Surface Structures of CaFe₂O₄ (001), (100), (110) and (111): A Density Functional Theory Study

Abstract

Calcium ferrite is the main binder phase in sinter, which affects the physical and metallurgical properties of sinter. It was also widely applied to photocatalysts, oxidation catalysts, photocathodes and gas sensors etc. The electronic, magnetic and chemical properties of it were investigated since its crystal structure was first reported. However, the right terminations, structures and relative stabilities of CaFe₂O₄ surface have not been systemically studied. Hence, the surface structures of CaFe₂O₄ (001), (100), (110) and (111) were calculated by using a generalized gradient approximation considering on-site Coulomb interaction of iron 3d electrons (GGA + U) in the paper. With U = 4.70 eV, the band gap for up-spin and down-spin energy of CaFe₂O₄ was calculated to be 1.91 and 1.81 eV, closed to the experimental value (1.90 eV). For the CaFe₂O₄ (001) surface, the O1 terminations were the most stable and the surface energy was 1.307 J·m⁻². In the case of CaFe₂O₄ (100) surface, the surface energies at Ca and O1 terminations were 1.278 and 1.568 J·m⁻², respectively. There were also two most stable CaFe₂O₄ (110) surfaces in close surface energies and terminated with the exposed Fe2 and O3 atoms. The surface energies of them were 1.489 and 1.570 J·m⁻², respectively. Among the fifteen CaFe₂O₄ (111) terminations, the surface energies at O2 (l) and O4 (f) terminations were the lowest and they were 1.421 and 1.455 J·m⁻². The calculated surface energies indicate that (100) was better than (001), (110) and (111) in thermodynamic, which agrees well with the experimental results.

1. Introduction

The calcium ferrite minerals as the dominant bonding phases in sinter, and the properties and behavior of them directly affect the physical and metallurgical properties of the sinter. The effects of atmosphere,¹⁾ CaO source²⁾ and components^3,4,5,6) on the formation behavior of calcium ferrite minerals were extensively studied during the sintering process. Moreover, the reduction behavior of calcium ferrite minerals under CO and H₂ were also investigated.^{7,8,9,10,11,12)} In addition, in recent research, it has been widely applied including as a thermal catalyst for decreasing the NOx emission in the sintering process,^13,14,15) as a photocatalytic for degradation of the organic wastewater^16,17) and volatile organic compounds,^18,19,20) as a photocathode for producing hydrogen in water splitting,^21,22,23) as a oxygen carrier for chemical looping gasification applications,^{24,25,26,27,28)} as a anode material for lithium-ion batteries²⁹⁾ and as a visible light activated gas sensor.³⁰⁾ The surface structure of calcium ferrite affects the strength of sinter and reduction reaction of it with CO and H₂ in the blast furnace. However, since CaFe₂O₄ was found as a p-type semiconductor and orthorhombic structure,³¹⁾ the stability of its surface structure has been few studied.

Recently, in order to clarify the structure of CaFe₂O₄, Obata³²⁾ and Sun³³⁾ calculated the electronic structure of CaFe₂O₄ based on the density-functional theory (DFT) and employed the generalized gradient approximation (GGA) by including the Hubbard parameter U = 5.2 eV (GGA + U). They found that the band gap energy was 1.9 eV and the valence band mainly consisted of the interaction Fe 3d and O 2p states while the conduction band was composed of Ca 3d states. At the same time, the theoretical total energies of the non-magnetic, ferromagnetic and antiferromagnetic of CaFe₂O₄ was also calculated. The result indicated the antiferromagnetic phase was the most stable among three phases, which was same to the calculated results by Das et al. with U = 4.0 eV.³⁴⁾ In addition, Wang et al.^35,36) calculated the surface energies of (040), (320), (201) and (401) surfaces, and they found the (401) surface had the lowest energy. However, the Hubbard parameter and the antiferromagnetic were not considered in the calculation process. At present, the right terminations, structures and relative stabilities of CaFe₂O₄ surface have not been systemically studied. Hence, in the paper, the structures and stabilities of the CaFe₂O₄ (001), (100), (110) and (111) surfaces at different terminations were calculated by using DFT + U method.

2. Computational Methods and Models

All DFT calculations were performed by using the Vienna Ab initio Simulation Package (VASP).^37,38,39) The pseudopotential was described with projector-augmented-wave (PAW) method.^40,41) The exchange-correlation energy was treated by adopting the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional.⁴²⁾ Because the strong on-site Coulomb interaction of iron 3d electrons , CaFe₂O₄ is strongly correlated. Strongly correlation, also known as strongly correlated electronic system, refers to the system in which the interaction between electrons cannot be ignored. In general, the electrons are independent in solids and the electrostatic interaction between electrons is usually ignored and does not appear in the Hamiltonian. However, the 3d electron orbitals overlap greatly in transition metal compounds and lanthanide compounds. The 3d-electrons are close to each other, and then the electrostatic energy is generated between electrons.^43,44) At this time, putting the electrostatic energy into the Hamiltonian, we get the strong correlation model (Hubbard model ). The first principles calculations within the local density approximation (LDA) or generalized gradient approximation (GGA) lead to considerable error in described the material properties of many transition metal compounds. The DFT + U approach has been proposed and developed in order to describe phenomena of the strong electron correlation. The U-value is usually chosen based on its accuracy in reproducing the electronic structures (i.e., experimental band gap) of the bulk materials. Therefore, we employed generalized gradient approximation (GGA) by including the Hubbard parameter U (GGA + U) calculations.⁴⁵⁾ For GGA + U calculation, the U value was carried out from 0.0 to 7.0 eV for bulk calculations. The cutoff energy for the plane wave expansion was set to be 400 eV. The force on each atom smaller than 0.03 eV/Å was set as convergence criterion. The Brillouin zone was sampled with the Monkhorst–Pack grid (MP).⁴⁶⁾ For integration within the Brillouin zone, specific k-points were selected using a 3 × 7 × 3 MP grid for the bulk calculation and a 3 × 4 × 1, 4 × 2 × 1, 2 × 3 × 1, 3 × 2 × 1 MP grid for (001), (100), (110) and (111) surface calculations.

The unit cell of CaFe₂O₄ was shown in Fig. 1. CaFe₂O₄ has an orthorhombic structure (Pnma, # 62) with lattice parameters a = 9.230 Å, b = 3.024 Å, and c =10.705 Å.^31,47) It consists of distorted FeO₆ octahedral sharing edges and corners with eight-coordinated calcium atoms. There are symmetrically two independent Fe atoms (Fe1 and Fe2), one Ca atom, and four independent O atoms (from O1 to O4) in the unit cell. Ca atom is surrounded by eight O atoms and Fe atoms are surrounded by six O atoms with a distorted octahedron shape.^32,33,48) In addition, CaFe₂O₄ has a magnetic property and the most stable structure is antiferromagnetic (AFM).^32,34,49) It is a consequence of the antiferromagnetic alignment of the magnetic moments on the two Fe atoms. Hence, the AFM of CaFe₂O₄ was also set in the present study.

Fig. 1.

Unit cell of CaFe₂O₄ (green, purple, khaki, gray, blue, red and azure balls represent the Ca, Fe1, Fe2, O1, O2, O3 and O4 atoms, respectively). (Online version in color.)

In this work, we considered several low-index planes: (001), (100), (110) and (111). There are seven, seven, seven and fifteen terminations on the (001), (100), (110) and (111) surfaces, respectively. To discuss these surfaces, the supercell was constructed by a (1 × 2) Ca₁₆Fe₃₂O₆₄ slab for (001) and (100) surfaces, and a (1 × 1) Ca₁₆Fe₃₂O₆₄ slab for (110) and (111) surfaces. In order to avoid the interaction between periodic structures, the vacuum gap was set as 20 Å. In our calculations, the topmost 5 Å layers of all slabs are relaxed, while the corresponding bottom layers are fixed. The dipole correction was employed to correct potential spurious terms arising from the asymmetry of the slabs.

For discussing and comparing the stability of surface terminations, we computed the surface energy γ_S, that describes the stability of a surface, is the energy required to cleave a surface from a bulk crystal. It is given by Eq. (1).⁵⁰⁾   

$${\gamma }_S=\frac{16.02}{2A}(E_S^{unrelax}-N{E}_b)+\frac{16.02}{A}(E_S^{relax}-E_S^{unrelax})$$(1)

where, γ_S is the surface energy, J/m²; A is the area of the surface considered, m²; $E_S^{relax}$ and $E_S^{unrelax}$ the energies of the relaxed and unrelaxed surfaces, eV, respectively; N the number of atoms in the slab and E_b the bulk energy per atom, eV/atom.

3. Results and Discussion

3.1. CaFe₂O₄ Bulk

In order to determine the Hubbard parameter for the correct description of CaFe₂O₄, benchmark calculations with U = 0 – 7 eV were carried out (Table 1). With the U parameter increasing, the band gap for up and down spin states of CaFe₂O₄ was linearly increased. Compared to the experimental band gap of 1.82 eV⁵¹⁾ or 1.90 eV,^32,52) the band gap for up and down spin states of CaFe₂O₄ is closed to that when U was 5 eV. Hence, we took U = 4.7 eV, the total electronic density of states (DOS) of CaFe₂O₄ bulk was shown in Fig. 2. It shows that CaFe₂O₄ is semi-conductor with the Fermi level sitting in the middle of the spin-down, and a band gap for up-spin and down-spin of 1.91 eV and 1.81 eV, close to value for CaFe₂O₄ bulk,^32,51,52) respectively. At U = 4.7 eV, the lattice parameters of CaFe₂O₄ were a = 9.302 Å, b = 3.044 Å, c =10.796 Å and the energy of the CaFe₂O₄ bulk was −188.347 eV, which were closed to the experimental value.^31,47)

Table 1. Band gap for up-spin and down-spin states of CaFe₂O₄ with different Hubbard parameter.

U/eV	0	1	2	3	4	5	6	7
Gapup/eV	0.30	0.60	1.00	1.39	1.70	2.12	2.44	2.76
Gapdown/eV	0.00	0.40	0.80	1.20	1.50	1.92	2.24	2.55

Fig. 2.

Total electronic density of states for CaFe₂O₄ obtained from the GGA + U approach (U = 4.7 eV). (Online version in color.)

3.2. CaFe₂O₄ (001) Surface Termination and Stability

Figure 3 shows the topmost layers of seven optimized terminations of the CaFe₂O₄ (001) surface. Figure 3(a) shows the first termination with O1 atom. Based on Bader charge analysis,^53,54) O1 and O2 atoms obtained 0.16 and 0.08 electrons while Ca lost 0.01 electrons, respectively. Ca atom moved toward the surface and the relaxations of Ca-O3 and Ca-O4 bonds were 15.2% and 17.2%, respectively. The angle of O1-Ca-O2 increased from 72.8 degree to 89.8 degree. The second termination with O2 atom was shown in Fig. 3(b). The bond of Fe1-O4 was break and O4 atom was up to the surface. Since O4 atom escaped from Fe1 atom, Fe1 and O2 atom obtained 0.33 and 0.11 electrons, respectively. The distance between Fe1 and O4 expanded from 0.202 nm to 0.324 nm. Meanwhile, O4 atom closed to the Fe2 and Ca atoms, which led to the 0.43 and 0. 02 electrons transfer from Fe2 and Ca atoms to the O4 atom, respectively. The relaxation of Ca-O4 was 12.8%. Meanwhile, the angle of O3-Fe2-O4 increased from 93.9 degree to 124.8 degree. The changes of the surface structure with Fe2 atom termination (Fig. 3(c)) were similar to those of the O2 atom termination, the O4 atom was exposed on the surface. Fe2 and Ca atoms lost 0.40 and 0.14 electrons, respectively, and Fe1 atom obtained 0.33 electrons. The bond lengths of Fe1-O4 and Ca-O4 changed to 0.345 and 0.221 nm. At the same time, the angle of O3- Fe2-O4 also increased from 93.9 degree to 125.2 degree. For the Fe1 termination (Fig. 3(d)), Fe1 atom moved downward, the O3 and O4 obtained 0.14 and 0.06 electrons. The bond length of Fe1-O4 decreased to 0.19 nm and the angles of O4-Fe1-O3 increased to 114.6 degree. In addition, the bond of Ca-O4 was break and the relaxation of Ca-O4 was 43.4%. O16 atom obtained 0.27 electrons. As can be seen from Fig. 3(e), the O3 atom closed to the Ca atom and obtained 0.22 electrons. The bond length of Ca-O3 decreased from 0.251 to 0.228 nm. Figure 3(f) shows the surface structure with O3 atom. It can be seen that the O3 atom was also closed to the Ca atom and obtained 0.04 electrons. The angle of O3-Ca-O4 also increased from 72.1 degree to 96.2 degree. In the case of O4 atom termination (Fig. 3(g)), it can be seen that O3 atom moved downward. O3 atom obtained 0.11 electrons from Fe1 atom. The band lengths of Fe1-O2 and Fe2-O2 were 0.191 and 0.195 nm, slightly shorter than those in the bulk (0.209 and 0.207 nm). At the same time, the angle of Fe1-O2-Fe2 expanded to 125.2 degree. Meanwhile, since the O4 atom obtained 0.13 electrons while the O2 atom lost 0.11 electrons, respectively, the Ca-O4 distances was 0.23 nm, short than the bulk values (0.257 nm).

Fig. 3.

Topmost layers of seven terminations for CaFe₂O₄ (001) surface (green, purple, khaki, gray, blue, red and azure balls represent the Ca, Fe1, Fe2, O1, O2, O3 and O4 atoms, respectively). (Online version in color.)

In addition, the surface energies at different terminations were calculated for the relative stability. Due to the same number of atoms (Ca₁₆Fe₃₂O₆₄ slab), the relative stability will be directly obtained by comparing the surface energies at different terminations. The surface energies at different terminations were shown in Table 2. The surface with O1 atom termination was the most stable and the surface energy of it was 1.307 J·m⁻² , the surface energies of other terminations were much higher between 1.885 and 3.798 J·m⁻².

Table 2. Surface energy of different terminations for CaFe₂O₄ (001) surface.

Terminations	O1	O2	Fe2	Fe1	Ca	O3	O4
$E_S^{relax}$/(eV)	−740.62	−736.65	−723.02	−730.94	−726.72	−724.65	−734.35
$E_S^{unrelax}$/(eV)	−737.09	−733.24	−719.50	−723.83	−723.68	−721.65	−732.12
γS/(J·m−2)	1.307	1.885	3.798	2.170	3.341	3.639	2.376

3.3. CaFe₂O₄ (100) Surface Termination and Stability

Figure 4 shows the topmost layers of seven optimized terminations of the CaFe₂O₄ (100) surface. Figure 4(a) shows the surface structure with O4 atom. It can be seen that O2 and O4 atoms were closed to Fe1 atom. Based on Bader charge analysis,^53,54) the O2 and O4 atoms obtained 0.06 and 0.01 electrons while Fe1 atom lost 0.04 electrons. In the optimized structure, Fe–O bond lengths of O2–Fe–O4 were 0.173 and 0.174 nm, respectively. Meanwhile, the angle of O2–Fe–O4 decreased from 160.1 degree to 153.4 degree. The second termination with Fe1 atom was shown in Fig. 4(b). The Ca-O2 bond was break and O2 atom exposed to the surface. Fe1 atoms lost 0.16 electrons while O2 obtained 0.05 electrons. The Fe–O bond lengths slightly decreased to 0.188 and 0.194 nm, respectively, and the angle of Fe1-O2-Fe1 increased from 98.5 degree to 117 degree. In addition, the Ca atom also obtained 0.03 electrons, the distance between Ca and O2 atoms increased from 0.24 to 0.381 nm. For the Fe2 (Fig. 4(c)), O2 (Fig. 4(d)) and O1 terminations (Fig. 4(e)), the Fe1 atom moved downward and it lost some electrons while O2 obtained electrons. The Fe–O bond lengths of O4-Fe1-O2 slightly decreased and the angle of O4-Fe1-O2 increased from 160.1 to about 177 degree. Figure 3(f) shows the surface structure with O3 atom. Since O3 atom obtained 0.20 electrons while Fe1 and O2 atoms lost 0.06 and 0.14 electrons. The O3 atom was closed to the Fe1 atom and the bond length of O3-Fe1 decreased to 0.175 nm. As can be seen from Fig. 3(g), the bond of O1-Fe2 was break and Oa1tom move upward. The Fe2 atom obtained 0.03 electrons while the Ca atom lost 0.05 electrons. The distance between O1 and Fe2 atom was expanded from 0.206 to 0.254 nm. Furthermore, the surface energies at different terminations were shown in Table 3. It can be seen that the surface with Ca atom termination was the most stable, and then the O2 termination. The surface energies of them were 1.278 and 1.568 J·m⁻², those of other terminations were over 2.10 J·m⁻².

Fig. 4.

Topmost layers of seven terminations for CaFe₂O₄ (100) surface (green, purple, khaki, gray, blue, red and azure balls represent the Ca, Fe1, Fe2, O1, O2, O3 and O4 atoms, respectively; atomic charges are marked in red). (Online version in color.)

Table 3. Surface energy of different terminations for CaFe₂O₄ (100) surface.

Terminations	O4	Fe1	Fe2	O2	O1	O3	Ca
$E_S^{relax}$/(eV)	−730.01	−729.09	−732.65	−732.39	−737.88	−731.88	−739.30
$E_S^{unrelax}$/(eV)	−726.87	−724.48	−729.19	−728.34	−735.24	−728.65	−735.71
γS/(J·m-2)	2.465	2.400	2.105	2.066	1.568	2.228	1.278

3.4. CaFe₂O₄ (110) Surface Termination and Stability

Figure 5 shows the topmost layers of seven optimized terminations of the CaFe₂O₄ (110) surface. Figure 5(a) shows the surface structure with O4 atom termination. As can be seen from Fig. 5(a), the O1 atom closed to Fe1 atom and got away from Fe2 atom. Based on Bader charge analysis,^53,54) the O1 atom got 0.25 electrons while Ca and Fe1 atoms lost 0.04 and 0.13 electrons, respectively. It led to the broken of O1-Fe1 and formation of the O1-Fe2 bond. The Fe1-O4 and Fe2-O4 distance were 0.214 and 0.40 nm. At the same time, the bond length of Ca-O4 increased from 0.256 to 0.285 nm. Figures 5(b) and 5(c) show the surface structures with O4 and Fe1 atom terminations, respectively. The O1-Fe2 bond was broke and the distance of it increased from 0.206 to 0.364 and 0.382 nm in Figs. 5(b) and 5(c), respectively. The electrons were transferred from the Ca and Fe2 atoms to the O1 atom. The bond length of O1-Fe2 decreased from 0.208 to 0.176 nm. For the Fe2 termination (Fig. 5(d)), the O1-Fe2 bond was broke while the new Ca-O1 bond formed. It may be the O1 obtained 0.44 electrons from the Ca and Fe2 atoms. The bond length of O1-Fe2 changed to 0.181 nm. Figure 5(e) shows the surface structures with O1 atom terminations. The O1 atom got 0.28 electrons while the Fe1 and Fe2 lost 0.19 and 0.06 electrons, respectively. Hence, the length of O1-Fe2 decreased from 0.209 to 1.83 nm and the new O1-Fe1 bond formed. The surface structures with O3 and Ca atoms terminations were shown in Figs. 5(f) and 5(g), respectively. The Ca-O4 bond was broke and the distance of it enlarged over 0.40 nm in Figs. 5(f) and 5(g). The Ca and O3 atom lost electrons and the relaxation of Ca-O3 was 5.7%.

Fig. 5.

Topmost layers of seven terminations for CaFe₂O₄ (110) surface (green, purple, khaki, gray, blue, red and azure balls represent the Ca, Fe1, Fe2, O1, O2, O3 and O4 atoms, respectively; atomic charges are marked in red). (Online version in color.)

In addition, the surface energies of CaFe₂O₄ (110) at different terminations were shown in Table 4. It shows that the surface energies of them were between 1.489 and 1.941 J·m⁻². The surface energy at Fe2 termination (Fig. 5(d)) was the lowest (1.489 J·m⁻²), which indicated the most stable structure. The surface energy at O3 termination was 1.570 J·m⁻², slightly below than that of Fe2 termination (Fig. 5(d)). However, the surface energy was the highest at Ca termination (Fig. 5(g)).

Table 4. Surface energy of different terminations for CaFe₂O₄ (110) surface.

Terminations	O4 (a)	O4 (b)	Fe1 (c)	Fe2 (d)	O1 (e)	O3 (f)	Ca (g)
$E_S^{relax}$/(eV)	−722.78	−720.64	−723.63	−726.37	−726.13	−727.69	−723.47
$E_S^{unrelax}$/(eV)	−717.55	−713.05	−716.58	−719.00	−720.40	−722.71	−719.16
γS/(J·m−2)	1.923	1.908	1.722	1.489	1.631	1.570	1.941

3.5. CaFe₂O₄ (111) Surface Termination and Stability

The topmost layers of fifteen optimized terminations of the CaFe₂O₄ (111) surface was shown in Fig. 6. Figures 6(a) and 6(b) shows the surface structure with O4 and Fe2 atoms termination, respectively. In Figs. 6(a) and 6(b), O4 atom closed to Fe1 atom and got away from Fe2, which indicated the length of O4-Fe bond decreased from 0.202 to 0.185 nm and the length of O4-Fe bond increased 0.210 nm. Because of the O4 atom got 0.08 electrons from Fe1 atom. Fe1 atom lost 0.21 and 0.16 electrons in Figs. 6(a) and 6(b), respectively. In addition, Ca atom also move toward O4 atom and lost 0.03 electrons. It led to the decreasing of the Ca-O4 bond length. As can be seen from Figs. 6(c)–6(f), the Ca-O3 bond was broke and a new Ca-O3 bond was restructured. The Ca atom lost about 0.03 electrons while the O3 atom obtained about 0.30 electrons in the new Ca-O3 bond. The distance between Ca and O3 was decreased from 0.346 to 0.230 nm. Moreover, Fe1 atom also closed to O3 atom and lost 0.04 electrons, which revealed the bond length of Fe1-O3 decreased from 0.208 to 0.185 nm. The surface structures with O3 atoms terminations were shown in Fig. 6(g). The Ca and two O4 atoms both lost 0.02 electrons, resulting in the breaking of the two Ca-O4 bonds and the distances between Ca and O4 were 0.287 and 0.346 nm, obviously longer than the bulk value (0.257 nm). Furthermore, 0.05 electrons transferred from Fe1 to O3 atom, which indicated the bond length of Fe1-O3 decreased to 0.185 nm. The surface structures with O2 atoms terminations were shown in Fig. 6(h). The relaxations of Ca-O3 and Ca-O4 bonds were 58.4% and 25.6%, which indicated the Ca-O3 and Ca-O4 bonds broke. The new Ca-O3 bond formed and the length of it was 0.224 nm. The O3 atom obtained 0.43 electrons. For Figs. 6(i)–6(l), since Fe1 and Fe2 snatched the electrons from O4 atom, the Fe1-O4 and Fe2-O4 decreased to 0.19 nm and the Ca-O4 bond broke. The distance between Ca and O4 were over 0.30 nm, significantly longer than the bulk value (0.257 nm). Figure 6(m) shows the surface structure with Fe1 atom termination. O2 and O3 atoms obtained 0.06 and 0.05 electrons from Fe1 atom. The bond lengths of O2-Fe1 and O2-Fe2 decreased from 0.201 and 0.208 nm to 0.182 and 0.190 nm, respectively. Figures 6(n) and 6(o) shows the surface structure with O4 atom termination. In Fig. 6(n), the O2, O3 and O4 obtained 0.10, 0.09 and 0.05 electrons. The lengths of the Ca-O2, Ca-O3 and Ca-O4 bonds decreased to 0.227, 0.233 and 0.242 nm. However, the bond length of Ca-O1 increased from 0.238 to 0.267 nm. In addition, the O3 atom got 0.08 electrons while Fe1, Fe2 and Ca atom lost 0.02, 0.04 and 0.02 electrons. Meanwhile, the lengths of O3-Fe1, O3-Fe2 and O3-Ca decreased from 0.208, 0.206 and 0.250 nm to 0.199, 0.190 and 0.233 nm.

Fig. 6.

Topmost layers of fifteen terminations for CaFe₂O₄ (111) surface (green, purple, khaki, gray, blue, red and azure balls represent the Ca, Fe1, Fe2, O1, O2, O3 and O4 atoms, respectively; atomic charges are marked in red). (Online version in color.)

The surface energies of CaFe₂O₄ (111) at different terminations were shown in Table 5. The surfaces with O2 (Fig. 6(l)) termination had the lowest surface energies (1.421 J·m⁻²).The surface energy at O4 (Fig. 6(o)) termination was 1.455 J·m⁻², slightly more than that of O2 (Fig. 6(l)). Hence, the CaFe₂O₄ (111) surface with O2 (Fig. 6(l)) termination were the most stable structures. Meanwhile, the surface energies of other terminations were much higher between 1.699 and 1.963 J·m⁻². The surface with Fe1 atom termination (Fig. 6(f)) had the highest surface energy.

Table 5. Surface energy of different terminations for CaFe₂O₄ (111) surface.

Terminations	O4 (a)	Fe2 (b)	O3 (c)	Ca (d)	Ca (e)	Fe1 (f)	O3 (g)	O2 (h)
$E_S^{relax}$/(eV)	−721.77	−723.07	−721.43	−723.18	−722.37	−719.68	−723.26	−722.11
$E_S^{unrelax}$/(eV)	−715.66	−716.70	−714.62	−716.48	−715.30	−712.77	−716.96	−714.45
γS/(J·m−2)	1.869	1.754	1.842	1.721	1.753	1.963	1.744	1.730

Terminations	Fe2 (i)	O1 (j)	O1 (k)	O2 (l)	Fe1 (m)	O4 (n)	O4 (o)
$E_S^{relax}$/(eV)	−720.59	−723.34	−725.03	−728.11	−724.47	−724.47	−727.93
$E_S^{unrelax}$/(eV)	−713.42	−716.79	−719.45	−722.25	−718.76	−718.76	−722.35
γS/(J·m−2)	1.877	1.721	1.669	1.421	1.699	1.699	1.455

3.6. Relative Stabilities of CaFe₂O₄ (001), (100), (110), and (111) Surfaces

The calculated surface energies for CaFe₂O₄ (001), (100), (110), and (111) surfaces at different terminations were shown in Tables 2, 3, 4, 5, respectively. Since the product between the number of k-points in any direction and the length of the basis vector in this direction was about 2.5 nm for all surfaces, the relative stability of all surfaces can be compared directly by the optimized surface energies mentioned above. According to Tables 2, 3, 4, 5, the surface energies of CaFe₂O₄ (001), (100), (110) and (111) surfaces were lowest at O1, Ca, O1 and O2 terminations, respectively. The smallest surface energies are 1.307, 1.278, 1.489 and 1.421 J·m⁻² for the (001), (100), (110) and (111) surfaces, respectively. The difference values of the smallest surface energies for different surfaces were small. In order to confirm the difference of the smallest surface energy between the different surfaces, the effects of the slab sizes, k-points and cutoff energy on the smallest surface energy for all surfaces were further studied during the calculation process. From Tables 6 and 7, the smallest surface energies of all surfaces remain almost unchanged with different slab sizes and k-points. The order of the smallest surface energy is still γ(100) <γ(001) <γ(110) <γ(111). It also indicated that surface energy was not affected by the periodic interaction with the (1 × 2) Ca₁₆Fe₃₂O₆₄ slab for (001) and (100) surfaces and (1 × 1) Ca₁₆Fe₃₂O₆₄ slab for (110) and (111) surfaces. In addition, it can be seen from Table 8, with the increasing of the cutoff energy, the smallest surface energies had a little increased for all surfaces. The order of the stabilities of the (001), (100), (110) and (111) surfaces was still constant at 375 eV of cutoff energy, while the stability of (110) surface was slightly higher than the (111) surface at 425 eV of cutoff energy . Overall, the (100) surface was the most stable. The method was also used to compare the relative stability of the low-index surfaces of LiFePO₄, α–Bi₂O₃ and VO₂ by Wang et al.,⁵⁵⁾ Lei et al.⁵⁶⁾ and Mellan et al.,⁵⁷⁾ respectively. Moreover, Wang et al.³⁵⁾ also observed the preferred orientation of the crystal for CaFe₂O₄ had the (100) surface by TEM.

Table 6. Surface energy of (001), (100), (110) and (111) surfaces with different slab sizes.

Surface	(001)		(100)		(110)		(111)
Slab	(1 × 2)	(2 × 4)	(1 × 2)	(2 × 4)	(1 × 1)	(2 × 2)	(1 × 1)	(2 × 2)
Terminations	O1	O1	Ca	Ca	Fe2 (d)	Fe2 (d)	O2 (l)	O2 (l)
K-points	3 × 4 × 1	1 × 2 × 1	4 × 2 × 1	2 × 1 × 1	2 × 3 × 1	1 × 1 × 1	3 × 2 × 1	1 × 1 × 1
Surface area/(Å2)	56.63	226.50	65.73	262.91	105.66	422.65	109.39	437.56
Cutoff energy/(eV)	400	400	400	400	400	400	400	400
$E_S^{relax}$/(eV)	−740.62	−2962.50	−739.30	−2957.27	−726.37	−2905.37	−728.11	−2912.01
$E_S^{unrelax}$/(eV)	−737.09	−2948.44	−735.71	−2942.88	−719.00	−2875.80	−722.25	−2890.54
γS/(J·m−2)	1.307	1.308	1.278	1.276	1.489	1.490	1.421	1.466

Table 7. Surface energy of (001), (100), (110) and (111) surfaces with different k-points.

Surface	(001)		(100)		(110)		(111)
Slab	(1 × 2)	(1 × 2)	(1 × 2)	(1 × 2)	(1 × 1)	(1 × 1)	(1 × 1)	(1 × 1)
Terminations	O1	O1	Ca	Ca	Fe2 (d)	Fe2 (d)	O2 (l)	O2 (l)
K-points	6 × 8 × 1	9 × 12 × 1	8 × 4 × 1	12 × 6 × 1	4 × 6 × 1	6 × 9 × 1	6 × 4 × 1	9 × 6 × 1
Surface area/(Å2)	56.63	56.63	65.73	65.73	105.66	105.66	109.39	109.39
Cutoff energy/(eV)	400	400	400	400	400	400	400	400
$E_S^{relax}$/(eV)	−740.61	−740.62	−739.30	−739.31	−726.37	−726.36	−727.69	−728.21
$E_S^{unrelax}$/(eV)	−737.09	−737.09	−735.70	−735.70	−719.00	−718.99	−722.14	−722.65
γS/(J·m−2)	1.308	1.307	1.278	1.275	1.490	1.490	1.476	1.436

Table 8. Surface energy of (001), (100), (110) and (111) surfaces with different cutoff energies.

Surface	(001)		(100)		(110)		(111)
Slab	(1 × 2)	(1 × 2)	(1 × 2)	(1 × 2)	(1 × 1)	(1 × 1)	(1 × 1)	(1 × 1)
Terminations	O1	O1	Ca	Ca	Fe2 (d)	Fe2 (d)	O2 (l)	O2 (l)
K-points	3 × 4 × 1	3 × 4 × 1	4 × 2 × 1	4 × 2 × 1	2 × 3 × 1	2 × 3 × 1	3 × 2 × 1	3 × 2 × 1
Surface area/(Å2)	56.63	56.63	65.73	65.73	105.66	105.66	109.39	109.39
Cutoff energy/(eV)	375	425	375	425	375	425	375	425
$E_S^{relax}$/(eV)	−741.30	−738.04	−739.97	−737.04	−727.00	−723.78	−728.73	−724.58
$E_S^{unrelax}$/(eV)	−737.77	−734.42	−736.36	−733.40	−719.61	−715.16	−723.01	−719.60
γS/(J·m−2)	1.210	1.657	1.196	1.547	1.440	1.592	1.387	1.745

4. Conclusions

(1) When the Hubbard parameter (U) was 4.7 eV, the band gaps of up-spin and down-spin of CaFe₂O₄ bulk were 1.91 and 1.81 eV, respectively. The lattice parameters were a = 9.302 Å, b = 3.044 Å, and c =10.796 Å. The relative error between the calculated and experimental values was less than 1%.

(2) The O1 termination of the CaFe₂O₄ (001) surface, the O1 and Ca terminations of the CaFe₂O₄ (100) surface, the Fe2 and O3 terminations of the CaFe₂O₄ (110) surface and the O2 (l) and O4 (o) terminations of the CaFe₂O₄ (111) surface were the most stable structures, respectively. The surface energies of them were closed and below 1.600 J·m⁻².

(3) The lowest surface energies of CaFe₂O₄ (001), (100), (110) and (111) surfaces were 1.307, 1.278, 1.489 and 1.421 J·m⁻², respectively. The strcuture of CaFe₂O₄ (100) suface with Ca termination was the most stability.

Acknowledgements

We thank Professors Liang CHEN and Ziqi TIAN of Ningbo Institute of Materials Technology and Engineering of Chinese Academy of Sciences for their helpful assistance during the calculation.

This work was supported by the National Natural Science Foundation of China (No. 51904127), the Natural Science Foundation of Jiangxi Province (No. 20192BAB216018), the PhD Project (No. 2018-YYB-05) and Generalized System of Preferences-One Type Project (No. 2018-XTPH1-05) of Jiangxi Academy of Sciences.

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