Topological insulators and metal-insulator transition in the pyrochlore iridates

The possible existence of topological insulators in cubic pyrochlore iridates $A_2{\text{Ir}}_2{\text{O}}_7$ ($A=\text{Y}$ or rare-earth elements) is investigated by taking into account the strong spin-orbit coupling and trigonal crystal-field effect. It is found that the trigonal crystal-field effect, which is always present in real systems, may destabilize the topological insulator proposed for the ideal cubic crystal field, leading to a metallic ground state. Thus the trigonal crystal field is an important control parameter for the metal-insulator changeover. We propose that this could be one of the reasons why distinct low-temperature ground states may arise for the pyrochlore iridates with different $A$-site ions. On the other hand, examining the electron-lattice coupling, we find that softening of the $\mathbf{q}=0$ modes corresponding to trigonal or tetragonal distortions of the Ir pyrochlore lattice leads to the resurrection of the strong topological insulator. Thus, in principle, a finite-temperature transition to a low-temperature topological insulator can occur via structural changes. We also suggest that the application of the external pressure along [111] or its equivalent directions would be the most efficient way of generating strong topological insulators in pyrochlore iridates.

Figure 1

(Color online) (a) Structure of the pyrochlore lattice. An Ir atom sits at each vertex of a tetrahedron. Four sites within a tetrahedral unit cell are numbered from 1 to 4. (b) The coordinate system used in this paper.

Figure 2

(Color online) Trigonal crystal field around an ${\text{Ir}}^{4+}$ ion. (a) Local geometry of an ${\text{IrO}}_6$ octahedron. A trigonal distortion is induced by compression or elongation of the surrounding oxygen octahedron along a $C_3$ symmetry axis for threefold rotation. (b) Splitting of degenerate $t_{2g}$ states under the trigonal crystal field. (c) Local $C_3$ axes for four Ir ions constituting a tetrahedron; each $C_3$ axis points to the body center of the tetrahedron.

Figure 3

(Color online) Local Ir-O-Ir configuration. Local octahedral axes for Ir atoms on the left-hand side $\left({\text{Ir}}_L\right)$ and the right-hand side $\left({\text{Ir}}_R\right)$ are not parallel to the global cubic axis adopted for the central oxygen atom.

Figure 4

(Color online) Tight-binding band structure of $\text{Ir}5d$ orbitals when there is no trigonal crystal-field effect. Here we have chosen ${\lambda }_{SO}=4.0$, $t_a=t_e=0.5$. Note that the upper four bands corresponding to the spin-orbit doublet $\left(j_{\text{eff}}=1/2\right)$ are well separated from the other eight bands which come from the spin-orbit quadruplet $\left(j_{\text{eff}}=3/2\right)$.

Figure 5

(Color online) Dispersions of $j_{\text{eff}}=1/2$ bands under trigonal crystal field with $t_a>t_e$. Here we have fixed ${\lambda }_{SO}=4.0$, $t_e=0.5$ while changing $t_a$. There is an accidental band touching between the two bands in the middle at the L point when $t_a/t_e=1.7$.

Figure 6

(Color online) Dispersions of $j_{\text{eff}}=1/2$ bands under trigonal crystal field with $t_a<t_e$. Here we have fixed ${\lambda }_{SO}=4.0$, $t_a=0.5$ but vary $t_e$. As $t_e/t_a$ increases the energy gap between the two bands in the middle reduces. When $t_e/t_a\approx 2.3$ band inversion occurs at the $\Gamma$ point and the system becomes metallic.

Figure 7

(Color online) Phase diagram as a function of the strength of the local trigonal crystal field represented by $t_e/t_a$. Note that a metallic phase occurs when $t_e/t_a$ becomes larger than the critical value of ${\left(t_e/t_a\right)}_c>1$. The red cross in the middle of two strong topological insulators (STI) indicates the point where accidental band touching occurs.

Figure 8

(Color online) Tetragonal lattice distortion induced by a $\mathbf{q}=0$ phonon mode with $E_g$ symmetry. Red solid and blue broken lines describe two inequivalent bond lengths under the lattice distortion.

Figure 9

(Color online) Trigonal lattice distortion induced by a $\mathbf{q}=0$ phonon mode with $T_{2g}$ symmetry. Red solid and blue broken lines describe two inequivalent bond lengths under the lattice distortion.

Figure 10

(Color online) Ground-state energy per unit cell for the pyrochlore iridate system coupled with a trigonal phonon mode along [111] direction. Here ${\lambda }_{SO}=4.0$, $t_a=0.5$, and $t_e/t_a=2.5$. We plot the change in the ground-state energy induced by the trigonal lattice distortion as a function of the hopping amplitude modulation ${\eta }_T$.

Figure 11

(Color online) Band-structure change induced by a trigonal lattice distortion, which comes from a $\mathbf{q}=0$ $T_{2g}$ phonon mode softening. A band gap opens at the $\Gamma$ point. The resultant insulating phase is a strong topological insulator.