Ab initio transport across bismuth selenide surface barriers

We investigate the effect of potential barriers in the form of step edges on the scattering properties of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$(111) topological surface states by means of large-scale ab initio transport simulations. Our results demonstrate the suppression of perfect backscattering, while all other scattering processes, which do not entail a complete spin and momentum reversal, are allowed. Furthermore, we find that the spin of the surface state develops an out-of-plane component as it traverses the barrier. Our calculations reveal the existence of quasibound states in the vicinity of the surface barriers, which appear in the form of an enhanced density of states in the energy window corresponding to the topological state. For double barriers we demonstrate the formation of quantum well states. To complement our first-principles results we construct a two-dimensional low-energy effective model and illustrate its shortcomings. Our findings are discussed in the context of a number of recent experimental works.

Figure 1

(a) The unit cell of the 3-QL slab leads used in the transport calculations, corresponding to a periodic quasi-two-dimensional system. The smaller (yellow) and larger (purple) spheres represent selenium and bismuth atoms, respectively. The slab is terminated on both sides by Se. (b) The band structure along the transport direction, $z$, is shown at $k_x=0$. The surface band in the energy window $[-0.05$, 0.30] eV has a helical spin texture, with the spin locked in the momentum direction. The transport setup for the scattering problem is shown for (c) a single barrier and (d) a double barrier. In both cases we add an extra single-QL-high barrier on the 3-QL-thick slab. Note that the same self-energies for semi-infinite 3-QL leads are attached on the left and right sides of the scattering region in (d), while different left and right electrodes, corresponding to 4-QL and 3-QL slabs, are needed in (c).

Figure 2

(a) Transmission across the surface barrier as a function of energy at different values of the $x$ component of the wave vector, orthogonal to the transport direction. Different curves correspond to different $k_x$, starting from $k_x=0$ up to $k_x=0.046335$ Å${}^{-1}$, in equal steps of 0.009 267 Å${}^{-1}$. Note the perfect transmission at $k_x=0$. At other incidence angles, $T$ is reduced. (b) The total transmission integrated over $k_x$ in the presence (black curve) and absence [gray (red) curve] of the barrier. (c) The transmission as a function of $k_x$, at different constant energy cuts in the energy region of the surface states. Note that $T$ is reduced from $T=2$ at a nonzero angle of incidence and, with sufficiently large $k_x$, drops down towards $T=1$. Upon further increase in $k_x$, the band edge for the Dirac cone at the bottom surface is reached, and the transmission abruptly goes to 0. Nonzero reflection at the barrier can be explained using the schematic in (d). At non-normal incidence there is a finite overlap between the spin projections of the forward- and backward-moving surface states. Backscattering at angles away from normal incidence is present even in the absence of time-reversal symmetry-breaking perturbations.

Figure 3

The DOS projected on the surface atoms along the scattering region at (a) $k_x=0$, (b) at $k_x=0.032$ Å${}^{-1}$, and (c) integrated over all $k_x$. At $k_x=0$ there are no oscillations. These start to emerge at $k_x=0.032$ Å${}^{-1}$ but are not visible in the average. Note in all panels the enhanced DOS on the left-hand side of the barrier. The second column of panels shows the Fourier transform of the projected DOS in the flat region adjacent to the barrier, at the corresponding $k_x$. The scattering vector resulting from backscattering at non-normal incidence is clearly shown in (b). The average, however, reveals no scattering. Here and henceforth warmer colors represent higher, and cooler colors indicate lower, values. The third column of panels shows the transmission as a function of energy for the three cases. For $k_x=0$ and $k_x=0.032$ Å${}^{-1}$, we also plot the band structure along the transport direction for comparison.

Figure 4

(a) Energy dispersion along $k_x$ (perpendicular to the transport direction) for a perfect periodic system comprising a 4-QL slab, (b) energy dispersion at the single barrier, and (c) energy dispersion 50 Å away from the single barrier. In (b)–(d) color plots show the projected density of states (PDOS) on the atom present at the barrier and an atom 50 Å away from the barrier and the PDOS on the atom at the double barrier (see Sec. 5 for a discussion). In (b) and (d) note the additional pair of interface states outside the Dirac cone, which merge with it around 0.2 eV.

Figure 5

The DOS projected on the bottom surface atoms along the scattering region at (a) normal incidence $k_x=0$ and (b) an oblique incidence $k_x=0.032$ Å${}^{-1}$. Note the absence of density oscillations in the bulk energy gap window, even at non-normal incidence. Panels on the right show the Fourier transform of the projected DOS in the flat region adjacent to the barrier. A comparison with Fig. 3 shows the absence of both bound states as well as the signature of dominant scattering vectors in the aforementioned energy range. This illustrates a small coupling between the two surfaces of the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ slab investigated in this work.

Figure 6

The spin-resolved local density of states (LDOS) for states incoming from the left-hand lead with an energy 0.175 eV above the Fermi level. We plot the spin projection along the (a) $x$, (b) $y$, and (c) $z$ directions at $k_x=0$. (d), (e), and (f) are the corresponding plots for $k_x=0.032$ Å${}^{-1}$. Here red represents positive values, while blue stands for negative ones. Scattering at the step edge even at $k_x=0$ allows the spin to rotate out of the plane of the slab, resulting in finite $y$ and $z$ components, in contrast to the unperturbed bottom surface, where these are negligible. At non-normal incidence $(k_x=0.032$ Å${}^{-1})$ the $z$ component of the spin-resolved LDOS becomes finite, while the step edge introduces a nonzero $y$ component. Insets: Zooms around the step edge.

Figure 7

(a) Potential profile for the Dirac model. The parameters are fitted to the ab initio results as follows: $V_1=V_4=0,V_2=-1.17$ eV, $d=20$ Å, and $L=60$ Å. The transmission as a function of energy is shown for (b) $V_3=-0.02$ eV and (c) $V_3=0.0$ eV. Different curves correspond to different $k_x$ points, with the same definition as used in Fig. 2.

Figure 8

(a) The transmission across the double surface barrier at different $k_x$. Similarly to the single-barrier case, $T=2$ for $k_x=0$ but then reduces at other angles of incidence. Note also the Fabry-Perot type oscillations in transmission, which are not present in Fig. 2. Different curves correspond to different $k_x$ values, from $k_x=0$ up to $k_x=0.03475$ Å${}^{-1}$, in equal steps of 0.002 317 Å${}^{-1}$. Inset: The integrated transmission obtained with (black curve) and without [gray (red) curve] the barriers. (b) The transmission as a function of $k_x$ at different constant energy cuts. These energies are chosen away from the resonances.

Figure 9

DOS projected on the surface atoms along the double-barrier scattering region at (a) $k_x=0$, (b) at $k_x=0.032$ Å${}^{-1}$, and (c) integrated over all $k_x$. Note the absence of density oscillations for $k_x=0$ and for the integrated DOS. The incidence at finite $k_x$ leads to density oscillations clearly seen in the long flat region adjacent to the barrier as shown in (b). The panels at the right are the corresponding Fourier transforms, which are featureless in (a) and (c), while clearly showing the presence of scattering in (b).

Figure 10

PDOS on the surface atoms for a double barrier of length 149.16 Å at (a) $k_x=0$ and (b) $k_x=0.032$ Å${}^{-1}$. Note the absence of quantum well states in (a). In (b) quantum well states interact with the bound state at the two barriers, leading to energy splitting of the bound state.