Neutral defects in SrTiO₃ studied with screened hybrid density functional theory

The calculations of neutral defect formation energies are based upon the formalism of Zhang and Northrup [41]. These values are generated using the equation

$$\begin{eqnarray} &&{E}_{\mathrm{f}}={E}_{\mathrm{T}}-[{E}_{\mathrm{T}}(\mathrm{perfect})-{n}_{\mathrm{Sr}}{\mu }_{\mathrm{Sr}}-{n}_{\mathrm{Ti}}{\mu }_{\mathrm{Ti}}-{n}_{\mathrm{O}}{\mu }_{\mathrm{O}}] \end{eqnarray} \tag{ 1 }$$

where E_T and E_T(perfect) are the calculated total energies of the supercells containing the point defect and the perfect bulk host materials, respectively. The number of each element removed from the perfect supercell is represented by n_x, while μ_x corresponds to the atomic chemical potentials in an SrTiO₃ crystal. Given the assumption that SrTiO₃ is always stable, the chemical potentials of the these elements can vary with the following restriction:

$$\begin{eqnarray} &&{\mu }_{\mathrm{Sr}}+{\mu }_{\mathrm{Ti}}+3{\mu }_{O}={\mu }_{{\mathrm{SrTiO}}_{3}~(\mathrm{bulk})}. \end{eqnarray} \tag{ 2 }$$

Atomic chemical potentials vary according to the sample composition and cannot be determined exactly. However, they can be varied to cover the whole phase diagram of SrTiO₃, by splitting into SrO, TiO, Ti₂O₃ and Ti₂O bulk phases. Hence, the calculated formation energies for the neutral point defects vary according to equilibrium positions; an example of this is O-rich versus O-poor points on the phase diagram. The calculated enthalpies of formation in idealized materials (non-relaxed structures) for phases containing Sr, Ti and O are summarized in table 2, and are compared to previous LSDA calculations [42] and experiment [43].

As a general trend, the formation enthalpies computed with HSE are close to the results from semilocal functionals LSDA and PBE (from this work), although the HSE values are slightly higher. The only exception is SrO, where both HSE and PBE tend to overestimate the formation enthalpies to the same extent, exceeding the LSDA values.

The formation energies of vacancy defects in STO as a function of its composition are plotted in figure 1 with the points A to G based⁵on the phase diagram in [42]. The plot shows that HSE's formation energies show a similar dependence on the chemical potentials as was observed for the LSDA results of [42]. However, both ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ and ${\mathrm{V}}_{\mathrm{Sr}}^{0}$ curves are shifted to higher formation energies, while the ${\mathrm{V}}_{\mathrm{O}}^{0}$ curve is slightly shifted downwards compared to the LSDA spectrum. At point A, ${\mathrm{V}}_{\mathrm{Sr}}^{0}$ has the lowest formation energy followed by ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ and ${\mathrm{V}}_{\mathrm{O}}^{0}$. At point B, ${\mathrm{V}}_{\mathrm{Sr}}^{0}$ remains the most stable but ${\mathrm{V}}_{\mathrm{O}}^{0}$ becomes more stable than ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ contrary to the LSDA predictions. Under low partial pressures of oxygen (O-poor conditions corresponding to points C to G), ${\mathrm{V}}_{\mathrm{O}}^{0}$ vacancies form more easily and dominate the spectrum while the formation energies of ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ and ${\mathrm{V}}_{\mathrm{Sr}}^{0}$ keep increasing.

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A quantitative comparison with previously published calculations using the same cell size formation energies [30, 42, 44] available at point B (oxygen rich conditions) reveals several issues of interest.

• For ${\mathrm{V}}_{\mathrm{O}}^{0}$, the semilocal functional PW91 [44] yields a formation energy of 8.56 eV, the global hybrid B3PW [30] predicted a higher formation energy of 8.74 eV, while LSDA [42] gave a lower value at 7.95 eV. Our HSE value is the lowest among these, yielding a formation energy of 7.43 eV. This decrease in the ${\mathrm{V}}_{\mathrm{O}}^{0}$ formation energy is one of the factors leading to a strong competition with ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ (see below).
• For ${\mathrm{V}}_{\mathrm{Ti}}^{0}$, the semilocal functional LSDA [42] yields a formation energy of 5.7 eV, making it far more stable than ${\mathrm{V}}_{\mathrm{O}}^{0}$. In contrast, with HSE we predict that E_f = 7.86 eV, meaning that ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ becomes less stable than ${\mathrm{V}}_{\mathrm{O}}^{0}$.
• ${\mathrm{V}}_{\mathrm{Sr}}^{0}$ remains the most stable defect under these conditions, with HSE providing a value of 2.81 eV compared to the 1.7 eV from LSDA.

Overall, the quantitative formation energy differences between our HSE results and previous LSDA results are substantial. These might originate from the enhanced accuracy we gain with HSE in the calculated valence band width (VBW) and band gap of bulk SrTiO₃ (see table 1) identified in [13] to reflect an enhanced precision in the defect formation energies.

It is also worth mentioning size effects. The formation energy of ${\mathrm{V}}_{\mathrm{O}}^{0}$ at point B in a larger supercell of 90 atoms, giving a 1.8% ${\mathrm{V}}_{\mathrm{O}}^{0}$ concentration, is 7.66 eV, which is 0.2 eV higher than the value obtained with the smaller supercell.

Isolated, neutral defects have been introduced into the crystal structure of cubic STO by removing one atom of either O, Sr or Ti, respectively, as depicted in figure 2. The structure was then fully relaxed using HSE/SZVP. Figure 3 illustrates the major displacements in each of the defect structures, namely ${\mathrm{V}}_{\mathrm{O}}^{0},{\mathrm{V}}_{\mathrm{Sr}}^{0}$ and ${\mathrm{V}}_{\mathrm{Ti}}^{0}$, relative to the idealized crystal. Distance decreases are depicted by black arrows, while increasing lengths are denoted by yellow arrows. In each case, the magnitude of displacement is implied by the size of the arrows: larger/thicker arrows indicate greater deviation from the defect-free structure.

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A 4% ${\mathrm{V}}_{\mathrm{O}}^{0}$ vacancy was created by removing one oxygen (figure 3(a)) from a 2 × 2 × 2 cell, inducing small, asymmetric bond length changes throughout the cell and an overall decrease in volume relative to the crystal. At the vacancy, the remaining O–Ti bonds all contract but to different extents. Near the vacancy, oxygens move toward nearby strontium, inducing O–Sr bond contractions of up to 5.66 pm. Further from the vacancy, in the regions of the cell behind the defect (and the plane of the figure), strontium appears to migrate inward, toward the vacancy, causing the more distant O–Sr bonds to lengthen by as much as 6.05 pm.

The ${\mathrm{V}}_{\mathrm{O}}^{0}$ structure contracts, and undergoes a tetragonal distortion where the lattice parameters are a = 3.879 and b = c = 3.890 Å, compared to the 3.902 Å in the crystal. A high concentration of oxygen vacancies was identified to be responsible for a similar tetragonal distortion reported experimentally [3] in STO doped with La. It is worth noting that with HSE/SZVP we are able to capture this small tetragonal distortion, corresponding to a c/a ratio of 1.0028. This distortion was also observed in previous LDA calculations [45] for supercells of the same size, but the c/a ratio was not reported.

As seen from figure 3(b), the atomic relaxation around the ${\mathrm{V}}_{\mathrm{Sr}}^{0}$ induces staggered bond lengths (vertically) and a general contraction of O–Sr bonds, with the large decrease being 1.55 pm, smaller than those observed for O–Sr in the presence of an oxygen vacancy. The alternating distances cause the linear Ti–O–Ti angle to decrease to 173.9°. Again, the defect structure contracts, with all lattice parameters decreasing: a = b = c = 3.880 Å.

The ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ structure depicted in figure 3(c) shows the axial oxygens experiencing a 9.2 pm decrease in length, while the medial oxygens move away from the vacancy by 0.6 pm. The Sr–O distances shrink over a range of 1.51–5.40 pm. While also asymmetric, these distortions have less of an effect on the cell, with the equilibrium lattice parameters being a = c = 3.892 and b = 3.894. The a/b ratio here is very small, so the structure remains effectively cubic following full relaxation.

In summary, the loss of metallic species results in smaller volumes with the loss of the larger metal, ${\mathrm{V}}_{\mathrm{Sr}}^{0}$, producing greater contraction. Removing the oxygen produces larger deviations from cubic symmetry in addition to overall shrinkage of the unit cell.

The electronic band structure and the projected densities of states (PDOSs) of the ideal and nonideal/defect STO supercells (doped with O, Sr and Ti neutral vacancies) are shown in figure 4. To allow a clear comparison between the different systems, the VBM was set as reference for all systems while the Fermi energy is depicted with a solid line. The PDOSs are plotted alongside with their corresponding band structure and rescaled in the same way.

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For the ideal 2 × 2 × 2 STO supercell the Brillouin zone folds, causing a direct band gap of 3.19 eV. The CBM in the ideal solid is triply degenerate at Γ and composed of a heavy electron band (he), a light electron band (le) and a spin–orbit band (so). The band degeneracy is lifted as we move far from the Γ point toward high symmetry directions X,M and R, giving rise to different values of the electron effective masses m_he,m_le,m_so in each direction. The electronic effective mass for each band i can be computed by the curves of energy versus $\vec{k}$ being fitted to a parabola [46]; $\vec{k}$ is taken from the Γ point up to 0.5% along Γ → X,Γ → M and Γ → R paths using the formula

$$\begin{eqnarray} &&\frac{1}{m}=\frac{2}{\hbar }\frac{{\mathrm{d}}^{2}E}{\mathrm{d}{k}^{2}}. \end{eqnarray} \tag{ 3 }$$

We estimate the Γ → X effective masses, without spin–orbit correction, ${m}_{h e}^{\Gamma \rightarrow \mathrm{X}}$ and ${m}_{l e}^{\Gamma \rightarrow \mathrm{X}}$, to be 7.3m_e and 0.5m_e respectively, where m_e is the free electron mass. This agrees well with previous non-relativistic calculations [23, 46, 47] using LSDA and HSE, which predicted he masses ranging from 6.1 to 7.3m_e and le masses of 0.4m_e. The remaining effective masses are ${m}_{h e}^{\Gamma \rightarrow \mathrm{M}}=1.1{m}_{e},{m}_{l e}^{\Gamma \rightarrow \mathrm{M}}=0.8{m}_{e},{m}_{h e}^{\Gamma \rightarrow \mathrm{R}}=0.9{m}_{e}$ and ${m}_{l e}^{\Gamma \rightarrow \mathrm{R}}=0.7{m}_{e}$.

Introducing ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ into STO, the defect band averaged in the Brillouin zone is located at 0.15 eV above the valence band maximum (VBM) and 3.06 eV below the conduction band maximum (CBM) (see table 3). The defect band has a VB character and is triply degenerate at Γ with a single band occupied by two electrons, while the upper two bands are empty (see figure 4). The crystal field resulting from the atomic relaxation around ${\mathrm{V}}_{\mathrm{Ti}}^{0}$ causes the triply degenerate CBM band to split in the following way: the heavy electron band (he) is followed by the light electron band (le) at 26 meV, while the spin–orbit band (so) is located 14 meV higher.

Table 3.  Location of the various defect bands averaged in the Brillouin zone with respect to the conduction band minimum (Δ_DB–CBM) and valence band maximum (Δ_DB–VBM). Data are for the 2 × 2 × 2 SrTiO₃ supercell computed with HSE/SZVP. Defect formation energies (in eV) at point B (oxidation condition) are reported as well.

	ΔDB–CBM	ΔDB–VBM	Ef
${\mathrm{V}}_{\mathrm{Ti}}^{0}$	3.06	0.15	7.86
${\mathrm{V}}_{\mathrm{Sr}}^{0}$	3.27	0.07	2.81
${\mathrm{V}}_{\mathrm{O}}^{0}$	0.44, 0.40a	2.82, 2.90a	7.43

^aExperimental estimations from [3, 48].

For ${\mathrm{V}}_{\mathrm{Sr}}^{0}$, a triply degenerate defect band appears at 0.07 eV above the VBM and 3.27 eV below the CBM at the Γ point. From figure 4, two bands are occupied by four electrons while the upper band is empty. The degeneracy of the defect band is lifted at the X point, where the two occupied bands remain degenerate while the unoccupied band is 600 meV higher in energy. The he and le bands in the CBM remain degenerate while the so band is located 1.5 meV higher in energy, mainly because the overall deformation ${\mathrm{V}}_{\mathrm{Sr}}^{0}$ introduces into the lattice is small.

Table 4.  Non-relativistic HSE/SZVP calculated heavy electron effective masses (in units of the free electron mass m_e) at the conduction maximum along different high symmetry directions in ideal (${m}_{h e}^{\mathrm{CBM}}$) and oxygen vacancy doped SrTiO₃ (${m}_{h e}^{\mathrm{DCBM}}$). k^∥ and k^⊥ represent the parallel and perpendicular planes to the direction of the compressive 100 strain resulting from ${\mathrm{V}}_{\mathrm{O}}^{0}$.

Direction	${m}_{h e}^{\mathrm{CBM}}$	${m}_{h e}^{\mathrm{DCBM}}$
Γ → X(100)
k∥	7.3	0.5
k⊥	7.3	11.2
Γ → M(110)
k∥	1.1	1.1
k⊥	1.1	1.1
Γ → R(111)
	0.9	0.9

Introducing ${\mathrm{V}}_{\mathrm{O}}^{0}$ into STO causes a defect band (DB) populated with two electrons to appear in the gap just below the bulk-like conduction band maximum labeled here as DCBM, indicating a donor band. The Fermi energy is shifted from the top of the valence band to the maximum of the defect band at the M point followed by the empty conduction band. The CBM conserves the bulk character as it remains empty and triply degenerate (clearly shown in figure 4), corresponding to the Ti t_2g states followed by a doubly degenerate e_g band. The above CBM conservation of degeneracy confirms that the band appearing underneath is a fully occupied non-degenerate defect band. The isosurface of the highest occupied state at the Γ special k-point (figure 5) shows that the electronic charge density is localized on ${\mathrm{V}}_{\mathrm{O}}^{0}$ occupying the Ti dangling bonds.

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We calculated the position of the defect state by averaging over the Brillouin zone. We found that the DB is located on average 0.44 eV below the CBM for the 40 atom supercell (see table 3) and 0.42 eV for the 90 atom supercell. This could be compared with the recent experimental measurements, which place the position of the defect level 0.4 eV below the CBM, causing the blue light photoluminescence of STO at room temperature [3, 48]. In the present calculations, self-defect interaction or the so-called size effects were minimized by keeping the defects neutral, and conserving the same high density of k-point sampling of the Brillouin zone [49]. Nevertheless, simulations using larger supercells and finite size scaling [7] are still needed for a more accurate comparison. With today's computational resources, a full relaxation of larger supercells using HSE/SZVP and the computational settings used here is computationally very expensive. This defect band seems to be shallower than the one computed with B3PW, which was located 0.79 eV below the CBM [30]. We attribute this to the B3PW functional overestimating the band gaps (indirect gap of 3.6 eV compared to 3.2 eV from experiment). Our HSE-predicted defect level for ${\mathrm{V}}_{\mathrm{O}}^{0}$ is deeper than that calculated with LDA; in those calculations the defect band can often not be distinguished from the CBM (resonant band) [45], although in some cases it lies as little as 0.08 eV below the CBM [42].

The atomic relaxation around ${\mathrm{V}}_{\mathrm{O}}^{0}$ leads to a tetragonal distortion of the supercell where c/a = 1.0028 (see section 3.2), corresponding to a −0.3% compressive strain along the X axis. This amount of strain is comparable to the values applied experimentally in La doped STO [24] and identified as causing a substantial increase of the electron mobilities (by a factor of 3.3) due to the appearance of light electron effective mass along the strain and the transport direction.

In our calculation, the effective mass ${m}_{h e}^{\Gamma \rightarrow \mathrm{X}}/{m}_{e}$ along the direction parallel to the compressive strain (k^∥ = π/a(100)) drops from the bulk value of 7.3 to 0.5 in defective SrTiO₃ (see table 4), indicating that the electron mobility increases substantially along the x-axis. However, ${m}_{h e}^{\Gamma \rightarrow \mathrm{X}}/{m}_{e}$ increases along the k^⊥ = π/a(010) = π/a(001) directions (perpendicular to the compressive strain) to 11.0, indicating a reduced electron mobility in these directions. Along Γ → M, the effective masses remain unchanged in the k^∥ = π/a(110) = π/a(101) containing the strain axis.

The substitution of one Sr atom by a La atom in our 2 × 2 × 2 STO supercell led to a dopant concentration of 12.5%, which is low enough to be compared to experimental data [3], where dopant concentrations as high as 15% have been used. The resulting Sr_0.875La_0.125TiO₃ compound relaxes to a cubic structure with lattice parameters a = b = c = 3.893 Å. This agrees well with experiments [3] which demonstrated that La doping conserves the cubic symmetry, and has a negligible effect on the c-axis lattice parameter.

Figure 6 shows the band structure of Sr_0.875La_0.125TiO₃ for spin up and spin down electrons. For the spin up electrons, the VBM is almost triply degenerate with a very small splitting of 7 meV; this is also the case of the lowest CB, a band which is populated with one extra electron and shows a small band broadening (13 meV). The energy difference between the last VB and the first populated CB is 3.16 eV. The next set of conduction bands, normally located at 200 meV above the CBM in the bulk STO, is split into one band 172 meV above the CBM followed by a doubly degenerate band 100 meV higher. Visualization of the isosurface of the highest occupied state at the Γ special k-point shows that the charge density is not localized on the defect site, but rather corresponds to the Ti t_2g states. The electronic density of states (not shown) also confirms that the CB remains dominated by Ti 3d states as it does in the STO bulk phase; La 4d starts to contribute to the CBM only about 2.3 eV above the Fermi energy.

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For the spin down electrons (bottom of figure 6), the band structure conserves most of the bulk band characteristics with triply degenerate CBM and VBM, but the band gap is larger than the bulk value, measuring 3.55 eV. The heavy electron effective masses relative to the free electron mass (${m}_{h e}^{\ast }/{m}_{e}$) of the lower band or heavy electron (he) band are isotropic and experience a decrease from the bulk value of 7.3 to 6.8 in the Γ → X direction, bringing the ${m}_{h e}^{\ast }/{m}_{e}$ ratio closer to the experimentally measured values [3] of 6 to 7.1 for 15% La doping. A smaller decrease is observed along the other high symmetry directions.