Bulk metals with helical surface states

In the flurry of experiments looking for topological insulator materials, it has been recently discovered that some bulk metals very close to topological insulator electronic states support the same topological surface states that are the defining characteristic of the topological insulator. First observed in spin-polarized angle resolved photoemission spectroscopy (ARPES) in Sb [D. Hsieh et al., Science 323, 919 (2009)], the helical surface states in the metallic systems appear to be robust to at least mild disorder. We present here a theoretical investigation of the nature of these “helical metals”—bulk metals with helical surface states. We explore how the surface and bulk states can mix, in both clean and disordered systems. Using the Fano model, we discover that in a clean system, the helical surface states are not simply absorbed by hybridization with a nontopological parasitic metallic band. Instead, they are pushed away from overlapping in momentum and energy with the bulk states, leaving behind a finite-lifetime surface resonance in the bulk energy band. Furthermore, the hybridization may lead in some cases to multiplied surface-state bands, in all cases retaining the helical characteristic. Weak disorder leads to very similar effects—surface states are pushed away from the energy bandwidth of the bulk, leaving behind a finite-lifetime surface resonance in place of the original surface states.

Figure 1

(Color online) An energy-momentum sketch of the fate of surface states when they overlap with a parasitic metallic bulk band. The solid (light green) regions mark the bulk density of states, the top and bottom bands being bulk states of a topological insulator, and a parasitic metallic band in the middle. The original surface-state branch (dashed line) overlaps with the middle (metallic) band. The surface-state branch hybridizes with the bulk, leaving a diffuse resonance in the bulk band energy range, as well as two sharp edge-state branches appearing above and below the parasitic metal energies. This demonstrates that the topological surface states, rather than being obliterated, are simply pushed away from overlapping with the bulk.

Figure 2

(Color online) A two-dimensional toy model for the investigation of a helical metal. The model is based on the Kane-Mele model (Refs. 1, 3) on the honeycomb lattice, which is denoted by dashed lines. The Kane-Mele model consists of nearest-neighbor hopping $t_1$ (green) between the two $\left(a,b\right)$ sublattices of the honeycomb lattice (denoted by empty and filled circles, respectively), and a complex second-neighbor hopping (a spin-orbit coupling term) with opposite sign when clockwise ($-i\lambda$, blue) and counterclockwise ($+i\lambda$, red), as indicated by the curved arrows in the figure. In addition to sites of the honeycomb lattice, we include a new set of sites $\left(c\right)$ at the centers of the hexagonal plaquettes of the honeycomb lattice, denoted by cross marks. The $c$ sites form a triangular sublattice, and nearest-neighbor hopping between them $t_2$ (brown) forms a metallic band. To explore the interplay between the topological insulators helical surface states and the bulk metallic band we mix the two systems by allowing hopping between the $c$ sites and the honeycomb lattice sites, $t_3$ (purple).

Figure 3

(Color online) Spectrum of the modified Kane-Mele model in Eq. (1), with a metallic (triangular) band added (Fig. 2). Here we use the parameters $t_1=1$, $\lambda =0.5$, $t_2=0.2$, $h=0.4$, and vary $t_3$. We show spectra for both [(a), (c), and (e)] zigzag and [(b), (d), and (f)] armchair edge strips, with mixing parameter values $t_3=0$, 0.03, and 0.3. The strips are finite in one direction, and periodic in the other, so the lattice is wrapped around a cylinder. The edge-state branches on different sides of the sample are denoted by thick lines, dark (blue), and light (orange), respectively. All bulk states are colored light gray. Helical edge-state branches can still appear (at least when the Fermi energy is shifted), as can be easily seen by counting the (odd) number of surface-state branches on one side of the strip crossing $E=0$ between $q=0$ and the BZ midpoint ($q=\pi$ for zigzag and $q=\pi /\sqrt{3}$ for armchair).

Figure 4

(Color online) The wave-function density ${\left|\psi \right|}^2$ of all 191 eigenstates identified as surface states in Fig. 3e. The graph shows all wave-function profiles superposed (the point here is to demonstrate that they all decay exponentially into the bulk) as a function of the coordinate $x$ perpendicular to the system boundary. All wave-function curves are plotted with the same (blue) color.

Figure 5

(Color online) Plot of the surface-state decay length vs the energy of the surface state. We use the same parameters as in Fig. 3e (shown in the inset). The energy range $0.6<ϵ<1$ is where the metallic bulk band appears, and hence a surface-state branch is absent. At all ends of a surface-state branch, where a bulk band appears, a divergence in the decay length is evident. In all cases, the decay lengths are no longer than 3, in a system of size $\sim 50$. The multivaluedness at the low- and high-energy extremes is due to the surface states curving a bit up in energy as one approaches the original topological insulator bands, as is shown in the inset.

Figure 6

(Color online) Typical spectral weight function of the Fano model. We take the spectral weight in Eq. (5), with parameters $E_1=-1$, $E_2=+1$, the surface-state energy $ϵ=0.5$ (within the bandwidth). For $\omega$ values outside the bulk bandwidth we take $\lambda ={10}^{-4}$ to approximate the delta function outside the bandwidth. For the continuous (red) curve we take ${\left|g\right|}^2\nu \left(E_F\right)=0.04$, and for the dashed (blue) curve we take ${\left|g\right|}^2\nu \left(E_F\right)=0.2$. At the lower coupling value (continuous curve), the widened peak near $\omega =0.5$ is the remnant of the original surface state, and the modest weight in one delta function already appears very near the band edge (the weight of the other delta function is too small to discern). For the stronger coupling value (dashed line) traces of the surface state are largely obscured in the bulk bandwidth but both the delta functions outside the bandwidth appear prominently.

Figure 7

(Color online) Numerical calculation of the surface-state spectral function. We take the surface-state eigenstates in the decoupled case $\left[{\psi }_0\left(\mathbf{q}\right)\right]$ and find their distribution among the eigenstates of the coupled system ${\varphi }_{\omega }\left(\mathbf{q}\right)$ by calculating the overlaps $A\left(\mathbf{q},\omega \right)={\left|⟨{\psi }_0\left(\mathbf{q}\right)|{\varphi }_{\omega }\left(\mathbf{q}\right)⟩\right|}^2$. This is expected to approximate Eq. (5). We use the same parameters as the strongly hybridized zigzag edge strip in Fig. 3e, and the overall spectrum is manifestly identical in the two images. We plot the surface-state spectral weight function versus transverse momentum (horizontal axis) and energy (vertical axis). Dark (blue) points have $A\left(\omega \right)>0.06$, intermediate gray (green) points have $0.06>A\left(\omega \right)>0.02$, and all points with $0.02>A\left(\omega \right)$ are very light gray. The exiled surface-state branches are manifest, and the ghost spectral resonance is identifiable as a faint peak overlapping with the bulk band, between the two exile branches. In addition, the lower spectral weight points map out the shadow of the full bulk spectrum.

Figure 8

The ghost surface resonance observed in surface probes. The quantity $\mathcal{J}$ measures the weight of each eigenstate ${\varphi }_{\omega }$ convolved with an exponentially decaying weight factor, mimicking the penetration depth of the experimental surface probes $\mathcal{J}\left(\omega ,\mathbf{q}\right)={\left|⟨{\varphi }_{\omega }\left(\mathbf{q}\right)|f\left(y\right)|{\varphi }_{\omega }\left(\mathbf{q}\right)⟩\right|}^2$. Here we plot $\mathcal{J}$ versus transverse momentum $\mathbf{q}$ and energy $\omega$, both in (a) 3D and (b) 2D plots. The grayscale color scheme is such that higher values are colored dark gray. We find significant values of $\mathcal{J}$ only for the surface states and the surface resonances, which are manifestly positioned in between the exiled surface states [most clearly in the 2D plot (b)]. In addition, the very light gray points map out the bulk states.

Figure 9

(Color online) All pictorial conventions are exactly the same as in Fig. 3, including the system size, and the criteria for identifying surface states. Here we use parameters $\lambda =0.3$, $\mu =0.8$, $t_1=1$, and ${\tilde{t}}_2=0.4$. We show the zigzag spectrum in (a) and the armchair edge spectrum in (b).

Figure 10

(Color online) All pictorial conventions are exactly the same as in Fig. 3, including the system size, and the criteria for identifying surface states. Here we use parameters $\lambda =0.08$, $t_2=-1$, $t_1=1$, $h=2$, and $t_3=0.5$. We show the spectrum of both zigzag and armchair edge strip geometries. The same expulsion of the surface states occurs, as in Fig. 3, and in addition we find the number of surface branches has tripled to three Kramer’s pairs on each wall of the strip.