Symmetry breaking and (pseudo)spin polarization in Veselago lenses for massless Dirac fermions

We study Veselago lensing of massless Dirac fermions by n-p junctions for electron sources with a certain polarization. This polarization corresponds to pseudospin for graphene and to real spin for topological insulators. Both for a point source and for injection into a sample through a narrow lead, we find that polarization leads to spatial symmetry breaking. For the Green's function, this results in a vertical displacement, or even complete vanishing of the main focus, depending on the exact polarization. For injection through a lead, it leads to a difference between the amounts of current emitted with positive and negative transversal momenta. We study both systems in detail using the semiclassical approximation. By comparing the results to the exact solutions, we establish that semiclassical methods provide a very effective way to study these systems. For the Green's function, we derive an easy-to-use analytical formula for the vertical displacement of the main focus. For current injection through a lead, we use semiclassical methods to identify two different scattering regimes.

Figure 1

The classical trajectories (red lines) for massless Dirac fermions that are emitted by a point source and are incident on an n-p junction at $x=0$ (dashed gray line). We see that the junction focusses the particles. The solid black line indicates the caustic, which is the envelope of the classical trajectories, and separates the region where each point lies on a single trajectory from the region where each point lies on three trajectories. It consists of twofold lines meeting into a cusp point at $(x_{\text{cusp}},0)$. (a) For $U_0>2E$, the cusp point $x_{\text{cusp}}>-x_s$ is the left-most point of the caustic. (b) When $U_0<2E$, the cusp point $x_{\text{cusp}}<-x_s$ is the right-most point of the caustic. (c) For $U_0=2E$, all trajectories are focused into a single point.

Figure 2

The density $\parallel \mathrm{\Psi }\parallel$ computed by numerically evaluating the exact wave function (16) for the dimensionless parameters $U_0=2.5$ and $h=0.0639$. For graphene, these numbers correspond to $E=100$ meV, $U_0=250$ meV, and $L=100$ nm. We consider three different polarizations. (a) For $({\alpha }_1,{\alpha }_2)=(1,1)/\sqrt{2}$, the density is symmetric about the $x$ axis. (b) When $({\alpha }_1,{\alpha }_2)=(1,0)$, we see that the symmetry is broken and that the maximum lies at $y<0$. (c) For $({\alpha }_1,{\alpha }_2)=(1,-1)/\sqrt{2}$, the density is symmetric again, but the central resonance has disappeared completely. The maximum of the color scale equals (a) 70, (b) 55, and (c) 22.

Figure 3

Sections of the norm $\parallel \mathrm{\Psi }\parallel$ of the exact wave function (16) on a line through the cusp point and parallel to the $y$ axis. The dimensionless parameters are equal to $U_0=2.5$ and $h=0.0639$. We clearly see that when the polarization ${\alpha }_2/{\alpha }_1$ decreases, the maximum is shifted to the left, while its size decreases. For ${\alpha }_2/{\alpha }_1=-1$, the wave function attains its minimum at the cusp point and the main focus has disappeared completely.

Figure 4

Comparison of different approximation schemes for the wave function near a caustic. For all figures, the dimensionless potential $U_0=2.5$, and the polarization $({\alpha }_1,{\alpha }_2)=(1,1)/\sqrt{2}$. The dimensionless parameter is different for each of the three rows, namely (a) $h=0.0639$, (b) $h=0.00639$, (c) $h=0.000639$. For graphene, these numbers correspond to $E=100$ meV, $U_0=250$ meV, and length scales of (a) $L={10}^2$ nm, (b) $L={10}^3$ nm, and (c) $L={10}^4$ nm. In the left column, we show a comparison along the $x$ axis, in which the position of the cusp is indicated by a vertical dashed gray line. Although the Pearcey approximation (28) typically gives too large values for the wave function, especially for large $h$, it correctly predicts the position of the maximum for all three values of $h$. In the middle column, we show a comparison along the line that is parallel to the $y$ axis and passes through the cusp point. The cusp point is again indicated by a vertical dashed gray line. In the right column, we show a comparison along a line perpendicular to one of the points on the fold line. The fold point is indicated by a vertical dashed gray line. In all three cases, the Airy approximation works rather well for a large range of values, whereas the WKB approximation only works far away from the fold point. In all of the plots, we have not only indicated the dimensionless coordinates but also the distance from the caustic in the relevant power of $h$, which is $h^{7/8}$ for the cusp caustic and $h^{5/6}$ for the fold caustic (see also Appendix pp2).

Figure 5

Comparison of different approximation schemes for the wave function on a line through the cusp point and parallel to the $y$ axis. The dimensionless semiclassical parameter is equal to (a) $h=0.0639$ and (b) $h=0.000639$, with $U_0=2.5$ for all figures. Left, middle, and right panels correspond to the polarizations $({\alpha }_1,{\alpha }_2)=(1,1)/\sqrt{2}, ({\alpha }_1,{\alpha }_2)=(1,0)$ and $({\alpha }_1,{\alpha }_2)=(1,-1)/\sqrt{2}$, respectively. In the left and middle panels, we clearly see that the Pearcey approximation ${\mathrm{\Psi }}_{c0}\left(\mathbf{x}\right)+{\mathrm{\Psi }}_{c1}\left(\mathbf{x}\right)$ correctly predicts the positions of the maxima. In the right panels, the Pearcey approximation ${\mathrm{\Psi }}_{c0}\left(\mathbf{x}\right)+{\mathrm{\Psi }}_{c1}\left(\mathbf{x}\right)+{\mathrm{\Psi }}_{c2}\left(\mathbf{x}\right)$ correctly predicts the position of the maximum and adequately reproduces the value of $\parallel \mathrm{\Psi }\parallel$ at $y=0$.

Figure 6

The density $\parallel \mathrm{\Psi }\parallel$ obtained from the uniform approximation (B36) for the dimensionless parameters $U_0=2.5$ and $h=0.0639$. We consider three different polarizations $({\alpha }_1,{\alpha }_2)$, namely (a) $(1,1)/\sqrt{2}$, (b) $(1,0)$, and (c) $(1,-1)/\sqrt{2}$. In all cases, the exact result shown in Fig. 2 is accurately reproduced. As in Fig. 2, the maximum of the color scale equals (a) 70, (b) 55, and (c) 22.

Figure 7

The density $\parallel \mathrm{\Psi }\parallel$ for the dimensionless parameters $U_0=2.5$ and $h=0.000639$. For graphene, these numbers correspond to $E=100$ meV, $U_0=250$ meV, and $L={10}^4$ nm. (a) The exact result obtained by numerically evaluating the exact wave function (16). (b) The result of combining the Pearcey approximation, the Airy approximation, and the WKB approximation. (c) The region in which each of the approximations is used. The Pearcey approximation is used inside the green ellipse. Between the two (dashed) purple lines the Airy approximation is used, and outside both these regions the WKB approximation is used. The black (dotted) line represents the caustic. The left, middle, and right panels correspond to three different polarizations $({\alpha }_1,{\alpha }_2)$, to wit $(1,1)/\sqrt{2}$; $(1,0)$ and $(1,-1)/\sqrt{2}$. The maximum of the color scale equals $30\times {10}^4$ for the left column, $23\times {10}^4$ for the middle column, and $5\times {10}^4$ for the right column.

Figure 8

The dependence of the position $y_{\text{max}}$ of the maximum on various parameters. We compare the numerically obtained maxima for the exact wave function (16) and the Pearcey approximation ${\mathrm{\Psi }}_{c0}\left(\mathbf{x}\right)+{\mathrm{\Psi }}_{c1}\left(\mathbf{x}\right)$ with the result (39) (labeled linear) and the solution of the third order equation (37) (labeled cubic). We consider graphene, where $\hslash v_F=3t{a}_{CC}/2$. (a) $E=100$ meV, $U_0=2.5E$, and ${\alpha }_2/{\alpha }_1=1$. The maximum only weakly depends on the length $L$. (b) The relative position $y_{\text{max}}/w$, where $w$ is the FWHM of ${\parallel \mathrm{\Psi }\parallel }^2$, approximately scales as $L^{-1/4}$. (c) $L=100$ nm, $U_0=2.5E$, and ${\alpha }_2/{\alpha }_1=1$. The position of the maximum $y_{\text{max}}$ approximately scales as $1/E$. (d) The relative position $y_{\text{max}}/w$ scales as $E^{-1/4}$ to a good approximation. (e) The dependence of $y_{\text{max}}$ on $U_0$ for ${\alpha }_2/{\alpha }_1=1, E=100$ meV, and $L=100$ nm. (f) The same as (e) for $L={10}^3$ nm. (g) The same as (e) for $L={10}^4$ nm. In all cases there is a local maximum at $U_0=2E$. The accuracy of the Pearcey approximation and the solution to the cubic equation improve as $L$ increases. Our result (39) generally performs well. (h) The dependence of the relative position $y_{\text{max}}/w$ on $U_0$ for $L=100$ nm. (i) The dependence of $y_{\text{max}}$ on the polarization for $E=100$ meV, $U_0=2.5E$, and $L=100$ nm. All results for $y_{\text{max}}$ coincide when ${\alpha }_2/{\alpha }_1⪆-0.4$. For smaller polarizations, only the Pearcey approximation stays close to the exact solution. (j) The dependence of $y_{\text{max}}$ on the position $x$ for $E=100$ meV, $U_0=2.5E, L=100$ nm, and ${\alpha }_2/{\alpha }_1=1$. The result is roughly constant, with the two lines intersecting close to $x_{\text{cusp}}$. (k) The relative position $y_{\text{max}}/w$ as a function of $x$.

Figure 9

The norm of the Fourier transform ${\overline{\mathrm{\Psi }}}_n^{\text{lead}}$ for various modes in a graphene lead of width 30 nm. (a) The energy $E=100$ meV and $\hslash v_F/\left(E_{0}w\right)=0.213$. There is only one mode in the lead, with a broad Fourier transform. (b) $E=400$ meV and $\hslash v_F/\left(E_{0}w\right)=0.0533$. There are six modes in the lead, which have a rather narrow Fourier transform. One can speak about characteristic momenta.

Figure 10

The norm $\parallel {\mathrm{\Psi }}_n\parallel$ of the wave function that results from current entering through a lead of width $w=30$ nm on the left side of a sample with $L=100$ nm. The n-p junction is located at $x=0$. (a) For $E=100$ meV, there is only one mode in the lead and the trajectories exit the lead at all angles. Since $U_0=200$ meV, they are focused in a single point, which makes the situation qualitatively similar to the one for the Green's function. (b) For $E=100$ meV and $U_0=250$ meV, we see a caustic. (c) When $E=250$ meV and $U_0=625$ meV, the interference pattern that results from the first mode in the lead is much less pronounced, and we are moving closer towards the deep semiclassical limit. (d) For $E=400$ meV, the lead contains six modes, each of which has a rather sharp Fourier transform and gives rise to a well-defined transversal momentum. Depicted here is the wave function that results from the first mode for $U_0=800$ meV. (e) We clearly see that for $E=400$ meV, the second mode carries a well-defined transversal momentum. For $U=1000$ meV, we see almost no interference pattern, implying that we are close to the deep semiclassical limit. (f) The total density ${\parallel \mathrm{\Psi }\parallel }_{\text{tot}}^2={\sum }_n{\parallel {\mathrm{\Psi }}_n\parallel }^2$, for $E=400$ meV and $U_0=1000$ meV. We see that we have a sharp focusing spot. The maximum of the color scale equals (a), (b) 0.18, (c), (d), (e) 0.25, (f) 0.135.

Figure 11

Comparison of the solution (42) with the stationary phase approximation (56). We consider ${\parallel \mathrm{\Psi }\parallel }_{\text{tot}}^2$, meaning that we sum over all modes in the lead. The lead width is $w=30$ nm and $L=100$ nm; the electron energy $E=400$ meV. (a) Comparison along the $x$ axis for $U_0=800$ meV. (b) Comparison along the line $x=L$ for the same potential. Both sections show quite good agreement. (c) Comparison along the $x$ axis for $U_0=1000$ meV. (d) Comparison along the line $x=190$ nm for the same potential. The agreement is not as good as for $U_0=2E$, but the stationary phase approximation still captures the essential features of the wave function.