Second-neighbor electron hopping and pressure induced topological quantum phase transition in insulating cubic perovskites

Perovskite structure is one of the five symmetry families suitable for exhibiting topological insulator phase. However, none of the halides and oxides stabilizing in this structure exhibit the same. Through density functional calculations on cubic perovskites (${\mathrm{CsSnX}}_3$; X = Cl, Br, and I), we predict a band insulator–Dirac semimetal–topological insulator phase transition with uniform compression. With the aid of a Slater-Koster tight binding Hamiltonian, we show that, apart from the valence electron count, the band topology of these perovksites is determined by five parameters involving electron hopping among the Sn-{$s,p$} orbitals. These parameters monotonically increase with pressure to gradually transform the positive band gap to a negative one and thereby enable the quantum phase transition. The universality of the mechanism of phase transition is established by examining the band topology of Bi based oxide perovskites. Dynamical stability of the halides against pressure strengthens the experimental relevance.

Figure 1

Spider chart illustrating the pressure induced topological phase transition in ${\mathrm{CsSnX}}_3$ in a configure space spanned by five one-electron tight-binding parameters $t^A$ and ${\epsilon }^A$ as defined through Eq. (1). The radially increasing equipressure contours are shown in black solid lines. The parameters are expressed in terms of their zero-pressure values.

Figure 2

Room temperature phonon band structure of ${\mathrm{CsSnCl}}_3$: (a) at zero pressure and (b) at 4.72 GPa ($\frac{V}{V_0}=0.84$). (c) The cubic lattice with ${\mathrm{SnX}}_6$ octahedra, and the bulk and surface Brillouin zone for ${\mathrm{CsSnX}}_3$. (d) The molecular orbital picture summarizing the effect of nearest neighbor chemical bonding [37]. (e)–(g) The band structure of ${\mathrm{CsSnX}}_3$ with and without compression. The compressed structures exhibit $s\text{−}p$ band inversion at the TRIM point $R$ ($\pi /a,\pi /a,\pi /a$).

Figure 3

(a)–(e) Bulk SOC band structure of ${\mathrm{CsSnCl}}_3$ in the vicinity of $E_F$ as a function of pressure. (f)–(j) The Sn-$s$ and $p$ orbital weights of band-1 (solid line) and band-2 or band-4 (dotted line). (k)–(o) The surface states, calculated using Wannier formalism, are plotted along the path $\overline{\mathrm{\Gamma }}\text{−}\overline{M}\text{−}\overline{\mathrm{\Gamma }}$. Together, the figures show that, as we increase the pressure, the bands yield a negative band gap associated with $s\text{−}p$ band inversion and thereby create surface Dirac-TI states.

Figure 4

(a) Schematic illustration of orbital manifestation in ${\mathrm{CsSnX}}_3$ under pressure. (b),(c) The variation of the energy eigenstates $E_i\left(R\right)$ of Eq. (3) with pressure for ${\mathrm{CsSnX}}_3$. The crossover between $E_1$ and $E_2$ and between $E_1$ and $E_{3,4}$ are associated with the topological phase transitions.

Figure 5

Relative increase in effective on-site energies and hopping parameters with increase in hydrostatic pressure. The superscripts “P” and “0” represent the values of the parameters at pressure P and at zero pressure (experimental equilibrium structure, respectively.)

Figure 6

Band structure of ${\mathrm{KBiO}}_3$ and ${\mathrm{BaBiO}}_3$ with volume expansion. While the equilibrium structure has $s\text{−}p$ band inversion at $R$, with expansion, the inversion vanishes and a normal band gap is realized. Here $E_F$ is set to zero.

Figure 7

Variation of valence band maximum ($E_1$) and conduction band minimum ($E_2$, $E_3$) at $R$ high symmetry $k$ point of ${\mathrm{KBiO}}_3$ and ${\mathrm{BaBiO}}_3$. At $V=V_0$, ${\mathrm{KBiO}}_3$ shows class-I TI state and ${\mathrm{BaBiO}}_3$ shows class-II TI states. With volume expansion of the unit cell, band inversion disappears, making it trivial insulation state.

Figure 8

Full basis TB band structure of ${\mathrm{CsSnI}}_3$ and DFT band structure.

Figure 9

Phonon band dispersions of ${\mathrm{CsSnX}}_3$ (X = Br, I). Here, $\gamma =\frac{V}{V_0}$.

Figure 10

First and second row show the bulk and surface band structure of ${\mathrm{CsSnBr}}_3$, respectively. Third and fourth row show the same for ${\mathrm{CsSnI}}_3$. With pressure the positive band gap transforms to a negative band gap, which induces $s\text{−}p$ band inversion. As a consequence topologically protected surface states are formed at point $\overline{M}$ (whose equivalent point is $R$ in bulk Brillouin zone). For details, see the main text.

Figure 11

Variation of energy gap and pressure with volume compression of different halide perovskites.

Figure 12

Band structure of ${\mathrm{ABiO}}_3$ as a function of volume expansion. The orbital contributions, around the high symmetry point $R$, to the lower two bands are indicated in the upper panel. With expansion, the $s\text{−}p$ band inversion vanishes to create a normal band gap.

Figure 13

Exchange-correlation functional dependent band gap of ${\mathrm{CsSnCl}}_3$ and ${\mathrm{CsSnBr}}_3$ as a function of lattice parameter. The slopes are found to be nearly the same for each functional. The HSE06 results are obtained using a pseudopotential method through the vasp simulation package [29, 30].