Topological phases in layered pyrochlore oxide thin films along the [111] direction

We theoretically study a multiband Hubbard model of pyrochlore oxides of the form $A_2$$B_2$O${}_7$, where $B$ is a heavy transition metal ion with strong spin-orbit coupling, in a thin-film geometry orientated along the [111] direction. Along this direction, the pyrochlore lattice consists of alternating kagome and triangular lattice planes of $B$ ions. We consider a single kagome layer, a bilayer, and the two different trilayers. As a function of the strength of the spin-orbit coupling, the direct and indirect $d$-orbital hopping, and the band filling, we identify a number of scenarios where a noninteracting time-reversal-invariant $Z_2$ topological phase is expected and we suggest some candidate materials. We study the interactions in the half-filled $d$ shell within Hartree-Fock theory and identify parameter regimes where a zero magnetic field Chern insulator with Chern number $\pm 1$ can be found. The most promising geometries for topological phases appear to be the bilayer which supports both a $Z_2$ topological insulator and a Chern insulator, and the triangular-kagome-triangular trilayer which supports a relatively robust Chern insulator phase.

Figure 1

(a) Schematic of the experimental geometry. The middle $A_2$$B_2$O${}_7$ thin-film layer is the region of physical interest. The capping regions $A_2$$B_2^{\prime }$O${}_7$ would ideally be nonmagnetic large gap band insulators of the same lattice constant as $A_2$$B_2$O${}_7$ and would only serve to structurally stabilize the thin-film region. The growth direction of the structure is along [111]. (b) The Brillouin zone of the underlying triangular Bravais lattice of the thin films. (c) Pyrochlore lattice structure showing alternating kagome and triangular lattice planes along the [111] direction. A kagome-triangular-kagome-triangular tetralayer is shown with green atoms in the kagome planes and gray atoms in the triangular planes.

Figure 2

Candidate thin films to be used in the $A_2$$B_2$O${}_7$ region shown in Fig. 1a. As shown in Fig. 1c, along the [111] direction the pyrochlore lattice is an alternating series of kagome and triangular lattice planes. Single, double, and triple layer stackings are shown in (a)–(d). Among these stackings, the bilayer and TKT trilayer are found to be the most promising for realizing robust (to model Hamiltonian parameters) $Z_2$ topological insulator and Chern insulator (zero magnetic field quantum Hall) phases.

Figure 3

The band structure of Eq. (1) along the high-symmetry directions [see Fig. 1b] of a single kagome layer with $t_p=-2t_s/3$; $t_s=-t,t$; and $\lambda =2t,4t$ as shown within the figure. Green (light gray) lines indicate filling fractions of a $Z_2$ TI. The thickness of the line indicates the size of the gap. In the cases shown $Z_2$ TIs occur at $t_{2g}$ filling fractions (a) $\frac{7}{9}$, $\frac{8}{9}$; (b) $\frac{7}{9}$, $\frac{8}{9}$; (c) $\frac{5}{9}$; (d) $\frac{5}{9}$, $\frac{7}{9}$, $\frac{8}{9}$.

Figure 4

The band structure of Eq. (1) along the high-symmetry directions [see Fig. 1b] of a bilayer with $t_p=-2t_s/3$; $t_s=-t,t$; and $\lambda =2t,4t$ as shown within the figure. Green (light gray) lines indicate filling fractions of a $Z_2$ TI. The thickness of the line indicates the size of the gap. In the cases shown $Z_2$ TIs occur at $t_{2g}$ filling fractions (a) $\frac{3}{4}$; (b) $\frac{3}{4}$; (d) $\frac{5}{6}$.

Figure 5

The band structure of Eq. (1) along the high-symmetry directions [see Fig. 1b] of a triangular-kagome-triangular layer with $t_p=-2t_s/3$; $t_s=-t,t$; and $\lambda =2t,4t$ as shown within the figure. Green (light gray) lines indicate filling fractions of a $Z_2$ TI. The thickness of the line indicates the size of the gap. In the cases shown $Z_2$ TIs occur at $t_{2g}$ filling fractions (a) $\frac{1}{3}$, $\frac{11}{15}$, $\frac{4}{5}$, $\frac{14}{15}$; (b) $\frac{1}{3}$, $\frac{11}{15}$, $\frac{4}{5}$, $\frac{14}{15}$; (c) $\frac{1}{3}$; (d) $\frac{4}{15}$, $\frac{1}{3}$, $\frac{3}{5}$, $\frac{13}{15}$.

Figure 6

The band structure of Eq. (1) along the high-symmetry directions [see Fig. 1b] of a kagome-triangular-kagome layer with $t_p=-2t_s/3$; $t_s=-t,t$; and $\lambda =2t,4t$ as shown within the figure. Green (light gray) lines indicate filling fractions of a $Z_2$ TI. The thickness of the line indicates the size of the gap. In the cases shown $Z_2$ TIs occur at $t_{2g}$ filling fractions: (a) $\frac{17}{21}$; (b) $\frac{2}{7}$, $\frac{17}{21}$; (c) $\frac{13}{21}$; (d) $\frac{13}{21}$, $\frac{15}{21}$.

Figure 7

Phase diagrams for the bilayer case in Fig. 2 at a $t_{2g}$ filling of 5/6 and a total $d$-shell filling of 1/2 for (a) $\lambda =2t$ and (b) $\lambda =4t$. The different colors represent different phases: TM is a topological metal (finite direct band gap, negative indirect gap, with nontrivial $Z_2$ index for fully occupied bands), M is a metal, I is a trivial insulator, and TI is a topological insulator. The black line is $t_p=-2t_s/3$. The $*$ represent ($-1$, 2/3) and the $\times$ represent (1, $-2/3$), corresponding to the two values of $t_s=\pm 1$ we considered in Figs. 3, 4, 5, 6.

Figure 8

The phases of the interacting model (1) and (3) with (a) $t_s=-t$, (b) $t_s=0.25t$, and (c) $t_s=t$ for a $t_{2g}$ filling of 5/6 (a total $d$-shell and $j=1/2$ filling of 1/2) in the limit of strong spin-orbit coupling. Here M is a metal, I is an insulator, TI is a topological insulator, MC is a magnetic metal, MI is a magnetic insulator, and CI is a Chern insulator (zero magnetic field quantum Hall state). The most robust Chern insulator is in the TKT structure for $t_s=-t$. Note that there is a small region of the Chern insulator in the bilayer system for $t_s=t$ around $U/t\approx 2.5$ near the end of the TI phase.

Figure 9

Top: The magnitude of the net magnetic moment, $|\langle {\mathbf{j}}_t\rangle |$, and the energy gap, $E_g/t$ ($E_{g,\mathrm{ind}}$ denotes the indirect gap and $E_{g,\mathrm{dir}}$ denotes the direct gap), as a function of $U/t$ for the bilayer system with parameters corresponding to Fig. 8. Bottom: The same for the TKT system.

Figure 10

The evolution of bands in a single kagome layer in the Hartree-Fock approximation when $t_s=t$ for (a) $U=1.90t$, (b) $U=1.95t$, (c) $U=2.10t$ and for a bilayer when $t_s=-t$ with (d) $U=0.65t$, (e) $U=0.85t$, (f) $U=1.05t$. For the single kagome layer this corresponds to the M$\to$MC$\to$MI transition in Fig. 8 and for the bilayer this corresponds to the I$\to$MC$\to$MI transition. Note the Kramers degeneracy at the time-reversal-invariant momenta $M$ and $\Gamma$ in (d) and the difference between $\Gamma K$ and $\Gamma K^{\prime }$ in (e) and (f).

Figure 11

The evolution of the bands when $t_s=t$ for a bilayer with (a) $U=2.53t$, (b) $U=2.55t$, (c) $U=2.56t$, (d) $U=2.58t$, and for a bilayer when $t_s=0.25t$ with (e) $U=1.62t$ and (f) $U=1.72t$. The figures clearly show that for $t_s=t$, the phase changes from (a) a TI to (b) a magnetic insulator (gap is too small to be visible) to (c) a Chern insulator and back to (d) a magnetic insulator. The red (gray) horizontal line in (c) reveals the gap of the Chern insulator. For $t_s=0.25t$, the (e) TI changes to a magnetic insulator (f) directly. Note the Kramers degeneracy at the time-reversal-invariant momenta $M$ and $\Gamma$ in the TI in (a) and (e) and the difference between $\Gamma K$ and $\Gamma K^{\prime }$.

Figure 12

The evolution of the bands when $t_s=-t$ for a TKT layer with (a) metal $U=0.04t$, (b) magnetic conductor $U=0.07t$, (c) Chern insulator $U=0.40t$, (d) Chern insulator $U=0.70t$. From (a) to (b), (c) the gap opens gradually, but later the gap closes again, and the tendency is revealed near the $\Gamma$ point in (d). The pattern of transitions is M$\to$MC$\to$CI.

Figure 13

Two kinds of magnetic configurations obtained within the Hartree-Fock approximation for a single kagome layer with (a) $t_s=-t,U=2.5t$, (b) $t_s=t,U=3.0t$ and a bilayer with (c) $t_s=-t,U=3.0t$, (d) $t_s=t,U=3.0t$. For the kagome layer, two configurations exist. One (a) is “rotating” and keeps the $C_3$ symmetry, while (b) is an antiferromagnetic phase without this property. For the bilayer, the $C_3$ symmetry is also retained for $t_s=-t$. In the other phase shown in (d), the magnetic moment 4 is antiparallel to moment 1, and moment 3 lies in the plane 413 while moment 2 lies in plane 412.

Figure 14

The two kinds of magnetic configurations for a TKT layer: (a) $t_s=-t,U=3.0t$, (b) $t_s=t,U=3.0t$ and for a KTK layer: (c) $t_s=-t,U=2.5t$, (d) $t_s=t,U=3.0t$. In (a), the moments 4, 5 are antiparallel to each other, and perpendicular to the 123 plane. In (b), the moments 4, 5 are antiparallel to each other, and parallel to the 123 plane. One moment in 123 will lie in the same plane as moment 4, 5, and the other two moments will point in an almost opposite direction. In (d) the moment 4 is parallel to the 123 plane.