Interplay between Rashba interaction and electromagnetic field in the edge states of a two-dimensional topological insulator

The effects of Rashba interaction and electromagnetic field on the edge states of a two-dimensional topological insulator are investigated in a nonperturbative way. We show that the electron dynamics is equivalent to a problem of massless Dirac fermions propagating with an inhomogeneous velocity, enhanced by the Rashba profile with respect to the bare Fermi value $v_F$. Despite the inelastic and time-reversal breaking processes induced by the electromagnetic field, no backscattering occurs without interaction. The photoexcited electron densities are explicitly obtained in terms of the electric field and the Rashba interaction, and are shown to fulfill generalized chiral anomaly equations. The case of a Gaussian electromagnetic pulse is analyzed in detail. When the photoexcitation occurs far from the Rashba region, the latter effectively acts as a “superluminal gate” boosting the photoexcited wave packet outside the light-cone determined by $v_F$. In contrast, for an electric pulse overlapping the Rashba region, the emerging wave packets are squeezed in a manner that depends on the overlap area. The electron-electron interaction effects are also discussed, for both intraspin and interspin density-density coupling. The results suggest that Rashba interaction, often considered as an unwanted disorder effect, may be exploited to tailor the shape and the propagation time of photoexcited spin-polarized wave packets.

Figure 1

The inhomogeneous profile (24) modeling a Rashba interaction region. The limit $W_R\ll l_R$ corresponds to a Rashba impurity, while the limit $l_R\ll W_R$ to a Rashba “barrier.” The insets represent a sketch of the local band dispersion $E=E\left(k\right)$ in the various regions. The Rashba interaction always leads to an increase of the velocity as compared to the bare Fermi velocity $v_F$ of the edge states [see Eq. (13)].

Figure 2

The behavior of the wave function of a right-moving electron approaching a Rashba interaction region, sketched by the grey area in suitably normalized units and modeled by the interaction profile (24), is shown at the electron energy $E=20\mathrm{meV}$. The real part of the spin-$\uparrow$ component ${\phi }_{E\uparrow }^{(+)}\left(x\right)$ [see Eq. (19)] (thin solid black curve), the spin-$\downarrow$ component ${\phi }_{E\downarrow }^{(+)}\left(x\right)$ [see Eq. (21)] (thick solid red curve) are shown, whereas the dashed curves represent the related absolute values. (a) The case of a Rashba impurity ($v_F=5\times {10}^5\mathrm{m}/\mathrm{s},g_R=10,x_R=0$ and $l_R=200\mathrm{nm},W_R=20\mathrm{nm}$). (b) The case of a Rashba barrier ($v_F=5\times {10}^5\mathrm{m}/\mathrm{s},g_R=10,x_R=0$ and $l_R=20\mathrm{nm},W_R=500\mathrm{nm}$). While away from the Rashba region a right-moving electron identifies a purely spin-$\uparrow$ component, when the Rashba region is approached a spin-$\downarrow$ component emerges over a length scale dependent on the smoothening length scale $l_R$ of the Rashba profile. Furthermore, the spatial period of the wave function increases as a result of the velocity enhancement (13) due to the Rashba interaction. The extremal points of the spin-$\uparrow$ component correspond to zeros of the spin-$\downarrow$ component.

Figure 3

The case of photoexcitation occurring far from the Rashba region. A Gaussian electric pulse (45) with $\mathrm{\Delta }=50\mathrm{nm}$ and $\tau =10\mathrm{fs}$ (spatial profile sketched by the dotted curve in suitable units) is applied far away from a Rashba interaction region (sketched by the grey area in suitable units) characterized by parameters $W_R=500\mathrm{nm},x_R=500\mathrm{nm},g_R=10$, and $l_R\to 0$ in (24). The photoexcited electron density $\mathrm{\Delta }n$ [see Eq. (44)] is shown at various snapshots: $t=0.05$ (thin solid curve), 0.5 (solid curve), 0.66 (dashed curve), and $1.0\mathrm{ps}$ (dash-dotted curve). Due to the increase of velocity (13) ascribed to the Rashba interaction, the right-moving wave packet is “boosted” by the Rashba barrier, as compared to its left-moving photoexcited partner that does not impact the barrier. The Rashba region effectively acts as a “superluminal gate” that enables the wave packet to access space-time regions beyond the light cone characterized by the bare Fermi velocity $v_F$.

Figure 4

The case of electron photoexcitation occurring inside the Rashba region. A Gaussian electric pulse (45) with $\mathrm{\Delta }=50\mathrm{nm}$ and $\tau =10\mathrm{fs}$ (spatial profile sketched by the dotted curve in suitable units) is applied inside a Rashba barrier with parameters $W_R=500\mathrm{nm},x_R=0,g_R=10$, and $l_R\to 0$ in (24) (grey area, normalized units). The photoexcited electron density $\mathrm{\Delta }n$ [see Eq. (44)] is shown at various snapshots: $t=0.01$ (thin solid curve), 0.04 (solid curve), 0.3 (dashed curve), and $1.0\mathrm{ps}$ (dash-dotted curve). When the wave packet emerges from the Rashba interaction region where photoexcitation occurs, it experiences a decrease of velocity from the value (13) to the bare Fermi velocity $v_F$, and its profile gets squeezed.

Figure 5

The case of photoexcitation partially overlapping with the Rashba region. A Gaussian electric pulse (45) with $\mathrm{\Delta }=100\mathrm{nm}$ and $\tau =10\mathrm{fs}$ (spatial profile sketched by the dotted curve in suitable units) is applied in the presence of a Rashba interaction barrier with parameters $W_R=100\mathrm{nm},x_R=-80\mathrm{nm},g_R=10$, and $l_R\to 0$ in (24) (grey area, normalized units). The photoexcited electron density $\mathrm{\Delta }n$ [see Eq. (44)] is shown at various snapshots: $t=0.01$ (thin solid curve), 0.3 (solid curve), 1.0 (dashed curve), and $1.5\mathrm{ps}$ (dash-dotted curve). The wave packets emerging from the Rashba region exhibit a fuzzy profile where two different Gaussian-like profiles coexist. The sharp peak stems from the photoexcitation occurring inside the Rashba barrier, whereas the broader one originates from the photoexcitation processes outside the Rashba barrier.

Figure 6

Possible realisation scheme of a setup for the photoexcitation in the presence of Rashba interaction. A reentrance etched on one edge of the quantum well in the topological phase and a dc voltage $V_g$ applied to a wedge gate enable one to control the Rashba interaction. Electromagnetic pulses $V\left(t\right)$ applied to finger gates and/or the wedge gate photoexcite electrons along the edge states. The different scenarios illustrated in Figs. 3, 4, and 5 can be realized, depending on the choice of the ac biased gate, the strength of the Rashba interaction, the size of the finger gates and the duration of the applied pulse. The shape of the spin-polarized photoexcited wave packets and their propagation time scales can thus be controlled, as discussed in the text.