Chiral states induced by symmetry-breaking in $α-T_3$ lattices: Magnetic field effect

TL;DR

This work analyzes how a perpendicular magnetic field influences chiral mid-gap states arising from sublattice-symmetry breaking in the $α$-$T_3$ lattice. Using a continuum $3\times3$ Hamiltonian with a kink mass profile, the authors first derive Landau levels for a uniform mass, revealing valley-dependent spacing and the necessity of a missing Landau level in the $θ=0$ limit. They then obtain analytical mid-gap edge states along a domain wall formed by the kink potential, showing that the dispersion is tunable with the lattice parameter $α$ (via $θ$) and with the mass amplitude $Δ_0$, all governed by the universal scaling parameter $δ_0=Δ_0 l_B/(\hbar v_F)$. Magnetic fields can break valley symmetry and induce chirality for $0<θ<π/4$, while at $θ=π/4$ the field generates dispersive mid-gap states from an originally flat band; for large $Δ_0$ the magnetic effects are suppressed. These results point to tunable valley-filter and valley-transport applications across the graphene–dice continuum.

Abstract

The sublattice-symmetry breaking in the $α-T_3$ lattice leads to a bandgap opening. A defect line in the substrate on which the $α-T_3$ lattice is deposited can be viewed as a topological change in the substrate that induces translational in-plane symmetry breaking, resulting in mid-gap states. These topologically protected states are confined along the defect line and exhibit preferential directional motion, with different signs for the different Dirac valleys. Within this context, we investigate how these unidirectional interface chiral states are affected in the presence of a perpendicular magnetic field and how they can be tuned by varying the controlling system parameter $α$. The latter tunes the $α-T_3$ structure from a honeycomb-like lattice ($α=0$) to a dice lattice ($α=1$). Our theoretical framework is based on the continuum approximation described by a $3\times 3$ matrix Hamiltonian with a sublattice symmetry-breaking term given by $Δ(x) diag(1,\quad -1,\quad 1)$, assuming $Δ(x)$ as a kink-like mass potential profile. Results for dispersion relations and wavefunction distributions for different $α$ parameters and magnetic field amplitudes are discussed. We demonstrate lifting of Landau levels degeneracy and of valley degeneracy. Our findings pave the way for proposing valley filter devices based on any evolutionary stage between the honeycomb-like and dice lattice structures of the $α-T_3$ phase, controlled by external fields.

Figures (10)

• Figure 1: (a) Schematic illustration of the $\alpha-T_3$ lattice, composed of three atomic sites (A in blue, B in red, and C in green), deposited on two half-infinite substrates (Substrate $1$ in pink for $x<0$ and Substrate $2$ in cyan for $x>0$), leading to a translational symmetry breaking along the $x$ direction viewed as a substrate-induced defect line along the $y$-direction. The hopping amplitudes that connect the atomic sites A--B and B--C are $t$ and $\alpha t$, respectively, where the $\alpha$ parameter can smoothly vary from $\alpha = 0$ for the honeycomb lattice to $\alpha = 1$ for the dice lattice. $a_0$ is the interatomic distance between nearest-neighbor sites. (b) Kink potential profile induced by breaking the sublattice symmetry in $\alpha-T_3$ lattice. Pink ($x<0$) and cyan ($x>0$) regions indicate the two regions along the $x$ direction with symmetry-breaking potentials that have flipped values. Blue, red, and green curves denote the $\Delta$ function associated with the A, B, and C sublattices, respectively.
• Figure 2: (Color online) Landau levels [Eq. \ref{['eq.landau']}] as a function of the (a) wave vector component $k_y$ and (b) magnetic field intensity $B_0$ are shown for charge carriers in the $K$ ($K^\prime$) valleys in dashed (dotted) curves. The shaded orange regions correspond to the free continuum energy bands [Eq. \ref{['eq.free.continuum.spectrum']}]. The green solid curve corresponds to the flat Landau level [Eq. \ref{['eq.landau.flat']}]. It was taken $\Delta_0=81.13$ meV for panels (a) and (b) and $\theta=\pi/6$ for panels (a), (b), and (c). $\Delta E_{n,\pm}^{\tau}=E_{n+1,\pm}^{\tau}-E_{n,\pm}^{\tau}$ [Eq. \ref{['eq.landau.B']}] is plotted in (c) as function of $\Delta_0$ for $n=0,1,2,3,4$ and $\tau=+1$ [dashed curves] ($\tau=-1$ [dotted curves]). The arrows indicate the increase in index state $n$ at the conduction (+) and valence (-) band Landau levels.
• Figure 3: (Color online) Energy spectrum as a function of the momentum $k_y$ for four different values of $\theta$ and two kink-like mass potential amplitudes. Blue (red) curves correspond to the mid-gap energies for charge carriers in the $K$ ($K^\prime$) valley. (a,b) Zero magnetic field ($B_0=0$): Panels show $\Delta_0=81.13$ meV (first column) and $\Delta_0=243.39$ meV (second column), respectively. Orange-shaded region: continuum of dispersive bulk states [Eq. \ref{['eq.dispersive']}]; flat orange curve: flat-band bulk states [Eq. \ref{['eq.flat']}]. (c,d) Finite magnetic field ($B_0 \neq 0$): Panels show $\delta_0=1$ and $\delta_0=3$ [Eq. \ref{['eq.delta0']}], corresponding to ($B_0=5$ T, $\Delta_0=81.13$ meV) and ($B_0=5$ T, $\Delta_0=243.39$ meV), respectively. Dashed (dotted) curves: Landau levels for charge carriers at the $K$ ($K^\prime$) valley [Eq. \ref{['eq.landau.B']} with $\tau=+1$ ($-1$)].
• Figure 4: (Color online) Energy spectrum as a function of $k_y$ for the kink-mass profile, taking four different $\theta$ values: (a) $\theta = 0$, (b) $\theta = \pi/12$, (c) $\theta=\pi/6$, and (d) $\theta = \pi/4$. Blue (red) curves correspond to the energies for charge carriers in the $K$ ($K^\prime$) valley. Here $\delta_0=3$ was kept fixed, which corresponds to $B_0\approx 0.55$ T and $\Delta_0=81.13$ meV. Black dashed (dotted) curves correspond to the Landau levels for charge carriers in the $K$ ($K^\prime$) valley, given by Eq. \ref{['eq.landau.B']} for $\tau=+1 (-1)$.
• Figure 5: (Color online) Group velocity as a function of $k_y$ momentum of the chiral states depicted in Fig. \ref{['fig.3']}. Blue (red) curves correspond to the energies for charge carriers in the $K$ ($K^\prime$) valley. Dotted, dot-dashed, dashed, and solid curves represent the results $v/v_F$ for the angles $\theta=0$, $\theta=\pi/12$, $\theta=\pi/6$, and $\theta=\pi/4$, respectively.