Effects of surface-bulk hybridization in three-dimensional topological metals

Identifying the effects of surface-bulk coupling is a key challenge in exploiting the topological nature of the surface states in many available three-dimensional topological “metals.” Here we combine an effective-model calculation and an ab initio slab calculation to study the effects of the lowest order surface-bulk interaction: hybridization. In the effective-model study, we discretize an established low-energy effective four-band model and introduce hybridization between surface bands and bulk bands in the spirit of the Fano model. We find that hybridization enhances the energy gap between bulk and Dirac surface states and preserves the latter's spin texture qualitatively, albeit with a reduced spin-polarization magnitude. Our ab initio study finds the energy gap between the bulk and the surface states to grow upon an increase in the slab thickness, very much in qualitative agreement with the effective-model study. Comparing the results of our two approaches, we deduce that the experimentally observed low magnitude of the spin polarization can be attributed to a hybridization-type surface-bulk interaction. We discuss evidence for such hybridization in existing ARPES data.

Figure 1

(a) Spectra of the model on a 300-layer-thick slab. The three chemical potentials ${\mu }_{\mathrm{TI}}$, ${\mu }_{\mathrm{TM}}$, and ${\mu }_{\mathrm{M}}$ are taken as representative points for the three regimes TI ($\mu \le E_c$), M ($\mu \gtrsim E_{\mathrm{M}}$), and TM ($E_c\le \mu \lesssim E_{\mathrm{M}}$) as defined in the text, respectively. (b–d) The corresponding unhybridized [dashed (blue) curve] and hybridized [solid (green) curve] spatial profiles of the pair of degenerate conduction Dirac states $|{\Psi }_{D,{\mathbf{k}}_{\parallel }}{\left(z\right)|}^2$ at ${\mathbf{k}}_{\parallel }={\mathbf{k}}_{\parallel ,\mu }$ with $\mu ={\mu }_{\mathrm{TI}}$, ${\mu }_{\mathrm{TM}}$, and ${\mu }_{\mathrm{M}}$, respectively. (e) Effect of hybridization on the spectra. (f) ARPES data on Bi${}_2$Se${}_3$ [3].

Figure 2

The effect of hybridization on the degree of spin-momentum locking in different regimes. Spin expectation values are calculated for a 300-layer-thick slab using a conduction Dirac state $|{\psi }_{D,k_x\widehat{x}}\rangle$ with $k_x=k_{x,\mu }$ at different representative chemical potentials $\mu ={\mu }_{\mathrm{TI}}$, ${\mu }_{\mathrm{TM}}$, and ${\mu }_{\mathrm{M}}$. (a) Spin polarization $\langle S_y\rangle \left(k_x\widehat{x}\right)$. (b) Total spin magnitude $S\left(k_x\widehat{x}\right)$ (defined in the text).

Figure 3

Thickness dependence of the hybridization effects. (a) Dimensionless measure of the bulk-Dirac energy gap $r={\Delta }_{DB}/{\Delta }_{BB}$ (defined in the text) at different slab thicknesses for $\mu ={\mu }_{\mathrm{TM}}$. (b) Spin polarization of a conduction Dirac state $\langle S_y\rangle \left(k_x\widehat{x}\right)$ at $\mu ={\mu }_{\mathrm{TM}}$.

Figure 4

(a) DFT-calculated band structure of a 6-QL slab of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. (b) DFT-calculated spin expectation values of the conduction Dirac state $\langle S_i\rangle \left(k_x\widehat{x}\right)$ for a 6-QL ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ slab.

Figure 5

(a) Ratio $r={\Delta }_{DB}/{\Delta }_{BB}$ within the TM regime, calculated at a fixed ${\mathbf{k}}_{\parallel }$. (b) $\langle S_y\rangle$ of the conduction Dirac state calculated using DFT as a function of the slab thickness $N$.