Origins of intrinsic p-type conductivity, p–n transition and substoichiometry in SrO

Abstract

To understand the complex electrical behaviour and deviations from ideal stoichiometry in strontium oxide we have investigated its defect chemistry using a hybrid quantum mechanical/molecular mechanical (QM/MM) embedded-cluster approach. Depending on the temperature and oxygen partial pressure, oxygen interstitials or strontium and oxygen vacancies are found to dominate, while strontium interstitials are rare. Notably, charge-neutral oxygen interstitials form a peroxy-like closed-shell configuration, which is the commonest native point defect in SrO under normal conditions explaining the Sr substoichiometry which is not electrically active. Formally charged double acceptors strontium vacancies prove to be the primary source of the hole excess over negative carriers supplied by donor species, contributing to the material's p-type conductivity. Based on our calculations, we predict that at ultralow oxygen partial pressure (P = 1.0 × 10⁻¹⁵ bar) and high temperatures (>1100 K) in SrO, the electron concentration surpasses the hole concentration, which has previously been reported in pure BaO (also at about 1100 K) and the double barium–strontium oxide (at 850 K) by D. W. Wright, (Nature, 1949, 4173, 714) with the oxygen split interstitial acting as a donor. On increasing the oxygen partial pressure, the hole concentration exceeds the electron concentration, resulting in effective p-type conductivity. Only under low oxygen pressures (e.g., 10⁻⁸ bar) and high extrinsic donor concentrations (>10¹⁷ cm⁻³) might SrO switch to n-type conductivity at high temperatures (>1250 K). This study provides essential insights into intrinsic defects and mechanisms of SrO's p-type conductivity, aiding in understanding and predicting other p-type materials.

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1. Introduction

Strontium oxide (SrO) belongs to the alkaline-earth oxides family, characterized by its highly symmetrical rock salt structure. It is a wide band gap semiconductor (5.9 eV) with a high exciton binding energy and electrochemical stability,¹ making it suitable for diverse applications including catalysis and microelectronics.^2,3 SrO could potentially serve as a candidate for diluted half-metallic ferromagnetic semiconductors (DMSs) when doped with non-magnetic elements like carbon⁴ and nitrogen⁵ or transition metals.⁶ Intrinsic defects in SrO play a crucial role in all these applications. The p-type conductivity of SrO observed in experiments⁷ may be attributed to the intrinsic defects.

Intrinsic defects in SrO have been widely explored experimentally. Kappers⁸ reported observation of the intrinsic V⁻ (a positive hole trapped at a cation vacancy) and V⁰ centers (two holes trapped at a cation vacancy) in single-crystal SrO on a sample x-irradiated at 77 K using electron spin resonance (ESR) spectroscopy. Tench et al.⁹ studied the formation of cation and anion vacancies in SrO after irradiation by 20 MeV protons, again using ESR. They found that the rate of formation of cation vacancies fits a model of interstitial-vacancy recombination. The efficiency of vacancy production is an order of magnitude less than that predicted for primary displacements confirming the importance of recombination between vacancies and those interstitials formed within a critical volume around the vacancy. Seeman et al.¹⁰ found by ESR a new type of V center in SrO, which consists of an O⁻ ion (localized hole) near a cation vacancy surrounded by two OH⁻ groups. They also found that there are several types of superoxide (O₂⁻ centers) in SrO in x-irradiated SrO single crystals using ESR. All these centers consist of an O₂⁻ molecule stabilized near various defects.¹¹ Johnson and Hensley¹² reported that the observation of the F centers (oxygen vacancies containing two electrons) at room temperature peaked at 2.5 eV and the F⁺ centers (oxygen vacancies containing one electron) peaked at about 3.1 eV. Feldott and Summers¹³ observed that the F⁺ band appears to be carried by electrons and has a thermal activation energy of 0.1 eV.

Intrinsic defects in SrO have also been widely studied using theoretical calculations. Yamamoto et al.¹⁴ investigated the V_o in SrO using first-principles PAW calculations with the Perdew–Burke–Ernzerhof (PBE) generalised gradient approximation (GGA) functional. The main conclusion was that the energy state of the neutral oxygen vacancy is located at 1.02 eV below the conduction band minimum (CBM). They also found that the formation of the oxygen vacancy is enhanced under a reducing atmosphere. Sun et al.¹⁵ also investigated the Sr vacancy in SrO using the GGA + U approach with U applied on O 2p orbitals. They found that the neutral and singly charged Sr vacancy could introduce a magnetic moment into the SrO crystal which leads to the material showing half-metallicity. The same authors¹⁶ reported that the optical absorption band of F and F⁺ color centers peak at 2.49 eV and 3.06 eV, respectively by using hybrid HSE calculations. Middleburgh et al.¹⁷ reported using GGA-PAW pseudopotentials that the accommodation of excess oxygen in SrO results in the formation of a peroxide ion centred on an oxygen site, rather than forming a single oxygen species.

Despite this extensive range of studies, there is no comprehensive understanding of all types of intrinsic defects in SrO nor of the reason for its native p-type conductivity. Previous theoretical studies have primarily utilized supercell approaches, which have limitations in modeling charged defects.^18,19 In this work, we investigated intrinsic defects in SrO using the hybrid quantum mechanical/molecular mechanical (QM/MM) embedded-cluster method,^20–23 which avoids spurious interactions between periodic images of charged defects and thus overcomes the disadvantage of the supercell approach. We examine oxygen and strontium vacancies, single oxygen and strontium interstitials, and a peroxide-like oxygen interstitial using the PBE0 (ref. 24) and BB1K²⁵ hybrid functionals. Formation energies of various defects are calculated, and the origin of the p-type conductivity related to these defects is investigated.

2. Computational details

2.1 QM/MM calculations

The hybrid QM/MM embedded-cluster method²¹ has been employed to model the relevant point defects in SrO within the ChemShell package.^23,26,27 In this approach, a Sr-centered 129-atom QM cluster was embedded within a large cluster with 7599 atoms in total. The QM cluster is treated by hybrid density functional theory (DFT) and the surrounding region is treated at the MM level of theory. The interface region between the QM and MM regions serves as the buffer layer to minimize the mismatch of the QM and MM levels of theory. Here, we use a specially designed local embedding pseudopotential on the cationic sites in the form of a linear combination of three Gaussian functions¹⁹ with the fitted parameters in atomic units as given below:

r²U_p(r) = −53.9247r e^−20.5435r2 + 19.9965r² e^−1.3535r2 +1.973r² e^−0.762r2 (1)

For the QM part of the calculations, we use Density Functional Theory (DFT) with triple-zeta plus polarization Gaussian basis set²⁸ on oxygen and strontium, including a 28-electron effective core potential (ECP) on Sr,²⁹ as implemented in the NWCHEM³⁰ software. To reduce the computational load, we have removed the f function from the oxygen basis set and some of the highly diffuse functions from the strontium basis sets, which do not contribute to ionic bonding. As mentioned, for electron exchange and correlation, the PBE0 (ref. 24 and 31) functional with 25% exact exchange, and BB1K²⁵ functional with 42% exact exchange are used.

For the MM part of the simulation, pairwise Buckingham interatomic potentials are used in the context of the shell model:³²

where r_ij is the distance between the two interacting ions and A, ρ, and C₆ are the parameters complemented by the Lennard-Jones potentials:where A′ and B′ are the parameters of the potential. The on-site ionic polarizability is treated by the shell model where the atomic core is connected to the shell via a harmonic spring with the constant κ₂: E_cs = ½κ₂(δr_i)². The sum of the core and shell charges on an ion equals its formal charge. The ionic polarizability α in vacuum in the shell model is given by , where Y is the shell charge. The electrostatic Coulomb interaction is calculated by with k_e, the dimensional Coulomb constant (14.3996 eV Å e⁻²).

The General Utility Lattice Package (GULP) code^33,34 is employed for the MM part of the calculation, which is divided into two regions. One is the MM-active region where all the atoms are allowed to relax, while the other is the MM-frozen region where all the atoms are kept fixed to simulate the bulk environment. The outermost layer of the whole cluster is fitted point charges, and its role is to eliminate the effects of surface termination and reproduce the Madelung potential of bulk SrO.

We employed the GULP code to develop a new shell-model interatomic potential for SrO. Our previous work has demonstrated the importance of explicit incorporation of relative ionic polarisabilities in fitting the shell-model potential for metal oxides.¹⁸ This approach significantly enhances the capability to accurately model the electrostatics and polarisation in defect and surface calculations.^18,35 To calculate the ionic polarisabilities of Sr²⁺ and O²⁻ in the SrO lattice, we employed an embedded-cluster model, where a single QM ion was surrounded by MM ions as the environment. DFT calculations were performed at the PBE0 level of theory with the same basis set as one used in defect QM/MM calculations using NWChem.³⁰ Computational details can be found in our previous work.¹⁸ Curiously, a consistent polarisability of 5.71 a.u. was obtained for both Sr²⁺ and O²⁻ ions. Therefore, we imposed a constraint to equalise the k₂ values on both ions in the potential fitting. Parameters of oxygen–oxygen repulsion potential were taken from ref. 36. By adjusting the shell charges and fitting other short-range parameters against experimental data, an optimised potential was determined for SrO, as shown in Table 1.

Table 1 The optimised shell-model interatomic potential for SrO. The cut-off of the short-range potentials was set to 15 Å

(a) Short range potentials
Interaction	A (eV)	ρ (Å)	C 6 (eV Å^6)	A′ (eV Å^12)	B′ (eV Å^6)	r min (Å)	r max (Å)
O^2− shell–Sr^2+ core	1865.3676	0.348774	0.0	10.0	43.0	0	15
O^2− shell–O^2− shell	22 764.3	0.149	0.0	10.0	43.0	0	15

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(b) Shell model
Species	Y (e)	k 2 (eV Å^−2)
Sr^2+ shell	−2.45	45.235768
O^2− shell	−2.45	45.235768

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The performance of the new potential set in modelling bulk SrO is shown in Table 2 in comparison to experiment and DFT results. In general, the new potential is able to accurately reproduce the lattice constant and dielectric constants, which are critical to QM/MM modelling. The calculated elastic properties, phonon frequencies, and lattice energy are also in good agreement with experimental and theoretical data.

Table 2 Performance of the newly developed shell-model potential in describing the structure and properties of SrO compared to experimental and DFT results. Observables include lattice constant (a₀), static and high-frequency dielectric constants (ε₀ and ε_∞), bulk modulus (B₀), elastic constants (C₁₁, C₁₂, and C₄₄), gamma-point phonon frequencies (TO_Γ and LO_Γ), and lattice energy (ΔH_L)

Observable	Shell model	Expt.	DFT
a 0 (Å)	5.1602	5.1615 (ref. 37)	5.17 (PBE0),^38 5.12 (PBEsol0)^38
ε 0	13.09	13.1 (ref. 39)
ε ∞	3.11	3.46 (ref. 39)	3.03 (PBE0),^38 3.09 (PBEsol0)^38
C 11	198.59	165,^40 175 (ref. 41)	170 (GGA)^42
C 12	70.82	49 (ref. 41)	48 (GGA)^42
C 44	70.82	56 (ref. 40 and 41)	59 (GGA)^42
B 0 (GPa)	113.41	86,^40 89 (ref. 41)	89 (GGA)^42
TOΓ (cm^−1)	244.93	224.2,^43 234.2 (ref. 44)	217.85 (LDA)^45
LOΓ (cm^−1)	502.91	485.0,^43 482.7 (ref. 44)	474.13 (LDA)^45
ΔHL (eV)	−34.16	−33.40 (ref. 46)

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2.2 Calculations of the intrinsic defects formation energies

The formation energy of a defect in the charge state q is defined aswhere E(X^q) and E₀ are the calculated total QM/MM energies of the defect and pure structures, n_i is the number of species that have been added (n_i > 0) or removed (n_i < 0) from the system to form the defect, μ_i is the chemical potential of species i, and E_F is the Fermi energy relative to the valence band maximum (VBM). An a posteriori correction²¹ is applied to correct for the long-range polarization effects outside the active region due to charged defects that extend to infinity, in the form:where R is the radius of the active region and Q is the net charge of the defect system. High frequency ε_∞ and static ε₀ dielectric constants are used for vertical and adiabatic processes, respectively.

The formation of charged defects also depends on the growth conditions of the materials. In the O-rich/Sr-poor conditions, the upper limit is determined by the formation of O₂ molecules, in which μ_O = 0 eV and μ_Sr = ΔH_f(SrO). In the O-poor/Sr-rich conditions, the lower limit of μ_O is determined by the SrO. In the QM/MM calculations, the reference energies for an O₂ molecule and a single Sr²⁺ ion are calculated using the NWCHEM with the corresponding basis set and functional. The sum of the first two ionization potentials of Sr (5.094 eV and 11.030 eV (ref. 46)) and the sublimation enthalpy of Sr⁴⁶ (ΔH_subl(Sr), 1.70 eV) are then used to obtain the reference energy of Sr(s). The SrO formation enthalpy (−6.182 eV) is calculated using the SrO cohesive enthalpy (10.45 eV (ref. 47)), the sublimation enthalpy of Sr, and the O₂ dissociation energy (5.136 eV (ref. 46)).

2.3 Oxygen chemical potential

As shown in eqn (4), the formation energy of a point defect is a function of the Fermi level and respective chemical potentials, which are therefore affected by the experimental conditions, in particular, the oxygen partial pressure P and temperature T. Reuter and Scheffler⁴⁸ give the relationship between the oxygen chemical potential μ_O(T,P) and the oxygen partial pressure P as following:

μ_O(T,P⁰) = ½[H(T,P⁰,O₂) − H(0 K,P⁰,O₂)] − ½T[S(T,P⁰,O₂) − S(0 K,P⁰,O₂)] (7)

where P⁰ = 1 atm is defined as the zero state, μ_O(0 K,P⁰) = = ½E_O2 = 0 eV, and H and S are enthalpy and entropy of the system respectively. The oxygen chemical potential μ_O can thus be calculated at various reaction conditions using the experimental enthalpy and entropy data.⁴⁹

2.4 Charge carrier and defect concentrations

The self-consistent Fermi energy level and the equivalent defect and carrier concentrations can be obtained using the defects formation energy as shown in eqn (4). The concentration of a defect X in charge state q is calculated by:where N_X is the density of sites where the defects may form, g_X^q is the degeneracy of the charge state, E_f(X^q) is the defect formation energy, and k and T are the Boltzmann constant and temperature.

The electron n₀ and hole p₀ concentrations can be calculated as:

f_h(E) = 1 − f_e(E) (12)

where f_e(E) is the Fermi–Dirac distribution function, ρ(E) is the density of states (DOS), E_g is the band gap and all energies are referenced to the top of VBM. The code SC-FERMI is used in the defects and charge carrier concentration calculations.⁵⁰

To calculate ρ(E) of SrO, the Vienna Ab initio Simulation Package (VASP)^51–54 code is employed using the PBE0 functional with the projector augmented wave (PAW)⁵⁵ method to model the interactions between the core and active, or valence electrons. There are ten and six valence electrons in Sr and O atoms. The crystallographic unit cell containing eight atoms is used in the DOS calculations. The plane wave cutoff energy is 650 eV and 9 × 9 × 9 Monkhorst–Pack⁵⁶k-points mesh is used. The convergence of the total energy and total force are 1 × 10⁻⁶ eV and 0.001 eV Å⁻¹, respectively.

3. Results and discussion

SrO is an archetypal rock-salt structured binary oxide with well-known basic defects, i.e., cation and anion vacancies and interstitials.^57,58 In the following sections, we focus on the results of our calculations of the energy of formation for these defects in their main charge states before we turn our attention to the problem of thermodynamic equilibrium between different defects and charge carriers in this system.

3.1 Formation energies of oxygen and strontium vacancies

We first consider the formation of the oxygen vacancy (V_o) and strontium vacancy (V_Sr). For V_o, the formation energies of the charge states of +2, +1, 0, and −1 are calculated. The charge neutral state can have both singlet and triplet configurations, for which calculations have been performed; for the −1 charge state, both the low and high spin configurations are investigated. For V_Sr, the formation energies of the charge states of −2, −1, 0, +1, and +2 are considered. For the neutral V_Sr, we found that the difference in energy between the singlet and triplet is 82 (121) meV using the PBE0 (BB1K) functional. For the charge state of +1, the high spin state is 19 (14) meV lower than that of the low spin state. For the charge state of +2, the spin configurations are singlet, triplet, and quintuplet. The energy of the quintuplet is 24 (6) meV lower than the triplet state, and 38 (7) meV lower than the singlet state. Thus, all the energies of different spin configurations in the V_Sr are close, so we did not include them in the formation energy diagram where only the one with the lowest energy is shown. The formation energies were calculated using the PBE0 and BB1K functionals, but there is little difference between the results for the two functionals as shown in the ESI.† Taking the neutral case of V_o and V_Sr, we found the formation energy difference of these two defects is only 0.2 eV. Therefore, we only show the results calculated using the PBE0 functional.

For the calculations of vacancies, we consider both the O-rich and O-poor conditions. In the O-poor conditions, the formation energy of V_o will shift down the enthalpy of formation of SrO which is measured at about 6.182 eV.⁵⁹ The oxygen vacancy has a lower formation energy under O-poor conditions indicating it will be easier to form.

The structure of the charge neutral V_o is shown in Fig. 1(a), and we found that there is little distortion around the oxygen vacancy. The formation energies of V_o as a function of the Fermi energy level relative to the VBM under O-rich are shown in Fig. 1(b). Importantly, for the V_o, the formation energy of the low spin configuration is more than 1 eV lower compared to the high spin configuration in both charge states 0 and −1. From Fig. 1(b), we also find that if the Fermi level lies between the VBM and the middle of the band gap, V_o with charge +2 has the lowest formation energy. Almost 1 eV above the middle of the band gap, the charge +1 V_o dominates. We use (n/m) to denote the charge of V_o changing from q = n to q = m, and for (+2/+1), the transition energy level is ∼2.85 eV above the VBM. For the remaining range close to the CBM, charge-neutral oxygen vacancies are favoured, and the transition energy level for (+1/0) is ∼4.4 eV above the VBM.

Fig. 1 (a) Structures of charge neutral oxygen vacancy, and (b) formation energies of oxygen vacancies, and (c) structures of the charge neutral strontium vacancies, and (d) formation energies of strontium vacancies, with the crossing point shown in (e). All the formation energies are plotted as a function of the Fermi energy level relative to the valence band maximum (VBM) under O-rich conditions.

The structure of the charge neutral V_Sr is shown in Fig. 1(c) and there is a some distortion of the oxygen atoms that originally bonded to the strontium. The formation energy diagrams for V_Sr under O-rich conditions are shown in Fig. 1(d) with the crossing point shown in Fig. 1(e). In contrast to V_o, the formation energies of V_Sr under O-poor conditions shifts up 6.182 eV (ref. 59) indicating it is favored under O-rich conditions. Close to the VBM, the charge +2 states of V_Sr have the lowest formation energy, and for the larger part of the band gap, the charge state −2 predominates. The energy transition level of (+2/−2) of V_Sr lies at ∼2.80 eV. Around 2.80 eV above VBM, there is a small region where all charge states (from −2 to +2) have similar formation energies.

3.2 Formation energies of oxygen and strontium interstitials

As noted, SrO is a cubic rock-salt structured material, and interstitial defects might be expected to occupy a special high-symmetry cubic site with fractional coordinates (0.25, 0.25, 0.25). For the charge neutral interstitial oxygen case, we found the geometry of the singlet and triplet configurations are quite different. In the triplet configuration, the interstitial oxygen sits in the center of the distorted cube as shown in Fig. 2(a). In contrast, for the singlet configuration, the interstitial oxygen atom forms an O–O bond with an oxygen ion of the cube as shown in Fig. 2(b) with a bond length of 1.40 Å and ∼2.2 eV lower in energy than the triplet.

Fig. 2 The geometry of interstitial defects in SrO. (a) Cubic interstitial oxygen, (b) the charge neutral split interstitial oxygen, (c) the split interstitial oxygen with charge state +1, (d) the split interstitial oxygen with charge state −1, (e) unstable split interstitial strontium, and (f) cubic interstitial strontium, Sr_i sitting in the center of the cube (in the ball and stick molecular diagram, green color is reserved for Sr, red for O, and the split interstitial O, peroxide ion is highlighted in blue).

Thus we identify two types of oxygen interstitial which are cubic interstitial (denoted as O_i) and split interstitial (denoted as O_ii). For the cubic oxygen interstitial O_i, charge states −2, −1, 0, and +1 were considered. For the split interstitial O_ii, except the charge neutral case, we also investigated the charge state +1 and −1. Interestingly, in the charge state +1, we found that the O–O points two neighbouring oxygen atoms with a bond length of 1.31 Å as shown in Fig. 2(c). This is reasonable, as this species carries a +1 charge and interacts with the oxygen atoms which take negative charges. Similarly, in charge −1, we found the O–O is aligned along [100] direction pointing towards strontium atoms as they take positive charge with the O–O bond length 1.87 Å shown in Fig. 2(d).

For the strontium interstitial Sr_i defect,⁶⁰ we did not observe a defect species with a Sr–Sr bond formed on the optimization. In order not to miss this kind of defect, we explore it in more detail as follows. Nine configurations are constructed by fixing the Sr–Sr bond length within the range of 2.4 to 3.3 Å. After optimizing these configurations, we obtained an energy curve which was fitted with a fourth-order polynomial, as shown in Fig. S1 in the ESI.† We found that the lowest energy position occurs when the Sr–Sr bond length is approximately 2.9 Å, as illustrated in Fig. 2(e). The total energy of this configuration is 0.56 eV higher than that of the normal interstitial structure. We then fully optimized this configuration. Unsurprisingly, it reverted to the normal interstitial structure as shown in Fig. 2(f). We conclude that the configuration with the Sr–Sr bond is unstable and may represent a transition state towards the normal interstitial structure. Therefore, we only have one type of cubic strontium interstitial which is shown in Fig. 2(f). Similarly to the strontium vacancies, in the charge neutral case, the singlet configuration is only 0.15 eV lower than that of the triplet configuration.

The formation energy diagram for oxygen interstitial under O-rich conditions is shown in Fig. 3(a) while for the O-poor conditions the energies will shift up 6.182 eV.⁵⁹ Under O-rich conditions, the formation energy of oxygen interstitial has a lower energy with the same charge state than that in O-poor conditions. For the cubic interstitial O_i in SrO, we found that the charge states of −2 have the lowest formation energies in the upper parts of the band gap. In the lowest part of the band gap, the split interstitial O_ii with charge +1 has the lowest formation energy. In the middle area of the band gap, the charge neutral split interstitial O_ii dominates. The corresponding transition energy level (+1/0) is ∼1.0 eV above the VBM, and above this level, the singlet charge neutral configuration O_ii is preferentially formed. The transition energy level (0/−2) is ∼3.9 eV above the VBM, and above this level, the cubic interstitial configuration O_i with charge −2 will dominate. For the charge state −1, it prefers to form close to the CBM. At this charge state, the formation energy of cubic interstitial O_i is 0.53 eV lower than that of the split interstitial O_ii. For the cubic interstitial O_i with charge state −1, the transition energy level (0/−1) lies at 4.3 eV. But for the split interstitial O_ii, the transition energy level (0/−1) is 4.9 eV. This indicates these species could transfer to each other under photoirradiation.

Fig. 3 Formation energies of (a) oxygen interstitial O_i and (b) strontium interstitial Sr_i as a function of the Fermi energy level relative to the VBM under O-rich conditions.

For Sr_i, from the formation energy diagram under O-rich conditions (for O-poor conditions, the energies will shift down 6.182 eV) shown in Fig. 3(b), we find across almost the whole band gap, the charge state of +2 is the most stable. There is only a small energy range below the CBM, where the charge state of +1 of Sr_i has a lower formation energy. The energy transition level of (+2/+1) is ∼5.3 eV above the VBM, which is close to the CBM, as Sr in a +1 charge state is too large to fit in the interstitial site. Only by losing the additional valence shell electron, as in the charge states +2, the ion can be stabilized at the cube site.

3.3 Charge carrier and defect concentrations

The equilibrium-state concentrations of charge carriers are determined by intrinsic defects in SrO which are also influenced by the oxygen partial pressure (P) and temperature. In this section, various realistic oxygen partial pressures P = 1.0 × 10⁻¹⁵, 1.0 × 10⁻⁸, 1.0 × 10⁻³, and 1 atm are considered. The equilibrium carrier and defect concentrations calculated under these conditions are shown in Fig. 4 where the code SC-FERMI^50,61 is used. We note that, alternative software exploiting similar methodological approaches like the DefAP code developed by Murphy et al.⁶² have also been reported in the literature.

Fig. 4 The equilibrium concentrations of holes p₀, electrons n₀, V_o, V_Sr, O_i and Sr_i under different oxygen partial pressure P, (a) P = 1.0 × 10⁻¹⁵ atm, (b) P = 1.0 × 10⁻⁸ atm, (c) P = 1.0 × 10⁻³ atm, and (d) P = 1 atm.

At lower oxygen partial pressure (P = 1.0 × 10⁻¹⁵ atm), we observe significant concentrations of three types of intrinsic defects (V_o, V_Sr, and O_i). The concentrations of V_o and V_Sr are comparable across a wide range of temperatures. However, below approximately 700 K, O_i exhibits the highest concentration. Between 700 K and 1100 K, the concentrations of defects V_o and V_Sr surpass that of O_i, leading to a rapid increase in the n₀ concentration with temperature. At 1100 K, the n₀ and p₀ concentrations are balanced. Above 1100 K, the electron concentrations exceed those of the holes, changing the system behavior to n-type. With an increase in oxygen partial pressure, such as at 10⁻⁸ atm, a similar trend is observed as with 10⁻¹⁵ atm. However, the intersection point of the electron and hole carrier concentrations is reached at around ∼1500 K. Upon further increasing the oxygen partial pressure to 10⁻³ atm, we observed that the concentration of O_i is highest below ∼1300 K, and the hole concentrations exceed the electron concentrations from 0 to 1500 K. Similarly, at P = 1 atm, a comparable trend is observed as with 10⁻³ atm. Under these four oxygen partial pressures, we observed that the concentration of Sr_i is very low due to its high formation energy, indicating its formation to be unfavorable in SrO.

Based on these calculated results, we confirm that SrO exhibits higher hole concentrations than the electron concentration under normal experimental conditions which leads to its p-type behavior. The origin of its p-type conductivity stems from the V_Sr with charge −2. The doubly charged oxygen vacancy at lower Fermi levels overtakes from the singly charged which successfully competes with holes as a charge compensating species for Sr vacancies. This leads to the concentrations of holes reduced and in turn, reduced the p-type conductivity.

3.4 Charge carrier concentrations under different acceptors and donors concentrations

We also studied whether the intrinsic defects respond to the presence of ionized impurities, both acceptors and donors. In SrO, silver (Ag) substituting on a Sr site commonly acts as an acceptor, while indium (In) substituting for Sr acts as a donor. The singly ionized acceptors and donors are denoted as Ag_Sr⁻, and In_Sr⁺, respectively. The concentrations of acceptors and donors chosen were 10⁹, 10¹¹, 10¹³, and 10¹⁷ cm⁻³. At an oxygen partial pressure P = 1 atm, the electron and hole concentrations under different acceptor and donor concentrations are shown in Fig. 5(a) and (b). We found that with higher concentrations of acceptors Ag_Sr⁻, the hole concentrations are higher while the electron concentrations are lower. For the donors In_Sr⁺, it is the opposite behaviour, as expected. The hole concentrations are, however, always higher than the electron concentrations regardless of the presence of acceptors Ag_Sr⁻ and donors In_Sr⁺ in the system at oxygen partial pressure 1 atm which indicates that the p-type conductivity is not suppressed or inverted by the ionized impurities. When the oxygen partial pressure is reduced to 10⁻³ atm, similar results are found.

Fig. 5 Electron (solid lines) and hole (dotted lines) concentrations in the presence of different acceptor and donor concentrations in unit with different oxygen partial pressures, (a) different Ag_Sr⁻ acceptor concentrations at P = 1 atm, (b) different In_Sr⁺ donor concentrations at P = 1 atm, (c) different Ag_Sr⁻ acceptor concentrations at P = 10⁻⁸ atm, and (d) different In_Sr⁺ donor concentrations at P = 10⁻⁸ atm.

On further reducing the oxygen partial pressure to 10⁻⁸ atm, the concentration of electrons and holes begin to show differences, as illustrated in Fig. 5(c) and (d). With acceptor concentrations Ag_Sr⁻ of 10⁹, 10¹¹ and 10¹³ cm⁻³, the hole concentration is higher than the electron concentration from 200 to 1400 K. At temperatures higher than 1400 K, the electron concentration approaches the hole concentration, indicating that the system may transition from p-type to n-type conductivity. However, at high acceptor concentrations, such as 10¹⁷ cm⁻³, the acceptors will accept more electrons and produce more holes. The hole concentration remains higher than the electron concentration, indicating the system maintains and enhances its p-type conductivity. For low concentrations of donors In_Sr⁺, such as 10⁹, 10¹¹ and 10¹³ cm⁻³, the effect on electron and hole concentration is small. Only above 1400 K does the electron concentration start to exceed the hole concentration. For the rest of the temperature range, the hole concentration remains higher than the electron concentration. At higher donor concentrations, such as 10¹⁷ cm⁻³, the donors produce more electrons, significantly affecting the carrier concentration. Around 1100 K, the electron concentration surpasses the hole concentration, indicating a change in the system's conductivity. We note that there is a change in the trend of hole concentration around T > 800 K in the presence of acceptors in Fig. 5(a) and (c), and the reason is that the concentration of V_Sr becomes higher at that temperature which contributes to the hole concentrations. All these results indicate that modifying the p-type conductivity of SrO is challenging. At normal oxygen partial pressure and low donor concentrations, the p-type conductivity remains dominant. Only under low oxygen partial pressure, such as 10⁻⁸ atm, and high donor concentrations, exceeding 10¹⁷ cm⁻³, might its p-type conductivity switch to n-type.

3.5 Discussion

To evaluate our calculated results, we compared them with experimental data and other theoretical studies. In our calculations, the Schottky defect pair (V_O²⁺ + V_Sr²⁻) was determined to have a formation energy of 4.47 eV using the PBE0 functional. The Frenkel defect pairs were calculated to have formation energies of 7.68 eV (V_O²⁺ + O_i²⁻) and 8.25 eV (Sr_i²⁺ + V_Sr²⁻). Similar values were obtained using the BB1K functional, as detailed in the ESI.† Our results indicate that Schottky defects are the dominant defect type. This is consistent with the work of Murarka and Swalin,^7,63 who reported that Schottky defects are predominant at high temperatures in conductivity experiments, with estimated formation energies of 3.7–4.0 eV, aligning well with our calculations. Ramani and Rao⁶⁴ reported the Schottky defect formation energy of 4.23 eV from Mott and Littleton model calculations employing two parameter repulsive potentials which is consistent with our calculated result. Mackrodt and Stewart⁶⁵ also confirmed that the Schottky pair is the most stable defect in SrO, but with a higher formation energy of 6.0 eV using their shell model potential calculations. Richardson⁶⁶ reported a similar higher value of 7.5 eV using a three-body potential. However, our calculations suggest that the charge-neutral oxygen interstitial (O_i) is the dominant defect in SrO under usual conditions, except under very low oxygen partial pressures. This type of defect, however, cannot be detected by conductivity measurements. A piece of possible experimental evidence could be referred to the 2.1 eV slopes reported in conductivity measurements by Copeland and Swalin⁷ at T > 1000 K and P = 10⁻¹² atm, attributed to the formation of O_i¹⁻ and a hole. Under such conditions, the chemical potential of oxygen (μ_O) is estimated to be at least −2.3 eV below the O-rich limit (μ_O = 0 eV),¹⁸ which increases the formation enthalpy of the oxygen split interstitial to at least 2.8 eV according to our PBE0 calculation. The increased entropy effect at high temperatures could account for the higher defect energies in our calculations compared to experimental measurements.

4. Conclusions

In summary, our calculations on intrinsic defects in SrO, using a hybrid QM/MM embedded-cluster approach have shown that oxygen vacancies exhibit the lowest formation energies in the +2, +1, and 0 charge states, with charge-neutral vacancies having the lowest energy in the singlet state. Strontium vacancies display the lowest formation energy in the −2 and +2 charge states. Similarly, oxygen interstitials with charge states of −2 and 0 exhibit the lowest formation energies. In the neutral singlet state, a peroxide-like oxygen interstitial emerges with the lowest formation energy, characterized by the interstitial oxygen forming an O–O bond with a lattice oxygen atom. In interstitial defects, the geometry involves the interstitial oxygen atom positioned at the center of the SrO cube. Strontium interstitials are predominantly present in the +2 charge state, exhibiting high formation energies, indicating them to be unfavorable in SrO. Our analysis suggests that dominant defects in SrO primarily consist of oxygen vacancies, strontium vacancies, and oxygen interstitials, and in particular the peroxide-like oxygen interstitial. Confirmation of these findings is evident in the charge carrier and defect concentration results. Under the lowest oxygen partial pressure (P = 1.0 × 10⁻¹⁵ atm), the peroxide-like oxygen interstitial exhibits the highest concentration, while oxygen vacancies and strontium vacancies remain balanced. The hole concentration stems from strontium vacancies with charge −2, and it is the origin of the p-type conductivity of SrO. Our results align well with experimental observations of SrO exhibiting p-type conductivity, with strontium vacancies with charge −2 being identified as the origin. Besides, the neutral oxygen interstitial is readily formed in SrO and has the largest concentration under normal oxygen partial pressure. In general, transparent conducting oxides of p-type character are rare compared to n-type, and understanding the fundamental limitations for their existence is important. Here the physics of SrO provides a valuable insight and indeed there are successful modifications of basic SrO which already led to new p-type conducting materials such as SrCuO.⁶⁷ This study offers valuable insights into the design of p-type conducting materials based on defect engineering.

Data availability

The data supporting this article has been included as part of the ESI.†

Conflicts of interest

The authors declare no competing financial interests.

Acknowledgements

This work was supported by The Key Technologies R&D Program of Henan Province (no. 242102521002) and the National Natural Science Foundation of China (grant # 22173026, 21703054). The authors acknowledge the use of the THOMAS, YOUNG, and ARCHER2 UK National Supercomputing Service (http://www.archer2.ac.uk) via membership of U.K.'s HEC Materials Chemistry Consortium, which is funded by EPSRC (Grant No. EP/P020194, EP/T022213, EP/R029431, EP/W014378 and EP/W014580).

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Footnotes

† Electronic supplementary information (ESI) available: (1) Energies curves as a function of the Sr–Sr bond length; (2) the data of formation energies of all the defect species using PBE0 and BB1K functionals. See DOI: https://doi.org/10.1039/d4ta07690g

‡ T. Liu and X. Zhang contributed equally to this work as co-first authors.

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