An electrical analogy to Mie scattering

Mie scattering is an optical phenomenon that appears when electromagnetic waves, in particular light, are elastically scattered at a spherical or cylindrical object. A transfer of this phenomenon onto electron states in ballistic graphene has been proposed theoretically, assuming a well-defined incident wave scattered by a perfectly cylindrical nanometer scaled potential, but experimental fingerprints are lacking. We present an experimental demonstration of an electrical analogue to Mie scattering by using graphene as a conductor, and circular potentials arranged in a square two-dimensional array. The tabletop experiment is carried out under seemingly unfavourable conditions of diffusive transport at room-temperature. Nonetheless, when a canted arrangement of the array with respect to the incident current is chosen, cascaded Mie scattering results robustly in a transverse voltage. Its response on electrostatic gating and variation of potentials convincingly underscores Mie scattering as underlying mechanism. The findings presented here encourage the design of functional electronic metamaterials.


Supplementary Tables
Supplementary Table 1 CNP

Supplementary Note 1
As described in the experimental part of the main text, our system shows the following characteristics: - Step-like feature is inverted depending on the local shift in energy in graphene (p-or n-type).
The relativistic scattering formalism based on wavelike behaviour as demonstrated above is able to explain the transverse voltage generation and its characteristics. We  Figures 11c and d).
We emphasize that even for extreme ratios (up to four orders of magnitude) between the resistances of the black and white discs the calculation shows no transversal potential differences. In addition, no transverse voltage difference is observe even in the case of using a larger network (50 x 100) or larger spacing between the local scattering potentials. Finally, we confirmed that moving from a squared canted 2DSL to the trivial case of a canted 1D periodic potential modulation we find a transverse voltage as was for the latter case previously demonstrated [17]. Our model is therefore fully consistent.
Another viewpoint is from a pure analytical perspective when a diffusive and nonrelativistic resistivity tensor is considered. Within this framework, the need for the absence of trans V in the previous resistor network model can be easily pinpointed.
The most general form of the off-diagonal resistivity tensor in diffusive and periodically modulated non-relativistic systems including a rotation angle 2DSL  in the modulation potential is [17] (   Table 1).

Supplementary Note 3
To compare quantitatively devices with canted periodic potentials (2DSLs) on different substrates and measured under different conditions (Fig. 1f main text), we calculate the mobility of the samples in order to estimate their mean free path. Our entire devices are always diffusive at the measured temperatures. Within a first approximation, the overall mobility of any of our devices would be composed of three contributions: the intrinsic contribution due to phonons, contribution due to charged impurities and contribution due to the imposed periodic potentials (circular dots). We note that we are mainly interested in the charged impurity contribution to the mobility since for a constant temperature the other two factors are equal for the devices we want to compare in Fig. 1f main text (i.e. devices with periodic potentials and 2DSL  = 30). Furthermore, the phonon contribution to mobility even at room temperature is estimated to give a mobility value of ~20 m 2 V -1 s -1 [9], implying that this factor is not the main contributor in our case (we have a much lower overall mobility). Regarding the circular metallic dots, it is predicted [3] and experimentally proved [10] [10][11][12][13][14].
The constant K has been measured to be between 5 x 10 15 V -1 s -1 [12] and 1.2 x 10 16 21 V -1 s -1 [15]. For our estimations, we take 1.2 x 10 16 V -1 s -1 which is the more conservative value to estimate the mobility in order to show that our devices work within the diffusive regime. In Supplementary Table 1 the charge neutrality point, metal composing the dots, charge impurity density, mobilities, Coulomb mean free path at the charge neutrality point (minimum mean free path), mean free path at 10 V away from the charge neutrality point, temperature, measuring environment and FoM of all the samples presented in Fig. 1f main text are shown.

Supplementary Note 4
The persistence of the non-monotonic transverse voltage also in the case of canted Ti-2DSL samples without surface hydrophobisation is illustrated in Supplementary   Figures 4a and 4b   The same qualitative behaviour is found for the sample with hydrophobicallyrendered surface (Supplementary Figures 5c and d). Strikingly, longitudinal and transverse resistances do not depend on the applied source-drain voltages. This is in contrast to all samples with canted 2DSL. This proves again that the transverse components measured in samples without 2DSL are due to the limit in the lithographical alignment precision in the sample-production process.

Supplementary Note 6
The impact of surface hydrophobisation on non-canted Ti-2DSL samples is illustrated in Supplementary Figure 3a [2,3]. If the device is appropriately gated, metallic impurities therefore create local p-n junctions (PNJs) within the monolayer [2,3]. Theoretically, the elastic scattering of relativistic carriers in graphene due to an individual circular PNJ has been recently addressed in Refs. and matching the reflected and transmitted waves at R r  to satisfy the continuity of the wavefunction [3][4][5][6].

5-8. The low-energy charge-carrier dynamics is calculated through the Dirac
This formalism is analogous to the description of Mie scattering in optics [6]. Known phenomena from optics appear in graphene in new guise to satisfy the absence of backscattering dictated by Klein tunnelling [3][4][5][6].
We are interested in the charge-carrier density n outside the metallic dot and the scattered current-density j , which are given by     n , and

In this equation,
N (equivalent to the modulus of the refractive index) is given by For a large diameter dot compared to the Fermi-wavelength of the Dirac-fermion, in the far-field shows a peaked forward scattering [5,6].
The far-field condition can be therefore considered as being already well-fulfilled at distances > R 2 away from the circular potential centre, that is, j has a dominating x j component there [4].
As shown by the foregoing calculations, the scattering by a single circular potential can be analytically determined by the generalization of the partial wave method [3][4][5][6] similar to Mie scattering [4]. In the 2DSL case, the scattering problem gets more complex depending on the additional parameter of the superstructure: the 2DSL tilting angle 2DSL  measured w.r.t. the overall current direction (i.e. propagation direction of incoming Dirac-fermion wavefunction). To extend the solution for the scattering by a single circular potential (dot) to the square 2DSL of circular potentials, we applied the transfer matrix method (TMM). These types of methods are extensively used for solving scattering problems in periodic (identical) potentials [8]. Within this formalism, we can connect the simple single dot scatterers as building block to generate square 2DSLs (Fig. 7) and determine the total transfer matrix by simple multiplication.
As discussed beforehand, within the far-field regime (i.e. distances > R 2 away from the circular potential centre; see also Ref. 4), a single circular potential acts on an incoming plane wave in the x direction in such a way that it generates a symmetric redistribution of the (charge-carrier) probability density in the transverse ( y ) direction. In addition, in terms of overall current, in the far-field limit only the current component in positive x direction is non-vanishing. These conditions can be expressed as follows: where the function ) ( y F is in general complex valued and normalised. We will see later that it represents the same redistribution as demonstrated in section (i) of the charge-carrier density , that is, the far-field scattering intensity is not spatially symmetric with respect to the transverse ( y ) direction. That is, an intensity imbalance establishes comparing the positive and the negative y -axis (see Fig. 9 and Fig 3a in  V , the same formalism as described in the previous paragraphs can be used. In here, we focus not only on the existence of an electronic density imbalance, but also its variation with gate V . For an evaluation, it is sufficient to consider the far-field scattering intensity (cf. Fig. 9) along the y-axis of the (cascaded) two-dot unit cell. Integrating the intensity independently for positive and negative y and taking the difference of these values provides an intensity imbalance and is a direct measure for the deviation of the incident electronic wave due to the canted geometry. Fig. 3a in the main text shows clearly that the intensity imbalance is maximal at a Fermi energy close to takes an inbetween value away from these two points (Fig 3a main text bottom).
Therefore, already the model for two dots considered in previous paragraphs is able to reproduce the characteristic step-like feature present in the experiments.  Fig. 10 and assuming an incoming electronic plane wave travelling in the x direction. As aforementioned, these systems are characterized by the absence of back-scattering and a peaked forward scattering [5][6][7][8], thus, we consider only transmitted and scattered probabilities in the far-field limit. We emphasise that this calculation is similar to the one leading to Fig. 9, able to produce the far-field scattering intensity along the y-axis for the two-dot case. Furthermore, we note that this approach implicitly implies the multiple Klein-phenomenon occurring at the circular potential [5][6][7][8]. For a given gate V we consider as the 'transmitted probability',  When increasing the gate voltage, it shows first a maximum around -10 V, a sudden decrease from -10 V until 7 V and a constant dependency afterwards. In contrast, the dark-blue curve in the bottom panel of Fig. 3b (main text)

V E 
, which corresponds to a decrease in scattering probability. Therefore, pot V is directly linked to the appearance of the relative maxima and the minima of trans V and to the multiple Klein tunnelling phenomena occurring in circular potentials, which is implicitly considered in the here applied model [3,4].