Electronic and magnetic properties of $\mathrm{T}{\mathrm{i}}_4{\mathrm{O}}_7$ predicted by self-interaction-corrected density functional theory

Understanding electronic properties of substoichiometric phases of titanium oxide such as Magnéli phase $\mathrm{T}{\mathrm{i}}_4{\mathrm{O}}_7$ is crucial in designing and modeling resistive switching devices. Here we present our study on Magnéli phase $\mathrm{T}{\mathrm{i}}_4{\mathrm{O}}_7$ together with rutile $\mathrm{Ti}{\mathrm{O}}_2$ and $\mathrm{T}{\mathrm{i}}_2{\mathrm{O}}_3$ using density functional theory methods with atomic-orbital-based self-interaction correction (ASIC). We predict a new antiferromagnetic (AF) ground state in the low temperature (LT) phase, and we explain energy difference with a competing AF state using a Heisenberg model. The predicted energy ordering of these states in the LT phase is calculated to be robust in a wide range of modeled isotropic strain. We have also investigated the dependence of the electronic structures of the Ti-O phases on stoichiometry. The splitting of titanium $t_{2g}$ orbitals is enhanced with increasing oxygen deficiency as Ti-O is reduced. The electronic properties of all these phases can be reasonably well described by applying ASIC with a “standard” value for transition metal oxides of the empirical parameter α of 0.5 representing the magnitude of the applied self-interaction correction.

Figure 1

Atomic structure of $\mathrm{T}{\mathrm{i}}_4{\mathrm{O}}_7$ crystal unit cell. Gray spheres represent titanium, and red spheres represent oxygen atoms. The numbering of Ti atoms is also shown.

Figure 2

The spin topology of the AF1 state (left panel) and the AF3 state (right panel) in $\mathrm{LT}-\mathrm{T}{\mathrm{i}}_4{\mathrm{O}}_7$. The blue and red dots represent spin up and spin down, respectively. Atomics spins of the atoms in the center unit cell and those connected to the cell are highlighted by brighter colors.

Figure 3

The evolution of energy difference ${\Delta }_2$ between the AF1 state and the AF3 state of $\mathrm{LT}-\mathrm{T}{\mathrm{i}}_4{\mathrm{O}}_7$ as a function of positive isotropic strain.

Figure 4

DOS of the competing AF1 and AF3 states, calculated by both the ASIC method (left panel) in SIESTA and the HSE06 hybrid functional in FHI-aims (right panel). The positive and negative DOS values represent spin up and spin down, respectively. The Fermi level (energy zero) is aligned with the conduction band minimum (CBM).

Figure 5

Differences between the ASIC ($\alpha =0.5$) total energy, $E_{\mathrm{T}}$, of the various spin configurations of $\mathrm{T}{\mathrm{i}}_4{\mathrm{O}}_7$ and the total energy of the nonmagnetic state, $E_{\mathrm{NM}}$, as a function of isotropic strain. Positive strain values correspond to tensile stress while negative strain values correspond to compressive stress. Total energies of the nonmagnetic states, $E_{\mathrm{NM}}$, vs applied strain are shown in the inserts.

Figure 6

The evolution of nonmagnetic density of states as a function of Ti-O stoichiometry. Upper panels: LDA results. Lower panels: ASIC results ($\alpha =0.5$). The black and red curves represent the projected density of states on Ti and O, respectively. For the metallic systems, the Fermi level (energy zero) position is uniquely defined, and for the semiconducting systems we align it to the CBM.

Figure 7

Predicted band gap vs empirical parameter α for $\mathrm{LT}-\mathrm{T}{\mathrm{i}}_4{\mathrm{O}}_7,\mathrm{Ti}{\mathrm{O}}_2$, and $\mathrm{T}{\mathrm{i}}_2{\mathrm{O}}_3$.