We first reproduce the experimentally determined properties of bulk SrFeO₂. As shown in Tables 1 and 2, the calculated lattice constants, band gap, and magnetic moment overall agree with the experiment and previous theoretical work.^[5] Then we study the strain effect on the bulk SrFeO₂. The SrFeO₂ thin film may be grown on various substrates. Except for the lattice-matched KTaO₃ substrate,^[1,2,19] the practical substrate lattice constants are smaller than 3.99 Å, such as 3.91 Å (SrTiO₃),^[19,28] 3.79 Å (LaAlO₃),^[29] 3.94 Å (DyScO₃),^[19,30] etc. Therefore, the compressive strains are introduced up to −0.06. The tensile strains are also introduced for comparison. To determine the magnetic orderings under strain, we consider the A-AFM, C-AFM, G-AFM, and FM. As shown in Fig. 2, for the G-AFM, both the intralayer nearest neighbor (NN) interactions and the interlayer NN interactions are AFM. For the A-AFM, the intralayer NN interactions are FM, while the interlayer NN interactions are AFM. For the C-AFM, the intralayer NN interactions are AFM while the interlayer NN interactions are FM. From Tables 1 and 2 we can see that the G-AFM has the lowest total energy, indicating that the G-AFM is the magnetic ground state in the strain range we studied. Table 1 also shows the bulk SrFeO₂ has the G-AFM at zero strain, which agrees with the experiment.^[1] From Fig. 3 we can see both the magnetic moment and the volume decrease with increasing strain. The change of volume indicates that SrFeO₂ does not have a perfect Poisson ratio, as a material with an ideal Poisson ratio has a constant volume with strain. This is also reflected in the lattice constants. Due to the high symmetric positions of Fe and O, the Fe–O bonds synchronously expand (reduce) with increasing (decreasing) strain, while the O–Fe–O bond angles keep constant (90°) on the strain range. Consistent with previous work,^[4,5] the density of states (DOS) in Fig. 4(b) shows that the bulk SrFeO₂ displays a Mott–Hubbard band gap, where the down Fe states locate right below the gap and other down 3d orbitals locate above it. The electronic configurations (see Fig. 4(b-2)) are (d_xy)¹(d_yzd_xz)²(d_x²−y²)¹(d_z²)¹ for the up-spin Fe²⁺(d⁶) ion while the sixth down-spin electron occupies the orbital. As shown in Figs. 4(a) and 4(c), when the tensile strain is imposed on SrFeO₂, the degenerate d_yzd_xz states in the conduction band (CB) are pushed into the higher energy levels, thus the insulator character is enhanced with increasing tensile strain.

Table 1.

Table 1.

Table 1. The calculated properties of bulk SrFeO2 without strain. For comparison, the experimental values and previous theoretical values are also provided. The units of a(b) and c are Å. The units of MFe and Egap are μB and eV, respectively. . Property a(b) c MFe Egap Present work 4.04 3.49 3.65 1.1 Experiment[1,27] 3.99 3.47 3.6 1.3 Theory[5] 4.01 3.42 3.6 1.1

Table 1. The calculated properties of bulk SrFeO2 without strain. For comparison, the experimental values and previous theoretical values are also provided. The units of a(b) and c are Å. The units of MFe and Egap are μB and eV, respectively. .

Table 2.

Table 2.

Table 2. The calculated total energies (meV/f.u.) of bulk SrFeO2 at different strains. EG, EA, EC, and EF denote the total energies of G-AFM, A-AFM, C-AFM, and FM, respectively. . Strain EG EA EC EF –0.06 0 116.86 20.13 115.88 –0.04 0 129.92 23.7 132.54 –0.02 0 124.32 20.19 128.13 0 0 106.66 19.7 113.97 0.02 0 91.33 19.1 99.67 0.04 0 76.78 18.53 84.01 0.06 0 61.84 17.88 71.28

Table 2. The calculated total energies (meV/f.u.) of bulk SrFeO2 at different strains. EG, EA, EC, and EF denote the total energies of G-AFM, A-AFM, C-AFM, and FM, respectively. .

Fig. 2.

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	Fig. 2. Schematic view of different magnetic orderings. For clarity the supercell of SrFeO2 is used.

Fig. 3.

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	Fig. 3. Evolution of structural and magnetic properties with strain in bulk SrFeO2. (a) The relaxed volume (vol) and magnetic moment. (b) Lattice constants. The dashed line shows the c-axis lattice constant (cst) for a material with an ideal Poisson ratio.

Fig. 4.

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	Fig. 4. (a-1), (b-1), (c-1), and (d-1) Total DOS and (a-2), (b-2), (c-2), and (d-2) orbital-resolved DOS of bulk SrFeO2.

On the contrary, the compressive strain makes the d_yzd_xz states in the CB come nearer with respect to the valence band maximum (VBM), thus the band gap decreases with increasing compressive strain. At the same time, the separation between the d_x²−y² states and the down d_z² states becomes larger around the Fermi level. When the compressive strain increases up to −0.06, the band gap nearly comes to zero (not shown), the d_x²−y² states also come nearer with respect to the Fermi level. A Mott-type insulator–metal transition (IMT), where the d_x²−y² orbital would cross the Fermi level, is expected if we continue to increase the compressive strain, which is similar to the case of IMT at hydrostatic pressure.^[6,31] Note that tests are performed at −0.08 and −0.1 strain, and we find the occupations of d_x²−y² states at the Fermi level, but the large strain values which may not be very practical are beyond the scope of the present work. Figure 5 shows the evolution of exchange interactions obtained using the classic Heisenberg spin model: H = −J₁∑S_i S_j−J₂∑S_i S_j − J₃∑S_i S_j, S_i(j) = ± 2.

Fig. 5.

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	Fig. 5. Exchange interactions with respect to the strain. The data (zero strain) of Ref. [5] are provided for comparison.

We consider the intralayer NN exchange interactions J₁, the interlayer NN exchange interactions J₂, and the interlayer next nearest neighbor (NNN) exchange interactions J₃. The calculated values at zero strain agree well with previous work.^[5] The exchange interactions show the anisotropic behavior where J₁ is about three times larger than J₂, indicating the dominant role of intralayer superexchange interactions. It is interesting to note that the strength (absolute value) of J₁ shows a decrease trend at −0.06 strain. This may imply the instability of G-AFM at very large strain as the AFM–FM transition has been found at high pressure.^[6,31]

In the tetragonal SrFeO₂ structure (see Fig. 1), the oxygen positions are equivalent to each other. Therefore, an arbitrary O atom is removed from the supercell to calculate the E_form of V_o. In extreme oxidation conditions (oxygen rich), the μ_o is subject to an upper bound given by the energy of O in an O₂ molecule, i.e., μ_o = (1/2)E_o2. In extreme reduction conditions (oxygen deficient), μ_Sr and μ_Fe are subject to an upper bound given by the energy of Sr and Fe in the bulk phases (μ_Sr = E_Sr,bulk, μ_Fe = E_Fe,bulk). Thus in this condition, μ_o = [E_{tot,bulk(f.u.)} − μ_Sr − μ_Fe]/2. The calculated V_o formation energy is shown in Fig. 6. We can see the E_form is relatively high. Thus, the V_o concentration is expected to be low in equilibrium condition. Although the E_form is positive (e.g., 5.1 eV, zero strain, oxygen rich), it is comparable with the E_form (zero strain, oxygen rich) of some typical TMOs, such as HfO₂^[32] (5.93 eV), ZnO^[33] (5.37 eV), ZrO₂^[34] (5.97 eV), etc. The V_o is often unintentionally introduced during the growth or annealing process of these TMOs.^[35,36] Thus, the V_o of SrFeO₂ should be stably formed under certain growth conditions.

Fig. 6.

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	Fig. 6. Formation energy of Vo as a function of strain.

Generally speaking, the increase in bond lengths and volumes associated with the tensile strain is likely to decrease the formation energies of V_o due to the decreased Coulomb interactions between ions.^[37] It is also believed that the remaining electrons caused by V_o would strengthen the pressure of the electron gas and thus induce an intrinsic compressive stress in the unrelaxed lattice.^[38] Therefore, the formation energy is expected to increase in the compressive strain range, and the tensile strain would relieve the intrinsic compressive stress and promote to create the V_o, which has been widely reported in previous similar work.^[39,40] As expected, the V_o formation energy decreases with increasing tensile strain, as shown in Fig. 6. However, we can see from Fig. 6 that the compressive strain also decreases the formation energy. Especially in the compressive strain range the formation energy shows a sharper decrease than that of the tensile strain range. To understand this phenomenon, we first examine the structural relaxations around V_o at each strain. As shown in Table 3 and Fig. 7(a), the removal of oxygen would unscreen the positive Fe ions around V_o by the Coulomb interactions, which results in significant outward relaxations for the two NN Fe ions. For the four NNN Fe ions, the distances between them (∼9 Å) are far larger than that of the two NN Fe ions, and the non-screen effect is smaller, thus the four NNN Fe ions show a slight inward relaxation. Although the relaxation patterns are similar for the compressive strain and the tensile strain, we note the outward relaxations are larger in the tensile strain range when the tensile strain is larger than 0.02, which may indicate the dominant role of the rapidly decreased Coulomb interactions in the tensile strain range. Noting that the decreased bond lengths and volumes caused by compressive strain would make the electronic structures more complex due to the enhanced overlap between orbitals, and thus change the electronic energies which may compensate the increased Coulomb interactions. To confirm this assumption, we decompose the V_o formation energy into its components in Fig. 8. We can see that in the compressive strain range the rapidly decreased band energies can overcompensate the increased remaining energies (mainly electrostatic), which induces the decrease of formation energy. On the contrary, in the tensile strain range the rapidly decreased remaining energies overcompensate the increased band energies when the tensile strain is larger than 0.02, indicating the dominant role of the electrostatic interactions in the tensile strain range. Therefore, the competitions between the band energies and the electrostatic interactions are the dominant mechanisms in determining the V_o formation.

Fig. 7.

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	Fig. 7. (a) Local structure around Vo. b1 and b2 are the bond lengths (Å). (b) Partial charge density of the defect levels shown in Fig. 9(b-2). The isosurface value is 0.015 e−/Bohr3.

Table 3.

Table 3.

Table 3. Structural relaxations around Vo. Δ denotes the changes of b1(b2) before and after relaxations. The units of b1(b2) are Å. . Strain b1 Δb1 b2 Δb2 before after before after –0.06 1.9 1.91 0.01 4.25 4.24 –0.01 –0.04 1.94 2.03 0.09 4.34 4.32 –0.02 –0.02 1.98 2.13 0.15 4.43 4.42 –0.01 0 2.02 2.05 0.03 4.52 4.51 –0.01 0.02 2.06 2.19 0.13 4.61 4.60 –0.01 0.04 2.1 2.29 0.19 4.7 4.69 –0.01 0.06 2.14 2.22 0.08 4.79 4.76 –0.03

Table 3. Structural relaxations around Vo. Δ denotes the changes of b1(b2) before and after relaxations. The units of b1(b2) are Å. .

Fig. 8.

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	Fig. 8. Decomposition of Eform into its contributions.

Fig. 9.

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	Fig. 9. (a-1), (b-1), (c-1), and (d-1) Total DOS and (a-2), (b-2), (c-2), and (d-2) orbital-resolved DOS of defect SrFeO2 with Vo.

As shown in Fig. 9, the V_o formation produces a large shift of the Fermi level. According to the crystal field theory, the removal of oxygen will make the Fe 3d orbitals with the y component fall in energy level. The d_x²−y² orbitals fall the most, as the electrons of d_x²−y² orbitals are concentrated in lobes along the directions of Fe–O bonds. Thus, the remaining electrons of V_o mainly occupy the empty down-spin d_x²−y² orbitals (see Figs. 7(b) and 9(b-2)) and the resulting donor levels locate right below the CBM, indicating the character of n-type conductivity. Comparing with the DOS (see Fig. 4(b-2)) of bulk SrFeO₂, the d_x²−y² electrons become delocalized, as can be seen from the wide distribution of more peaks in the energy range. Therefore, the mobility of the conduction electrons is expected to be large. With the increase of compressive strain, the intralayer superexchange interactions between d_x²−y² states through O p_σ increase, which lifts the energy levels of the d_x²−y² orbitals. Thus, the occupations of the conduction electrons rapidly decrease when the compressive strain is imposed (Fig. 9(c)). When the compressive strain increases to −0.06, the Fermi level is occupied mainly by the Sr d electrons while the down-spin d_x²−y² states are fully pushed into the CB. The tensile strain (Fig. 9(a)) displays the behavior which favors the occupations of d_x²−y² orbitals for the conduction electrons. This is because the decreased intralayer superexchange interactions further decrease the energy level of d_x²−y² orbitals. Even so, the separation between the donor level and the CBM becomes larger and the donor level approximately shifts to the middle of the enlarged gap at 0.06 strain. Thus, the n-type conductivity becomes poor under tensile strain. Finally, based on the above discussion, we propose a possible explanation for the observations of the Hall experiment. As reported in Fig. 6, the V_o concentration is expected to be low in SrFeO₂. In Ref. [19], Katayama et al. found the n-type conductivity on the lattice-matched KTaO₃ substrate which corresponds to the case of zero strain. The DyScO₃ substrate (3.94 Å) and the SrTiO₃ substrate (3.91 Å) would introduce −0.01 and −0.04 compressive strain. As discussed in Fig. 9, the delocalized d_x²−y² electrons are rapidly pushed into the CB once the compressive strain is applied. Although the DOS at −0.06 strain displays the metallic character, the Fermi level is mainly occupied by the localized Sr d electrons. Therefore, no n-type conductivity is found in the strained films.