Diffraction catastrophes and semiclassical quantum mechanics for Veselago lensing in graphene

We study the effect of trigonal warping on the focusing of electrons by n-p junctions in graphene. We find that perfect focusing, which was predicted for massless Dirac fermions, is only preserved for one specific lattice orientation. In the general case, trigonal warping leads to the formation of cusp caustics, with a different position of the focus for graphene's two valleys. We develop a semiclassical theory to compute these positions and find very good agreement with tight-binding simulations. Considering the transmission as a function of potential strength, we find that trigonal warping splits the single Dirac peak into two distinct peaks, leading to valley polarization. We obtain the transmission curves from tight-binding simulations and find that they are in very good agreement with the results of a billiard model that incorporates trigonal warping. Furthermore, the positions of the transmission maxima and the scaling of the peak width are accurately predicted by our semiclassical theory. Our semiclassical analysis can easily be carried over to other Dirac materials, which generally have different Fermi surface distortions.

Figure 1

(a) Simulation setup with an injector and collector lead (red) and drain leads on each side (blue). (b) Classical trajectories for the massless Dirac Hamiltonian at $U_0=2E$. (c)–(g) Classical trajectories (red) and caustics (black) for the Hamiltonian including trigonal warping. Unless otherwise indicated, $E=0.4$ eV. (c) $K$ valley, $U_0=0.8$ eV, $\theta =0$; (d) $K^{\prime }$ valley, $U_0=0.8$ eV, $\theta =0$; (e) section of the butterfly caustic. $K^{\prime }$ valley, $E=0.6$ eV, $U_0=1.18$ eV, $\theta =0$; (f) $K^{\prime }$ valley, $U_0=0.795$ eV, $\theta =\pi /12$; (g) $U_0=0.8$ eV, $\theta =\pi /6$.

Figure 2

(a)–(c) Results of the tight-binding simulations with $L_1=100$ nm. The density $|{\mathrm{\Psi }}_{\text{av},\alpha }{|}^2$ is averaged over sublattices and summed over lead modes in valley $\alpha$. (a) $K^{\prime }$ valley, $E=0.6$ eV, $U_0=2E, W_i=7.5$ nm; (b) $K^{\prime }$ valley, $E=0.6$ eV, $U_0=1.18$ eV, $W_i=7.5$ nm; cf. the classical trajectories in Fig. 1; (c) $K^{\prime }$ valley, $E=0.4$ eV, $U_0=2E, W_i=40$ nm. (d)–(f) Position, on the $x$ axis, of the caustic (dashed and dashed-dotted lines), semiclassical maximum (solid lines), and simulated maximum (symbols) for varying energy $E$, lattice orientation $\theta$, and $L_1$. The dashed gray lines indicate the Dirac result. The parameters equal (e),(f) $E=0.4$ eV, (d),(f) $\theta =0$, (d),(e) $L_1=100$ nm, (d),(f) $W_i=40$ nm, and (e) $W_i=50$ nm. In all cases $U_0=2E$.

Figure 3

(a)–(e) Intravalley transmission as a function of potential strength obtained from Kwant (light colors, wavy lines) and a billiard model (darker colors, smooth lines). (a) $E=0.4$ eV, $L_1=L_2=100$ nm; (b) $E=0.4$ eV, $L_1=L_2=300$ nm; (c) $E=0.6$ eV, $L_1=L_2=200$ nm; (d) $E=0.4$ eV, $L_1=100$ nm, $L_2=300$ nm. (e) $K$ valley, $L_1=100$ nm, $L_2=140$ nm for various energies (in eV). For $E=0.24$ eV, ${\partial }^4{S}_{np}/\partial p_y^4$ vanishes at $U_0=0.5$ eV and $x=L_2$. (f) Valley polarization $P=(T_K-T_{K^{\prime }})/(T_K+T_{K^{\prime }})$ as a function of potential for $E=0.4$ eV and various $L_1=L_2$ (in nm). (g) Maximum of the simulated intensity as a function of energy for $L_1=L_2=200$ nm. We also plot the potentials at which $x_{\text{cusp},\alpha }$ and the semiclassical (SC) $x_{\text{max},\alpha }$ equal $L_2$. For all figures $W_i=W_c=50$ nm.

Figure 4

(a) Density $|{\mathrm{\Psi }}_{\text{av},K^{\prime }}{|}^2$ from tight-binding simulations for $E=0.4$ eV, $m_{\text{lead}}=0.375$ eV, and $W_i=40$ nm. (b) Fourier transform (for each lead mode) of the wave function from (a) at $x=-L_1$. (c) Symmetry breaking as a function of $L_2$ for various $m_{\text{lead}}$ (in eV). $K^{\prime }$ valley, $E=0.1$ eV, $W_i=50$ nm. For all figures $L_1=100$ nm and $U_0=2E$.