Band structure engineering of topological insulator heterojunctions

We investigate the topological surface states in heterostructures formed from a three-dimensional topological insulator (TI) and a two-dimensional insulating thin film, using first-principles calculations and the tight-binding method. Utilizing a single Bi or Sb bilayer on top of the topological insulators $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3, \mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3, \mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_2\mathrm{Se}$, and $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$, we find that the surface states evolve in very peculiar but predictable ways. We show that strong hybridization between the bilayer and TI substrates causes the topological surface states to migrate to the top bilayer. It is found that the difference in the work function of constituent layers, which determines the band alignment and the strength of hybridization, governs the character of newly emerged Dirac states.

Figure 1

A perspective (a) side view and (b) top view of atomic structure of single Sb and Bi bilayers on a TI surface. (c) Calculated binding energy ($E_{\mathrm{b}}$) of bilayer on each TI substrate. (d) Calculated work function ($W$) of TI substrates and pristine Sb and Bi bilayers on each TI substrate lattice.

Figure 2

Calculated band structures of Sb bilayer on (a) $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$, (b) $\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3$, (c) $\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_2\mathrm{Se}$, and (d) $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ along the Γ-$K$ direction. Shaded (gray) regions are the projection of the bulk band structure of 3D TI into the 2D surface Brillouin zone. The color bar indicates the origin of the states. The band originating from the Sb bilayer (upper 2QL of TI) is colored in magenta (blue). Emerging Dirac states labeled by ${\mathrm{D}}^{*}$ and ${\mathrm{D}}_{\mathrm{R}}$ are the intrinsic and the Rashba-split states, respectively.

Figure 3

Calculated band structure for Bi bilayer on (a) $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3$, (b) $\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3$, (c) $\mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_2\mathrm{Se}$, and (d) $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ along the Γ-$K$ direction. Shaded (gray) regions are the projection of bulk band structures of 3D TI into the 2D surface Brillouin zone. The bands originating from the Bi bilayer (upper 2QL of TI) are colored in red (blue).

Figure 4

Calculated spin texture of (a)–(d) Sb bilayer and (e)–(f) Bi bilayer on $\mathrm{B}{\mathrm{i}}_2\mathrm{S}{\mathrm{e}}_3, \mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_3, \mathrm{B}{\mathrm{i}}_2\mathrm{T}{\mathrm{e}}_2\mathrm{Se}$, and $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$, respectively, along the Γ-$K$ direction. The color bar indicates the $x$ component of the spin (red for up and blue for down spin). The new Dirac cone ${\mathrm{D}}^{*}$ that emerges almost solely from the Sb or Bi bilayer exhibits a helical spin texture.

Figure 5

Calculated band structures of (a) Sb/BTS and (b) Bi/BTS at artificially increased interfacial distance between the bilayer and the TI surface from the equilibrium position ($\mathrm{\Delta }d$) by 4, 2, 1, and 0.5 Å, from left to right. The parameter δ is the valence band maximum energy relative to the Dirac point.

Figure 6

Evolution of the spin texture in (a) Sb/BTS and (b) Bi/BTS. The interfacial distance between the bilayer and the TI surface is increased from equilibrium by $\mathrm{\Delta }d=4$, 2, 1, and 0.5 Å (from left to right). The color bar indicates the $x$ component of the spin with up in red and down in blue. We observe the helical spin texture migrating to the bilayer.

Figure 7

Evolution of the orbital character in (a) Sb/BTS and (b) Bi/BTS. The interfacial distance between the bilayer and the TI surface is increased from equilibrium by $\mathrm{\Delta }d=4$, 2, 1, and 0.5 Å (from left to right). The states near the Fermi level originate from the Sb or Bi bilayer, having mostly $p$-orbital character, as marked by colored dots ($p_{\mathrm{x}}, p_{\mathrm{y}}$ in red and $p_{\mathrm{z}}$ in blue). We observe the $p_{\mathrm{z}}$-character states in Sb but $p_{\mathrm{x}}, p_{\mathrm{y}}$, and $p_{\mathrm{z}}$ states in Bi bilayer, which form new topological surface states.

Figure 8

Band structures from the model Hamiltonian for bilayer/TI heterostructure. (a) Sb bilayer and (b) Bi bilayer on BTS in the absence (left panel) and presence (right panel) of interfacial interactions. The energy zero is set to the Dirac point of the BTS surface states in the absence of the hopping term. The bands in magenta and red are the bilayer states and the linear band in blue shows the BTS surface states.

Figure 9

Calculated band structures of $\mathrm{BT}{\mathrm{S}}_{1\mathrm{QL}}/\mathrm{BT}{\mathrm{S}}_{5\mathrm{QL}}$ with the interfacial distance increased by $\mathrm{\Delta }d=4$, 1, 0.5, and 0 Å (from left to right) from the equilibrium distance.

Figure 10

(a) Schematic representation of the band alignment in BI/TI interfaces. The band alignment is determined by the work function difference between TI ($W_{\mathrm{TI}}$) and BI ($W_{\mathrm{BI}}$). For $W_{\mathrm{TI}}\approx W_{\mathrm{BI}}$, the band is aligned at a similar level (I); if $W_{\mathrm{TI}}>W_{\mathrm{BI}}$, the valence band of the BI hybridizes with the surface states of the TI (II); if $W_{\mathrm{TI}}<W_{\mathrm{BI}}$, the conduction band of the BI hybridizes with the TI surface states (III). The arrows in blue indicate the electron flow. (b) Schematic illustration of the hybridization of the BI band (in red) with the topological surface state (in blue). Shaded regions denote the bulk states.

Figure 11

(a) The band structures of a free-standing $\mathrm{B}{\mathrm{i}}_{1-x}\mathrm{S}{\mathrm{b}}_x$ alloy bilayer with $x=1$, 0.8, 0.6, 0.4, and 0, from left to right. The band gap is closed at $x=0.4$, indicating a topological phase transition from trivial insulator to quantum spin-Hall insulator. (b) Calculated band structures of $\mathrm{B}{\mathrm{i}}_{1-x}\mathrm{S}{\mathrm{b}}_x/\mathrm{BTS}$ heterostructure. The bands highlighted in blue represent the states originating from the alloy bilayer. The critical Sb composition for the topological phase transition is now shifted to $x\sim 0.67$ due to the interaction with the BTS substrate. We observe a gradual change in the band structure from the Sb-bilayer ($x=1$) to the Bi-bilayer case ($x=0$).