Strain tuning of Dirac states at the SnTe (001) surface

The topological crystalline insulator SnTe belongs to the recently discovered class of materials in which a crystalline symmetry ensures the existence of topologically protected surface states. The bulk band structure of SnTe is characterized by a band inversion at the four equivalent $L$ points, giving rise to a mirror Chern number of $n_{\mathcal{M}}=-2$. The (001) surface exhibits four Dirac cones which lie at non-time-reversal-invariant points close to $\overline{X}$ and $\overline{Y}$ and are protected by the $\left(\overline{1}10\right)$ and (110) mirror symmetries. In contrast to topological insulators, this symmetry can be broken via deformations of the crystal. This opens up new possibilities for manipulating the Dirac states and inducing a controllable gap. Here, we have employed density-functional theory to investigate the response of the Dirac states to applied distortions from first principles. Our calculations show that a local gap of up to $\approx 30\mathrm{meV}$ can be introduced via lattice deformations that break at least one of the underlying mirror symmetries. It is formed at either all four or just two cones, depending on the direction of the displacement vector, making it possible to create either a global gap or a state where opened and intact Dirac cones coexist. Notably, applying these deformations at the surface only can already induce the gap. If the complete slab is distorted, bulk bands are pushed into the gap making the whole system metallic.

Figure 1

(a) Rock-salt structure of SnTe and its $\left(\overline{1}10\right)$ mirror plane. (b) Reciprocal face-centered-cubic BZ with its (001) surface BZ. The $L$ points are projected onto the $\overline{X}$ and $\overline{Y}$ points. These points lie on the projection of the $\left(\overline{1}10\right)$ and (110) mirror planes (dashed green lines).

Figure 2

(a) GGA and LDA bulk band structures of SnTe around the $L$ point, showing a slightly indirect inverted band gap for the experimental lattice constant of $a_{exp}=6.327\AA$. As in all the following band structures, the Fermi energy is set to zero. (b) Direct band gap [i.e., energy difference of the conduction-band minimum (CBM) and the valence-band maximum (VBM)] at the $L$ point as a function of the lattice constant. Negative values indicate an inverted band gap.

Figure 3

GGA bulk band structure with indicated localization of the wave function on the Sn (blue) or Te (red) atom. (a) Inverted gap at the experimental lattice constant. (b) Lifted band inversion at $a=6.588\AA \approx 1.04a_{exp}$. (c) Calculation at $a_{exp}$ without the inclusion of SOC. The bands are inverted at $L$, but the gap is closed.

Figure 4

(a) Spin-polarized (001) surface states. The size of the red (green) dots is proportional to a positive (negative) Rashba component, the in-plane part of the spin polarization perpendicular to $\mathit{k}$ [36]. Remark: Along $\overline{X}\overline{M}$ the spin polarization points along the $y$ direction, so the Rashba component remains rather small. (b) The (001) surface-state dispersion for various slab thicknesses. (c) Hybridization between the surface states on both sides of the slab leads to a nonzero band gap at the Dirac point, which decreases with increasing thickness. Slabs with even and odd numbers of layers show different decay behaviors because the bottom and top layers are the same for even layers and are different for odd layers. In both cases the gap slightly oscillates around the overall exponential decay, indicated by the dashed lines.

Figure 5

(a) Localization of the (001) surface states. The width of the blue (red) bands is proportional to the fraction of charge resting on the Sn (Te) atoms in the first six layers, calculated within a Mulliken analysis [37]. Along the mirror plane $\overline{\Gamma }\overline{X}$ two bands with different orbital characters and localizations form a Dirac point, away from $\overline{X}$. Along $\overline{X}\overline{M}$, they couple and open a gap. The charge density of the marked Sn- and Te-like states is shown in (${\mathrm{a}}_i$) and (${\mathrm{a}}_{ii}$) for a cut along the $\left[\overline{1}10\right]$ direction. The Sn (Te) atoms are blue (red). (b) and (c) Quantitative analysis of the localization of the two marked (b) Sn- and (c) Te-like states, showing the fraction of charge resting on the different atoms per layer [see inset (c)]. As one can see, the Dirac states are reaching far into the slab. The connecting lines are a guide for the eye.

Figure 6

(a) Comparison of experimental ARPES results taken from Tanaka et al. [10] (points with error bars, see Ref. [10] for experimental details) with our DFT-GGA band structure (black lines, the gray parts are the projected bulk bands). The calculated Dirac bands are fitted in their linear regime, indicated by the vertical blue lines, yielding the red dashed lines. The crossing point of these two lines is placed over the experimentally anticipated Dirac point $D_{exp}$ for a direct comparison. (b) The (001) surface BZ indicating the $\mathit{k}$-space position of the experimental (purple dot) and the calculated (blue dot) Dirac cones, which lie closer to $\overline{X}$. Measurements [10] were performed along the green arrow using different photon energies indicated by the color of the points. To enable a comparison, the $\mathit{k}$-space path was shifted such that it cuts through the theoretical Dirac cone (green dashed arrow).

Figure 7

(a) Side view of the topmost layers of the (001) surface, the middle atoms lie in the second row in the $y^{\prime }$ direction. The top layers are displaced in the $x^{\prime }$ direction, destroying the $y^{\prime }{z}^{\prime }$ mirror plane while leaving the $x^{\prime }{z}^{\prime }$ plane intact. (b) Corresponding BZ where only the $\overline{X}\overline{\Gamma }\overline{X}$ line remains a projection of a mirror plane. Along $\overline{Y}\overline{\Gamma }\overline{Y}$ the topological protection is lifted, and a gap can be opened at $D_{1,3}$. (c) If the layers are displaced in the $\left[x^{\prime }{y}^{\prime }{z}^{\prime }\right]=\left[110\right]$ direction, both mirror symmetries are broken, and a gap can be opened at all cones. (d) Surface band structure of a 51-layer slab with Rashba component of the spin polarization for a displacement of the top eight layers in the $x^{\prime }$ direction by $0.224\AA$ each relative to the one below, which equals a tilting angle of $\approx 4^{\circ }$. This opens up a gap (26 meV) at $D_{1,3}$, leaving $D_{2,4}$ closed. The dashed blue lines are the bands of the undisturbed system. (e) Top eight layers being displaced in the $\left[x^{\prime }{y}^{\prime }{z}^{\prime }\right]=\left[110\right]$ direction by $0.224\AA$ each. Now a gap of 23 meV is opened at all cones. Only $\overline{Y}\overline{\Gamma }$ is shown since the $\overline{\Gamma }\overline{X}$ line is equivalent.

Figure 8

Quantitative analysis of the correlation between the induced gap at the Dirac points and the applied displacement for various layers. The displacement is given as the length of the vector each layer is shifted relative to the one below for two different directions. (a) Displacement along the $x^{\prime }$ direction, opening a local gap at $D_{1,3}$. (b) Displacement along $\left[x^{\prime }{y}^{\prime }{z}^{\prime }\right]=\left[110\right]$, opening a global gap at all four Dirac points. The connecting lines are a guide to the eye.

Figure 9

(a) Band structure for a deformation of the whole slab in the $x^{\prime }$ direction of $0.224\AA$ between adjacent layers. At $D_{1,3}$ a band gap of 30 meV is opened, but bulk bands close the gap around $\overline{X}$ and the bulk system becomes metallic. (b) Blue graph: Dependence of the local band gap at $D_{1,3}$ as a function of the deformation. (b) Red graph: Linear dependence of the bulk band gap which is already closed at around $0.1\AA$. (c) Band structure for a displacement of $0.089\AA$ in the [110] direction, breaking both mirror symmetries. A surface band gap of 9 meV is opened. (d) Blue graph: Dependence of the surface band gap at all Dirac points as a function of deformation. (d) Red graph: Linear dependence of the bulk band gap which is closed at around $0.13\AA$.