Interplay between boundary conditions and Wilson's mass in Dirac-like Hamiltonians

Dirac-like Hamiltonians, linear in momentum $k$, describe the low-energy physics of a large set of novel materials, including graphene, topological insulators, and Weyl fermions. We show here that the inclusion of a minimal $k^2$ Wilson's mass correction improves the models and allows for systematic derivations of appropriate boundary conditions for the envelope functions on finite systems. Considering only Wilson's masses allowed by symmetry, we show that the $k^2$ corrections are equivalent to Berry-Mondragon's discontinuous boundary conditions. This allows for simple numerical implementations of regularized Dirac models on a lattice, while properly accounting for the desired boundary condition. We apply our results on graphene nanoribbons (zigzag and armchair), and on a PbSe monolayer (topological crystalline insulator). For graphene, we find generalized Brey-Fertig boundary conditions, which correctly describe the small gap seen on ab initio data for the metallic armchair nanoribbon. On PbSe, we show how our approach can be used to find spin-orbital-coupled boundary conditions. Overall, our discussions are set on a generic model that can be easily generalized for any Dirac-like Hamiltonian.

Figure 1

(a) Hexagonal cell of monolayer graphene with the $A$ (blue) and $B$ (red) sublattices emphasized. The $K$-point basis functions ${\phi }_{\mu }\left(\mathit{r}\right)$ (with $\mu =A,B$) for each sublattice is composed by $p_z$ orbitals multiplied by the Bloch phases indicated at each site, with $\varphi =2\pi /3$. For $K^{\prime }, {\phi }_{{\mu }^{\prime }}\left(\mathit{r}\right)$ are composed replacing $\varphi \to -\varphi$. (b) Projections of the $K, K^{\prime }$, and $\mathrm{\Gamma }$ points of the Brillouin zone into its one-dimensional counterparts for zigzag ($k_y$) and armchair ($k_x$) nanoribbons. Lattices for (c) zigzag and (d) armchair nanoribbons with $N_D$ ($N_A$) dimers (atoms) from edge to edge. Their primitive vectors ${\mathit{a}}_{Z/A}$ and unit cells are highlighted. The carbon-carbon distance is $a/\sqrt{3}\approx 0.142$ nm. The empty circles at the edges indicate the lattice sites that were removed to form each nanoribbon, defining the effective dimensions $W=W_0+2a/\sqrt{3}$ and $L=L_0+a$, where $W_0=(3N_D/2-1)a/\sqrt{3}$ and $L_0=(N_A-1)a/2$.

Figure 2

Zigzag bands ($N_D=24$) from the $k^2$ model for different boundary parameters $\eta$ and Wilson's masses $m_Z$. (a) For $\eta =0$ and finite $m_Z$, the generalized BF boundary condition returns the expected zigzag band structure. (b) For smaller $m_Z\to 0.15m_Z$ the doublers appear near $\overline{X}$ as hybridized extra cones. (c) For $\eta =1$ (original BF) the doublers at $k_y^{\text{doubler}}\approx \pm [\overline{K}-2m_Z/\left(\hslash v_F{\delta }_x^2\right)]$ cannot be eliminated (dotted lines). The arrows in (a) and (c) point to the state $(E,k_y)$ used to plot the envelope functions $\mathit{F}\left(x\right)$. Near $\overline{K}, \mathit{F}\left(x\right)$ for both (d) $\eta =0$ and (e) $\eta =1$ are similar and smooth (solid lines). (e) The $\mathit{F}\left(x\right)$ for the doublers (dotted lines) show nonphysical phase oscillations on the scale of the numerical discretization.

Figure 3

(a) The DFT gaps (circles) as a function of $L$ (log scale) fall into straight lines on a wide range $3\le N_A\le 50$. The dashed lines are fits used to obtain the linear coefficients that yield the constraints between (b) the Fermi velocity and the correction $\mathrm{\Delta }\theta$ to the BF boundary conditions. (c) Comparison between the gaps in DFT (circles), $k$-linear (dotted lines) and $k^2$ models (solid lines). The dotted match the solid ones, but are shifted for clarity.

Figure 4

Comparison between the DFT band structures and (a) the $k$-linear, (b) tight-binding, and (c) $k^2$ models for $N_A=48$. At low energy all models agree, while for $\left|E\right|\gtrsim 0.5$ eV parabolic corrections become relevant and broken chirality starts to develop. The envelope densities (colored symbols) match well the DFT data for (d) the first and (e) second conduction bands at $k_x=0$.

Figure 5

Lattices of PbSe ribbons of types (a) A, (b) B, and (c) C, as defined in Ref. [58]. (d)–(f) PbSe band edges at $k_x=0$ varying the boundary condition parameters $\rho$ and $\theta$. (d) For $m>0$, the system is trivial, there are no states within the gap $|E/\mathrm{\Delta }|<1$. (e) For $m<0$, a pair of degenerate topological Dirac crossings appear, and its crossing point at $k_x=0$ is controlled by $\rho$: (e1) for $\rho >0$ the crossing is downshifted, and (e2) for $\rho <0$ it shifts up in energy. (f) A finite $\theta$ or ${\mathrm{\Delta }}_C$ splits the Dirac crossings as shown in (f1).

Figure 6

Comparison between the DFT band structures and (a) the $k$-linear, (b) tight-binding, and (c) $k^2$ models for armchair nanoribbons with (a1)–(c1) $N_A=49$, (a2)–(c2) $N_A=50$, and (a3)–(c3) zigzag nanoribbons with $N_D=48$. The envelope densities (colored symbols) match well the DFT data for (d) the first and (e) second conduction bands at $k_x=0$ in all cases.