## Abstract

The focusing of electric current by a single *p-n* junction in graphene is theoretically predicted. Precise focusing may be achieved by fine-tuning the densities of carriers on the *n*- and *p*-sides of the junction to equal values. This finding may be useful for the engineering of electronic lenses and focused beam splitters using gate-controlled *n-p-n* junctions in graphene-based transistors.

There are many similarities between optics and electronics. Rays in geometrical optics are analogous to classical trajectories of electrons, whereas electron de Broglie waves can interfere. The electron microscope is one example of the technological implementation of this similarity. The analogy with optics may also hold considerable potential for semiconductor electronics. In optics, transparent interfaces between materials are used in lenses and prisms to manipulate light beams. So far, interfaces have played a rather different role in semiconductors, where the central place was, for a long time, occupied by the *p-n* junction (PNJ). As a result of a depletion region near the contact between two semiconductors with different types of charge carriers (and a large energy gap), conventional PNJs are not suitable for precision manipulation of electron beams, which, if realized, may lead to a new functionality in microelectronics. From this perspective, a lot of promise is offered by the recently discovered (*1*) two-dimensional (2D) gapless semiconductor, graphene (*2*). Fine-tuning of the carrier density in graphene by means of electrical gates (*3*–*5*) or doping of the underlying substrate (*6*) was demonstrated, thus paving the way toward controllable ballistic PNJs. On the one hand, the PNJ in graphene is highly transparent for the charge carriers (*7*, *8*). On the other, the transmission of electrons through the *p-n* interface may resemble optical refraction (*9*) at the surface of metamaterials with negative refractive index (*10*–*12*): The straight interface is able to focus electric current, whereas a ballistic stripe of *p-*type graphene separating two *n-*type regions acts as a lens.

A feature of the band structure of graphene [monolayer of graphite (*2*, *13*)] is that its valence band (π) and conduction band (π*) touch each other. In the absence of doping, the Fermi level in graphene is at the energy that belongs to both bands and corresponds to the Bloch states in the corners of the hexagonal Brillouin zone of this 2D honeycomb crystal. For the states with a small quasi-momentum ħ**k** counted from the corresponding corner of the Brillouin zone, the dispersion ϵ(**k**) and group velocity **V** = *d*ϵ/*d*(ħ**k**) of electrons are given by ϵ_{c}(**k**) = ħ*vk*, **V**_{c} = *v***k**/*k* in conduction band and ϵ_{v}(**k**) = –ħ*vk*, **V**_{v} = –*v***k**/*k* in valence band (ħ is Planck's constant, *h/*2π). Figure 1 illustrates such a dispersion for electrons in *n-*type graphene (on the left) and *p-*type graphene (on the right). In a split-gate structure (Fig. 1), voltages ±*U* applied to the two gates shift the degeneracy point of the electron dispersion cones down by ħ*vk*_{c} on the left and up by ħ*vk*_{v} on the right, thus forming a PNJ separating the *n-*region with the density of electrons ρ_{e} = *k*^{2}_{c}/π and the *p-*region with the density of holes ρ_{h} = *k*^{2}_{v}/π. Here, *k*_{c(v)} is the radius of the Fermi circle in the conduction (valence) band.

The transmission of charge through the PNJ bears resemblance to the refraction of light by left-handed metamaterials (*10*–*12*) with refractive index equal to –1. As a wave enters such a material, the relative direction of its group velocity **V** and the wave vector **k** of the wave reverses from parallel (in vacuum) to antiparallel. Therefore, upon refraction, the sign of the tangential velocity component of the propagating wave inverts, and the normal component remains the same. As a result, rays that diverge in vacuum become convergent after entering the metamaterial (*9*). For electrons in the PNJ, the Fermi momentum *k*_{c(v)} plays the same role as the refractive index in geometrical optics, with the sign determined by the type of band—positive for the conduction band and negative for the valence band.

Indeed, let us consider, in Figs. 2A and 3A, a de Broglie wave of an electron approaching the PNJ from the *n-*side with the velocity **V** = (*v*cosθ_{c},*v*sinθ_{c}) and the wave vector **k** = (*k*_{c}cosθ_{c},*k*_{c}sinθ_{c}). At the interface, this wave is partly reflected to the state with the wave vector **k**′ =(–*k*_{c}cosθ_{c},*k*_{c}sinθ_{c}) and partly transmitted to the valence band state with the velocity **V** = (*v*cosθ_{v},*v*sinθ_{v}) and the wave vector **k** = (–*k*_{v}cosθ_{v}, –*k*_{v}sinθ_{v}) on the *p-*side. For a sharp PNJ, the probability of the transmission is cos^{2}θ_{c}/cos^{2}(½θ_{c} + ½θ_{v}) (*7*, *8*). Because the component of the electron momentum along a straight interface should be conserved, *k*_{c}sinθ_{c} = –*k*_{v}sinθ_{v}, which determines Snell's law for transmitted electrons. (1)

The negative sign of *n* in Eq. 1 implies that the PNJ transforms a divergent flow of electrons emitted by a source on the *n-*side into a convergent flow on the *p-*side. This results in focusing (Fig. 2A) for a symmetric junction, ρ_{h} = ρ_{e} corresponding to *n* = –1. Under the latter condition, electrons injected at (–*a*,0) in the *n-*region at the Fermi energy meet again in a symmetric spot at (*a*,0).

In an asymmetric junction (Fig. 3A) for, e.g., *n* = –0.82, which corresponds to ρ_{h}/ρ_{e} =0.67, a sharp focus transforms into a pair of caustics that coalesce in a cusp—a singularity in the density of classical trajectories. Similar singularities in ray and wave optics were investigated and classified (*14*) with general catastrophe theory (*15*). Ballistic trajectories of electrons in the *p-*region of an asymmetric PNJ are rays *y* = *a*tanθ_{c} + *x*tanθ_{v}, where θ_{v} is related to θ_{c} by Eq. 1. The condition for a singularity, δ*y*/δθ_{c} = 0, determines the form of caustics *y*_{caust} (*x*) as well as the position *x*_{cusp} of the cusp. (2)

To detect focusing by a single flat interface in graphene, one can use a small electric contact as a source of electrons while another local probe located on the *p*-side plays the role of a detector. Electric conductance between the two contacts would reflect the probability for a carrier to get from the source to the probe. When the concentration of carriers is low, their de Broglie wavelength is big enough to envisage contacts smaller than the wavelength. To study electron transmission in a phase-coherent system between contacts of such a small size, the above-described classical picture should be complemented with the analysis of quantum interference pattern of electron de Broglie waves. Figures 2B and 3B show the result of full quantum mechanical calculations of the current of electrons emitted at (–*a*,0) and detected by a point contact near the focal point in the symmetric PNJ (Fig. 2B) and in the vicinity of a cusp (Eq. 2) that appears when the symmetry ρ_{h} = ρ_{e} is lifted off (Fig. 3B). The calculation was performed by applying the Kubo formula to the single-particle Dirac-like Hamiltonian (*2*, *7*, *8*) of electrons in graphene. It is assumed that the sample is ballistic, that is, that both the elastic and inelastic mean free path of electrons is larger than the typical size of the structure. Around, but not too close to the focus (*k*_{v}*r* ≫ 1), the analytically calculated current is stands for the distance from the probe to the focus]. The anisotropy of the current distribution is caused by the dependence of the transmission coefficient on the incidence angle and is smeared at shorter distances *k*_{v}*r* < 1. The current map calculated in the vicinity of the cusp for ρ_{h} ≠ ρ_{e} shows characteristic patterns described by the canonical diffraction function for this type of wave catastrophe (*14*). The maximum of the current would be when the probe is at the tip of the cusp, (|*n*|*a*,0). The width *y*_{*} of the bright spot near the cusp (Fig. 3B) or the focus (Fig. 2A) in the *y* direction can be estimated as *y _{*}k*

_{v}∼max[1,(½

*ak*

_{c}|

*n*

^{–1}–

*n*|)

^{¼}]. For a junction with |

*n*|>1(ρ

_{h}> ρ

_{e}), the pattern near the cusp is mirror-reflected as compared with that shown in Fig. 3B for |

*n*|<1.

It has been discovered (*16*) in the scanning tunneling microscopy (STM) studies of elliptically shaped corals on the surface of copper that the presence of an impurity at one focus of the ellipse is reflected by the STM map in the vicinity of the other focus. Therefore, oscillations of the local density of states of electrons formed around a static local perturbation (*17*) can be replicated through focusing by a carefully engineered fence of atoms. Similarly, focusing of electrons by a sharp PNJ in graphene could create a “mirage” that mimics the effect of a perturbation on the opposite side of the *n-p* interface. Consider, for example, a small island of a bilayer (*5*, *18*), which locally distinguishes between two sublattices (A and B) of the honey-comb lattice for electrons in the surrounding sheet [due to Bernal stacking of two adjacent monolayers (*19*, *20*)]. It induces a change in the local electron density of states (LDOS), which is different on sublattices A and B. The long-range oscillations of the alternating LDOS can be detected, with STM, as a difference of between the tunneling current from the STM tip to the A and B sites. Figure 2C shows the results (obtained with the Green's functions technique) of a quantum-mechanical analysis of oscillations of δ*j*_{A-B} around the mirage image of a bilayer island formed on the other side of symmetric PNJ in the monolayer sheet. To compare Fig. 2D shows the calculated mirage image of a spike of electrostatic potential (smooth at the scale of the lattice constant in graphene), which induces LDOS oscillations equal on the two sublattices. The difference between these two images is caused by the lack of backscattering off A-B symmetric scatterers specific to graphene (*21*).

Unlike the ideal left-handed metamaterial (*10*), focusing in the PNJ is not perfect. In symmetric junctions, it occurs only for electrons exactly at the Fermi level, and it is spread into caustics for electrons excited to higher energies. Therefore, the sharpness of electron focusing decreases with temperature. If the focused electron flow is detected by a contact of size *d* ≫ λ_{F}, a pronounced signal in the focus will persist up to *T* ∼ħ*v k*_{c}*d*/*a*. For example, in a ballistic structure with *a* ∼1 μm, *d* ∼0.1 μm, and ρ_{e} = ρ_{p} ∼3 × 10^{12} cm^{–2}, focusing may persist up to the nitrogen temperature. The interference effects shown in Figs. 2 and 3 are washed outat a much smaller temperature scale, *T* ∼ħ*v*/*a*.

Focusing of electrons by a sharp *p-n* junction in graphene can be used to turn the *n-p-n* junction into a Veselago lens for electrons. In such a device (Fig. 4A), the density of charge carriers in the *p-*region (with width *w*) can be controlled by the top gate. If the densities in the *n-* and *p-*regions are equal (ρ_{h} = ρ_{e}), charge carriers injected into graphene from the contact *S* shown in Fig. 4A would meet again in the focus at the distance 2*w* from the source (contact *D*_{3} in Fig. 4A). Varying the gate voltage over the *p-*region changes the ratio *n*^{2} = ρ_{h}/ρ_{e}. This enables one to transform the focus into a cusp displaced by about 2(|*n*| –1)*w* along the *x* axis and, thus, to shift the strong coupling from the pair of leads SD_{3} to either SD_{1} (for ρ_{h} < ρ_{e}) or SD_{5} (for ρ_{h} > ρ_{e}). Fig. 4, B and C, illustrate another graphene-based device in which a prism-shaped top-gate may be used as a focusing beam splitter. For example, electrons emitted from contact *B* (Fig. 4B) are distributed between the contacts *b* and β, whereas the signal sent from contact *A* (Fig. 4C) is replicated into the pair of contacts *a* and α. Graphene has recently been brought into contact with a superconducting metal, and the Josephson proximity effect through graphene has been observed (*22*). Consequently, a beam splitter (Fig. 4, B and C) can be used to experiment with Einstein-Podolsky-Rosen (*23*) pairs of particles.