Ωlimq→01q2∑c,v,k2wkδϵck−ϵvk−ω×uck+eαquvkuck+eβquvk∗,where c and v denote the conduction and valence band states, respectively, and uck 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. The real part of the dielectric tensor is obtained via a Kramers–Kronig transformation:(2)εαβ1ω=1+2πP∫0∞εαβ2ω′ω′ω′2−ω2+iηdω′,where P denotes the principal value of the integral. Details of the method can be found in [27]. The 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.


## 2.2. Crystalline Phases at Room Temperature and Below

The present work focuses on the following crystalline phases at room temperature and below: the rhombohedral FEN 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 carried out for the high-temperature simple cubic perovskite phase (Pm3¯m, SG 221, Z = 1). All four phases have been initially set up using experimental data reported by Jiang et al. [28] for the Pm3¯m 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. The 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 eVÅ−1. In addition, for the simple cubic perovskite Pm3¯m 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). The volume dependence of the total energies for the four different phases gives access to the bulk modulus B0, defined as(3)B0=V0∂2Etot∂V2V=V0,where Etot is the total energy and V0 is the equilibrium bulk volume. For cubic crystals, the bulk modulus can also be expressed in terms of the elastic moduli C11 and C12 [29–31]:(4)B0=13C11+2C12.Figure 1
Crystalline phases of NaNbO3 at room temperature and below in comparison to the cubic perovskite phase (Pm3¯m, SG 221) (d). The rhombohedral R3c phase (SG 161) (a), the orthorhombic Pbcm phase (SG 57) (b), and the monoclinic Pm phase (SG 6) (c) are shown. Pictures have been generated using VESTA [32].
(a)
(b)
(c)
(d)Typically, the total energies are fitted to the Murnaghan equation of state [33]:(5)EV=EV0+B0VB0′V0/VB0′B0′−1+1−B0V0B0′−1,giving access to the bulk modulus B0 and its pressure derivative B0′ as well as the ground state unit-cell volume V0, respectively.


## 3. Results and Discussion

### 3.1. Simple Cubic PerovskitePm3¯m Phase

The structural properties for the simple cubic perovskitePm3¯m phase of NaNbO3 (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. The 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.Figure 2
NaNbO3 total energy curves calculated with different 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 Pm3¯m 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. The vertical black dashed lines correspond to the experimental ground state volumes.From the total energy curves, the ground state lattice constants and bulk moduliB0 have been obtained and are compared to available experimental data in Table 1.Table 1
Structural properties of the simple cubic perovskitePm3¯m phase of NaNbO3 in comparison to experimental data.
P
m
3
¯
m
PBE
PBEsol
AM05
PBE0
Experiment
a (Å)
3.975
3.944
3.947
3.931
3.9507 [28]
V
0 (Å3)
62.807
61.349
61.490
60.745
61.6626 [28]
B
0 (GPa)
176.6
188.8
185.6
204.0
136.7a
193.02 [15], 206.3b
B
0
′ (GPa)
4.4
4.6
4.6
4.3
—
E
KS
dir (eV)
2.404
2.386
2.383
4.567
—
E
KS
ind (eV)
1.652
1.639
1.642
3.756
3.29 [34]
Given are the lattice parametera, equilibrium unit-cell volume V0, the bulk modulus B0 and its pressure derivative B0′, and the direct EKSdir (Γ-Γ) and indirect EKSind (M-Γ) Kohn–Sham energy gaps, respectively, calculated with different exchange-correlation functionals. aCalculated from experimental elastic constants C11=230 GPa and C12=90 GPa [11] using (4); bcalculated from theoretical elastic constants C11=447 GPa and C12=86 GPa [13] using (4).It can be seen from Figure2(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 Pm3¯m phase of NaNbO3. The 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 [29–31]. As has been shown recently for oxide semiconductors, this might improve with a self-consistent determination of the amount of Hartree–Fock exact-exchange 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 Pm3¯m phase with the zero-temperature DFT calculations. Their GGA calculations yielded a lattice constant a=3.9516 Å and a bulk modulus B0=193.02 GPa, 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 Figure3. The most obvious result is that all three GGA functionals yield very similar electronic band structures. This 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, and the AM05 functional, respectively, and very similar valence-band bandwidths. The 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. The 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.Figure 3
NaNbO3 electronic band structures calculated for the simple cubic perovskite Pm3¯m phase employing different exchange-correlation 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).
(a)
(b)
(c)
(d)Based on the obtained relaxed ground state structures for the simple cubic perovskitePm3¯m phase, we calculated the optical properties. The real (orange) and imaginary (green) parts of the dielectric function are shown in Figure 4 for the different exchange-correlation functionals. Similarly to the electronic band structures shown in Figure 3, the three GGA functionals yield nearly indistinguishable dielectric functions. The onsets in the imaginary parts reflect the similar bandgaps already seen in the electronic band structures, and the heights and widths of the different peaks reflect 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. The most striking difference in the PBE0 calculated dielectric functions is the shifted onset in the imaginary part, reflecting 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. The 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.Figure 4
NaNbO3 dielectric functions calculated using different exchange-correlation potentials. The 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].


### 3.2. MonoclinicPm Phase

The total energy curves for the monoclinicPm phase are shown in Figure 2(c), calculated employing different GGA functionals. It can be seen that the difference between AM05 and PBEsol functional are larger compared to the simple cubic perovskite Pm3¯m phase and that the conventional PBE functional again overestimates the unit-cell volume. While the PBEsol functional yields a slightly smaller ground state volume compared to the experiment, the AM05 functional ground state volume lies slightly above. The obtained ground state structural properties are given in Table 2.Table 2
Structural properties of the monoclinicPm phase of NaNbO3.
P
m
PBE
PBEsol
AM05
Experiment
a (Å)
5.576
5.495
5.513
5.50453 [5]
b (Å)
15.731
15.504
15.554
15.52975 [5]
c (Å)
5.640
5.559
5.576
5.56779 [5]
V
0 (Å3)
494.719
473.596
478.138
475.95682 [5]
B
0 (GPa)
99.8
117.1
109.4
—
B
0
′ (GPa)
3.3
6.5
6.4
—
E
KS
dir (eV)
2.330
2.290
2.291
—
Δ
E (meV)
−111
−109
−94
—
Given are the lattice parametersa, b, and c, the equilibrium unit-cell volume V0, the bulk modulus B0 and its pressure derivative B0′, and the direct (Γ-Γ) Kohn–Sham energy gaps EKSdir, respectively, calculated with different exchange-correlation functionals. The last row gives the energy difference ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3¯m phase.The electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state volume is shown in Figure5(c). The 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 Pm3¯m 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 Pm3¯m 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.Figure 5
NaNbO3 electronic band structures for the crystalline phases at room temperature and below calculated using the PBEsol functional revised for solids [18]. The 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.The real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure6(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.Figure 6
NaNbO3 dielectric functions for the crystalline phases at room temperature and below calculated using the PBEsol functional revised for solids [18] (solid lines). The real (green) and imaginary (orange) parts of the dielectric functions 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. For easier comparison, the dielectric functions of the high-temperature cubic perovskite Pm3¯m phase (SG 221) are included in (d). In addition, PBE0 hybrid functional calculations based on the PBEsol ground state structures are shown with dashed lines.


### 3.3. OrthorhombicPbcm Phase

The total energy curves for the orthorhombicPbcm phase are shown in Figure 2(b), calculated employing different GGA functionals. It can be seen that the difference between the AM05 and PBEsol functional is again larger compared to the simple cubic perovskite Pm3¯m phase and similar to the monoclinic Pm phase. The 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. The obtained ground state structural properties are given in Table 3.Table 3
Structural properties of the orthorhombicPbcm phase of NaNbO3.
P
b
c
m
PBE
PBEsol
AM05
Experiment
a (Å)
5.558
5.496
5.514
5.505 [37]
5.506 [6]
b (Å)
5.625
5.562
5.580
5.569 [37]
5.566 [6]
c (Å)
15.670
15.494
15.545
15.519 [37]
15.520 [6]
V
0 (Å3)
489.903
473.632
478.290
475.771 [37]
475.632 [6]
B
0 (GPa)
105.1
117.1
108.9
157.5 [38]
B
0
′ (GPa)
5.4
6.8
6.9
4.0 [38]
E
KS (eV)
2.360
2.298
2.298
3.45 [34]
Δ
E (meV)
−108
−109
−94
—
Given are the lattice parametersa, b, and c, the equilibrium unit-cell volume V0, the bulk modulus B0 and its pressure derivative B0′, and the direct (Γ-Γ) Kohn–Sham energy gaps EKSdir, respectively, calculated with different exchange-correlation functionals. The last row gives the energy difference ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3¯m phase.The electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state structure is shown in Figure5(b). The 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 Pm3¯m phase, but much larger than its indirect Kohn–Sham energy gap of 1.639 eV. However, the valence-band bandwidth is again very similar to the simple cubic perovskite Pm3¯m 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.The real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure6(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 difference in the structural properties (different orientations of the oxygen octahedra) has only little influence on the electronic and subsequently the optical properties, respectively.


### 3.4. RhombohedralR3c Phase

The total energy curves for the rhombohedralR3c phase are shown in Figure 2(a), calculated employing different 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. The obtained ground state structural properties are given in Table 4.Table 4
Structural properties of the rhombohedralR3c phase of NaNbO3.
R
3
c
PBE
PBEsol
AM05
Experiment
a (Å)
5.556
5.489
5.506
5.481 [10]
c (Å)
13.888
13.720
13.762
13.685 [10]
V
0 (Å3)
371.274
357.990
361.314
356.060 [10]
B
0 (GPa)
100.8
114.9
107.9
—
B
0
′ (GPa)
6.8
10.2
6.9
—
E
KS
dir (eV)
2.819
2.660
2.709
—
Δ
E
−120
−111
−100
—
Given are the lattice parametersa and c, the equilibrium unit-cell volume V0, the bulk modulus B0 and its pressure derivative B0′, and the direct (Γ-Γ) Kohn–Sham energy gaps EKSdir, respectively, calculated with different exchange-correlation functionals. The last row gives the energy difference ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3¯m phase.The electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state volume is shown in Figure5(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. The 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 Pm3¯m 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. The valence-band bandwidth, however, is again very similar to the simple cubic perovskite Pm3¯m, 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.The real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure6(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.


## 3.1. Simple Cubic PerovskitePm3¯m Phase

The structural properties for the simple cubic perovskitePm3¯m phase of NaNbO3 (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. The 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.Figure 2
NaNbO3 total energy curves calculated with different 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 Pm3¯m 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. The vertical black dashed lines correspond to the experimental ground state volumes.From the total energy curves, the ground state lattice constants and bulk moduliB0 have been obtained and are compared to available experimental data in Table 1.Table 1
Structural properties of the simple cubic perovskitePm3¯m phase of NaNbO3 in comparison to experimental data.
P
m
3
¯
m
PBE
PBEsol
AM05
PBE0
Experiment
a (Å)
3.975
3.944
3.947
3.931
3.9507 [28]
V
0 (Å3)
62.807
61.349
61.490
60.745
61.6626 [28]
B
0 (GPa)
176.6
188.8
185.6
204.0
136.7a
193.02 [15], 206.3b
B
0
′ (GPa)
4.4
4.6
4.6
4.3
—
E
KS
dir (eV)
2.404
2.386
2.383
4.567
—
E
KS
ind (eV)
1.652
1.639
1.642
3.756
3.29 [34]
Given are the lattice parametera, equilibrium unit-cell volume V0, the bulk modulus B0 and its pressure derivative B0′, and the direct EKSdir (Γ-Γ) and indirect EKSind (M-Γ) Kohn–Sham energy gaps, respectively, calculated with different exchange-correlation functionals. aCalculated from experimental elastic constants C11=230 GPa and C12=90 GPa [11] using (4); bcalculated from theoretical elastic constants C11=447 GPa and C12=86 GPa [13] using (4).It can be seen from Figure2(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 Pm3¯m phase of NaNbO3. The 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 [29–31]. As has been shown recently for oxide semiconductors, this might improve with a self-consistent determination of the amount of Hartree–Fock exact-exchange 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 Pm3¯m phase with the zero-temperature DFT calculations. Their GGA calculations yielded a lattice constant a=3.9516 Å and a bulk modulus B0=193.02 GPa, 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 Figure3. The most obvious result is that all three GGA functionals yield very similar electronic band structures. This 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, and the AM05 functional, respectively, and very similar valence-band bandwidths. The 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. The 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.Figure 3
NaNbO3 electronic band structures calculated for the simple cubic perovskite Pm3¯m phase employing different exchange-correlation 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).
(a)
(b)
(c)
(d)Based on the obtained relaxed ground state structures for the simple cubic perovskitePm3¯m phase, we calculated the optical properties. The real (orange) and imaginary (green) parts of the dielectric function are shown in Figure 4 for the different exchange-correlation functionals. Similarly to the electronic band structures shown in Figure 3, the three GGA functionals yield nearly indistinguishable dielectric functions. The onsets in the imaginary parts reflect the similar bandgaps already seen in the electronic band structures, and the heights and widths of the different peaks reflect 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. The most striking difference in the PBE0 calculated dielectric functions is the shifted onset in the imaginary part, reflecting 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. The 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.Figure 4
NaNbO3 dielectric functions calculated using different exchange-correlation potentials. The 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].


## 3.2. MonoclinicPm Phase

The total energy curves for the monoclinicPm phase are shown in Figure 2(c), calculated employing different GGA functionals. It can be seen that the difference between AM05 and PBEsol functional are larger compared to the simple cubic perovskite Pm3¯m phase and that the conventional PBE functional again overestimates the unit-cell volume. While the PBEsol functional yields a slightly smaller ground state volume compared to the experiment, the AM05 functional ground state volume lies slightly above. The obtained ground state structural properties are given in Table 2.Table 2
Structural properties of the monoclinicPm phase of NaNbO3.
P
m
PBE
PBEsol
AM05
Experiment
a (Å)
5.576
5.495
5.513
5.50453 [5]
b (Å)
15.731
15.504
15.554
15.52975 [5]
c (Å)
5.640
5.559
5.576
5.56779 [5]
V
0 (Å3)
494.719
473.596
478.138
475.95682 [5]
B
0 (GPa)
99.8
117.1
109.4
—
B
0
′ (GPa)
3.3
6.5
6.4
—
E
KS
dir (eV)
2.330
2.290
2.291
—
Δ
E (meV)
−111
−109
−94
—
Given are the lattice parametersa, b, and c, the equilibrium unit-cell volume V0, the bulk modulus B0 and its pressure derivative B0′, and the direct (Γ-Γ) Kohn–Sham energy gaps EKSdir, respectively, calculated with different exchange-correlation functionals. The last row gives the energy difference ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3¯m phase.The electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state volume is shown in Figure5(c). The 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 Pm3¯m 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 Pm3¯m 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.Figure 5
NaNbO3 electronic band structures for the crystalline phases at room temperature and below calculated using the PBEsol functional revised for solids [18]. The 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.The real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure6(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.Figure 6
NaNbO3 dielectric functions for the crystalline phases at room temperature and below calculated using the PBEsol functional revised for solids [18] (solid lines). The real (green) and imaginary (orange) parts of the dielectric functions 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. For easier comparison, the dielectric functions of the high-temperature cubic perovskite Pm3¯m phase (SG 221) are included in (d). In addition, PBE0 hybrid functional calculations based on the PBEsol ground state structures are shown with dashed lines.


## 3.3. OrthorhombicPbcm Phase

The total energy curves for the orthorhombicPbcm phase are shown in Figure 2(b), calculated employing different GGA functionals. It can be seen that the difference between the AM05 and PBEsol functional is again larger compared to the simple cubic perovskite Pm3¯m phase and similar to the monoclinic Pm phase. The 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. The obtained ground state structural properties are given in Table 3.Table 3
Structural properties of the orthorhombicPbcm phase of NaNbO3.
P
b
c
m
PBE
PBEsol
AM05
Experiment
a (Å)
5.558
5.496
5.514
5.505 [37]
5.506 [6]
b (Å)
5.625
5.562
5.580
5.569 [37]
5.566 [6]
c (Å)
15.670
15.494
15.545
15.519 [37]
15.520 [6]
V
0 (Å3)
489.903
473.632
478.290
475.771 [37]
475.632 [6]
B
0 (GPa)
105.1
117.1
108.9
157.5 [38]
B
0
′ (GPa)
5.4
6.8
6.9
4.0 [38]
E
KS (eV)
2.360
2.298
2.298
3.45 [34]
Δ
E (meV)
−108
−109
−94
—
Given are the lattice parametersa, b, and c, the equilibrium unit-cell volume V0, the bulk modulus B0 and its pressure derivative B0′, and the direct (Γ-Γ) Kohn–Sham energy gaps EKSdir, respectively, calculated with different exchange-correlation functionals. The last row gives the energy difference ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3¯m phase.The electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state structure is shown in Figure5(b). The 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 Pm3¯m phase, but much larger than its indirect Kohn–Sham energy gap of 1.639 eV. However, the valence-band bandwidth is again very similar to the simple cubic perovskite Pm3¯m 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.The real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure6(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 difference in the structural properties (different orientations of the oxygen octahedra) has only little influence on the electronic and subsequently the optical properties, respectively.


## 3.4. RhombohedralR3c Phase

The total energy curves for the rhombohedralR3c phase are shown in Figure 2(a), calculated employing different 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. The obtained ground state structural properties are given in Table 4.Table 4
Structural properties of the rhombohedralR3c phase of NaNbO3.
R
3
c
PBE
PBEsol
AM05
Experiment
a (Å)
5.556
5.489
5.506
5.481 [10]
c (Å)
13.888
13.720
13.762
13.685 [10]
V
0 (Å3)
371.274
357.990
361.314
356.060 [10]
B
0 (GPa)
100.8
114.9
107.9
—
B
0
′ (GPa)
6.8
10.2
6.9
—
E
KS
dir (eV)
2.819
2.660
2.709
—
Δ
E
−120
−111
−100
—
Given are the lattice parametersa and c, the equilibrium unit-cell volume V0, the bulk modulus B0 and its pressure derivative B0′, and the direct (Γ-Γ) Kohn–Sham energy gaps EKSdir, respectively, calculated with different exchange-correlation functionals. The last row gives the energy difference ΔE per functional unit cell with respect to the high-temperature simple cubic perovskite Pm3¯m phase.The electronic band structure calculated using the PBEsol functional and based on the PBEsol ground state volume is shown in Figure5(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. The 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 Pm3¯m 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. The valence-band bandwidth, however, is again very similar to the simple cubic perovskite Pm3¯m, 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.The real (green) and imaginary (orange) parts of the dielectric function calculated using the PBEsol functional (solid lines) are shown in Figure6(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.


## 4. Conclusions

In summary, we presented a detailed DFT investigation on the structural properties of the crystalline phases of NaNbO3 at room temperature and below, end member in the solid solution sodium potassium niobate (Na,K)NbO3, and a promising lead-free alternative to PZT. The calculations assessed the performance of different GGA exchange-correlation functionals, namely, the conventional PBE parametrisation, the AM05 parametrisation, and the PBEsol parametrisation. For the simple cubic perovskite Pm3¯m 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 approximation. This is in line with similar investigations for the other end member KNbO3 [21] of the solid solution sodium potassium niobate (Na,K)NbO3. 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. This can serve as a basis for future theoretical works, calculating the spontaneous polarisations in the different phases of NaNbO3 at room temperature and below or to investigate the influence of strain effects on the structural, electronic, and optical properties, respectively.



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*Source: 6416057-2018-03-07.xml*