Defect, Di ﬀ usion and Dopant Properties of NaNiO 2 : Atomistic Simulation Study

: Sodium nickelate, NaNiO 2 , is a candidate cathode material for sodium ion batteries due to its high volumetric and gravimetric energy density. The use of atomistic simulation techniques allows the examination of the defect energetics, Na-ion di ﬀ usion and dopant properties within the crystal. Here, we show that the lowest energy intrinsic defect process is the Na-Ni anti-site. The Na Frenkel, which introduces Na vacancies in the lattice, is found to be the second most favourable defect process and this process is higher in energy only by 0.16 eV than the anti-site defect. Favourable Na-ion di ﬀ usion barrier of 0.67 eV in the ab plane indicates that the Na-ion di ﬀ usion in this material is relatively fast. Favourable divalent dopant on the Ni site is Co 2 + that increases additional Na, leading to high capacity. The formation of Na vacancies can be facilitated by doping Ti 4 + on the Ni site. The promising isovalent dopant on the Ni site is Ga 3 + .


Introduction
In the field of energy storage, a significant amount of research activity has been devoted to Li-ion batteries [1][2][3][4][5][6][7][8][9][10]. As the global distribution of lithium is limited and inhomogeneous, there is a necessity to find an alternative to the lithium ion battery for the next-generation of high capacity energy storage systems, particularly in the hybrid electric vehicles.
Rechargeable sodium ion batteries have become promising for the application in large-scale energy storage devices due to the remarkable abundance of sodium on the earth. The performance of a sodium ion battery (SIB) relies on several factors. Developing cathode materials exhibiting cheap, safe and high energy density is one of the key steps for constructing promising rechargeable SIBs. A variety of sodium-based cathode materials have been examined experimentally and a limited number of theoretical works have been reported in the literature [4,5,[11][12][13][14][15][16][17][18][19][20]. For portable applications, a few of them can be promising as the ionic radius of Na is much larger than the ionic radius of Li. Larger size of Na will also affect Na + ion intercalation and diffusion leading to degradation in the specific capacity and rate capacity [21]. Electrochemical stability would be difficult due to the volume expansion caused by Na + insertion. Furthermore, low potential and large atomic weight of Na lead to the specific energy reduction in the battery [4].

Computational Methods
Classical pair-wise potential calculations for the bulk and defective NaNiO 2 structures were carried out using the General Utility Lattice Program (GULP) code [49]. In this method, total energy (lattice energy) of the system is calculated using long range (Coulombic) and short range (electron-electron repulsive and attractive intermolecular forces). Buckingham potentials (refer to Supplementary Materials) were used to model short range forces. Full structural relaxations (both ion positions and lattice constants) were performed using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [50]. In all relaxed configurations, forces on each atoms were <0.001 eV/Å. Point defects were modelled using the Mott-Littleton method [51]. Full charge with spherical shape of ions at dilute limit can overestimate the defect formation enthalpies. Nevertheless, the trend will be the same. Isobaric parameters were used in the present simulation to calculate the formation and migration energies. In our previous theoretical work, we have discussed the detailed thermodynamic relations associated with isobaric parameters [52][53][54].
Equilibrium lattice constants were obtained by performing full geometry optimisation (both ion positions and lattice constants). The calculated lattice constants are reported in Table 1. There is a good agreement between the expertiment and calculation ensuring the choice of potentials used in this study.

Intrinsic Defect Processes
Intrinsic defects play an important role in the migration of ions in materials. Here we considered point defects in the form of vacancies, interstitials, and anti-sites. They were then combined to calculate formation energies for Frenkel and Schottky defects. Anti-site defect energies were calculated by interchanging the cations (Ni on Na and Na on Ni). The reaction energies for those defects are written by using Kröger-Vink notation [55].
Na/Ni antisite (cluster): Na + Ni → {Na : Ni •• } X (7) Figure 2 shows the defect reaction energies per defect. The present calculations show that the Na-Ni anti-site defect cluster is the most favourable defect (1.86 eV). This suggests that there is a possibility of a small amount of Na on Ni sites (Na ) and Ni on Na sites (Ni ⦁⦁ ). This defect has been identified theoretically as a most promising defect in a variety of oxide materials [31][32][33][34][35][36][37][38][39][40][41][42]. During the preparation of as-prepared battery materials and the charge-discharge process, the presence of this defect was noted [35,[56][57][58][59]. The Na Frenkel is the second most thermodynamically favourable defect process and the defect energy is only 0.16 eV higher than the anti-site defect energy. In the Na Frenkel process, Na vacancies would be created and this process would facilitate vacancy mediated Na-ion diffusion in NaNiO2. The reaction energy for the formation of Na2O Schottky (relation 5) is 2.16 eV/defect. This process is also competitive as the defect energy is close to the defect energy calculated

Intrinsic Defect Processes
Intrinsic defects play an important role in the migration of ions in materials. Here we considered point defects in the form of vacancies, interstitials, and anti-sites. They were then combined to calculate formation energies for Frenkel and Schottky defects. Anti-site defect energies were calculated by interchanging the cations (Ni on Na and Na on Ni). The reaction energies for those defects are written by using Kröger-Vink notation [55].
Schottky : Na X Na Na/Ni antisite (cluster) : Na X Na + Ni X Ni → Na Ni : Ni •• Na X (7) Figure 2 shows the defect reaction energies per defect. The present calculations show that the Na-Ni anti-site defect cluster is the most favourable defect (1.86 eV). This suggests that there is a possibility of a small amount of Na on Ni sites (Na Ni ) and Ni on Na sites (Ni •• Na ). This defect has been identified theoretically as a most promising defect in a variety of oxide materials [31][32][33][34][35][36][37][38][39][40][41][42]. During the preparation of as-prepared battery materials and the charge-discharge process, the presence of this defect was noted [35,[56][57][58][59]. The Na Frenkel is the second most thermodynamically favourable defect process and the defect energy is only 0.16 eV higher than the anti-site defect energy. In the Na Frenkel process, Na vacancies would be created and this process would facilitate vacancy mediated Na-ion diffusion in NaNiO 2 . The reaction energy for the formation of Na 2 O Schottky (relation 5) is 2.16 eV/defect. This process is also competitive as the defect energy is close to the defect energy calculated for the anti-site and the Na Frenkel. The formation of further V Na and V •• O can be enhanced by the Na 2 O Schottky process at moderate temperatures.

Sodium Ion Diffusion
As Na-ion diffusion rate strongly influences the cathode performance, it is necessary to examine the sodium ion diffusion pathways together with activation energies. Experimental characterisation of diffusion pathways can be quite challenging. Current simulation techniques enabled us to identify the Na-ion diffusion pathways with activation energies at the atomic level. We identified a promising Na local hop with the Na-Na seperation of 2.95 Å. Its activation energy of migration is calculated to be 0.67 eV. Figure 3 shows the long range Na-ion difussion pathway constructed by connecting local Na hops and the corresponding energy profile diagram for the local Na hop. The long range Na-ion diffusion pathway is in the ab plane exhibiting a non-linear pattern with overall activation energy of 0.67 eV. The present simulation results show that Na-ion diffusion will be significant in this material.

Sodium Ion Diffusion
As Na-ion diffusion rate strongly influences the cathode performance, it is necessary to examine the sodium ion diffusion pathways together with activation energies. Experimental characterisation of diffusion pathways can be quite challenging. Current simulation techniques enabled us to identify the Na-ion diffusion pathways with activation energies at the atomic level. We identified a promising Na local hop with the Na-Na seperation of 2.95 Å. Its activation energy of migration is calculated to be 0.67 eV. Figure 3 shows the long range Na-ion difussion pathway constructed by connecting local Na hops and the corresponding energy profile diagram for the local Na hop.

Sodium Ion Diffusion
As Na-ion diffusion rate strongly influences the cathode performance, it is necessary to examine the sodium ion diffusion pathways together with activation energies. Experimental characterisation of diffusion pathways can be quite challenging. Current simulation techniques enabled us to identify the Na-ion diffusion pathways with activation energies at the atomic level. We identified a promising Na local hop with the Na-Na seperation of 2.95 Å. Its activation energy of migration is calculated to be 0.67 eV. Figure 3 shows the long range Na-ion difussion pathway constructed by connecting local Na hops and the corresponding energy profile diagram for the local Na hop. The long range Na-ion diffusion pathway is in the ab plane exhibiting a non-linear pattern with overall activation energy of 0.67 eV. The present simulation results show that Na-ion diffusion will be significant in this material. The long range Na-ion diffusion pathway is in the ab plane exhibiting a non-linear pattern with overall activation energy of 0.67 eV. The present simulation results show that Na-ion diffusion will be significant in this material.

Solution of Divalent Dopants
Additional Na in the as-prepared NaNiO 2 material would increase the capacity and the rate of Na-ion diffusion. The later is due to the shorter Na-Na distances arising from additional Na in the material than those present in the defect-free structure. Divalent doping on the Ni site is a promising stratergy to introduce Na interstitials as explained in the equation 8. A similar stratergy was applied to Li, Na, and Mg-ion battery materials in our previous theoretical studies [36][37][38][39][40][41][42][43][44][45][46][47]. The solution of MO (M = Mg, Co, Fe, Ca, Sr and Ba) was considered using the following reaction: Solution enthalpies are reported in Figure 4. In all cases solution enthalpies are endoergic suggesting that this process should be carried out under thermal condition. The lowest solution enthalpy is calculated for Co 2+ (3.89 eV). It should be noted that the solution enthalpies of MgO (3.93 eV) and FeO (4.00 eV) are closer to the values calculated for CoO indicating that Mg 2+ and Fe 2+ are also candidate dopants. The possible formula for the Co-doped composite is NaCo x Ni 1−x O 2 (0.00 < x < 1.00). The exact value of x can be determined by experiments. There is a gradual increase in the solution enthalpies from Ca to Ba. Solution enthalpies of CaO and SrO are 5.01 eV and 7.36 eV respectively. Solution enthalpy of BaO is highly positive (~10 eV) meaning that this process cannot be executed at operating temperatures.

Solution of Divalent Dopants
Additional Na in the as-prepared NaNiO2 material would increase the capacity and the rate of Na-ion diffusion. The later is due to the shorter Na-Na distances arising from additional Na in the material than those present in the defect-free structure. Divalent doping on the Ni site is a promising stratergy to introduce Na interstitials as explained in the equation 8. A similar stratergy was applied to Li, Na, and Mg-ion battery materials in our previous theoretical studies [36][37][38][39][40][41][42][43][44][45][46][47]. The solution of MO (M = Mg, Co, Fe, Ca, Sr and Ba) was considered using the following reaction: Solution enthalpies are reported in Figure 4. In all cases solution enthalpies are endoergic suggesting that this process should be carried out under thermal condition. The lowest solution enthalpy is calculated for Co 2+ (3.89 eV). It should be noted that the solution enthalpies of MgO (3.93 eV) and FeO (4.00 eV) are closer to the values calculated for CoO indicating that Mg 2+ and Fe 2+ are also candidate dopants. The possible formula for the Co-doped composite is NaCoxNi1−xO2 ( 0.00 < x < 1.00). The exact value of x can be determined by experiments. There is a gradual increase in the solution enthalpies from Ca to Ba. Solution enthalpies of CaO and SrO are 5.01 eV and 7.36 eV respectively. Solution enthalpy of BaO is highly positive (~10 eV) meaning that this process cannot be executed at operating temperatures.

Solution of Trivalent Dopants
Here, a wide range of trivalent dopants (M = Al, Ga, Fe, In, Sc,Y, Gd, and La) were considered at the Ni site. These dopants are isovalent to Ni because the oxidation state of Ni in NaNiO2 is +3. Isovalent doping process is important as it can control the point defects and tune the electronic properties of this material. The following reaction equation was used to calculate the solution enthalpy.

M O + 2Ni → 2M + Ni O
The results reveal that Ga 3+ is the most favourable dopant at the Ni site (refer to Figure 5). Its solution enthalpy is 0.05 eV. The second most favourable dopant is Al 3+ having the solution enthalpy of 0.12 eV. We further predict that Fe 3+ is also worth investigating as its solution enthalpy (0.14 eV) is very close to the solution enthalpies calculated for Ga 3+ and Al 3+ . The solution enthalpy for In2O3 is 1.17 eV. There is a reduction in the solution enthalpy for Sc2O3 though the ionic radius of Sc is larger than that of In. There is a gradual increase in the solution enthalpy from Y to La. As they exhibit relatively high solution enthalpies, they are highly unlikely to substitute the Ni.

Solution of Trivalent Dopants
Here, a wide range of trivalent dopants (M = Al, Ga, Fe, In, Sc, Y, Gd, and La) were considered at the Ni site. These dopants are isovalent to Ni because the oxidation state of Ni in NaNiO 2 is +3. Isovalent doping process is important as it can control the point defects and tune the electronic properties of this material. The following reaction equation was used to calculate the solution enthalpy.
The results reveal that Ga 3+ is the most favourable dopant at the Ni site (refer to Figure 5). Its solution enthalpy is 0.05 eV. The second most favourable dopant is Al 3+ having the solution enthalpy of 0.12 eV. We further predict that Fe 3+ is also worth investigating as its solution enthalpy (0.14 eV) is very close to the solution enthalpies calculated for Ga 3+ and Al 3+ . The solution enthalpy for In 2 O 3 is 1.17 eV. There is a reduction in the solution enthalpy for Sc 2 O 3 though the ionic radius of Sc is larger than that of In. There is a gradual increase in the solution enthalpy from Y to La. As they exhibit relatively high solution enthalpies, they are highly unlikely to substitute the Ni.

Solution of Tetravalent Dopants
We also considered tetravalent dopants (M = Si, Ge, Ti, Sn, Zr, and Ce) at the Ni site. This process required the creation of Na vacancies according to the following reaction equation: The formation of Na vacancies would facilitate Na ion migration in this material. Figure 6 reports the solution enthalpies of MO2. The promising candidate for this process is Ti 4+ as this dopant exhibits the lowest solution enthalpy (+0.05 eV). As there is a very small enthalpy difference between the Ge 4+ , and Ti 4+ , Ge 4+ can also be considered for experimental verification. High solution enthalpy for SiO2 can be due to the smaller radius of Si 4+ (0.40 Å) ion compared to that of Ni 3+ (0.56 Å). Solution enthalpy increases with ionic radius from Sn to Ce. The highest solution enthalpy (5.38 eV) is observed for Ce implying it could be an important dopant only at high temperatures.

Diffusion of Na-Ion in the Presence of Dopants
Here we calculate the activation energies for Na-ion diffusion when promising dopants (Co 2+ , Ga 3+ and Ti 4+ ) are present on the Ni site. Figure 7 shows the energy profile diagrams for Na hops in each cases.

Solution of Tetravalent Dopants
We also considered tetravalent dopants (M = Si, Ge, Ti, Sn, Zr, and Ce) at the Ni site. This process required the creation of Na vacancies according to the following reaction equation: The formation of Na vacancies would facilitate Na ion migration in this material. Figure 6 reports the solution enthalpies of MO 2 . The promising candidate for this process is Ti 4+ as this dopant exhibits the lowest solution enthalpy (+0.05 eV). As there is a very small enthalpy difference between the Ge 4+ , and Ti 4+ , Ge 4+ can also be considered for experimental verification. High solution enthalpy for SiO 2 can be due to the smaller radius of Si 4+ (0.40 Å) ion compared to that of Ni 3+ (0.56 Å). Solution enthalpy increases with ionic radius from Sn to Ce. The highest solution enthalpy (5.38 eV) is observed for Ce implying it could be an important dopant only at high temperatures.

Solution of Tetravalent Dopants
We also considered tetravalent dopants (M = Si, Ge, Ti, Sn, Zr, and Ce) at the Ni site. This process required the creation of Na vacancies according to the following reaction equation: The formation of Na vacancies would facilitate Na ion migration in this material. Figure 6 reports the solution enthalpies of MO2. The promising candidate for this process is Ti 4+ as this dopant exhibits the lowest solution enthalpy (+0.05 eV). As there is a very small enthalpy difference between the Ge 4+ , and Ti 4+ , Ge 4+ can also be considered for experimental verification. High solution enthalpy for SiO2 can be due to the smaller radius of Si 4+ (0.40 Å) ion compared to that of Ni 3+ (0.56 Å). Solution enthalpy increases with ionic radius from Sn to Ce. The highest solution enthalpy (5.38 eV) is observed for Ce implying it could be an important dopant only at high temperatures.

Diffusion of Na-Ion in the Presence of Dopants
Here we calculate the activation energies for Na-ion diffusion when promising dopants (Co 2+ , Ga 3+ and Ti 4+ ) are present on the Ni site. Figure 7 shows the energy profile diagrams for Na hops in each cases.

Diffusion of Na-Ion in the Presence of Dopants
Here we calculate the activation energies for Na-ion diffusion when promising dopants (Co 2+ , Ga 3+ and Ti 4+ ) are present on the Ni site. Figure 7 shows the energy profile diagrams for Na hops in each cases. In the case of Co 2+ there is a reduction (by 0.10 eV) in the activation energy. This is due to the reduction in the Na-Na distance (2.91 Å) compared to that of pure crystal structure (2.95 Å). This perturbation in the distance can be due to the Co 2+ occupying the Ni 3+ (charge mismatch). Doping of Ga on the Ni site does not affect the activation energy as there is no change in the Na-Na distance. This is due to the similar charge (+3) of Ga and Ni. There is a slight elongation noted in the Na-Na distance when Ti 4+ is present on the Ni site. This reflects in the increment of activation energy by 0.14 eV. The ionic radii of Ni 3+ , Co 2+ , Ga 3+ and Ti 4+ are 0.56 Å, 0.58 Å, 0.47 Å and 0.61 Å respectively. It is also noted that the contribution of ionic radius mismatch in the activation energy is less significant.
Considering the solution enthalpies and activation energies for the promising dopants, it is noted that Ga 3+ doping at the Ti site is the most favourable as this dopant exhibits the lowest solution enthalpy of 0.05 eV and the same activation energy of 0.67 eV for Na-ion diffusion calculated in the un-doped NaNiO2. The second most favourable dopant is Ti 4+ as its solution enthalpy is 0.06 eV and this dopant increases the activation energy by 0.14 eV. Though there is a reduction in the activation energy (by 0.10 eV) upon Co 2+ doping, the solution enthalpy is significantly high (3.89 eV) meaning that this dopant is not promising.

Conclusions
The present atomistic simulation study enabled us to gain insights into the defect chemistry, Naion diffusion pathways with activation energies, and dopant properties in NaNiO2. The lowest energy defect is the Na-Ni anti-site, in which Na and Ni would exchange their positions at low concentration. The Na Frenkel is the second most favourable defect process. Long range Na-ion diffusion is in the ab plane with the activation energy of 0.67 eV. This suggests that there will be a significant Na-ion conduction in this material. Doping of Co 2+ and Ti 4+ at the Ni site would increase the concentration of Na interstitials and Na vacancies respectively. The most favourable isovalent dopant at the Ni site is Ga 3+ . Considering the solution enthalpies and activation energies for the promising dopants, it is found that Ga 3+ doping at the Ti site is the most favourable. This theoretical prediction should motivate future experimental work on this potentially important material.
Supplementary Materials: The following are available online at www.mdpi.com/xxx/s1, Table S1: Interatomic potential parameters used in the atomistic simulations of NaNiO2.
Funding: This research was financially supported by European Union's H2020 Programme under Grant Agreement no 824072-HARVESTORE.
In the case of Co 2+ there is a reduction (by 0.10 eV) in the activation energy. This is due to the reduction in the Na-Na distance (2.91 Å) compared to that of pure crystal structure (2.95 Å). This perturbation in the distance can be due to the Co 2+ occupying the Ni 3+ (charge mismatch). Doping of Ga on the Ni site does not affect the activation energy as there is no change in the Na-Na distance. This is due to the similar charge (+3) of Ga and Ni. There is a slight elongation noted in the Na-Na distance when Ti 4+ is present on the Ni site. This reflects in the increment of activation energy by 0.14 eV. The ionic radii of Ni 3+ , Co 2+ , Ga 3+ and Ti 4+ are 0.56 Å, 0.58 Å, 0.47 Å and 0.61 Å respectively. It is also noted that the contribution of ionic radius mismatch in the activation energy is less significant.
Considering the solution enthalpies and activation energies for the promising dopants, it is noted that Ga 3+ doping at the Ti site is the most favourable as this dopant exhibits the lowest solution enthalpy of 0.05 eV and the same activation energy of 0.67 eV for Na-ion diffusion calculated in the un-doped NaNiO 2 . The second most favourable dopant is Ti 4+ as its solution enthalpy is 0.06 eV and this dopant increases the activation energy by 0.14 eV. Though there is a reduction in the activation energy (by 0.10 eV) upon Co 2+ doping, the solution enthalpy is significantly high (3.89 eV) meaning that this dopant is not promising.

Conclusions
The present atomistic simulation study enabled us to gain insights into the defect chemistry, Na-ion diffusion pathways with activation energies, and dopant properties in NaNiO 2 . The lowest energy defect is the Na-Ni anti-site, in which Na and Ni would exchange their positions at low concentration. The Na Frenkel is the second most favourable defect process. Long range Na-ion diffusion is in the ab plane with the activation energy of 0.67 eV. This suggests that there will be a significant Na-ion conduction in this material. Doping of Co 2+ and Ti 4+ at the Ni site would increase the concentration of Na interstitials and Na vacancies respectively. The most favourable isovalent dopant at the Ni site is Ga 3+ . Considering the solution enthalpies and activation energies for the promising dopants, it is found that Ga 3+ doping at the Ti site is the most favourable. This theoretical prediction should motivate future experimental work on this potentially important material.
Funding: This research was financially supported by European Union's H2020 Programme under Grant Agreement no 824072-HARVESTORE.