Fig. 1. Temperature-dependent resistivity of Fe₃O₄ sample I with FeO∶Fe₂O₃ = 1∶1.025 and sample II with FeO∶Fe₂O₃ = 1∶1.08^[1].

Fig. 2. Spinel unit cell: (a) Stacking pattern of sub-lattices; (b) crystal structure^[24].

Fig. 3. Relationship between the unit cells referred to the structure with space group , P2/m, P2/c and Cc^[35].

Fig. 4. Schematic diagram of (a) trimeron and (b) distribution of trimeron^[23].

Fig. 5. TEM results of ((a)−(d)) Fe₃O₄(001) and ((e)−(h)) Fe₃O₄(111) films. White Miller index above (below) T_V are marked with cubic (monoclinic) Fe₃O₄. Yellow Miller index in Fig. (g) indicates Al₂O₃. Brown and red spheres in Fig. (b) and Fig. (f) represent Fe and O^[41,43].

Fig. 6. {110} APB defects in Fe₃O₄: (a) The ideal cubic Fe₃O₄ structure; (b) APB-I; (c) APB-II. The APB crystal translations are indicated by green vectors. Red, blue and gray spheres represent the oxygen atoms, tetrahedral Fe and octahedral Fe atoms^[48].

Fig. 7. Symmetrically distinct crystallographic relationships between cubic and monoclinic phases of magnetite^[49].

Fig. 8. (a) STM image of Fe₃O₄(100) surface at 78 K; (b) profile along the line marked in red of (a); (c) the monoclinic unit cell of Fe₃O₄; (d) two mirrored monoclinic cells with opposite monoclinic c axis at a twin boundary^[51].

Fig. 9. Spin-polarized DOS of Fe₃O₄ twin boundaries (TBs): (a) Type I TB; (b) Type II TB; (c) Type III TB. E_F is represented by the red dashed lines. The relaxed atomistic models are also given for reference. The DOS suggest that the magnetic coupling across the type I TB is ferromagnetic and those across the type II and III TBs are antiferromagnetic^[52].

Fig. 10. Sketch map of the electronic ground state of Fe 3d electrons and magnetic couplings in Fe₃O₄^[53].

Fig. 11. Temperature dependent Anisotropy constant K₁ of Fe₃O₄^[27].

Fig. 12. Models for electron localization on Fe_B sites of Fe₃O₄: (a) Verwey’s tetragonal model of Fe²⁺/Fe³⁺ charge order; (b) an Anderson tetrahedron of two Fe²⁺ and two Fe³⁺ ions; (c) bond-dimerization in the Fe_B4 tetrahedron, where the electrons are localized in two shortened Fe_B-Fe_B distances, shown as bold lines^[53].

Fig. 13. DOS of Fe₃O₄ with the monoclinic structure projected onto the Fe_Bd orbitals. Fermi level E_F is set at 0 eV^[54].

Fig. 14. Isovalue of the confidence factor. The best agreement is obtained for δ₁₂ = 0.12 ± 0.025 electrons and δ₃₄ = 0.10 ± 0.06 electrons, where the charge occupancies of Fe₁ and Fe₄ are 5.38, 5.62 and 5.40, 5.60, respectively^[63].

Fig. 15. Temperature dependent resistivity of 150 and 660-nm thick Fe₃O₄ films in the temperature range of 60–350 K. The temperature dependent magnetization of 660-nm thick film at a magnetic field of 300 Oe^[71].

Fig. 16. Magnetoresistance of 660 nm thick Fe₃O₄ films at (a) 70 K and (b) 115 K; (c) Temperature dependent magnetoresistance of 660-nm thick Fe₃O₄ film at the magnetic fields of 0.5, 1, 2, 4 T. The dotted lines are simulations using Mott’s formula^[71].

Fig. 17. Temperature dependent (a) magnetization and (b) zero-field resistivity of Fe₃O₄ single crystal and films with the thickness of 200, 50 and 15 nm^[73].

Fig. 18. Spin orientation of two ferromagnetic chains with antiferromagnetic coupling at an atomically sharp boundary at a magnetic field^[77].

Fig. 19. AMR of the (a) 67 nm thick Fe₃O₄ film and (b) Fe₃O₄ single crystal at a magnetic field of 5 T^[81].

Fig. 20. AMR of the epitaxial Fe₃O₄(100) film: (a) Temperature-dependent AMR at a 50 kOe magnetic field; AMR at (b) 110 K and (c) 80 K^[85].

Fig. 21. (a) Schematic of the measurements; (b) relation between AMR and distribution of in-plane trimeron of Fe₃O₄(100) film at 80 K and 50 kOe; (c) relation between AMR and distribution of in-plane trimeron of Fe₃O₄(111) film at 110 K and 10 kOe. The trimeron is shown in the upper right corner^[85].

Fig. 22. (a) Dielectric hysteresis loop of Pd/Fe₃O₄/Nd:SrTiO₃ heterostructure^[92]; (b) ionic structure of Fe octahedral sites with P2/c (left) and Cc (right) space groups. Orange and blue balls represent the Fe²⁺ and Fe³⁺ ions. Electric dipole moments caused by charge shifts are indicated by red arrows^[93].

Fig. 23. (a) Ferroelectric polarization along the x and z axes; (b) strain dependent total energy^[94].