Electron spin inversion in gated silicene nanoribbons

We study locally gated silicene nanoribbons as spin active devices and we solve the quantum scattering problem in the atomistic tight-binding formalism. Particular attention is paid to the low energy range for which only four subbands appear at the Fermi level. We find that the gated segments of zigzag nanoribbons can be used for spin inversion. The strong intrinsic spin-orbit coupling in the presence of an external vertical electric field provides a fast spin precession around the axis perpendicular to the silicene plane. The spin inversion length can be as small as 10 nm. On the other hand, in the armchair nanoribbons the spin inversion occurs via the Rashba effect which is weak and the spin inversion lengths are of the order of microns.

Figure 1

Sketch of the device. The Fermi level electrons propagate from the source (S) to the drain (D). The voltage $V_G$ applied to the top gate produces a perpendicular electric field $F_z$ on length $L$ [see Eq. (1)]. The role of the source (S) and drain (D) is played by homogeneous semi-infinite silicene ribbons outside the gated area.

Figure 2

Band structure of a zigzag silicene nanoribbon with 28 atoms across its width. Color in the upper (a)–(d) [lower (e)–(h)] row denote the spin $S_z$ ($S_x$) component in $\hslash /2$ units. An external magnetic field of $b_0=1\mu \mathrm{T}$ oriented in the $z$ and $x$ direction was applied: in (a), (b), (e), and (f) $\mathbf{B}=(0,0,b_0)$ and in (c), (d), (g), and (h) $\mathbf{B}=(b_0,0,0)$, respectively. Plots (a), (e), (c), and (g) represent band structures when no external electric field is applied while on the (b), (f), (d), and (h) the $F_z=100$ mV/Å. (i) Zoom of a band structure marked by rectangle in plot (d).

Figure 3

Spin $S_z$ component maps for a zigzag nanoribbon 28 atoms wide for the intrinsic SO coupling constant ${\lambda }_{\text{SO}}=0$ (a) and ${\lambda }_{\text{SO}}=3.9$ meV (b). In between dashed horizontal lines the external electric field $F_z=1$ V/Å is applied. The first propagating mode with initial spin up has been chosen at $E_F=258$ meV.

Figure 4

Band structure of an armchair silicene nanoribbon with 19 atoms width for electron with spin polarization in $z$ direction. Colors in upper (lower) row represent spin $S_z$ ($S_x$) component in $\hslash /2$ units. The left panels correspond to external electric field $F_z=0$ and the right panels to $F_z=1$ V/Å.

Figure 5

Total conductance $G$ (blue) and the spin-flipping $G_{ud}$ contribution (red) for an armchair silicene nanoribbon with the length of the gated area $L=842$ nm and the external electric field $F_z=1$ V/Å. The basis of spin states perpendicular to the ribbon is considered here. The black dots marked the peaks described in Table 1.

Figure 6

The lowest conduction subbands for zigzag nanoribbon and $F_z=100$ mV/Å. The color scale denotes the spin $S_z$ component in $\hslash /2$ units. The difference between two opposite-spin right-going subbands at Fermi energy $E_F=24.2$ meV is marked as $\mathrm{\Delta }k=1.02\frac{1}{2a}$.

Figure 7

Total conductance $G$ (blue), the spin flipping $G_{ud}$ (red), and spin conserving $G_{uu}$ (orange) contributions. Here $G_{ud}$ denotes the flip from $S_x=1$ to $S_x=-1$ in $\hslash /2$ units. The width of the zigzag ribbon was set to 28 atoms, the length is $L=5.5$ nm and $F_z=100$ mV/Å. The two peaks marked by black dots correspond to dashed lines in band structure in Fig. 6.

Figure 8

Total conductance $G$ (blue) and conductance with spin flip $G_{ud}$ (in $x$ direction) with variable length $L$ on which external electric field is applied. For (a) and (b) the $F_z=100$ mV/Å and (c) and (d) $F_z=1000$ mV/Å. In two cases (b) and (d) the ${\lambda }_{\text{SO}}$ was increased 10 times to measure the contribution of the intrinsic spin-orbit coupling to the offset arising. For black triangles in (a) the first mark from the left corresponds to the system described in the text, the middle and the last one to Figs. 9 and 9, respectively. The applied Fermi energies: $E_F=24.2$ meV (a), $E_F=38.386$ meV (b), $E_F=231.653$ meV (c), and $E_F=252.989$ meV (d).

Figure 9

$S_x$ spin projection maps for $L$ (a) 19.3 nm and (b) 21.6 nm [see the length marked by triangles in Fig. 8] for the solution of the scattering problem with spin-up electron incident from the side of negative $y$. (c) Zoom of the area where one period of the spin rotation is visualized. (d) Scattering density for the system shown in plot (b). The horizontal dashed lines limit the area of the external electric field $F_z=100$ mV/Å. The electron density of the input lead with $F_z=100$ mV/Å as above for two incoming modes $m_1$ (e) and $m_2$ (f).

Figure 10

Schematic view of a spin detector with FM placed above the left (a) and right (e) lead of silicene nanoribbon with gate voltage $F_z=100$ meV/Å. (b) and (d) Correspond to pristine silicene and (c) is a gated area where precession occurs ($L=5.5$ nm). Below each section (a)– (e) band structure of its infinite counterpart has been drawn. Horizontal line denotes the energy $E_F=24.2$ meV accordingly to the previous spin-flip length. $L_0$ is the spacer between the gated region and the leads.

Figure 11

Total conductance $G$ (light colored lines) as a function of the spacer length $L_0$ [see Figs. 10 and 10] compared to the $G_{ud}$ (dark blue) in nanoribbon without FM leads (see Fig. 7).

Figure 12

Conductance $G$ with spin conserving $G_{uu}$ and spin-flipping $G_{ud}$ parts in disordered system for a random fluctuation of the vertical field $U_z$ including the left edge (a), and with no fluctuation on the left edge (c). Map of on-site energies that correspond to $|V_z|\equiv e(F_z+U_z)\left|{\ell }_k\right|$ has been presented for each case (b) and (d), respectively. Magnitude of the fluctuating field $U_z$ in both cases is comparable to $F_z$ and corresponds to energy $\approx 20$ meV.