DFT+U and Quantum Monte Carlo study of electronic and optical properties of AgNiO₂ and AgNi_1-xCo_xO₂ delafossite

Previous DFT studies of AgNiO₂ delafossites assumed that on-site Coulomb interactions were not important and so did not use DFT+U. Wawrzyńska et al. (2007, 2008) One of the aims of our work is to examine the role of on-site Coulomb interactions in detail in order to ascertain their importance. Because there are no previously reported values for an optimal value of U, $U_{\rm opt}$, we first estimated $U_{\rm opt}$. We used a procedure established in previous worksFoyevstova et al. (2014); Luo et al. (2016); Shin et al. (2017, 2018); Saritas et al. (2019) that has proven to be a reliable and unbiased way to estimate $U_{\rm opt}$ for transition-metal oxides. In this procedure, we minimize the DMC total energy of the PBE+U trial wavefunction as a function of U. Because the DMC total energy obeys a variational principle, this energy will exhibit a minimum. Specifically, in DMC, the minimization of the energy with respect to U is a one-parameter optimization of the many-body wavefunction nodal surface. For simplicity we assume the same U value for all Ni atoms, regardless of their local charge states. Figure 2 shows the DMC total energy for the AFM AgNiO₂ unit cell as function of the value of U in the PBE+U trial wavefunction. Using a quartic fit, we estimated an optimal U value of $U_{\rm opt}=4.4(1)$ eV for Ni, which is close to the DMC $U_{opt}=4.7(2)$ eV found in AFM NiO. Shin et al. (2017) We use the value of U = 4.4 eV obtained from DMC for all subsequent PBE+U calculations in this study.

To investigate how varied $p$-$d$ hybridization within the DFT+U scheme may change the electronic properties of AgNiO₂, we first compare the electron density-of-states (DOS) obtained using PBE and PBE+U. As expected, the DOS for 2H-AgNiO₂ clearly exhibits metallic features with filled states at the Fermi level both for PBE and PBE+U (Fig. 3). For both levels of theory, we can see a small gap beginning about 1 eV above the Fermi level, which suggests the possibility of tuning metallic 2H-AgNiO₂ to a semiconductor by tuning this gap to open at the Fermi level through, e.g., hole doping. This gap is wider and closer to the Fermi level in PBE+U, as shown in Fig. 3(b). In addition, we confirmed that the Hubbard U leads to more semimetallic electronic properties of AgNiO₂ as the conduction band minimum in PBE+U is closer to the Fermi level with lower DOS than in PBE. This suggests that localized Ni 3$d$ orbitals induce a semimetallic nature in AgNiO₂, and that AgNiO₂ possesses an intriguing potential of tuning the band gap to a semiconductor or insulator.

For further analyses of the effects of U on semimetallic AgNiO₂, we compared total charge and spin densities obtained from PBE and PBE+U. Figure 4(a) and (b) show significant differences in both charge and spin densities between PBE+U and PBE near the Ni sites – accumulation and depletion can be found near the Ni sites in both the charge and spin density differences. The charge density differences between PBE+U and PBE induced by the Hubbard U are mainly located on the octahedral NiO₆ structures, with no significant changes near the Ag sites. Within the NiO₂ layers, there is a rather pronounced charge density redistribution induced by Hubbard U on the Ni-O bond. This shows that the Hubbard U strongly affects the $p$-$d$ hybridization of the Ni-O bonds, even though AgNiO₂ is in a semimetallic phase. Among the Ni sites, there is a large charge accumulation on the Ni²⁺ sites (Ni2) that also possess large magnetic moments. This indicates that there is discrepancy between the magnetic moments obtained by PBE and PBE+U, as the Hubbard U affects the magnetic moment on Ni. In addition to the charge density difference, we can also see that PBE underestimates the spin density on Ni sites relative to PBE+U (see Fig. 4), which is analogous to results obtained in an earlier DMC study of AFM NiO,Shin et al. (2017), although the spin density difference is smaller for AgNiO₂ than for insulating NiO. As assumed, it is clear that influence of the Hubbard U is not as large in semimetallic AgNiO₂ compared to its effect in insulating NiO; however, we conclude that the existence of localized 3$d$ orbitals is still leads to moderate effects in AgNiO₂ because of the large density differences between PBE and PBE+U.

In order to further accurately assess the electronic properties of 2H-AgNiO₂, we performed DMC calculations of AgNiO₂ using a PBE+U trial wavefunction with the optimal value of U. We estimated the cohesive energy of AgNiO₂ by computing $E(AgNiO_{2})-E(Ag)-E(Ni)-2E(O)$, where $E(AgNiO_{2})$, $E(Ag)$, $E(Ni)$, $E(O)$ are the DMC total energy of AgNiO₂ and that of atomic Ag, Ni, and O, respectively. The computed DMC AgNiO₂ cohesive energy with full incorporation of the finite-size analysis is 14.23(3) eV/f.u., which is significantly smaller than the PBE result of 15.21 eV/f.u. but consistent with PBE+U one of 14.21 eV/f.u.. Significantly larger PBE cohesive energy than DMC seems to be related with overestimation of NiO cohesive energy compared to corresponding experimental result within PBE functionals.Shin et al. (2017) Although experimental values of the AgNiO₂ cohesive energy are not available to the best of our knowledge, the large differences in cohesive energy clearly shows a large discrepancy between the DMC, DFT, and DFT+U schemes in dealing with the electronic structure of AgNiO₂. The charge density difference between DMC and PBE+U, $\rho({\rm DMC})-\rho({\rm PBE+U})$, shows a charge density accumulation on Ni-O complexes in DMC relative to PBE+U, somewhat similar to the charge density difference $\rho({\rm PBE+U})-\rho({\rm PBE})$, but the charge accumulation in $\rho({\rm DMC})-\rho({\rm PBE+U})$ is concentrated on specific Ni-O pairs in the $yz$ plane, while density difference $\rho({\rm PBE+U})-\rho({\rm PBE})$ is more spread out over the entire NiO₆ layer. From this anisotropic density accumulation in DMC relative to PBE+U, we suspect there is a similar symmetry-breaking in the Ni-O bonds to that already seen in DMC studies of NiO and HfO₂. Shin et al. (2017); Chimata et al. (2019) In Fig. 4(d) and (f), we see strong spin accumulation and depletion only on the AFM Ni sites. This tells us that magnetic moment on the Ni sites is significantly underestimated in both PBE and PBE+U compared to DMC. In order to compare the DMC and DFT magnetic moments, we computed the magnetic moments on Ni sites as function of U. Figure 5 shows that the DFT magnetic moment increases monotonically on the AFM Ni sites Ni2 as U increases. However, even at large values of U, up to 6 eV where PBE+U magnetic moment shows the largest value, the PBE+U moment is still smaller than DMC magnetic moment. The estimated DMC magnetic moment on the AFM Ni sites is 1.71(1) $\mu_{B}$, which is slightly larger but in the good agreement with the reported local magnetization of Ni, 1.552(7) $\mu_{B}$. Wheeler et al. (2009) We see that PBE+U magnetic moment shows empirically closest result with the experimental one in U $\sim$ 2 eV with 1.58 $\mu_{B}$ while PBE without U exhibits smaller value of 1.46 $\mu_{B}$. From this analysis, we conclude that the Hubbard U significantly affects the band gap and magnetic moment of 2H-AgNiO₂, and the addition of a Hubbard U is necessary in order to achieve reasonably accurate magnetic moment and charge density within DFT.

The PBE+U and DMC results for AgNiO₂ described in the previous section indicate the possibility of a metal-insulator transition based on the observations both in PBE+U and DMC of a small of electron density from states in the conduction band just below the Fermi level. This suggests that the introduction of dopants or other defects into stoichiometric AgNiO₂ may provide a path to move the conduction band minimum above the Fermi level. Various transition-metal doped delafossites have in fact been studied previously as transition-metal doping has been known to enhance $p$-type semiconductor properties. Among various transition metal candidates, we consider here Co as a dopant and investigate how Co doping influences the electronic properties and band gap opening in 2H-AgNiO₂. When studying these intermetallics, it is crucial first to obtain an accurate structure as the equilibrium structure and geometry vary with the concentration of dopants and with their locations, and the electronic structure in turn depends strongly on the geometry of the structure. We attempted to obtain a good quality lattice structure for AgNi_1-xCo_xO₂ by considering both the pure AgNiO₂ geometry but with dopants on Ni1 sites, and a fully relaxed structure within DFT+U framework. To compare these two geometries and to choose an energetically stable geometry for the intermetallic, we estimated the DMC total energy for these structures. The result is that the pure 2H-AgNiO₂ structure with Co on the Ni1 sites exhibits a lower FN-DMC energy than the fully relaxed structure. Therefore, we used the pure 2H-AgNiO₂ as a structure for the intermetallic AgNi_1-xCo_xO₂. Details in FN-energy comparison between different geometries are in Supplemental information.^† In order to optimize the trial wavefunction for AgNi_1-xCo_xO₂, we determined an optimal U-value of 4.0(1) eV for the Co dopants by minimizing the DMC total energy for the 2H-AgCoO₂ structure (see Supplemental Information).^† Among potentially available Co concentrations of AgNi_1-xCo_xO₂, we first considered AgNi_0.66Co_0.33O₂ structure. Although the existence of MIT can be expected on AgNi_0.66Co_0.33O₂ since it is experimentally reported on $x\sim 0.3$,Shin et al. (1993) the energetic stability of phases that result from random Co substitutions at various Co concentration is unclear. In order to investigate the relative stability of various random phases and the dependencies of their electronic and optical properties on substitutional sites, we considered additional phases of AgNi_0.66Co_0.33O₂ wherein four Ni⁺² Ni1 sites out of a total of eight are replaced by Co dopants. We did not consider substitution of Co on the AFM Ni2 sites because calculations showed that this leads to a collapse of the magnetic order and an energetically unstable structure of the mixture. Because there are too many AgNi_0.66Co_0.33O₂ configurations with four Co atoms on eight possible sites to make comprehensive DMC calculations practical, we selected only four different phases, shown in Fig. 6.

We first compute the PBE+U density-of-states of these four phases in order to compare their optical properties. As can be seen in Fig. 7, the optical properties of the AgNi_0.66Co_0.33O₂ mixture depends strongly on which of the Ni1 sites are substituted with Co. Phases 1 and 4 show completely closed band gaps and metallic densities-of-states; however, phases 2 and 3 exhibit open band gaps. Because of the completely different electronic properties of the four phases, with phases 1 and 4 metallic and phase 2 and 3 semiconductor-like, and the very large differences in densities-of-states near the Fermi level, we conclude that the electronic and optical properties vary strongly with the specific sites used for Co-substitution, and the detailed properties of AgNi_0.66Co_0.33O₂ can potentially be controlled by selectively choosing the sites for substitution.

Because there are many possible metallic and semiconducting phases of AgNi_0.66Co_0.33O₂, it is important to find the most stable one. We estimated the DMC total energy of four candidates based on symmetry. Figure 8 shows the PBE+U and the DMC total energy differences between the four phases with the energy (PBE+U and DMC, respectively) of phase 1 as reference at zero total energy. As can be seen in the figure, the semiconducting phases 2 and 3 have lower DMC total energy than the metallic phases 1 and 4, indicating that the semiconducting phases are more energetically favored and stable than metallic ones for the AgNi_0.66Co_0.33O₂ mixture. There is a large DMC energy difference between the metallic phases 1 and 4 in DMC and a smaller PBE+U energy difference, but a relatively small energy difference between phases 2 and 3 both for PBE+U and DMC. The much smaller PBE+U energy difference between the metallic and semiconducting phases than the DMC energy difference, about 0.05 eV/f.u. and 0.21(1) eV/f.u., respectively, strongly suggests that the semiconducting phases driven by Co-substitution are due to electron correlations between the Co and Ni sites, effects that are well accounted for in DMC but not as accurately in DFT or DFT+U. From lower PBE+U and DMC total energies on semiconducting phases than semimetallic ones confirmed the existence of MIT, transiting favored phase from semimetallic on pristine AgNiO₂ to semiconducting phase on AgNi_0.66Co_0.33O₂. In addition, since coexistence of semimetallic and semiconducting phase is observed at the concentration of $x=0.33$, we assume that the critical Co concentration of MIT is located nearby $x=0.33$, which is consistent with the experimental measurement of MIT on $x=0.3$. Shin et al. (1993) On the other hand, we see the metallic phase in higher Co concentration on the single NiO₆ layer than $x=0.33$ as seen in both phase 1 and 4. Since the varied structure are in-layer density fluctuations and each of these contain a layer at higher Co concentration, these results leads us to suspect the existence of reentrant phase to the metallic phase on high Co concentration over $x=0.33$.

In order to investigate optical properties on high Co concentration for AgNi_1-xCo_xO₂, we additionally considered high Co concentration of $x=0.66$, substituting all non-magnetic Ni1 sites in a hexagonal pattern (see Fig. 9). This $x=0.66$ seems a hypothetical structure at a concentration where no experimental result for the electronic and optical properties has been reported.

The PBE+U density-of-states of AgNi_0.33Co_0.66O₂ (see Fig. 10(a)) shows that Co-doping moves the valence band edge very close to the Fermi level, and at the conduction band edge, 3$d$-Co states have fully replaced $d$-Ni ones. Although the valence band edge still lies above Fermi level, the closeness of the band edge to the Fermi levels suggests that Co-substitution on the Ni^+3.5 Ni1 sites results in the electronic properties of AgNi_0.33Co_0.66O₂ moving from those of a semimetal closer to those of an insulator.

Figure 3(b) shows the DMC spin density difference between AgNi_0.33Co_0.66O₂ and AgNiO₂. The figure shows a density changes on the AFM Ni sites Ni2, but density change is opposite in sign to the induced AFM magnetic moments. This tells us that the magnetic moments on Ni sites on AgNi_0.33Co_0.66O₂ are smaller than in pristine AgNiO₂, and this is confirmed by a DMC estimate of the magnetic moment if 1.60(2) $\mu_{b}$ for AgNi_0.33Co_0.66O₂, which is smaller than the moment of 1.81 $\mu_{b}$ for the same Ni2 sites in AgNiO₂. These results suggest that reentrance to the metallic phase can be possible at high Co doping. This behavior is also consistent with phase 1 and 4 in Fig. 6 for AgNi_0.66Co_0.33O₂, and confirms that reentrance to the metallic phase from insulator can be possible in high Co concentration.

The thermodynamic stability of the AgNiO₂ delafossite and its doped variants depends strongly on their formation energies. In order to estimate optimal conditions for formation of AgNi_1-xCo_xO₂, we calculated the enthalpy of formation for AgNiO₂ and its doped variants under different growth conditions. The enthalpy of formation of AgNiO₂ can be estimated by computing $\Delta H^{AgNiO_{2}}_{f}=\Delta\mu_{Ag}+\Delta\mu_{Ni}+2\Delta\mu_{O}$ where $\mu_{X}$ indicates the chemical potential for given atom $X$. In order to prevent formation of competing phases and phase separation or decomposition of AgNiO₂, these chemical potentials should be constrained as follows:

One phase boundary to AFM Co₃O₄ is avoided by respecting a constraint on Co-doping in AgNi_1-xCo_xO₂. We computed the enthalpy of formation for AgNiO₂, its decompositions, and Co₃O₄ using DFT+U reference energies for solid Ag, Ni, Co, and for gas-phase molecular O₂. We did not try to estimate DMC formation energies because of previously reported difficulties in computing an accurate reference energy for ferromagnetic bulk Ni using a single Slater-determinant trial wavefunction. Shin et al. (2017)

Based on the computed $\Delta H_{f}$ using PBE+U, the phase diagram of AgNiO₂ can be illustrated as a function of the allowed ranges of the chemical potentials of Ag, Ni, and O, given the constraints on them, as shown in Fig. 11. Persson et al. (2005); Walsh et al. (2008) Within the boundaries given by the constraints on the enthalpy of formation of AgO, Ag₂O, and NiO, we obtained the following chemical potentials of ($\Delta\mu_{Ag},\Delta\mu_{Ni},\Delta\mu_{O}$) under different growth conditions: Ag-poor:Ni-poor:O-rich A(-0.10, -2.85, -0.05), Ag-rich:Ni-poor:O-poor B(0.00, -2.75, -0.15), Ag-poor:Ni-rich:O-rich C(-0.08,-2.78,-0.10), and Ag-rich:Ni-rich:O-poor D(0.00, -2.71, -0.17). Using these chemical potentials, the formation energy of a Co defect is given by

where $E_{AgNi_{1-x}Co_{x}O_{2}}$ $E_{AgNiO_{2}}$, $n_{i}$, and $E_{i}$ are the total energy of AgNi_1-xCo_xO₂ and AgNiO₂, the number of added ($n_{i}<0$) and removed ($n_{i}>0$) atoms for substitutions, and the reference energy from the standard solid or gas-phase reference states for the constituent elements, respectively.

Table 1 summarizes the computed formation energies of Co dopants in AgNi_0.66Co_0.33O₂ and AgNi_0.33Co_0.66O₂. As is seen in the Table, PBE+U predicts spontaneous formation of Co-defects for all phases and growth conditions for which the formation energy of the defect is negative. It has previous been reported that PBE+U tends to underestimate the formation energy of defects in transition metal oxide systems Santana et al. (2015); Shin et al. (2017); Chimata et al. (2019); Ichibha et al. (2023), so spontaneous defect formation may not occur. The PBE+U results lead us to confirm that the Ag-rich:Ni-poor:O-poor growth condition is the most favorable one for pure AgNiO₂ and AgNi_1-xCo_xO₂ with the lowest formation energy within the given constraints. Although PBE+U does not provide quantitatively accurate formation energies for Co doping, a qualitative comparison between various growth conditions does give guidelines for the best growth conditions for synthesizing AgNiO₂ and AgNi_1-xCo_xO₂.

In order to compare stability of AgNi_1-xCo_xO₂ with the binary oxides, we compute PBE+U formation energy against elemental solids and binary oxides under stoichiometric conditions. In Table 2, we see large formation energy gap of $\sim$ 8 eV between one relative with elemental solids and the binaries. With comparison of formation energies table 1 and table 2, we see that formation energies of AgNi_1-xCo_xO₂ under the chemical potential constraints are significantly closer to formation energies from the binary oxides than those from the elemental solids in table 2, which tells us formation of AgNi_1-xCo_xO₂ is almost energetically consistent with the ideal formation against binary oxides. In addition, smaller formation energies in x = 0.66 than x = 0.33 in all growth conditions lead us to conclude relative difficulty of AgNi_0.33Co_0.66O₂ synthesis.