Introduction

Single-phase multicomponent metal and ceramic solid solutions have recently generated significant excitement as the large configurational entropy associated with these materials enables the stabilization of single phases of materials that are otherwise immiscible with remarkable physical properties1,2,3,4,5,6,7,8. Regarding ceramics, these so-called high-entropy or entropy-stabilized phases possess a large configurational entropy created through chemical disordering of one or more cation sublattices while (typically, but not necessarily9) the anion sublattice remains monoatomic2. Since their recent discovery, high-entropy oxides have demonstrated exceptional electrochemical and electronic functional properties such as a high ionic mobility10, improved lithium storage and cyclic stability11, efficient thermochemical water splitting at high temperature12, magnetic exchange coupling13, and colossal dielectric properties14. The proposed underlying mechanisms for these enhanced properties may involve intrinsic point defects. For example, proton15 and Li+ ion conduction is thought to involve vacancy-mediated diffusion16,17,18, oxygen vacancies can influence light absorption and charge separation for photoelectrochemical water splitting19, uncompensated surface spins can result from vacancies20, and dielectric responses can be tailored through acceptor, donor, and trap states created by vacancies and defect complexes21. Clearly, an understanding and control of defect formation in these materials is critical for understanding their functional properties, may open doors to the discovery of, e.g., filamentary-based memristor materials22, and shed light on diffusion-mediated processes23,24. Hence, the situation leads to the question: In an oxide with a large configurational entropy, how is thermodynamic defect formation influenced by the significant stereochemical disorder?

Considering an ideal chemically ordered solid, the increase in the configurational entropy is the thermodynamic driving force for the formation of point defects. In the case of multicomponent alloys with large configurational entropy, the contribution of alloy configurational entropy on the vacancy concentration remains controversial as different derivations for the equilibrium vacancy concentration are obtained using different assumptions on the variables that are conserved during vacancy formation25,26. A more complete description must also consider the possible variation of the formation enthalpy due to different local configurations and interactions among constituent elements, making defect formation in high-entropy materials distinct compared to low-entropy compounds27,28.

The situation is more complicated for defects in high-entropy oxides where two distinct sublattices and charge compensation are involved. In the case of (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O, five cations randomly populate the cation sublattice while only oxygen populates the anion sublattice. Yet, the oxygen sublattice is expected to have diverse local bond configurations since the nearest neighbors to oxygen consist of locally different combinations of the five constituent cations, while prior work has found that the oxygen sublattice experiences distortions from the ideal rocksalt structure29,30. Thus, point-defect formation in high-entropy oxides must consider the random local configuration in addition to the different constituent elements.

Here, we apply density functional theory calculations and statistical analysis based on linear regression to investigate point-defect formation in the entropy-stabilized oxide (ESO) (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O, and its compositional variants, and relate it to the local configuration and structural distortion. We also determine that charge compensation and localization in the ESO is the origin of its intrinsic electrically insulating behavior. Our results reveal the effect of stereochemical disorder, local composition fluctuations, and charge compensation on point-defect formation, which enable the engineering of the electronic and electrochemical properties of ESOs.

Results

Formation energy of cation vacancies

We first calculate the formation energy of cation vacancies in their neutral charge state when no free carriers are released or consumed by holes (Methods). Here, 8 cation vacancies with different local environments are calculated for each of the 5 cation elements (thus 40 cation vacancies in total). As seen in Fig. 1a, b, we find that the formation energy of cation vacancies decreases with increasing average strain of the 6 bonds for each cation atom relative to the binary oxides (\(\epsilon _{BO}^{rel}\)), but it does not correlate with the average local strain relative to the ESO (\(\epsilon _{ESO}^{rel}\)) defined as:

$$\epsilon _{BO}^{rel} = \frac{1}{6}\mathop {\sum }\limits_{i = 1}^6 \frac{{d_{i,ESO} - d_b}}{{d_b}} \times 100\left( {{{\mathrm{\% }}}} \right),$$
(1)
$$\epsilon _{ESO}^{rel} = \frac{1}{6}\mathop {\sum }\limits_{i = 1}^6 \frac{{d_{i,ESO} - d_{ESO}}}{{d_{ESO}}} \times 100\left( {{{\mathrm{\% }}}} \right),$$
(2)

where di,ESO is the local cation-oxygen bond length in ESO, db is the cation-oxygen bond length in the corresponding rocksalt binary oxide, and dESO is the average cation-oxygen bond length in the ESO. Except for Co, cation-oxygen bonds in the ESO are under tensile strain, which increases in the order of Mg, Ni, Zn, and Cu. This trend follows the average bond length of the corresponding binary oxides (Fig. 1c). We attribute the trend of decreasing cation vacancy formation energy with increasing tensile strain to the weakening of chemical bonds as the bond length increases, thus reducing the energy needed to create vacancies.

Fig. 1: Local-structure dependence of cation vacancy formation in (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O and comparison with binary oxides.
figure 1

a, b The DFT-calculated formation energies of cation vacancies in (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O as a function of a local strain relative to the average ESO bond length and b local strain relative to the bond length in corresponding binary oxide. The formation energies of cation vacancies decrease with increasing local strain relative to binary oxide. c The average formation energy of cation vacancy in ESO as a function of the average cation-oxygen bond length in corresponding binary oxides showing that the formation of cation vacancy is determined by the shift of the cation-oxygen bond distance from its equilibrium position in the binary oxides. d The comparison between the cation vacancy formation energy for binary oxides and the average cation vacancy formation energy for the ESO.

Full size image

We also compared the average vacancy formation energies for each cation in the ESO to those in their binary rocksalt phases (i.e., the same crystal structure as the ESO) (Fig. 1d). We note that the Cu vacancy formation energy in CuO is negative because rocksalt CuO is thermodynamically unstable. Interestingly, the formation energies of cation vacancies in the ESO do not correlate to the values in their corresponding binary oxides. The formation energies of all cation vacancies decrease under tensile strain compared to their binary oxide phases. In a previous study, Daigle et al. evaluated the effect of chemical disorder on cation vacancy formation energy in a high-entropy diboride and related the cation vacancy concentration to the atom pair affinities and composition of cation near neighbors31. In our work, however, we found that both the sign as well as the magnitude of the local strain (that arises from the deviation of the metal-oxygen bond length in ESO alloys relative to the corresponding binary oxide) are decisive factors in determining the formation of cation vacancies in multicomponent crystals.

Formation energy of oxygen vacancies

Next, we calculate the formation energy of oxygen vacancies (VO) in the neutral charge state as a function of different first-nearest-neighbor (1NN) shell compositions (Fig. 2a). Due to the finite supercell size, 40 different combinations of the 210 possible combinations of the 1NN shell are calculated. VO formation energies range from 0.17 eV to 2.54 eV and strongly dependent on the local configuration. To evaluate the contribution of each 1NN cation to the formation of the VO, the calculated formation energies (EfDFT(VO)) are fitted using linear regressions over the five input variables (NNi, NMg, NCo, NZn, NCu) that describe the number of atoms of each cation species surrounding the oxygen site:

$$E_{LR}^f\left( {V_O} \right) = E_{ave}^f + \Delta E_{1NN,Ni}^f \cdot N_{Ni} + \Delta E_{1NN,Mg}^f \cdot N_{Mg} + \Delta E_{1NN,Co}^f \cdot N_{Co} + \Delta E_{1NN,Zn}^f \cdot N_{Zn} + \Delta E_{1NN,Cu}^f \cdot N_{Cu},$$
(3)

where EfLR(VO) denotes the site-dependent formation energy of the ESO that depends on the local cation composition (not a linearly averaged formation energy for the entire system), Efave the intercept term underlying the average VO formation energy in the ESO, and ∆Ef1NN the model parameter value assigned to each variable. The correlation between EfDFT(VO) and EfLR(VO) is shown in Supplementary Fig. 1a. The correlation is determined by the root-mean-square error (RMSE) between EfDFT(VO) and EfLR(VO), which is computed to be 0.235 eV. ∆Ef1NN for each cation is plotted in Fig. 2b. We find positive ∆Ef1NN values for Mg and Ni, and a large negative value for Cu, implying that oxygen vacancies are more (less) likely to form in a Cu- (Mg- or Ni-) rich local environment. We obtain negative ∆Ef1NN values for Zn and Co, however, the values are relatively small. Using our linear regression model, the distribution of Ef(VO) is obtained for all 210 possible combinations of 1NN shells (Supplementary Fig. 1b), which shows a large variation of the Ef(VO) (from –0.8 eV to 2.4 eV) resulting from local configurational disorder.

Fig. 2: Dependence of the oxygen-vacancy formation energy on the local configuration and structural distortion in (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O.
figure 2

a DFT-calculated formation energy of oxygen vacancy (VO) as a function of first-nearest-neighbor (1NN) shell composition. Each column represents a different oxygen atom in the cell and the cation combination for each 1NN shell is illustrated by the colored bars in the background. The data is ordered along the horizontal axis according to Ef from our linear regression analysis. b The contribution of each 1NN cation to the oxygen vacancy formation, obtained by fitting the DFT-calculated VO formation energies to the number of each 1NN cation with the linear-regression scheme. The formation energy of VO strongly depends on the local atomic configuration: it increases with an increasing number of Mg and Ni 1NN cations, and it decreases with more Cu 1NN cations. c Schematic of the octahedral arrangement of O in ESO, where the six 1NN cation sites are determined by the combination of the five cation species. Different combinations of cations lead to different local bond-angle deviations, Δθ (°) d The values of the average M1-O-M2 bond-angle deviation from 90°, where M1 and M2 indicate metal cations in the ESO. Cu (Mg) contributes the most (least) to the distortion of bond angle. e The formation energy of VO as a function of average bond-angle deviation and average bond strain. The local structural distortion, determined by local bond strain and bond-angle deviation, decreases the formation energy of VO’s and increases their formation probability.

Full size image

The contribution of a 1NN cation to the VO formation energy correlates with the local lattice distortion, which is characterized by the local bond-angle deviation from 90° (the ideal bond angle for the rocksalt structure) and bond strain. Figure 2d illustrates the average cation-oxygen-cation bond angle deviation for the various combinations of cations calculated from the relaxed DFT structure of (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O. The local bond angle distorts the most for Cu, followed by Zn, Co, Ni, and Mg 1NN cations, in decreasing order, where the maximum average bond-angle deviation is observed for the Cu-O-Cu bond (3.97 °) and the minimum is for Mg-O-Mg (1.12 °). Figure 2e plots Ef(VO) against the average bond-angle deviation and the average bond strain of the six bonds in an octahedron, revealing the correlation between local lattice distortion and Ef(VO). Ef(VO) is linearly fitted with the average bond-angle deviation and the average bond strain, separately, in Supplementary Fig. 1c, d. The correlation defined by the root mean square error between Ef(VO) and the average bond angle deviation/strain is computed to be 0.4567 eV/0.4092 eV, which corresponds to a relative error of 19%. Our results therefore demonstrate that the formation energy of oxygen vacancies is related both to the local bond-angle deviation and to the local strain, which stem from the strain of the cation-oxygen bonds in chemically disordered ESOs relative to the binaries.

Thermodynamic transition levels of vacancies

We further investigate the electronic properties of vacancies in equimolar Mg0.2Ni0.2Co0.2Cu0.2Zn0.2O. Among various structural configurations, we chose oxygen and cation vacancies with the lowest formation energies and calculate their stable charge states. Figure 3a shows the formation energy (Ef) of vacancies in the ESO in their dominant charge states as a function of the Fermi energy (EF) level at the characteristic growth condition for bulk synthesis (1000 °C in air)2. We find that VO is a deep donor defect with the +2/0 thermodynamic transition level at EF = 1.239 eV below the CBM. VCu, VZn, and VCo are stable in the 0, –1, or –2 charge states while VMg and VNi are only stable in the 0 or –2 charge states, depending on the Fermi level. All cation vacancies are deep acceptors with acceptor ionization energies spanning from 0.658 eV (for VCu) to 1.271 eV (for VMg).

Fig. 3: Charge compensation and localization from point defects in (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O.
figure 3

a Formation energy of oxygen and cation vacancies as a function of Fermi level assuming bulk growth conditions (1000 °C, air). An oxygen vacancy is a deep donor while cation vacancies are deep acceptors. b The histogram and fits of the change in Bader charge of 1NN cations before and after the formation of an oxygen vacancy. The inset illustrates the local atomic structure of the entropy-stabilized oxide with the lowest-formation-energy oxygen vacancy and the isosurface of the band-decomposed charge density at the conduction band minimum (CBM). The value of the isosurface is chosen to be 50% of the charge density. The charge distribution of excess electrons from an oxygen vacancy (yellow) is localized on the nearest Cu atoms. c Calculated Fermi-level pinning energy as a function of oxygen chemical potential. The characteristic oxygen chemical potential and Fermi level pinning energy under bulk (1000 °C, air) synthesis conditions as well as standard ambient temperature and pressure (25 °C, air) are specified. Under typical growth conditions, the Fermi energy is pinned at a deep energy level within the band gap, in agreement with the observation of the electrically insulating state of (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O. d Calculated equilibrium Fermi energy level as a function of Cu mole fraction. The mole fraction for the non-variable cations equally composes the remaining mole fraction. Increasing Cu composition leads to increased charged oxygen vacancy formation, which enhances the equilibrium Fermi level toward CBM. e Calculated equilibrium vacancy concentration of oxygen and cations as a function of Cu mole fraction. By utilizing the local configuration effect on oxygen vacancy formation energy and the subsequent cation vacancy formation for charge neutrality, the vacancy density of ESO can be tuned by cation mole fraction.

Full size image

We also investigate the charge distribution of electrons bound to oxygen vacancies in the neutral charge state. The inset of Fig. 3b illustrates the band-decomposed charge density for electrons bound to the oxygen vacancy with the lowest formation energy. The electrons localize around the 1NN Cu atoms. In order to extend our analysis to all oxygen vacancies in our supercell, we calculated the change of the Bader charges of the 1NN cations before and after the formation of VO (Fig. 3b)32,33,34,35. We find that the electron density around the neighboring Cu changes the most, by 0.47 e on average, while the electron density around Mg remains relatively unchanged (0.00037 e on average), and the charge states of Zn, Co, and Ni change by 0.04 e, 0.07 e, 0.09 e on average, respectively. Since the Cu 3d orbitals contribute the most (91%) to the conduction band minimum (CBM) wave function of the ESO, we anticipate that the conduction mechanism of the ESO to be dominated by variable-range electron hopping between localized Cu states.

Equilibrium Fermi level and vacancy concentration

The equilibrium Fermi level is determined by the interplay of extra electrons released by positively charged VO and consumed by negatively charged cation vacancies. We identify the Fermi level pinning position (EFpin) to be the crossing point between VO2+ and VCu2− in the Ef vs EF diagram, which are the two dominant vacancies in the ESO. Since the concentration of vacancies at different charge states is affected by the synthesis conditions, with EFpin changing accordingly, we calculate EFpin according to the oxygen chemical potential (µO) and plot it in Fig. 3c. We also specify the µO at the conditions for typical bulk ESO synthesis (1000 °C in air) and standard temperature and pressure. Reducing µO (oxygen-poor conditions) lowers the formation energy of VO2+ and increases EFpin while the opposite is found for increasing values of µO. However, we determine EFpin is a deep level within the band gap for the µO values under typical synthesis conditions, indicating that (Mg0.2Ni0.2Co0.2Cu0.2Zn0.2)O is intrinsically electrically insulating.

We then calculate the equilibrium oxygen and cation vacancy concentration as a function of cation composition, particularly the Cu composition. As the formation energy of a cation vacancy is cation-type specific and does not show strong correlation with local configuration, we select the lowest vacancy formation energy for each cation species to calculate the equilibrium cation-vacancy concentrations. We assumed element-specific vacancy types and negligible contribution from mixing entropy26, and calculated each cation vacancy concentration using the Maxwell-Boltzmann equation:

$$n\left( {V_{cation}} \right) = N_{cation} \cdot P \cdot \exp \left( { - \frac{{E^f(V_{cation})}}{{k_BT}}} \right),$$
(4)

where P is the chemically weighted site density by treating each cation vacancy as a separate species.

On the other hand, we applied our linear regression model (\(E_{LR}^f\left( {V_O} \right)\)) to model Ef(VO0) as well as Ef(VO2+). As our calculations show that the charge transition level (Et) of VO varies relatively little (<0.2 eV) depending on the local configuration, we assume the charge-transition level of VO is the same for all O sites and modeled Ef(VO2+) according to:

$$E^f\left( {V_O^{2 + }} \right) = E_{LR}^f(V_O) + 2(E_F - E_t),$$
(5)

By assuming configurational homogeneity without local ordering (as previously demonstrated in ref. 29) and considering all possible local configurations, the equilibrium oxygen vacancy concentration in both the neutral and the +2 charge state in compositionally varied ESOs (e.g., Mg0.25(1-x)Co0.25(1-x)Ni0.25(1-x)CuxZn0.25(1-x)O) is calculated using:

$$n\left( {V_O} \right) = \mathop {\sum }\limits_i N_O \cdot P_i \cdot {{{\mathrm{exp}}}}\left( { - \frac{{E_{LR,i}^f(V_O)}}{{k_BT}}} \right),$$
(6)

where NO is the number of oxygen sites, Pi is the probability of the ith 1NN configuration (see Methods), EfLR,i(VO) is the calculated oxygen vacancy formation energy for the ith 1NN configuration, kB is the Boltzmann constant, and T is the synthesis temperature. Vacancies with 1NN configurations that result in a negative formation energy (e.g., 93% of the vacancy configurations with more than four 1NN Cu) are assumed to form spontaneously. Then, the charge-neutrality condition is applied to calculate the equilibrium Fermi energy (EFeq), and the equilibrium vacancy concentrations of oxygen and cations are calculated as a function of Cu mole fraction, xCu. Figure 3d plots EFeq a function of xCu and Fig. 3e plots the equilibrium vacancy concentrations of oxygen and cations a function of xCu, while the mole fraction of the other cations is 0.25(1–xCu), for T = 1000 °C. The VO concentration increases significantly as the Cu mole fraction increases. Also, though the increased VCu with increasing xCu introduces more holes, EFeq increases toward the CBM as the larger number of electrons are introduced from oxygen vacancies with increasing xCu. This result shows that increasing the Cu mole fraction increases both the O vacancy and the Cu vacancy concentrations at an approximately constant ratio, independent of alloy composition, due to the charge compensation between these two dominant vacancy species.

Disscussion

Among the wide-ranging applications, ESOs have been particularly studied as energy-storage materials owing to the high capacity retention and Li-extraction ability of the compound11,36,37. According to the proposed mechanism, Li+ insertion/extraction occurs via the conduction path which involves oxygen vacancies10, and Li+/Li conversion occurs inside the compound via the redox reaction with some of the cations in ESOs (e.g., Co2+, Cu2+)11. A defect-rich structure allows the formation of Li2O nuclei while keeping the structure of the host matrix intact throughout cycling, which ensures the reversibility of the reaction11. Defects can also facilitate Li superionic conduction, which allows high discharge capacity. The proposed mechanism indicates that selecting/tuning the composition to design ESO with abundant oxygen vacancies and redox-active species by, e.g., increasing the Cu composition, is an effective strategy to improve the electrochemical performance of ESOs38.

In addition, cation-valence engineering by tuning the concentration of oxygen vacancies can affect the functional properties of ESOs. Particularly, variations to the Co charge state directly influence the magnetic transitions of ESOs as Co2+ (d7) is a magnetic cation whereas Co3+ (d6) is magnetically inert39. Kotsonis et al.40 observed an increase in Co3+ by reducing the film-growth temperature, which leads to magnetic dilution of the system. According to our analysis, the observed charge-state change of Co is expected to be associated with reduced oxygen vacancy concentration at lower growth temperatures.

To conclude, we investigated the thermodynamic and electronic properties of cation and oxygen vacancies in entropy-stabilized oxides. We uncovered design rules for controlled vacancy formation by tuning the cation composition. We found that the local compositional variations in ESOs give rise to structural distortions and strain, arising from the deviation of the cation-oxygen bond lengths relative to the binaries, play a decisive role in the formation of both cation and oxygen vacancies. Based on a linear-regression analysis of the first-principles results, we developed a model to predict the oxygen vacancy density as a function of cation composition. Our results provide useful guidance to identify thermodynamically preferred defect sites and engineer the vacancy formation in multi-component alloys.

Method

Density functional theory calculation

We performed first-principles calculations based on density functional theory using the projector augmented wave (PAW) method and generalized gradient approximation (GGA) functionals as implemented in the Vienna Ab initio Simulation Package (VASP)41,42,43. The PAW pseudopotentials including the Mg 3s2, Co 3d74s2, Ni 3d84s2, Cu 3d104s1, Zn 3d104s2 and O 2s22p4 electrons as the valence state were employed, and the plane-wave cut-off energy was set to 400 eV. To better predict the localized characteristics of the d-electron shell, on-site Coulomb interaction corrections were applied for the transition metal atoms using the DFT + Hubbard U formalism44. In this approach, the Hubbard U parameter is chosen from the Automatic-Flow (AFLOW) standard for high-throughput construction of materials science and electronic structure database45, which is 5, 5.1, 4.0, and 7.5 eV for Co, Ni, Cu, and Zn, respectively. All structures are relaxed using a conjugate-gradient algorithm. The maximal force criterion of 0.01 eV/Å was used for the ionic relaxation and a maximal energy criterion of 10−8 eV was used for the electronic self-consistent loop.

We modeled the (Mg0.2Co0.2Ni0.2Cu0.2Zn0.2)O entropy-stabilized oxide and its compositional variants using 80-atom supercells and Γ-centered 2 × 2 × 2 Brillouin-zone sampling grids. The alloy randomness is modeled using Special Quasi-random Structures (SQS)46, generated with the Alloy Theoretic Automated Toolkit47 to match the pair correlation functions of the random solid solution up to a radius of 6 Å. As the magnetic structure of the rocksalt (MgCoNiCuZn)O is antiferromagnetic in its ground state, we also considered the arrangement of atomic spins where (111) planes contain magnetic moments alternatively parallel and antiparallel with the spin direction along [211]13,48,49. Each cation is equally populated in the two magnetic sites: (\({{{\mathrm{Mg}}}}_{0.2}{{{\mathrm{Co}}}}_{0.1}^ \uparrow {{{\mathrm{Co}}}}_{0.1}^ \downarrow {{{\mathrm{Ni}}}}_{0.1}^ \uparrow {{{\mathrm{Ni}}}}_{0.1}^ \downarrow {{{\mathrm{Cu}}}}_{0.1}^ \uparrow {{{\mathrm{Cu}}}}_{0.1}^ \downarrow {{{\mathrm{Zn}}}}_{0.2}\))O. The lattice constant is determined from the energy minimum. The band gap is corrected using a linear correction scheme we designed that is based on the scissor-shift method; the valence-band maximum (VBM) and the conduction-band minimum (CBM) for the constituent binary oxides (rocksalt structure for MgO, CoO, NiO, and ZnO, and tenorite structure for CuO) are calculated using the GGA + U and the HSE0650 functionals, and the average energy difference between the VBM (or CBM) states calculated with GGA + U and with HSE06 is used to correct the VBM (or CBM) energy of the ESO (Supplementary Fig. 2). Although we expect that the presence of alloy disorder can lead to band bowing and the reduction in the band gap compared to our linear correction scheme, our results on Mg0.5Zn0.5O alloy shows that our method gives an error bar for the VBM and CBM corrections by less than 0.1 eV. Since both cation and oxygen vacancies in ESOs are predicted to be deep defects with ionization energies higher than 0.65 eV, we expect that the bowing of the band edges by an amount on the order of a few 100 meV does not affect their predicted deep nature. We also construct 64-atom supercells for rocksalt MgO, CoO, NiO, ZnO, and CuO to calculate the formation energy of cation vacancies in binary oxides. The metastable rocksalt structure CuO was calculated by relaxing the unit-cell volume and ionic positions while maintaining the symmetry.

The formation energy of a point defect D in charge state q is defined by51

$$E^f\left( {D^q} \right) = E_{{{{\mathrm{tot}}}}}\left( {D^q} \right) - E_{{{{\mathrm{tot}}}}}\left( {{{{\mathrm{bulk}}}}} \right) - {\sum} {n_i} \left( {E_i + \mu _i} \right) + {{{\mathrm{q}}}}\left( {E_F + E_v} \right) + E_{{{{\mathrm{corr}}}}}\left( {D^q} \right),$$
(7)

where \(E_{{{{\mathrm{tot}}}}}\left( {D^q} \right)\) is the total energy of a supercell with a point defect, \(E_{{{{\mathrm{tot}}}}}\left( {{{{\mathrm{bulk}}}}} \right)\) is the total energy of a reference supercell without a point defect, ni is the number of defect atoms added to or removed from the supercell, Ei is the energy per atom in its elemental phase, μi is the chemical potential, EF is the Fermi level, Ev is the valence-band maximum, and \(E_{{{{\mathrm{corr}}}}}\left( {D^q} \right)\) is the correction energy arising from defect-defect interaction due to the finite supercell size, which is calculated with the SXDFECTALIGN52 code. For the correction energy, we set the static dielectric constant of the ESO to ε0 = 17.99, which was determined by density functional perturbation theory53.

The chemical potential of the defect species depends on the growth temperature and pressure. We choose typical bulk synthesis conditions of ESO (T = 1000 °C in air) and calculate the oxygen chemical potential, μO = –2.306 eV, using the following equation:

$$\mu \left( {P,T} \right) = k_BT\;{{{\mathrm{In}}}}\left( {{{{{n}}}}{\it{V}}_{\it{Q}}} \right) = k_BT\;{{{\mathrm{In}}}}\left( {\frac{P}{{RT}}V_Q} \right)$$
(8)

where kB is the Boltzmann constant, n is the density of O2, VQ is the quantum volume of O2, and R is the ideal gas constant. The chemical potentials for the cation species are also limited by the formation of secondary phases such as MgO, Co3O4, NiO, ZnO, and Cu2O. The derivation and actual values for the chemical potentials are described in Supplementary Tables 1 and 2.

Probability of the local configuration

There are 210 combinations that the five distinct cations can occupy the 6 nearest neighbor sites of oxygen. The probability of the local configuration is determined by:

$$P = \frac{{6!}}{{n_{Ni}! \cdot n_{Mg}! \cdot n_{Co}! \cdot n_{Zn}! \cdot n_{Cu}!}}x_{Ni}^{n_{Ni}} \cdot x_{Mg}^{n_{Mg}} \cdot x_{Co}^{n_{Co}} \cdot x_{Zn}^{n_{Zn}} \cdot x_{Cu}^{n_{Cu}}$$
(9)

where \(n_{Ni}\), \(n_{Mg}\), \(n_{Co}\), \(n_{Zn}\), and \(n_{Cu}\) are the number of atoms of the Ni, Mg, Co, Zn, and Cu cation constituents in the nearest neighbor sites to the oxygen atom, and \(x_{Ni}\), \(x_{Mg}\), \(x_{Co}\), \(x_{Zn}\), and \(x_{Cu}\) are the Ni, Mg, Co, Zn, and Cu mole fractions in the ESO, respectively.