Bond ordering and phase transitions in ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ under high pressure

The Kitaev model of spin-1/2 on a honeycomb lattice supports degenerate topological ground states and may be useful in topological quantum computation. ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ with a honeycomb lattice of Ir ions has been extensively studied as a candidate for the realization of this model, due to the effective $J_{\text{eff}}=1/2$ low-energy excitations produced by the spin-orbit and crystal-field effect. As the eventual realization of the Kitaev model has remained evasive, it is highly desirable and challenging to tune the candidate materials toward such an end. It is well known that external pressure often leads to dramatic changes in the geometric and electronic structure of materials. In this Rapid Communication, the high-pressure phase diagram of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ is examined by first-principles calculations. It is found that ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ undergoes a sequence of structural and magnetic phase transitions, from a magnetically ordered phase with space group $C2/m$ to two bond-ordered nonmagnetic phases. The low-energy excitations in these high-pressure phases can be well described by the $J_{\text{eff}}=1/2$ states.

Figure 1

(a) Crystal structure of $C2/m$, viewed from slightly off the $\mathit{b}$ direction. (b) Nearly ideal Ir honeycomb lattice of $C2/m$, viewed from the direction perpendicular to the $ab$ plane. Green, blue, and magenta bonds are $x$, $y$, and $z$ bonds, respectively. The bond length relation is $l_x=l_y\lesssim l_z$. The Ir honeycomb lattice of (c) $P\overline{1}$ with shorter $x$ bonds, $l_x<l_y\approx l_z$, (d) $P{2}_1/m$ with shorter $x_1$ and $y_2$ bonds, $l_{x_1}=l_{y_2}<l_{x_2}=l_{y_1}\approx l_z$, and (e) $C2/m$-zz with shorter $z$ bonds, $l_z<l_x=l_y$. Note that all short Ir-Ir bonds in (c)–(e) are highlighted by thicker lines.

Figure 2

Calculated static lattice enthalpy of different structures relative to (a) the $C2/m$-zigzag structure and (b) the $P{2}_1/m$ structure, respectively. (c) Calculated bond lengths vs pressure, which reflects the changes of bond ordering. (d) Calculated total moment magnitude and angle to the $\mathit{a}$ axis of the $C2/m$-zigzag structure under various pressures.

Figure 3

(a) The static lattice enthalpy of $C2/m$-zigzag, $P\overline{1}$, and $P{2}_1/m$ structures with zero-point energy corrections. The inset in (a) shows the results without zero-point energy corrections for comparison. (b) The pressure vs temperature phase diagram considering the phonon free energy.

Figure 4

(a) Calculated phonon density of states of the $P\overline{1}$ and $P{2}_1/m$ structures under different pressures. (b) The differences of integrated zero-point vibration energies of the $P\overline{1}$ and $P{2}_1/m$ structures under different pressures. (c) The evolution of the vibration energies of the dimerization modes with pressure. The insets in (c) are the dimerization modes of the $P\overline{1}$ and $P{2}_1/m$ structures, where only the vibrations of Ir atoms are shown. The gray dashed lines in the insets are guides to the eye for the Ir honeycomb lattice.

Figure 5

Calculated tight-binding band structures (four Ir per unit cell) with SOC of (a) $C2/m$-zigzag, (b) $P\overline{1}$, and (c) $P{2}_1/m$, at 48 GPa, respectively. Fat bands are plotted for band structures with SOC (without Hubbard $U$), where the linewidth represents the weight of the $J_{\text{eff}}=1/2$ states. The high-symmetry $\mathbf{k}$ points are $\mathrm{\Gamma }(0,0,0)$, $Y(0,1/2,0)$, $X(1/2,0,0)$, $M(1/2,1/2,0)$, $m(1/2,1/2,1/2)$, $x(1/2,0,1/2)$, $y(0,1/2,1/2)$, and $g(0,0,1/2)$.