Local versus extended deformed graphene geometries for valley filtering

The existence of two inequivalent valleys in the band structure of graphene has motivated the search of mechanisms that allow their separation and control for potential device applications. Among the several schemes proposed in the literature, strain-induced out-of-plane deformations (occurring naturally or intentionally designed in graphene samples), ranks among the best candidates to produce separation of valley currents. Because the valley filtering properties in these structures are, however, highly dependent on the type of deformation and setups considered, it is important to identify the relevant factors determining optimal operation and detection of valley currents. In this paper, we present a comprehensive comparison of two typical deformations commonly found in graphene samples: local centrosymmetric bubbles and extended folds/wrinkles. Using the Dirac model for graphene and the second-order Born approximation, we characterize the scattering properties of the bubble deformation, while numerical transmission matrix methods are used for the foldlike deformations. In both cases, we obtain the dependence of valley polarization on the geometrical parameters of deformations and discuss their possible experimental realizations. Our study reveals that extended deformations act as better valley filters in broader energy ranges and present more robust features against variations of geometrical parameters and incident current directions.

Figure 1

Schematics of the out-of-plane Gaussian bump (a) and Gaussian fold (b).

Figure 2

Strain induced pseudomagnetic field $\mathbf{B}$ and $\mathrm{\Phi }\left(\mathbf{r}\right)$ for bump and fold. $\mathbf{B}$ for bump (a), and fold (b). $\mathrm{\Phi }\left(\mathbf{r}\right)$ for bump (c) and fold (d). Parameters: $\eta =0.1$ and $b=15\mathrm{nm}$.

Figure 3

Polar plots of the differential cross section (in angstroms) for $K$ (red solid) and $K^{\prime }$ (blue dotted) valleys with different incident angles indicated by the arrows and the total cross section as a function of incident angles. [(a) and (b)] ${\theta }^{\prime }=0^{\circ }$, [(c) and (d)] ${30}^{\circ }$, and [(e) and (f)] ${60}^{\circ }$. (g) and (h) show the total cross section vs the incident angle ${\theta }^{\prime }$. The left column corresponds to energy $E=20\mathrm{meV}$, while the right column corresponds to energy $E=300\mathrm{meV}$. Other parameters: $b=15\mathrm{nm}$, $\eta =0.1$, and $E_b\approx 44\mathrm{meV}$.

Figure 4

Schematics of classical trajectories of electron motion in different magnetic field distributions. Top (bottom) panel corresponds to antisymmetric (symmetric) field distribution with respect to the incident direction, which is chosen as $\widehat{x}$. Left (right) column corresponds to $K$ ($K^{\prime }$) valley.

Figure 5

Polarization $P$ (black), ${\sigma }_{D}P$ for $K$ (red solid), and $K^{\prime }$ (blue dotted) valley corresponding to [(a)–(d)] in Fig. 3. The spikes in polarization plots correspond to directions with zero values of the second-order correction, around which an abrupt change of sign in the polarization occurs as the angle of incidence is changed.

Figure 6

${\sigma }_{\mathrm{tot}}$ and ${\sigma }_{\mathrm{trans}}$ vs incident energy $E$ for the incident angle (a) $0^{\circ }$ and (b) ${30}^{\circ }$. (c) Differential cross section for $0^{\circ }$ incidence with $E=80$ meV. (d) Differential cross section for ${30}^{\circ }$ incidence with $E=93$ meV. (e) Polarization $P$ and ${\sigma }_{D}P$ corresponding to data in (c). (f) Polarization $P$ and ${\sigma }_{D}P$ corresponding to the data in (d).

Figure 7

Polar plots of the differential cross section (Å) for contributions of scalar (pink dotted) and pseudovector (green solid) potentials for valley $K$ and incident angle ${\theta }^{\prime }=0^{\circ }$. (a) corresponds to energy $E=20\mathrm{meV}$. Inset shows a zoom in to visualize the magnitude of the pseudovector potential contribution. (b) corresponds to energy $E=300\mathrm{meV}$. (c) and (d): differential cross section per valley at low and high energies, respectively, for the total potential $V^{\tau }+\mathrm{\Phi }$. (e) and (f) show the corresponding polarization $P$ and ${\sigma }_{D}P$ for the two valleys. Other parameters: $b=15\mathrm{nm}$, $\eta =0.1$, and $g_s=3\mathrm{eV}$.

Figure 8

Transmission probability vs incident angle $\theta$ for $K$ (left column) and $K^{\prime }$ (right column) valleys. Results are shown as a function of energy $E$ in (a) and (b), as a function of the strain strength $\eta$ in (c) and (d), and as a function of the fold width $b$ in (e) and (f) for each valley, respectively.

Figure 9

Transmission probability vs incident angle $\theta$ for $K$ valley. Results are shown as a function of energy $E$. Panel (a): transmission for a single barrier potential. Panel (b): transmission for a double square barrier potential. Inset: solid (black) line: profile of $\mathbf{A}\left(y\right)$, dashed (magenta) line: profile of double square potential well. Parameters: $b=50\mathrm{nm}$, $\eta =0.1$.

Figure 10

Valley polarization $P$ (left column) and $T_{K}P$ (right column) vs incident angle $\theta$ and energy $E$, strain strength $\eta$, and fold width $b$.

Figure 11

Results for Gaussian fold with axis oriented at an angle $\gamma ={25}^{\circ }$ with respect to the zigzag direction. [(a) and (b)] Transmission $T$ for $K$ and $K^{\prime }$ valley, (c) polarization $P$, and (d) $TP$ for $K$ valley. Other parameters: $b=50\mathrm{nm}$ and $\eta =0.1$.

Figure 12

Transmission $T$ due to (a) only the scalar potential and (b) both scalar and vector potentials for $K$ valley. The scalar potential contribution is valley independent, while (b) is mirror symmetric with respect to $\theta =\pi /2$ for $K^{\prime }$ valley. (c) Valley polarization $P$ and (d) $TP$ for $K$ valley.

Figure 13

Scheme of valley filtering and detection of polarized beams with Gaussian folds (yellow areas). The local gates (grey areas) are employed to tune the Fermi levels so that the dependence of transmission probability on incident energy can be measured. In the middle trench setup (see text), multiple reflection and transmission events between the two folds can generate highly polarized electron beams at the end of the trench.

Figure 14

Scheme for discretization of the pseudomagnetic field into constant slices (black rectangles) and corresponding gauge field segments (red dashed lines). Thick black and blue curves represent continuous pseudomagnetic field $B_z$ and the gauge fields $A_x/b$, respectively.

Figure 15

Transmission probability vs $\hslash v_F{k}_x$ and energy $E$ for (A) $K$ and (B) $K^{\prime }$ valley. Parameters: $b=50\mathrm{nm}$ and $\eta =0.1$. The black lines correspond to the left ($K$) and right ($K^{\prime }$) boundaries of Eq. (32).

Figure 16

A Gaussian fold (yellow shaded area) along angle $\gamma$ with respect to the zigzag direction. The zigzag and armchair directions define the original $x_{0}o{y}_0$ reference frame. The rotated frame $xoy$ is defined with the $x$ axis lying along the fold axis.