Enhanced asymmetric valley scattering by scalar fields in nonuniform out-of-plane deformations in graphene

We study the electron scattering produced by local out-of-plane strain deformations in the form of Gaussian bumps in graphene. Of special interest is taking into account the scalar field associated with the redistribution of charge due to deformations on the same footing as the pseudomagnetic field. Working with the Born approximation approach, we show analytically that even when a relatively small scalar field is considered, strong backscattering and enhancement of the valley-splitting effect could arise as a function of the energy and angle of incidence. In addition, we find that the valley polarization can reverse its sign as the incident energy is increased. These behaviors are totally absent if the scalar field is neglected or screened. Interestingly, we find that there is a further possibility of controlling the valley scattering polarization purely by electrical means through the presence of external scalar fields in combination with strain fields. These results are supported by quantum dynamical simulations of electron wave packets. Results for the average trajectories of wave packets in locally strained graphene clearly show focusing and beam-splitting effects enhanced by the presence of the scalar field that could be of interest in the implementation of valleytronic devices.

Figure 1

(a) Pseudomagnetic field and (b) scalar field at valley $K$ produced by a Gaussian bump of height $h_o=10\text{nm}$, width $b_o=50\text{nm}$, and scalar field coupling constant $g=3\text{eV}$.

Figure 2

Differential scattering cross section ${\sigma }_D^{\pm }$ for valley $K$ (blue) and $K^{\prime }$ (orange) as a function of the outgoing angle ${\theta }_2$ for horizontal incidence (first row) and vertical incidence (second row); in the left column $g=0$, and for the right column $g=3\text{eV}$.

Figure 3

Total scattering cross section ${\sigma }_T$ as a function of the incident angle ${\theta }_1$ for (a) $g=0$ and $g=3$ eV for valleys (b) $K$ and (c) $K^{\prime }$.

Figure 4

Valley polarization efficiency $\mathcal{P}$ as a function of the incident angle ${\theta }_1$ for incident energy (a) $E=15$ meV and (b) $E=60$ meV. (c) $\mathcal{P}$ as a function of the incident energy for different incident angles ${\theta }_1$. We take $g=3$ eV, except for the dashed line, for which $g=0$ eV.

Figure 5

Probability density ${\left|\mathrm{\Psi }(x,y)\right|}^2$ of an incident wave packet coming from the left $[(x_0,y_0)=(-150,0)\text{nm}]$ with $E=110\text{meV}$ at time $t=350\text{fs}$ with $g=0$ (first row) and $g=3$ eV (second row). Different columns correspond to different valleys: (a) and (c) correspond to valley $K$, and (b) and (d) correspond to valley $K^{\prime }$.

Figure 6

Trajectories of $〈\mathit{r}〉$ (black) of an incident wave packet coming from the left with $E=110\text{meV}$ for different impact parameters plotted on top of the pseudomagnetic field profile for valleys $K$ (left column) and $K^{\prime }$ (right column) with (a) and (b) $g=0$ and (c) and (d) $g=3\text{eV}$.

Figure 7

Trajectories of $〈\mathit{r}〉$ (black) of an incident wave packet coming from the bottom with $E=110\text{meV}$ for different impact parameters plotted on top of the pseudomagnetic field profile for valleys $K$ (left column) and $K^{\prime }$ (right column) with (a) and (b) $g=0$ and (c) and (d) $g=3\text{eV}$.

Figure 8

Relation between the deflection angle and the impact parameter $Y_0$ ($X_0$) for horizontal (vertical) incidence for valley $K$ (blue curve) and valley $K^{\prime }$ (yellow curve). (a) and (c) Cases without scalar field and (b) and (d) cases with finite scalar field.