Electronic and magnetic properties of ${\mathrm{GaFeO}}_3$: Ab initio calculations for varying Fe/Ga ratio, inner cationic site disorder, and epitaxial strain

In this study we present ab initio density-functional theory calculations on stoichiometric, cation-doped, and strained $\mathrm{GaFeO}{}_3$. We start with a detailed discussion of the origin of the antiferromagnetic (AFM) superexchange in stoichiometric $\mathrm{GaFeO}{}_3$ and give a molecular orbital description of the exchange mechanism derived from our calculations. In addition, we study the properties of the Fe-O-Fe bonds for different geometries to underline the angle and distance dependence of the AFM coupling as formulated in the Goodenough-Kanamori rules. We describe the AFM ground state of $\mathrm{GaFeO}{}_3$ as a result of two intrinsic Fe-O-Fe chains that meander through the crystal along the $c$ direction. The magnetocrystalline anisotropy energies are calculated for the stoichiometric phase with and without inner cationic site disorder, and the presence of a sublattice-dependent anisotropy is examined. Furthermore, we perform our studies of ${\mathrm{Ga}}_{2-x}\mathrm{Fe}{}_x{\mathrm{O}}_3$ for varying Fe concentrations $\mathit{x}(0.0\le \mathit{x}\le 2.0)$ where at a value of $\mathit{x}=0.0$ and $\mathit{x}=2.0$ it transforms into the isomorphic $\varepsilon \text{−}\mathrm{Ga}{}_2{\mathrm{O}}_3$ and $\varepsilon \text{−}\mathrm{Fe}{}_2{\mathrm{O}}_3$ phases, respectively. The effect of strain was also studied. Incorporating dopants and applying strain to the simulation cell changes the intrinsic geometry and thus the magnetic properties of gallium ferrite.

Figure 1

Cell structure of pure GFO. The Ga1, Ga2, Fe1, and Fe2 sublattices are indicated.

Figure 2

Panel (a) shows the total DOS (black), the overall Fe $d$ states (red), and the O $p$ states (blue) of pure GFO. Panel (b) shows the projected DOS of an Fe1-O-Fe2 complex. Plotted are the corresponding Fe1 (red area) and Fe2 (yellow area) $d$ states and the O $p$ state (blue area). All DOS are calculated within the HSE approximation.

Figure 3

The upper part shows the MO diagram of an Fe1-O-Fe2 complex inside GFO. The lower part of the figure sketches the corresponding DOS. Marked in red and blue are the Fe $d$ and the O $p$ states, respectively. Pale red denotes the additional Fe electron which becomes shifted below the Fermi energy by the Fe-O interaction.

Figure 4

Unit cell with a single Fe1-O-Fe2 (125.${84}^{\circ })$ complex. Bond length and bond angle are varied to consider the alteration on the AFM superexchange.

Figure 5

Plots (a) and (b) show the distance and the angle dependency of the AFM coupling strength, respectively. Increasing the distance and decreasing the angle leads to a decrease in AFM coupling strength. All energies are given per 40-atom simulation cell.

Figure 6

[(a)–(d)] Simulation cells with different single Fe1-O-Fe2 complexes. These complexes differ in their enclosed angle and bond length. The respective enclosed Fe1-O-Fe2 angles are 122.${17}^{\circ }$ for (a) and (b) and 125.${84}^{\circ }$ for (c) and (d). Case (e) is a modification which contains all complexes (a)–(d) in a single simulation cell. The total AFM coupling strength for configuration (e) turns out to the be close to the sum of the individual coupling strengths.

Figure 7

Dominant Fe1-O-Fe2 chain along the $c$ axis in GFO.

Figure 8

Panel (a) shows the magnetization density (spin up minus spin down) of an Fe1-O-Fe2 couple inside the GFO simulation cell. Denoted in red and blue tones are the positive and negative magnetization values, respectively. Panel (b) shows the magnetization density of an Fe2 atom being part of two Fe1-O-Fe2 complexes. The left O atom is part of a strong AFM coupling strength Fe1-O-Fe2 (122.${17}^{\circ })$ complex and the right O atom of a weak Fe1-O-Fe2 (102.${52}^{\circ })$ configuration.

Figure 9

Shown are the respective simulation cells of ${\mathrm{Ga}}_{2-x}\mathrm{Fe}{}_x{\mathrm{O}}_3$ for a doping concentration of $\mathit{x}=0.0,\mathit{x}=1.0$, and $\mathit{x}=2.0$. These are equivalent to the crystal structures of $\varepsilon$-GO, GFO, and $\varepsilon$-FO.

Figure 10

We show the AFM coupling strength $E_{\mathrm{diff}}$ (blue) and the corresponding ground state magnetic moment (red) as a function of the doping concentration $x$. In the range of $\mathit{x}=0.0$ to $\mathit{x}=0.5$ the first Fe1-O-Fe2 chain in formed in GFO denoted by “ch1” (chain 1). In the “ch2” highlighted area the second Fe1-O-Fe2 chain is formed, where $E_{\mathrm{diff}}$ for “ch1” and the “ch2” shows the same behavior. In the area “oct” (octahedral) all Ga2 sites are substituted by Fe atoms. Here the slope is constant, indicating the additive character of the AFM coupling strength of the formed Fe1-O-Fe(Ga2) configurations. In the area “tet” (tetrahedral) all Ga1 sites become stepwise occupied by Fe atoms. As before we find an almost-linear increase in the coupling strength.

Figure 11

We show the defect formation energy $E^{\mathrm{form}}$ (blue) as a function of the doping concentration $x$ and the difference $\mathrm{\Delta }{E}_{\mathrm{form}}$ (red) (see text).

Figure 12

AFM coupling strength $E_{\mathrm{diff}}$ as a function of the strained lattice constant in the $a$ (red), $b$ (green), and $c$ (blue) directions.

Figure 13

Straining the ${\mathrm{Ga}}_{2-x}\mathrm{Fe}{}_x{\mathrm{O}}_3$ simulation cell in the $c$ direction for a doping concentration of $\mathit{x}=0.9$ to $\mathit{x}=1.4$, including the case of inner cationic site disorder. Plotted is the AFM coupling strength $E_{\mathrm{diff}}$ as a function of strain.