Veselago lens and Klein collimator in disordered graphene

We simulate electron transport through graphene nanoribbons of realistic size containing a p–n junction patterned by electrostatic gates. For a sharp p–n interface, Klein tunneling leads to refocusing of a divergent beam forming a Veselago lens. Wider transition regions allow only electrons with near-perpendicular incidence to pass the junction, forming a Klein collimator. Using a third nearest neighbor tight binding description we explore the influence of interface roughness and bulk disorder on guiding properties. We provide bounds on disorder amplitudes and p–n junction properties to be satisfied in order to experimentally observe the focusing effect and compare our predictions to very recent realizations.


Introduction
Graphene nanostructures [1][2][3][4] continue to attract considerable attention mainly due to their potential applications in high mobility electronics [5] and solid state quantum information processing [6,7]. Intensive research has been triggered by the unique electronic properties of graphene [8,9] including the gapless linear dispersion, and the relativistic Landau level (LL) spectrum [10,11]. Graphene nanostructures also promise to allow exploration of phenomena related to massless Dirac fermions in reduced dimensions [12][13][14] in the presence of confinement. In particular, relativistic Klein tunneling [15] can give rise to unconventional transmission properties of p-n junctions in graphene [16][17][18] such as the Veselago lens effect [19,20]. Accordingly, an ideally sharp interface separating an electron and a hole cone acts for fermions as a material with negative index of refraction similar to recently discovered metamaterials for electromagnetic radiation [21][22][23], resulting in effects such as 'cloaking' [24]. Electrostatically patterned nanoribbons can thus be viewed as the starting point for graphene-based twodimensional electron optics with unconventional properties. Recent advances in fabricating high-quality graphene-hexagonal boron nitride (hBN) sandwich structures [25,26] have overcome intrinsic difficulties in creating tunneling barriers and confining electrons in graphene, opening up the pathway towards graphene-based electron optics. Electrostatic gates on top of the hBN layer can now be placed much closer to the graphene sheet enabling sharper p-n junctions. While exper imental realization of such electron 'beam' shaping and focusing devices has remained a challenge, first experimental observations have become available [27]. In this work we present theoretical simulations for electron transmission through a realistic p-n nano-scale graphene junction, and investigate its potential for electron-optical applications. Our focus is on the destructive influence of disorder on the focusing and collimation effects. We consider both a corrogated interface of the p-n junction and bulk disorder on the n-and p-side. Previous investigations of Klein tunneling in disordered p-n junctions [29][30][31] show that disorder improves transmission of the p-n junction. We show quantitatively that both Veselago lens effects and beam (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. collimators are robust in the presence of moderate interface roughness and weak bulk disorder, and give quantitative constraints for experimental realizations.

Klein tunneling
The linear band crossing in graphene at the K-point, the socalled Dirac point creates a double-cone structure that closely mimics the dispersion relation of massless Dirac fermions (see figure 1(a)), described by the Dirac-like Hamiltonian, where p x (p y ) denote the momentum operators and we have set E(k K ) = 0. The approximation (1) ignores both the length scale of the graphene lattice constant a = 1.4 Å and the broken rotational symmetry of the cone due to the hexagonal lattice structure, an effect known as trigonal warping [8]. The symmetric electron-like (hole-like) dispersion relation of equation (1) above (below) the Dirac point allows to locally tune the Fermi energy to create n-doped (electron-like) or p-doped (hole-like) regions of carrier density by an external potential. At a finite-width potential step V(x) (by, e.g. a back gate) [8] (see figure 1) between an n-doped and a p-doped region, electrons may tunnel from the n-region into the p-region (see figure 1(a)). In the limit of a sharp interface with the transition half-width d small compared to the de Broglie wavelength λ D , λ d D , tunneling occurs with near unit probability due to the electronhole symmetry of H (equation (1)), a phenomenon known as Klein tunneling [15,32]. Since the group velocity is reversed when switching from the upper to the lower cone, i.e. from the n to the p region, the wave originally propagating in the ( ) k k , x y direction in the n-region is transmitted into the p region with wavevector ( ) ( , , x y x y due to flux conservation at the interface. The resulting scattering kinematics corresponds to the optical analogue of a metamaterial with a negative index of refraction (figure 1). Consequently, a diverging ray of trajectories emanating from a source point (S) will be focused by an ideal p-n interface onto the point F on the p side. Such an electron-optical lens could be created in graphene simply by applying a discontinuous potential step. If, however, the transition from n-to p-region is gradual instead of sudden, i.e. if d is of the order of λ D , new effects appear. For grazing incidence with k k y x at the p-n interface, k y may exceed the local F causing total reflection rather than transmission. Consequently, partial transmission through the p-n interface is restricted to near-normal incidence [28] and no distinct focal point exists. The p-n junction operates in this regime as filter that only transmits on near-normal incidence.

p-n junction in graphene nanoribbons
Realizing electron-optical elements such as lenses and filters suggested by the ideal massless Dirac fermion picture (equation (1)) in graphene structures must account for the discrete honeycomb lattice structure with lattice constant a made up by two interleaved triangular sublattices (A and B). It can be described in tight-binding approximation by the Hamiltonian [33] ∑ ∑ where the sum (i,j) extends over pairs of lattice sites, φ j s , is the tight-binding orbital with spin s at lattice site j, V i is a locally varying potential which includes in the present case the potential step (equation (2)), and γ i j , are the hopping matrix elements between lattice sites i and j. We omit physical spin in the following. In contrast to the Dirac Hamiltonian of equation (1), the electronic structure of graphene features a weakly broken electron-hole symmetry accounted for in the present simulations by including third-nearest-neighbor coupling (for details see [34]). Furthermore, the hexagonal symmetry of the graphene lattice distorts the perfectly circular Dirac cone at energies farther away from the Dirac point. This so-called trigonal warping [8,37] is also included in our third-nearest neighbor description.
We explore the consequences of this symmetry breaking for a graphene nanoribbon that extends to infinity to the left and right, ( , containing a single p-n transition. Note that a flake of finite size would be unsuitable to realize focusing of rays as bound state effects of the finite-size flake would overshadow the propagation. We therefore use open boundary conditions within the framework of an effective Hamiltonian The width W of the nanoribbon plays a key role for observing lens or collimator effects. Only for sufficiently large width λ W D (for more quantitative estimates see below) pronounced lens effects appear while for smaller W dist ortions by reflection at the boundary largely mask the effects. The simulation of such wide ribbons (in the following we consider W between 120 and 300 nm) represents a numerical challenge. The simulation of such wide ribbons (1200 lattice sites in transverse direction for W = 120 nm) includes several million carbon orbitals. We consider a point source (S) on the n side (figure 2) and calculate the propagated wavefunction at the position x in the ribbon, where x S is the source point of current injection. We use a parallelized, distributed matrix inversion [35,36] suitable for the efficient description of large-scale graphene nanodevices. We first probe the Veselago lens effect predicted for massless Dirac fermions in a graphene nanoribbon with an ideally sharp p-n junction ( → d 0, see figure 3(a)). The source point x S of the incoming electric current is located at the center of the ribbon at a distance x S from the interface larger than the de Broglie wavelength λ D . For a doping of E = V 0 = 0.5 eV corresponding to a p-n step height of 2V 0 = 1 eV, λ D follows from the relation for which pronounced focusing effects are already expected. In order to prevent the reduction of the visibility of focusing by boundary reflections, W should be large compared to x S , W x S . Combining these two constraints implies λ W D indicating that only in wide p-n junctions Veselago lens and Klein collimation are clearly visible. We observe focusing of the ray down to the diffraction limit with the lateral width s of the focal spot at F close to λ ≈ s 2 D . We have, furthermore, numerically verified that distributing the source point over a small area S of size λ D 2 does not significantly change the focusing pattern. Likewise, a finite energy resolution ∆E, as long as ∆ E E, does not substantially affect the quality of the focusing. However, even in this ideal limit of a sudden potential step (i.e. d = 0 in  equation (2)) small deviations from the Dirac picture such as trigonal warping [8,37] become visible. Consequently, when changing from the electron to the hole cone, the conservation of k y is not perfect, leading to a slight asymmetry in the distance of focal point F and source S from the p-n junction.

Measures for focusing and collimation
In order to quantitatively characterize in the following focusing and collimation properties in realistic p-n graphene junctions we introduce as a measure for finding the electron inside the spot width (−s, s) of an ideal Veselago lens with λ = s 2 D (see figure 3(b)) the probability where the normalization constant A(x) is determined by P(x; s = W/2) = 1. We furthermore introduce the contrast C, characterizing the visibility of focusing regardless of its location within the p domain. Equation (7) remains applicable also in cases where the focal point is displaced relative to the position F predicted for the ideal Veselago lens. The normalization ′ A in equation (8) will be kept fixed at C(x) = 1 for the ideally sharp p-n junction to allow a direct comparison between the contrast for different realizations of p-n junctions.

Finite transition width
As a first step towards a realistic scenario for a p-n junction we consider a finite transition (half)width d of an otherwise ideal junction. We observe a gradual change from the Veselagolens type focusing (for 2d = 0.5 nm λ D , figure 4(a)) to the low-divergence beam filter (for 2d = 15 nm >λ D , figure 4(d)) predicted by Cheianov et al [28]. This beam shaping and collimation is not an immediate consequence of the negative index of refraction itself but of the filtering out of large-angle momentum components incident on the junction, i.e. filtering due to imperfect Klein tunneling at larger angles of incidence [18]. For the linear potential step (equation (2)), the filtering out of larger angles of incidence α (α = 0 denotes perpendicular incidence) can be analytically estimated in terms of an effective tunnel probability of [28] ( ) The range of transmitted angles is thus controlled by the ratio /λ d D . For small α, equation (9) will yield probabilities close to one even for large d (indeed, at α = 0, any finite d will lead to perfect transmission). For larger angles of incidence, however, only p-n junctions with small transition widths will allow for transmission, leading to the beam-like pattern we observe in our numerical simulations. While the beam formed by such a Klein collimator is well focused with width λ ≈ s 2 D , its total amplitude is, of course, reduced with increasing transition width d in the regime λ d D , leading to a reduction of the absolute contrast (see figure 4(e)). Spatially extended p-n junctions thus allow for the formation of a tightly focused electron beam with interesting applications such as collisionfree ballistic transport of the confined beam due to the suppression of interactions with (rough) edges of nanoribbons. The potential height V 0 provides a second tuning parameter for the p-n junction. For optimal Klein tunneling the Fermi level E F is tuned by doping to coincide with V 0 (figure 1). For smaller V 0 , the increased electron wavelength decreases the resolution of the interference structures (compare figures 5(a)-(d)). Moreover, for very small V 0 and correspondingly large λ D the focal spot width increases resulting in a decrease of P and C ( figure 5(e)). On the other hand for V 0 too large (⩾ 0.7 eV), effects of the nonlinear band bending far away from the Dirac point decrease the focusing efficiency and the contrast ( figure  5(e)). Simultaneously decreasing the transition width d, V 0 and E F , while keeping the ratio / λ d D constant, improves transmission in disorder-free junctions [16]: at small energies, and thus large λ D , a soft junction (featuring a transition length of the order of λ D , yet large compared to the lattice constant) transmits better than a sharp one. However, experimental realizations require robustness of focusing against finite long-range disorder (as induced, e.g. by the substrate) providing a lower bound for V 0 .

Disorder and interface roughness
Creating a potential step with a strong gradient in the electronic potential that varies by a fraction of an eV over a few nanometers for a p-n junction in a realistic device still poses a considerable challenge. This requires fabrication techniques beyond simple back-gate voltages, e.g. etching of contacts very close to the graphene membrane, or top-gate approaches [14]. Such approaches, however, invariably introduce some level of roughness, i.e. deviation from a straight interface perpendicular to the ribbon axis (see figure 2). Simulation of a realistic scenario for the realization of such nanoscale electron-optical structures requires the inclusion of corrugation of the p-n interface. Such interface roughness ∆x limits focusing and contrast unless it is negligibly small compared to the de Broglie wavelength λ D . The focusing parameter P and the contrast C remain high for λ ∆x D but rapidly decrease for larger ∆x ( figure 6(a)). Clearly, the effect of interface roughness could be reduced by increasing λ D . The latter can be achieved by reducing the energy separation V 0 between the n-and p-regions. However, as the size of the focal spot also scales with λ D , suppression of interface roughness would come at the expense of a reduced contrast.
A second limitation for a realistic graphene nanoribbon is bulk disorder, in particular due to interaction with the substrate. Disorder limits electron mobility and can cause a transport gap in quantum dot measurements [4]. To investigate the influence of such a disorder on electron-optical properties, we include a bulk disorder potential in the n-and p-regions (see figure 2(d)). We use a correlated disorder potential ( ) V x D , with an amplitude V D and an autocorrelation length ξ in both the n-and p-regions. A well-defined junction requires of course that local potential variations are smaller than the potential step of the junction, V V D 0 . As expected, the effect of dis order is strongest when the correlation length ξ matches the de Broglie wavelength, ξ λ ≈ D, while it is reduced for both longer and shorter correlation lengths. The latter applies only as long as the correlation length is still large compared to the lattice spacing, ξ λ a . For very short correlation lengths of the order of the lattice spacing characteristic for lattice point defects, strong inter-valley scattering, a peculiarity of graphene, can severely impede well-defined electron-optical properties. It should be noted that Klein tunneling itself is not directly affected by bulk disorder but the lensing effect is. As a result, we observe several local maxima rather than a single focal point (see figure 6(b)). Accordingly, the probability P of finding the electron in the focal region is significantly reduced with increasing potential strength ⟨ ⟩ V D , while the contrast C decreases only slowly (see figure 6(b)). Up to a disorder potential of the order of ⟨ ⟩ ≈ V 0.1 D eV corre sponding to the contrast is still above the 50% level. At the same time, the height of the potential step should be smaller than the characteristic energies associated with the interface corrugation, Combining the constraints (10) and (11) provides a criterion for the admissible potential for which focusing and collimation should be observable. We emphasize that these bounds apply to fully phase-coherent single-electron transport.
In the present simulation we do not explicitly treat sublattice-breaking disorder σ ∝ σ V z , where the Pauli matrix σ z acts  (2)) and a step height of V 0 = 500 meV. The source point x S is 25 nm away from the onset of the p-n junction. (e) Contrast C (equation (8)) and focusing probability P (equation (7)) of an ideal p-n junction as a function of transition width for different distances  S of a p-n junction of width W = 120 nm for different potential step heights V 0 (see insets, equation (2)) at fixed transition width of 2d = 5 nm. Source point at a distance of 25 nm away from the p-n junction. (e) Contrast C (equation (8)) and focusing probability P (equation (7)) of the p-n junction as a function of step height V 0 . (a) V 0 = 100 meV.
on the sublattice degree of freedom. The potential σ V implies different local potentials on the two trigonal sublattices of graphene. Such short-range fluctuations are induced, e.g. by substrates such as hexagonal boron nitride. Depending on the relative alignment of the graphene layer with respect to the substrate, the two sublattices of graphene positioned, e.g. over a boron and a nitrogen atom may see quite different potentials [38]. The resulting σ V opens a band gap at the Dirac point. We have verified numerically that potentials of the order of 10% of the step height V 0 do not strongly affect the focusing properties of the p-n junction.
Testing our predictions (equation (12)) for smaller disorder strength ⟨ ⟩ V and step heights V 0 (and thus larger wave lengths) requires larger system sizes. We simulate a junction of width W = 300 nm (i.e. about a factor three wider than the p-n junctions discussed above). For these simulations, we solve the Dirac equation (1) in the continuum limit rather than the TB Schrödinger equation for numerical simplicity thereby neglecting effects due to the discrete lattice structure. As expected, the small assymetry of the transmitted wave due to trigonal warping disappears entirely. Consequently, these results can give only an upper bound for the quality of focusing for realistic graphene. We find the same qualitative trends as before: the focusing probability P decreases with increasing diso rder strength (figure 7). Klein collimation for larger d is more robust to disorder (compare bottom and top row of figure 7). Efficient focusing at larger disorder amplitudes requires larger step heights. While larger disorder even increases the overall transmission of the p-n junction, in line with previous results [31], the focusing decreases rapidly (see figure 8).
A comparison to recent experiment further corroborates our analysis. Lee at al. realized a transition width of ≈ d 12 nm using a top gate made of few-layer hexagonal boron nitride [27]. They consider a series of two junctions (i.e. a  (7)) as a function of averaged disorder strength ⟨ ⟩ V for three different step heights V 0 of the p-n junction (see insets), evaluated for a junction with narrow transition region (d = 0.2 nm, top row) and a wide transition region (d = 7 nm, bottom row). P is evaluated at distance x = 50 nm (left column) and x = 90 nm (right column) from the p-n junction. Width W of the ribbon is 300 nm, calculation based on the Dirac approximation of equation (1). S near a p-n junction of step height V 0 = 500 meV, width W = 120 nm with a rough interface ∆ = x 5 nm. The inset shows the evolution of focusing probability P (solid line, see equation (7)) and the contrast C (dashed line, see equation (8)) as a function of interface roughness ∆x. (b) Same as (a) for a p-n junction with bulk disorder potential ( ) V x D , ⟨ ⟩ = V 0.2 eV. The inset shows P and C as a function of mean disorder potential amplitude V . p-n-p and n-p-n transition), and find a small but discernable enhancement of transmission of the order of 5% due to the Veselago lens effect. Estimating the step height in this experimental device with a gate coupling η ≈ × 7.2 10 10 cm −2 V −1 yields for a gate voltage V G of 10 V a doping level of , or a Fermi level of [27] ħ i.e. the value used in our calculations for figure 8. The experiment estimates disorder by a mean free path Λ mfp of µ Λ ≈1.7 mfp m. To compare to the disorder scale of our calculations we determine numerically a mean free path associated with our randomly correlated disorder. We find µ Λ ≈1.7 mfp m for a disorder strength of ⟨ ⟩ = V 20 meV using a fit to averaged transmission coefficients [39,40]. Our calculations yield a P value of 0.45 for this disorder strength (see lowest curve in figures 7(a and c)), suggesting an upper bound for the focusing probability of two junctions of 20% in fair agreement with the experimental findings of a 5% enhancement [27]. Testing our relation for the energy scales equation (12) (14) confirms that the experiment is within our suggested bounds for observing the Veselago lens effect. Indeed, the tight constraint ⩽ ⟨ / ⟩ V V 5 D 0 might explain the comparatively small enhancement of the transmitted current found in the experiment. We thus conclude that the quantitative bounds we provide are a good indicator for the suitability of an exper imental setup to exploit electron focusing by p-n junctions.

Conclusions
We have investigated electron-optical focusing and filtering in realistic graphene devices generalizing the notion of a Veselago lens based on Klein tunneling of massless Dirac fermions. We find that the focusing effect is robust against moderate disorder and give quantitative upper bounds for the distortion by bulk disorder and interface roughness. The major challenge in experimental realization remains achieving sharp potential steps necessary for Klein tunneling. Comparing junctions of different step height, we find that lower step heights show better focusing for zero bulk disorder, yet suffer more strongly from finite disorder. Comparing our prediction with recent experiment indicates an optimized setting when the junction height is about an order of magnitude larger than the average bulk disorder.