Valley filtering in 8-$Pmmn$ borophene based on an electrostatic waveguide constriction

Materials with tilted Dirac cones, such as $8\text{−}Pmmn$ borophene, are being explored for valleytronic applications as the tilting direction is different for nonequivalent valleys. In this paper, a valley-filtering device based on electrostatic waveguides is proposed. First, these waveguides are examined from a theoretical point of view. An inner product is defined starting from the probability current density along the waveguide axis. It is shown that the bound modes with real eigenvalues are mutually orthogonal and orthogonal with respect to all radiating modes. In a next step, by exploiting these orthogonality properties, a simulation procedure is introduced based on an explicit, symplectic partitioned Runge-Kutta time-stepping method specifically adapted for this problem. Finally, this approach is applied to the situation of a waveguide nanoconstriction and it is demonstrated that this structure can function as a valley filter. Within a certain window in the energy domain, transmission is practically zero for one valley, while being almost perfect for the other one. The effect of several design variables, such as length and width of the constriction, is carefully investigated. Moreover, the effect of misalignment between the tilting direction and the waveguide axis is assessed, showing that the proposed valley-filtering design is robust against deviations up to several tens of degrees. In addition, the simulation results reveal that the dispersion relation of the waveguide modes is not necessarily monotonic, which can give rise to oscillations in the transmission function due to interference effects.

Figure 1

Plot of the two nonequivalent Dirac cones of $8\text{−}Pmmn$ borophene for $V=0$. The left (right) plot relates to the valley index $\tau$ equal to 1 ($-1$).

Figure 2

(a) Eigenvalues of (3) distributed in the complex plane for valley ${\mathit{k}}_D$. The hyperbolic secant potential (14), with $V_0=0.45\mathrm{eV}$ and constant $W=10\mathrm{nm}$, was chosen together with an energy $E$ equal to $-0.02\mathrm{eV}$. (b) Plot of the real part of $u$ as a function of the position $x$ for the bound eigenmode associated with $k_y=-1.54$ ${\mathrm{nm}}^{-1}$ [indicated by a large blue cross in (a)]. (c) Plot of the real part of $u$ as a function of the position $x$ for the radiating eigenmode associated with $k_y=(0.023+0.096i)$ ${\mathrm{nm}}^{-1}$ [indicated by a large red dot in (a)].

Figure 3

Illustration of the simulation setup of an electrostatic waveguide with nanoconstriction. The middle region (black) represents a two-dimensional plot of the potential profile. The simulation domain is terminated by an absorbing layer (gray region). The constriction is of length $L$. The source is indicated by a red line, while the blue dashed line represents the observer.

Figure 4

(a), (b) Plots of the eigenvalues $k_y$ of the bound modes versus energy $E$ for the hyperbolic secant potential (14) with $W=10\mathrm{nm}$ and $V_0=0.45\mathrm{eV}$. (a) Gray dots represent the eigenvalues corresponding to valley ${\mathit{k}}_D$, while in (b) the eigenvalues associated with valley $-{\mathit{k}}_D$ are displayed. (c), (d) Plots of the eigenvalue $k_y$ of the lowest-order, forward-propagating bound mode as a function of energy $E$ for a set of values of the full width at half maximum $W$ for the valleys ${\mathit{k}}_D$ and $-{\mathit{k}}_D$, respectively. For (a)–(d) the dispersion relation in the case of zero potential is given by the full black line. The dashed black line is the dispersion relation in the case of a constant potential $-V_0$. The eigenvalues $k_y$ were calculated by means of the numerical procedure outlined in Sec. 3.

Figure 5

(a)–(f) Snapshots of the probability density, demonstrating the valley-filtering effect of a nanoconstriction. Plots (a)–(c) are attributed to valley ${\mathit{k}}_D$ and (d)–(f) to $-{\mathit{k}}_D$. The potential profile corresponds to (14) with $V_0=0.45\mathrm{eV}$ and with $W$ varying with the position $y$ according to (15). The full width at half maximum of the broad part of the potential $W_0$ is $10\mathrm{nm}$, while the full width at half maximum of the narrow section $W_1$ equals $4\mathrm{nm}$. The length of the narrow section $L$ is $50\mathrm{nm}$ and the length of the transition region is characterized by $L_T=5\mathrm{nm}$. The spectral content of the incoming mode is centered around $E=-0.06\mathrm{eV}$.

Figure 6

Transmission through the nanoconstriction as a function of the energy $E$ for various design parameters. The transmission for the valleys ${\mathit{k}}_D$ and $-{\mathit{k}}_D$ is depicted in (a) for $V_0=0.45\mathrm{eV}$, $W_0=10\mathrm{nm}$, $W_1=4\mathrm{nm}$, $L=50\mathrm{nm}$, and $L_T=5\mathrm{nm}$. The effect of the width of the narrow section $W_1$, the length of the narrow section $L$, and the length of the transition region are illustrated by (b), (c), and (d), respectively. In (b)–(d) the same parameters as in (a) are employed except for the one that is varied. Only valley ${\mathit{k}}_D$ is considered in (b)–(d).

Figure 7

Effect of the parameter $L_T$ on the transmission through the nanoconstriction as a function of the energy $E$ for the valley ${\mathit{k}}_D$. The employed parameters are $V_0=0.45\mathrm{eV}$, $W_0=10\mathrm{nm}$, $W_1=4\mathrm{nm}$, and $L=100\mathrm{nm}$.

Figure 8

(a) Detail of the dispersion relation of the lowest-order forward-propagating bound mode in a hyperbolic secant potential (14), with parameters $W=4\mathrm{nm}$ and $V_0=0.45\mathrm{eV}$. The eigenvalues $k_y$ were calculated by means of the numerical procedure outlined in Sec. 3. The considered valley is ${\mathit{k}}_D$. The two eigenmodes indicated with a red and blue dot correspond to an energy $E$ of $-0.034\mathrm{eV}$. They are plotted as a function of the position $x$ in (b) and (c), respectively. (b) corresponds to $k_y=0.142$ ${\mathrm{nm}}^{-1}$, while $k_y$ is equal to 0.200 ${\mathrm{nm}}^{-1}$ for the eigenmode presented in (c).

Figure 9

Effect of the parameter $\theta$, the angle between the axis of the waveguide and the tilting direction, on the transmission through the nanoconstriction as a function of the energy $E$ for valley ${\mathit{k}}_D$ (a) and valley $-{\mathit{k}}_D$ (b). The employed parameters are $V_0=0.45\mathrm{eV}$, $W_0=10\mathrm{nm}$, $W_1=4\mathrm{nm}$, and $L=50\mathrm{nm}$.

Figure 10

Energy value $E_{T=0.5}$ for which the transmission function equals 0.5 as a function of the mismatch angle $\theta$ for both valleys ${\mathit{k}}_D$ and $-{\mathit{k}}_D$. The employed parameters are $V_0=0.45\mathrm{eV}$, $W_0=10\mathrm{nm}$, $W_1=4\mathrm{nm}$, and $L=50\mathrm{nm}$.