Topological phase transition in a topological crystalline insulator induced by finite-size effects

We study electronic states and topological invariants of (001) films of the topological crystalline insulator (TCI) ${\mathrm{Pb}}_x{\mathrm{Sn}}_{1-x}\mathrm{Te}$. Gapless surface Dirac cones on bulk TCIs become gapped in thin films due to the finite-size effect, which is the hybridization between the states on the top and bottom surfaces. We clarify that the TCI film has a strong finite-size effect as compared to three-dimensional topological insulators such as ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Moreover, the energy gap oscillates with the thickness of film. The oscillation stems from topological phase transitions in two dimensions. The obtained data of the topological invariants and energy gap serve as guide to TCI-device applications.

Figure 1

Crystal structure of SnTe, the first Brillouin zone, and its projection onto the (001) plane. The mirror symmetry with respect to (110) plane illustrated in the figure protects the surface Dirac cones.

Figure 2

Energy dispersion for SnTe near the $\overline{X}$ point in slab geometry shown in the inset. The arrow denotes the Dirac point of the surface states. The number of layers is set to $N_z=45$. The parameters for SnTe are taken from Ref. [59].

Figure 3

Finite-size effect induced energy gap for thin films of nontopological PbTe [(green) square], TCI SnTe [(red) circle], and TI Bi${}_2$Se${}_3$ [(blue) triangle] as a function of the thickness. The solid and dashed lines denote the magnitude of the energy gap for the odd and even numbers of layers, respectively.

Figure 4

Energy gap in thin film of Pb${}_x$Sn${}_{1-x}$Te for various compositions $x$ for the odd numbers of layers.

Figure 5

Period $d$ of the oscillation of the energy gap in the film in units of the number of layers as a function of $x$ (a) and of $1/\sqrt{E_{\mathrm{g}}\left(\mathit{L}\right)}$ (b) with $E_{\mathrm{g}}\left(\mathit{L}\right)$ the bulk band gap at the $L$ point.

Figure 6

Dimensional crossover (a) from three-dimensional (3D) to two-dimensional (2D) insulators in a Pb${}_x$Sn${}_{1-x}$Te film. The two-dimensional gapless surface states (b) have a large gap in a thin film of $N_z<\xi \sim 10$ (c). The mirror-Chern phase $|N_{\mathrm{M}}|=2$ supporting the mirror-chiral gapless edge states and the $|{\zeta }_{\mathrm{M}}|=2$ phase, which is defined by Eq. (11), supporting the mirror-helical gapless/gapful edge states, is realized for the odd and even numbers of layers, respectively. Energy dispersions of the edge states are schematically shown. The (red) solid and (blue) dashed line denote those of edge states in the mirror-even and mirror-odd sectors. In the thin limit, the system falls into a trivial insulator. The two-dimensional and the projected one-dimensional BZs are illustrated in (d).

Figure 7

Topological phase diagram in odd numbers of layers of Pb${}_x$Sn${}_{1-x}$Te. TCI with $|N_{\mathrm{M}}|=2$ is realized in the shaded regions. The energy gap is denoted by the open circles, closed squares, and open squares for a nontopological insulator $N_{\mathrm{M}}=0$, TCI with $N_{\mathrm{M}}=2$, and TCI with $N_{\mathrm{M}}=-2$, respectively.

Figure 8

The edge charge ${\rho }_{+,\mathrm{c}}(k_y,E)$ (a) and spin ${\rho }_{+,\mathrm{s}}(k_y,E)$ (b) spectral functions and the spin-resolved edge DOS ${\rho }_{+,\uparrow }\left(E\right)$ and ${\rho }_{+,\downarrow }\left(E\right)$ (c) for the mirror-even sector. The thickness is set to $N_z=7$. All the vertical axes denote energy $E$ (eV) in the same scale.

Figure 9

Edge spectral function ${\rho }_{\pm ,\mathrm{c}}(k_y,E)$ for $N_z=23$ [(a) and (e)], 25 [(b) and (f)], 27 [(c) and (g)], and 29 [(d) and (h)]. The mirror eigenvalue is given by $M=+i$ for (a)–(d) and $M=-i$ for (e)–(h). The mirror Chern number is obtained to be $N_{\mathrm{M}}=2$ for $N_z\le 25$ and $N_{\mathrm{M}}=-2$ for $N_z\ge 27$.

Figure 10

Topological phase diagram in even numbers of layers of Pb${}_x$Sn${}_{1-x}$Te. Energy gap and topological invariant ${\zeta }_{\mathrm{M}}$ are shown. $|{\zeta }_{\mathrm{M}}|=2$ is denoted by the closed square in the shaded region.

Figure 11

One-dimensional gapless edge states along the (100) edge in a SnTe film with $N_z=20$ for (a) the mirror-even and (b) both sectors. The degeneracies denoted by the arrows are protected by mirror-reflection (MRS) and time-reversal (TRS) symmetries, respectively.

Figure 12

Evolution from gapless to gapped edge states induced by perturbation. The energy dispersions correspond to that in Fig. 11.

Figure 13

Edge spectral function for $N_z=38$ [(a) and (c)] and $N_z=40$ [(b) and (d)]. The mirror eigenvalue is given by $M=+i$ for (a) and (b) and $M=-i$ for (c) and (d).

Figure 14

Edge spectral function in the vicinity of the band gap. (a)–(d) correspond to those in Fig. 13, respectively.

Figure 15

$2\pi$ [(a) and (b)] and $4\pi$ [(c) and (d)] periodicities of the energy dispersion in each mirror sector. Edge spectral function for $N_z=19$ [(a) and (b)] and for $N_z=20$ [(c) and (d)] of SnTe film. (a) and (c) [(b) and (d)] show those around $k_y=-\pi$ ($k_y=\pi$).

Figure 16

Domain of integration for topological numbers. The dashed and solid lines indicate the projected Brillouin zone and that in each mirror sector for the even numbers of layers, respectively. The $\overline{\Gamma }$ $(k_x=0,k_y=0)$ and $\overline{M}$ $(2\pi ,0)$ points are spatial-inversion invariant momenta (SIIM) and the $\overline{X}$ point $(\pi ,\pi )$ is a time-reversal invariant momentum (TRIM). Regions I–VI denote the domain reduced by symmetries. The topological numbers defined in regions III and V are the $\mathbb{Z}$ (${\zeta }_{\pm }$) and ${\mathbb{Z}}_2$ (${\nu }_{\pm }$) numbers, respectively.

Figure 17

Evolution of Wannier center in the mirror-even sector of SnTe films for $N_z=12$ (${\zeta }_{\mathrm{M}}=0$) [(a) and (b)] and $N_z=14$ ($|{\zeta }_{\mathrm{M}}|=2$) [(c) and (d)].

Figure 18

Evolution of Wannier center in the mirror-even sector of SnTe films for $N_z=38$ ($|{\zeta }_{\mathrm{M}}|=2$) [(a) and (b)] and $N_z=40$ (${\zeta }_{\mathrm{M}}=0$) [(c) and (d)].