Band-gap formation and morphing in $\alpha \text{−}{T}_3$ superlattices

Electrons in $\alpha \text{−}{T}_3$ lattices behave as condensed-matter analogies of integer-spin Dirac fermions. The three atoms making up the unit cell bestow the energy spectrum with an additional energy band that is completely flat, providing unique electronic properties. The interatomic hopping term, $\alpha$, is known to strongly affect the electronic spectrum of the two-dimensional (2D) lattice, allowing it to continuously morph from graphenelike responses to the behavior of fermions in a dice lattice. For pristine lattice structures the energy bands are gapless, but small deviations in the atomic equivalence of the three sublattices will introduce gaps in the spectrum. It is unknown how these affect transport and electronic properties such as the energy spectrum of superlattice minibands. Here we investigate the dependency of these properties on the parameter $\alpha$ accounting for different symmetry-breaking terms, and we show how it affects band-gap formation. Furthermore, we find that superlattices can force band gaps to close and shift in energy. Our results demonstrate that $\alpha \text{−}{T}_3$ superlattices provide a versatile material for 2D band-gap engineering purposes.

Figure 1

Schematic of the $\alpha \text{−}{T}_3$ lattice where the sites of the three sublattices are colored differently. The limit $\alpha =0$ corresponds to the honeycomb lattice (graphenelike), and $\alpha =1$ corresponds to the dice lattice. The hopping amplitude between the different atoms is indicated. The region bounded by the gray lines corresponds to the unit cell.

Figure 2

Energy spectrum of massless Dirac fermions in the $\alpha \text{−}{T}_3$ lattice (a) in the full first Brillouin zone, and (b) around the $K$ point.

Figure 3

Energy spectrum of Dirac fermions for arbitrary values of the parameter $\theta$ in the $\alpha \text{−}{T}_3$ lattice when the symmetry-breaking term $\widehat{U}={\widehat{U}}_1$ is used in Eq. (4). (a) Full first Brillouin zone, and (b) spectrum around the $K$ point.

Figure 4

Energy spectrum of Dirac fermions in the $\alpha \text{−}{T}_3$ lattice for different values of $\theta$ when the symmetry-breaking term $\widehat{U}={\widehat{U}}_2$ is used in Eq. (4). The full first Brillouin zone is shown at the top and below the energy spectrum around the $K$ point for (a) $\theta =0$ (graphenelike case), (b) $\theta =\pi /12$, (c) $\theta =\pi /6$, and (d) $\theta =\pi /4$ (dice case).

Figure 5

Schematic representation of the superlattice potential in the $x\text{−}y$ plane. Dark regions denote the barrier region with height $V\left(x\right)=V_b$ and the white region represents the well with zero potential. The angles ${\varphi }_w$ and ${\varphi }_b$ in the inset, respectively, denote the angles of the carriers in the wells and barriers regions. The profiles of the 1D periodic potential are given by the figure at the bottom.

Figure 6

Electronic band structures at $K_{x}L=0$ for (a) $\theta =0$ (graphenelike case), (b) $\theta =\pi /12$, (c) $\theta =\pi /6$, (d) $\theta =\pi /4$ (dice case) with $V_b=7E_L, W_w=W_b=L/2$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$.

Figure 7

Density of states for $\theta =0$ (black solid curve), $\theta =\pi /12$ (blue dashed curve), $\theta =\pi /6$ (red dash-dotted curve), and $\theta =\pi /4$ (magenta dotted curve) for the same parameters as in Fig. 6.

Figure 8

Valence and conduction bands of the spectrum of a superlattice considering $\theta =0$ (graphenelike), and $\theta =\pi /4$ (dice) with $V_b=21E_L, W_w=W_b=L/2$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$.

Figure 9

Electronic band structures for $K_{x}L=0$ with $\theta =0$ (black solid curve), $\theta =\pi /6$ (red dashed curve), and $\theta =\pi /4$ (blue dot-dashed curve) for $K_{x}L=0$ with $V_b=21E_L, W_w=W_b=L/2$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$.

Figure 10

Group velocity along the $k_y$ direction around the main Dirac point (I), and around the extra Dirac point (II) indicated in Fig. 9.

Figure 11

Electronic band structures at $K_{x}L=0$ for (a) $\theta =0$ (graphenelike case), (b) $\theta =\pi /12$, (c) $\theta =\pi /6$, (d) $\theta =\pi /4$ (dice case) with $V_b=7E_L, W_w=W_b=L/2, \mathrm{\Delta }=0.1V_b$, and $\widehat{U}={\widehat{U}}_1$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$ in all cases.

Figure 12

Density of states for $\theta =0$ (black solid curve), $\theta =\pi /12$ (blue dashed curve), $\theta =\pi /6$ (red dash-dotted curve), and $\theta =\pi /4$ (magenta dotted curve) for the same parameters as in Fig. 11.

Figure 13

Electronic band structures at $K_{x}L=0$ for (a) $\theta =0$ (graphenelike case), (b) $\theta =\pi /12$, (c) $\theta =\pi /6$, (d) $\theta =\pi /4$ (dice case) with $V_b=7E_L, W_w=W_b=L/2, \mathrm{\Delta }=0.4V_b$ for $\widehat{U}={\widehat{U}}_1$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$ in all cases.

Figure 14

Density of states for $\theta =0$ (black solid curve), $\theta =\pi /12$ (blue dashed curve), $\theta =\pi /6$ (red dash-dotted curve), and $\theta =\pi /4$ (magenta dotted curve) for the same parameters as in Fig. 13.

Figure 15

Valence and conduction bands of the spectrum of a superlattice considering $\theta =0$ (graphenelike), and $\theta =\pi /4$ (dice) with $V_b=21E_L, W_w=W_b=L/2, \widehat{U}={\widehat{U}}_1$, and $\mathrm{\Delta }=0.4V_b$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$.

Figure 16

Electronic band structures at $K_{x}L=0$ for $\theta =0$ (black solid curve), $\theta =\pi /6$ (red dashed curve), and $\theta =\pi /4$ (blue dot-dashed curve) with $V_b=21E_L, W_w=W_b=L/2, \mathrm{\Delta }=0.4V_b$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$.

Figure 17

Electronic band structures at $K_{x}L=0$ for (a) $\theta =0$ (graphenelike case), (b) $\theta =\pi /12$, (c) $\theta =\pi /6$, (d) $\theta =\pi /4$ (dice case) with $V_b=7E_L, W_w=W_b=L/2, \mathrm{\Delta }=0.1V_b$ when $\widehat{U}={\widehat{U}}_2$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$ in all cases.

Figure 18

Density of states for $\theta =0$ (black solid curve), $\theta =\pi /12$ (blue dashed curve), $\theta =\pi /6$ (red dash-dotted curve), and $\theta =\pi /4$ (magenta dotted curve) for the same parameters as in Fig. 17.

Figure 19

Electronic band structures at $K_{x}L=0$ for (a) $\theta =0$ (graphenelike case), (b) $\theta =\pi /12$, (c) $\theta =\pi /6$, (d) $\theta =\pi /4$ (dice lattice) with $V_b=7E_L, W_w=W_b=L/2, \mathrm{\Delta }=0.4V_b$ when $\widehat{U}={\widehat{U}}_2$, where $L=1200$, and $E_L=\hslash v_F/L$ in all cases.

Figure 20

Density of states for $\theta =0$ (black solid curve), $\theta =\pi /12$ (blue dashed curve), $\theta =\pi /6$ (red dash-dotted curve), and $\theta =\pi /4$ (magenta dotted curve) for the same parameters as in Fig. 19.

Figure 21

Valence and conduction bands of the spectrum of a superlattice considering $\theta =0$ (graphenelike), and $\theta =\pi /4$ (dice) with $V_b=21E_L, W_w=W_b=L/2, \widehat{U}={\widehat{U}}_2$, and $\mathrm{\Delta }=0.4V_b$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$.

Figure 22

Electronic band structures at $K_{x}L=0$ for $\theta =0$ (black solid curve), $\theta =\pi /6$ (red dashed curve), and $\theta =\pi /4$ (blue dot-dashed curve) with $V_b=21E_L, W_w=W_b=L/2, \mathrm{\Delta }=0.4V_b$, where $L/a_0=1200$, and $E_L=\hslash v_F/L$.