Super-geometric electron focusing on the hexagonal Fermi surface of PdCoO2

Geometric electron optics may be implemented in solids when electron transport is ballistic on the length scale of a device. Currently, this is realized mainly in 2D materials characterized by circular Fermi surfaces. Here we demonstrate that the nearly perfectly hexagonal Fermi surface of PdCoO2 gives rise to highly directional ballistic transport. We probe this directional ballistic regime in a single crystal of PdCoO2 by use of focused ion beam (FIB) micro-machining, defining crystalline ballistic circuits with features as small as 250 nm. The peculiar hexagonal Fermi surface naturally leads to enhanced electron self-focusing effects in a magnetic field compared to circular Fermi surfaces. This super-geometric focusing can be quantitatively predicted for arbitrary device geometry, based on the hexagonal cyclotron orbits appearing in this material. These results suggest a novel class of ballistic electronic devices exploiting the unique transport characteristics of strongly faceted Fermi surfaces.

Quantitative energy dispersive X-ray spectroscopy (EDS), displayed in figure S1a, was used to confirm the elemental composition of the delafossite crystals using the AZtec software platform from Oxford Instruments. Typically, the oxygen concentration is severely underestimated due to a wrong carbon coating thickness, since carbon has an absorption edge near oxygen and heavily absorbs oxygen x-rays. Therefore, if the oxygen is fixed by stoichiometry to 2 ions, Pd and Co are found in equal atomic concentration.
Additionally, we present the Shubnikov-de Haas oscillations measured in a FIB-defined PdCoO2 transport bar, with current flowing along the crystallographic a-axis in an out-of-plane magnetic field (B||c) at 2K. The clear observation of quantum oscillations demonstrates that the high crystalline quality of the crystals after FIB sample fabrication. The observed frequencies F1 = 28.9 kT and F2 = 30.2kT, and masses m1 = 1.65me and m2 = 1.73me ± 0.02 agree well with the reported values 14 for bulk samples.
The residual resistance ratios extracted from the data in Fig. 2d are 457, 459 and 355 for VB, VC and VD respectively. Although the size of the overall device exceeds the mean free path, ballistic effects at low temperatures may lead to a correction of the measured non-local voltage.
Supplementary   During this investigation, a total of 7 PdCoO2 crystal platelets were structured into TEF devices. These included 5 nozzle sets oriented along the 2-beam direction, 2 nozzle sets oriented along the 3-beam direction and 3 nozzle sets oriented 11 degrees away from the 2-beam and 3-beam direction respectively. The measurement results of these devices reproduce the data shown in figure 3. The following image series displays the device fabrication process used for a single crystal into which one set of nozzles oriented along the 2-beam direction was cut. The fabrication approach described here can be used to fabricate nozzle along any desired in-plane direction of the crystal.
Supplementary Figure 2: Step-by-step overview of the fabrication of a ballistic delafossite device using a Ga-based FIB. a) A PdCoO2 crystal is fixed onto a sapphire substrate (1.6 x 1.6 x 0.4 mm 3 ) with 5 min araldite ® rapid epoxy, which is cured for 1 hour at 100 °C on a hot plate. Care is taken to select a crystal with as little step edges as possible and with well-defined hexagonal growth edges, such that the crystallographic orientation can be directly determined.
b) The crystal is thinned down in the center to a final thickness of less than 1 µm (here 700 nm), using Ga 2+ ions at 30 kV, cutting a rectangle pattern with a current of 65 nA, 1 µs dwell time and the "dynamic all directions" scan option. Thinning down the crystal is a necessary step to fabricate narrow, closely spaced nozzles later on. Further ~10 µm wide rectangular cuts are made using 65 nA through the remaining thick parts of the crystal to define current and voltage contacts. A small gap is left to reduce re-deposition in the central area.
c) Rectangular cuts are patterned with 2.5 -9.3 nA in the central region, which define a rectangular measurement region. The sides of the rectangle are polished with 2.5 nA under an angle of +1 degree with respect to the normal milling direction to obtain flat boundaries. d) In order to ensure a homogenous current flow between all palladium layers despite having a top current injection, holes are patterned through the entire depth of the crystal with a current of 47nA and 2ms dwell time. At the inner edge of these 'root'-like features, the amorphous FIBdamage layer and re-deposition couples the individual layers together and increases the interlayer conductance. Roots are also milled into the central part of the device using a current of 2.5 nA and 2 ms dwell time.
e) The constrictions leading up to the nozzles are patterned with 80 pA. Making long and thin constrictions is favorable, as they act as long flexures and reduce mechanical cracking of then nozzles due to strain from differential thermal contraction while cooling down.
f) The nozzles are cut using an array of cleaning cross section (CCS) cuts at 40 pA, cut under an angle of 1degree. Initially the nozzles to a width of about 500nm and are then sequentially thinned down with CCSs until the final width of the nozzle is achieved. g) Overview of the final device. If the nozzles are thinner than 350 nm, a second layer of 5 min araldite epoxy is added on top of the finished device and cured at room temperature for 24 hours. This reduces the substrate strain during cooldown and avoids nozzle fracture.
h) Final device on sapphire substrate. Silver wires were attached using Epotek EE129-4 silver epoxy and cured at 100 °C for 1 hour. A 100 nm thick layer of sputtered Au connects the preevaporated gold leads on the substrate with the crystal device.

Supplementary Note 3. Current dependence
As a result of the extremely high in-plane conductivity of PdCoO2 at low temperatures, comparatively large measurement currents are necessary in order to record clean signatures of the TEF effect. The typical measurement current used for the devices presented here is on the order of 1mA. In order to exclude the occurrence of self-heating effects, we have performed the same TEF measurement for a range of current spanning nearly two orders of magnitude (25µA, 100µA, 1mA, 2mA), thus varying the input power P nearly four orders of magnitude. As presented in figure S3a, the resulting position, height and width of the TEF peaks do not undergo any measurable change. We therefor conclude that the system is in the linear response regime.
Supplementary In the following we compare the measured TEF signal for 3 distinct devices fabricated from 3 different crystals. In order to compare the measured signals, we divide the voltage by the applied current as well as the device thickness, summarized in the table below. The observed differences in peak height and position (see fig. S4a) can partially be accounted for taking the varying nozzle widths into consideration. Additionally, the shape of the nozzle influences the degree of collimation the ejected electron beam and can therefore change the height of the focusing peak. We conclude that the key signatures of the TEF signals are highly reproducible, while details may depend on the specific device geometry implemented. a) The devices corresponding to the data presented here are displayed in figure 2 (blue), S2 (red) and S3 (cyan) respectively. The peaks of the TEF signals, indicated by dashed lines, are located at 3.22T, 4.25T, 6.28T and 11.86T, for nozzle separations of 4, 3, 2, and 1µm. Here we present the TEF signal of the furthest spaced nozzle pair which we have fabricated. The distance between these nozzles (1 and 8 from figure 2) is 35µm, which amounts to a path length of approximately 53µm through the sample (sketched in blue in panel a). As a comparison, we also show the TEF signal of a nozzle pair which is separated by a distance of 15µm. The measurements performed at 2K are presented in panel b, in which up to 8 TEF peaks are clearly identifiably. This demonstrates the length scale over which ballistic effects are still observable. in the 3-beam direction compared to the 2-beam direction, corresponding to the difference between inradius and circumradius of a hexagon. The diameter of the circular FS is expressed relative to the circumradius of the hexagon and was chosen smaller for clarity. Compared to a circular FS the 3-beam direction has an enhanced peak, while the main peak along the 2-beam direction is reduced and has a second broad hump. b) In a circular FS the simulated focusing spectrum (shaded purple) diverges. For the mathematical derivation c.f. methods S10. c) Comparison of the TEF spectra of a perfect hexagon and a hexagon with warped sides and rounded corners inferred from the FS of PdCoO2 along the 3-beam direction. The perfectly flat edges lead to a geometrical enhancement of height of the focusing peak (so called supergeometric focusing or 'sfocusing'). In a warped hexagon, sfocusing still leads to an increased TEF peak. The amplitude ratio of subsequent TEF peaks, q = An/An+1, is often believed to be a direct probe of the specularity of the boundaries. In this simple picture, the decay of the peak height is equated to the boundary specularity coefficient p. This, however, neglects corrections due to Fermi surface shape. To identify the relation between q and p, we have therefore performed ballistic simulations for a hexagonal FS (3-beam and 2-beam direction) as well as a circular FS. a) Simulated TEF spectrum along the 3-beam direction for a total of 100'000 particles with an isotropic incident angle distribution. The specularity of the boundary is p = 0.9, meaning that 90% of the electrons are specularly reflected upon impact with the boundary and the remaining 10% are assigned a random angle. During the simulation, all points of impact with the boundary for all particles are saved and displayed in the histogram above. For further analysis, the number of impacts in the orange shaded regions, corresponding to the TEF peak areas, is extracted.

Supplementary
b) The counts in the orange shaded area in panel a as a function of TEF peak number are plotted in color for a wide range of simulated specularity coefficients p between 0 and 0.9. The black line is a fit of the form * + to the data, where A scales the overall amplitude, q is the extracted "experimental specularity coefficient" and c is a background offset.
c) The extracted experimental amplitude ratio q as a function of the specularity coefficient p for the 3-beam (orange) and 2-beam (blue) direction as well as a circular FS (purple). The black line indicates where q=p.
The main result of these simulations is that the assumption of q = p, identifying the true surface specularity p with the measured power law coefficient q is not strictly applicable, even for a circular Fermi surface. Hence the amplitude ratio is an indicator, but not a perfect measure of the specularity of the boundary. The physical reason for this is two-fold. In the case of fully specular reflection the peak width grows with the number of peaks and due to their convolution with a finite nozzle size the measured voltage decreases with increasing peak number. In the opposite limit, even in the case of completely diffusive boundary scattering (p=0), a large number of TEF peaks are expected to arise from focusing. Therefore, a simple analysis will extract a significant q value for p=0. Indeed, we find q≈0.33 for fully diffuse scattering (figure S7c). This value has a simple physical interpretation. Due the 3 main directions of propagation, approximately 1/3 of the electrons will be scattered into the direction that will be focused again. This statistical mechanism will lead to an apparent specularity of the boundary despite a completely diffusive scattering process. This is an alternative formulation of the super-geometric focusing properties of PdCoO2.

Supplementary Note 8. Peak position analysis
In the following we analyze the ratio between the fields B1 at which the first TEF maxima along the 2-beam and 3-beam direction occur, both for the experimentally measured and simulated spectra displayed in figure 3. For a mathematically hexagonal Fermi surface, no focusing is expected in the 2-beam-direction as discussed in the main text. Instead, here the ratio of the longest possible trajectories gives a measure of the range of electron transmission (Fig. S8). In the case of the ideal hexagon, the ratio of travel distances is geometrically given by the ratio of the inner to outer diameter of a hexagon, √$ % ≈ 0.87. This simple argument is already quite close to the experimental results. When the rounded corners of the realistic Fermi surface of PdCoO2 are taken into account, a macroscopic density of states will be focused at the same field, and a focusing peak occurs even in the 2-beam direction. The ratio of the furthest travelled distance in both directions increases to 0.91 for the realistic Fermi surface parameters obtained by ARPES and quantum oscillations. This happens as the distortions increase $789:; and decrease %789:; . In a circular Fermi surface, this ratio is naturally one and all TEF is independent of the crystal orientation. Note that a ratio closer to one, however, does not indicate a deformation to a more circular orbit, but merely reflects the geometric properties of the slightly star-shaped Fermi surface. This is exemplified in Fig. S8, as the realistic Fermi surface is distorted distinctly into a star-shape and not a more circular object. Yet the ratio increases. This back-of-the-envelope estimate already quite accurately reflects the ratio of the measured focusing fields. To further improve the modelling, we take both the realistic Fermi surface and the finite nozzle width into account in the Monte-Carlo simulations. Here we obtain a ratio of 0.96 for the 2µm nozzle, in good quantitative agreement with the measured ratio of 0.99. The main reason for the increase of the ratio is that finite nozzles allow focusing trajectories over a small range of cyclotron radii, a geometric property that a full kinematic simulation naturally takes into account. The main factors attributing to the subtle deviation between experiment and tuning-parameter-free simulations (~3%) are measurement errors in the exact nozzle geometries in the SEM images; nozzle-to-nozzle deviations of their width during fabrication; and the emission characteristic of the nozzles. The latter were assumed to be isotropic in k-space while in reality the ballistic connections to the nozzles themselves may lead to deviations in the emission spectrum.  Figure 9: Geometrical model of TEF on a circular FS.
The further two nozzles are spaced apart along the edge of the sample, the longer the path length s of an electron traveling through the bulk of the device, which increases the chances of being scattered away from its ballistic orbit. As pointed out by Tsoi et al. 34 , the amplitude A1 of the first TEF peak is proportional to 7= > ⁄ . The amplitude, however, also depends upon the ratio of / , where b is the width of the accepting nozzle and L is the distance between the nozzles. Assuming a point-like injection source and only the accepting nozzle having a width b, we find + = , where L is the maximum distance an electron can travel at a fixed field ( = 2 D ). In a x L θ b injection nozzle crystal system with a circular FS, the travel distance x of an electron injected under an angle θ can be found by trigonometry to be = 2 D cos , where D = ℏK L 9M denotes the cyclotron radius and O is the Fermi momentum. With that and by Taylor expanding cos ≈ 1 − R S % for small , we find Δ = 2 ≈ 2U %8 V . Accordingly, the amplitude of the first peak will decrease with increasing nozzle distance as W = 2U %8 V 7= > ⁄ , where the path length is given by = π D .
In the case of a hexagonal Fermi surface, the amplitude is similarly dependent on 7= > ⁄ as well as the ratio / . For fitting the peak decay and extracting the mean-free-path λ in Fig. 3c we use the form W (L) = A 7= > ⁄ U 8 V + , with = 0.3µ and %89:; = $ % , $89:; = √3 are the path lengths for the 2-beam and 3-beam directions respectively. In addition to λ, the free variables are A, which sets the overall amplitude, and t, which takes the geometrical deviations from a non-circular FS into account. In the 3-beam direction the path-length is ill-defined due to the very nature of the super-geometric focusing effect. We choose the average between the longest and shortest path possible. The fit results are summarized in table 1. We note that this analysis is only valid for 8 V ≪ 1; once the nozzle width becomes comparable to the nozzle spacing the description breaks down. Further, particularly noticeable in the regime where ~, but true in general, is that the maximum of the focusing peak does not occur at strictly = 2 D but at lower magnetic fields when a nozzle of finite width can collect the maximum number of electrons.  Fig. 3c) with the form A W (L) = A e 7d e ⁄ U f g + t. The small value of t in the 2-beam direction reflects the fact that the focusing in this orientation originates from the rounded corners of the hexagon which can be locally approximated by a circle. In the super-geometric focusing configuration the flat sides of the hexagon no longer resemble a circle leading to a larger t value.