Flat-Lens Focusing of Electron Beams in Graphene

Coupling electron beams carrying information into electronic units is fundamental in microelectronics. This requires precision manipulation of electron beams through a coupler with a good focusing ability. In graphene, the focusing of wide electron beams has been successfully demonstrated by a circular p-n junction. However, it is not favorable for information coupling since the focal length is so small that the focal spot locates inside the circular gated region, rather than in the background region. Here, we demonstrate that an array of gate-defined quantum dots, which has gradually changing lattice spacing in the direction transverse to propagation, can focus electrons outside itself, providing a possibility to make a coupler in graphene. The focusing effect can be understood as due to the gradient change of effective refractive indices, which are defined by the local energy band in a periodic potential. The strong focusing can be achieved by suitably choosing the lattice gradient and the layer number in the incident direction, offering an effective solution to precision manipulation of electron beams with wide electron energy range and high angular tolerance.

graphene for manipulation of propagation of electron waves. Some effects such as quasibound states, the resonant transport and the defect states, similar to those in photonic crystals, have been studied in graphene quantum dot superlattices [27][28][29][30][31] . Here, we calculate the energy-momentum dispersion relation of electrons in a periodic potential in graphene, and on this basis, design a graded array acting as a flat lens to focus a parallel electron beam. Similar to photonic crystals 32,33 , we can also define an effective refractive index (ERI) in periodic potentials for electron waves by its energy-momentum dispersion relation. The gradually changing ERI, which is realized by the gradually changing lattice spacing, will result in the focusing of a parallel electron beam. We will see that the focal spot locates outside the array and a coupler could be designed based on the structure.

Results
In the scattering problem of electron waves with energies E close to the Dirac point, the Hamiltonian can be written as where Θ (R − r) is the Heaviside function with R the radius of the circular potential, V is the applied constant bias of the gated region and σ = (σ x , σ y ) are Pauli matrices. We employ reduced units, i.e., ħ = 1 and the Fermi velocity v F = 1 throughout. The focusing is realized with an array of gate-defined quantum dots which share an identical radius R = 1 in graphene and are applied a constant bias V = 5. Here all quantities with displacement dimension are in unit of the radius R since it is set to be 1. At the same time, the incident electron beam has unit electron density at the beam center for the convenience of comparison when the electron density is plotted in all figures. The unmodified structure is a square-lattice crystal with lattice constant d = 5. In order to achieve the focusing effect, a lattice gradient in the transverse direction (perpendicular to the incident wave vector) is added to the original. We study an n t × n l array which is illuminated by a Gaussian beam along the + y direction, where n t and n l denote the layer number of quantum dots in the transverse (x) and longitudinal (y) direction, respectively. In the two edges of the array, there is the equal lattice spacing in the x and y direction and then the lattice spacing in the x direction will linearly increase from the two edges to the center with the increment g = d i+1 − d i , as is schematically shown in Fig. 1(a). The focusing effect is simulated in an array with n t = 20 and n l = 6 and the energy of the incident electron E = k = 0.5 in Fig. 1. The spatial distribution of electron density is plotted in Fig. 1(c,d) for the array without (with) the lattice gradient. The obvious focusing effect can be seen in Fig. 1(d) when the lattice gradient exists. While the propagation of electron waves is hardly affected by the unmodified square-lattice array, as shown in Fig. 1(c). In Fig. 1(b), we plot the electron density along the x-directional line across the focal spot (in Fig. 1(d)) for the cases in Fig. 1(c,d) and for the initial incident wave, respectively. One can see that for the non-gradient case, both the intensity and the distribution of the electron density only have a negligibly change, compared to the incident wave. When the array has gradient, the electron density at the focal spot increases by 2.29 times and correspondingly the waist radius obviously decreases. The comparison indicates that the focusing effect is realized due to the existence of the lattice gradient. In Fig. 1(d), the focal spot lies outside the array, showing that the structure can be used as a coupler. When considering the practical realization, one can choose a typical radius R = 5 nm, and thus the longitudinal length of the array in Fig. 1(d) is 125 nm with the energy E = 65.6 meV and the bias V = 656 meV.

Discussion
The focusing of electron waves can be similarly understood as that of light through optical lenses. In optics, an optical system for focusing satisfies the principle of equal optical path according to Fermat's principle 34 . Since optical path is determined by the spatial distance light travels and the refractive index, the principle of equal optical path requires that one of both must vary when light is focused by a converging lens. For conventional convex lenses, the propagation distance will continuously change due to its curved surface. But for a flat lens, the principle of equal optical path has to depend on the change of refractive index. Similarly, we also need to define an ERI for electron waves in quantum-dot arrays in graphene by using the concept of the local energy band. To this end, we respectively calculate the energy band structure of electrons for two periodic structures of quantum dots with square lattice in graphene in Fig. 2. The black (circle) line corresponds to the array with lattice constant d = 5 while the red (triangle) line d = 7. One can see that the energy band exhibits a noticeable variation when the lattice constant is changed. For the dispersion relation of light in photonic crystals, we have dω/dk ∝ 1/n where n is the ERI and thus the slope of the optical band is related with the ERI 35,36 . Correspondingly, we define an ERI n eff for electron waves in graphene with dE/dk ∝ 1/n eff based on the same principle. In Fig. 2 each dispersion band is approximately a straight line for the energy E ranging from 0.3 to 0.5, so their different slopes illustrate that the larger lattice constant produces the larger ERI. This means that the gradient of lattice constant will lead to the gradient of the ERI n eff . Thus the array probably satisfies the requirement of the principle of equal optical path due to the ERI gradient and has ability to focus electron waves.
Just like an optical lens, the focussing property of electron waves is also related with the geometrical property of the array, such as its thickness and the gradient. We first keep the increment g = 0.4 and n t = 20 unchanged and see the effect of the thickness, i.e., the layer number in the longitudinal direction n l , on focal length and the electron density at focal spot. Figure 3(a) displays the electron density at focal plane which increases with n l enhancing from 3 to 18. It seems that the more the layer number the electron wave travels is, the more strongly it is focused. In order to verify the assumption, we plot the full width at half maximum (FWHM) along the x-directional line at focal plane to characterize the size of focal spots in Fig. 3(a). We see that the FWHM decreases with n l increasing, showing that the electron wave is better focused with the n l increasing. Figure 3(b) shows that the focal length becomes small as n l increases. Namely, the focus spot is more and more close to the array with the layer number increasing. This means that the array with larger thickness is more like an optical converging lens with smaller radius of curvature. One can also note that the focal spot always locates outside the array for arbitrary n l , meeting the requirement of a coupler. To illustrate these features of the flat lens, we plot the electron density distribution in a large spatial region in Fig. 4(a,b) for n l = 8 and n l = 16, respectively. We can find more details about the focusing from them. First, the electron wave is hardly reflected by the arrays. The fact that all of waves can arrive at the transmission side ensures that our design has a high focusing efficiency. Second, a strong focusing shown in Fig. 4(b) can be realized with the layer number n l larger than 16.
The gradient is another factor that makes an effect on the focusing property. In Fig. 3(c,d), the lattice increment g is changed from 0.2 to 1.0 and the layer number in the longitudinal direction n l = 6 and in the transverse direction n t = 20 remains unchanged. One can see that the electron density at focal spot tends to achieve the maximum value when the increment g is in the vicinity of 0.3 and the FWHM has the minimum value at the same time, implying that a good focusing occurs. This demonstrates that the increment g = 0.3 is an optimal value of the gradient for the array to product the focusing behavior. To understand the existence of the optimal value, we plot the electron density distribution in a spatial region for the arrays with the increment g = 0.2 and g = 1.0 in    4(c,d), respectively. The obvious electron scattering in other directions can be seen in Fig. 4(d), preventing a part of electron waves being transmitted and focused. Since the change of the increment will result into the varying ERI according to the energy band plot in Fig. 2, we can owe the degradation of the focusing to the inappropriate ERI when g is far away from 0.3. In the language of light, too small change of ERI is not enough to deflect the wave, while too large change will lead to the scattering of waves in other directions (as the case in Fig. 4(d)). In other words, the principle of equal optical path will be difficult to be satisfied due to the inappropriate ERI. We also note that although the FWHM has an optimal value, the focal length in Fig. 3(d) always becomes larger when the gradient enhances. Therefore the array with a small gradient is like an optical lens with a small radius of curvature. Again, the array has the enough large focal length for arbitrary g to make sure that the focal spot locates outside itself.
Based on the above analysis for the focusing properties, we can realize the strong focusing by designing an array with n l larger than 16 and g in the vicinity of 0.3, as shown in Fig. 4(b). The electron density at focal spot is 4.4 times as large as that of the incident Gaussian wave, and the FWHM is 21% smaller than that of the incident wave. In addition, we can also achieve the desirable focal length by suitably choosing the longitudinal layer number n l and the gradient.
Until now, we only study the focusing of electron waves for the normal incidence case. When considering practical applications as a coupler in graphene-based microelectronics, we expect that the focusing is also robust for non-normal incident angles. In Fig. 5(a-c), we display the focusing effect by an array with n l = 6 and g = 0.4 when the angle of incidence is θ i = 5°, 10°, and 15°, respectively. One can see that electron density at focal spot and the FWHM do not obviously change with the angle of incidence enhancing. It shows that the focusing effect has a high angle tolerance.
Once a quantum-dot array with the definite layer number and gradient is patterned in graphene, it is not practical to change its geometric structure. Therefore, it is expected that electron waves with different energy can be focused by it. For the same array as in Fig. 4(a), the electron density distribution displayed in Fig. 5(d) shows that the wave with electron energy E = 0.4 still can be well focused. The further calculation demonstrates that such an array can focus waves with electron energy E ranging from 0.32 to 0.5. The energy band plot in Fig. 2 clearly shows that the ERI has only a slight variation in a wide energy range with nearly the same slope. Therefore, the focusing by the graded array can occur in this wide range.
In summary, electron waves in graphene can be focused using a quantum dot array with gradually changing lattice constant in the transverse direction. With the total transmission of electron waves after passing through the array, the strong focusing can be even achieved by choosing the appropriate gradient and the layer number in the longitudinal direction. The large focal length ensures that focal spot lies outside the array and the structure can be used as a coupler. The design paves a new way for precision manipulation of electron beams in the graphene electronics with the high angular tolerance and the wide energy range.

Methods
The simulation is done by first solving the scattering problem of a single quantum dot based on Mie scattering method in graphene 14,16,37 . The incident and scattered waves are expanded as follows: , J m and H m (1) are, respectively, the mth order Bessel function and Hankel function of the first kind, φ in is the angle of incidence. Similarly, the inner field inside the rod can be written as where k 0 and k s is respectively the wave vector in the background region and the gated region and k s = nk 0 with the refractive index n = |V − E|/E. After imposing the boundary condition at the surface of the circular potential with its radius R, the scattering coefficient a m is given    (5) and (6) by a scattering matrix S.
In a system of multiple gated potentials, the incident wave that strikes the surface of potential j consists of two parts: (1) the initial incident wave and (2) the scattered waves of all other potentials according to multiple scattering theory 38 (also known in solid-state physics as the K. K. R. method 39,40 ). Thus it can be written as