Abstract
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 PdCoO_{2} gives rise to highly directional ballistic transport. We probe this directional ballistic regime in a single crystal of PdCoO_{2} by use of focused ion beam (FIB) micromachining, defining crystalline ballistic circuits with features as small as 250 nm. The peculiar hexagonal Fermi surface naturally leads to enhanced electron selffocusing effects in a magnetic field compared to circular Fermi surfaces. This supergeometric 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.
Introduction
Electronic conduction in metals is typically well captured by Ohm’s law as frequent collisions of the electrons lead to diffusive motion and locally defined conductivity. An essential prerequisite to this common transport regime is a momentumrelaxing meanfree path, λ, that is much smaller than the size of the conductor. In extremely clean metals, however, λ may exceed the size of the sample, and a ballistic description of charge transport becomes appropriate. The diffusive motion underlying Ohm’s law is replaced by ballistic trajectories, akin to the motion of billiard balls. In such situations, geometric electronbeam optics should be achievable in solidstate devices. Significant progress has been made in twodimensional electron systems (2DESs), where essential elements of electron optics (familiar from a freespace context) have been demonstrated in highpurity semiconducting heterostructures and graphene. These include collimated electron sources^{1,2}, lenses^{3,4,5}, waveguides^{6,7}, beam splitters^{8}, transistors^{9}, refractive^{10,11}, and reflective^{12,13} elements. Electronic solidstate devices based on high carrier density metals differ from freespace electronbeam applications, as they operate in a quantum regime of a Fermi gas at temperatures far below their Fermi energy. Therefore, the accessible electron states for conduction are locked to the Fermi energy and the Fermi momentum, unlike the freespace electronic beams where all energies and momenta are accessible.
To date, such device concepts rely on straightline electron trajectory segments, or in a magnetic field, circular orbits associated with small, isotropic Fermi surfaces (FS) as in graphene or 2DESs. In principle, unlike the free electron case, a solid offers the opportunity to engineer dispersion relations E(k) via Bragg scattering off the lattice. By tuning the hopping integrals, the shape of the FS can be made strongly noncircular defining preferred directions of electronic motion in the solid. Here, we report a striking directionality of ballistic electron motion in the material PdCoO_{2}, arising from its almost perfectly hexagonal FS that defines three preferred directions of motion.
The metallic delafossite PdCoO_{2} is an extraordinarily clean conductive oxide that exhibits a meanfree path of ∼20 µm at low temperatures^{14}, rendering the delafossite metals class the most conductive oxides known^{15,16,17,18}. The quasi2D crystal consists of highly conductive, triangular coordinated palladium sheets separated by layers of insulating CoO_{2} octahedra. Only a single, halffilled band crosses the Fermi level, resulting in a large, cylindrical FS consisting of a nearly perfectly hexagonal cross section (k_{F}^{min} = 0.90 Å^{−1}, k_{F}^{max} = 1.00 Å^{−1})^{14}, which is weakly warped along k_{z}^{14}. Unusual transport characteristics, such as strong momentumconserving scattering processes have recently attracted attention, as they have been argued to lead to hydrodynamic transport^{19}.
Here we focus on the exotic ballistic regime of PdCoO_{2} at low temperature (2 K) and in a range of applied outofplane magnetic fields (B ≤ 14 T), arising from transverse electron focusing (TEF) in combination with the hexagonal Fermi surface. In particular, we demonstrate how this peculiar Fermi surface shape leads to unusual TEF trajectories owing to the physical fact that mostly three discrete directions of group velocity are available to the quasiparticles. The large, flat sections of a hexagon are shown to significantly enhance the TEF signals, a situation we refer to as supergeometric focusing. In addition, we explore how the strong deviations of the Fermi surface from a circle lead to highly crystal directiondependent TEF.
Results
Transverse electron focusing (TEF) geometry
Two narrow contacts resembling nozzles connecting along the same crystal edge and two far away large electrodes form the electric connections to the device (Fig. 1a). A uniform magnetic field B is applied perpendicular to the surface, causing electrons to follow cyclotron orbits. If the nozzle separation L is smaller than or comparable with the shortest microscopic meanfree path in the problem, one expects undisturbed cyclotron motion between the nozzles. Similar to the circular trajectories of electrons in freespace, electrons in a lattice move on orbits defined by the geometry of their FS. When an integer multiple n of the cyclotron diameter matches the contact spacing L, an excess of electrons arrives at the distant contact, leading to an increase in its chemical potential. As the cyclotron radius, \(r_{\mathrm{c}} = \hbar k_{\mathrm{F}}/eB\), is inversely proportional to the magnetic field B, and electrons have a finite probability to be reflected specularly back into the channel, a linearly spaced train of peaks in measured potential can be observed at fields \(B_{\mathrm{n}} = \frac{{2n\;\hbar k_{\mathrm{F}}}}{{eL}}\), where \(\hbar\) is the reduced Planck constant, k_{F} is the Fermi vector, and e is the electronic charge. These correspond to trajectories with n–1 bounces off the surface. First observed and studied in elemental conductors (Bi^{20}, Sb^{21}, W^{22}, Cu^{22}, Ag^{23}, Zn^{24}, and Al^{25}), the TEF effect was further investigated in 2DESs^{12} and graphene^{26,27} and has recently been employed to spatially separate and detect electron spins in a spin–orbit coupled system^{28}.
The role of the hexagonal Fermi surface
In PdCoO_{2}, the TEF signal strongly deviates from that expected for free electrons due to the hexagonal shape of the FS. In the presence of a magnetic field, the charge carriers revolve around the FS in kspace in a plane perpendicular to the applied field. In real space, the shape of the cyclotron motion is given by a 90° rotation of the kspace orbit. In materials with circular FSs, for instance singlelayer graphene or 2DESs, the electron trajectories are circular and their velocity distribution is isotropic in real space (see Fig. 1b). Interesting deviations from circular Fermi surface shapes can be observed in bi and trilayer graphene exhibiting trigonal warping^{26}, as well as graphene superlattices^{27}. A peculiar situation emerges in such strongly faceted FSs as in PdCoO_{2}, in which a macroscopic proportion of states all move in the same direction as the group velocity is locally perpendicular to the FS. This large weight of parallel moving states leads to a strong geometric enhancement of the focusing capabilities of a material, which results in a highly anisotropic velocity distribution in the palladium planes with three preferred directions of motion.
On a circular FS, a simple geometric model can be used to understand the shape of the TEF peak. An electron injected at an angle \(\theta\) away from normal incidence will be focused at a distance \(x = 2r_{\mathrm{c}}\,{\mathrm{cos}}\theta\) away from the nozzle. By assuming an isotropic angular distribution of electron injection, we can find the distribution of distances from the injection nozzle at which electrons return to the edge, by calculating the classical probability density function, which is given by \(n\left( x \right) = \frac{2}{\pi }\frac{1}{{\sqrt {(2r_{\mathrm{c}})^2  x^2} }}\) (see Supplementary Note 10 for derivation). The divergence at \(x = 2r_{\mathrm{c}}\) describes the focusing effect on a circular FS; even when electrons are injected evenly in all directions, those entering the device under a small angle \(\theta\) will all be focused onto nearly the same spot by a magnetic field (purple shaded region in Fig. 1c). This focusing occurs due to the presence of a welldefined Fermi surface in the metal, and is hard to achieve in freespace electron beams as it depends on highly monochromatic electrons.
The hexagonal FS of PdCoO_{2}, however, hosts large flat sheets along which electrons are naturally focused onto the same point (orange shaded region in Fig. 1c). Such flat faces lead to strong deviations of the electron trajectory compared to curved ones. While the magnetic field constantly transfers momentum to the quasiparticle changing its state in kspace, the group velocity in real space remains constant. This leads to a macroscopic number of states being focused onto the same spot, and we refer to this situation as supergeometric focusing to set it apart from the geometric focusing of curved Fermi surfaces. This intuitive picture of enhanced focusing is corroborated by numerical TEF simulations on the experimentally determined hexagonal FS of PdCoO_{2} (Fig. 1d, see Supplementary Fig. 11 and Supplementary Note 11 for details on the simulations).
Focused ion beam microstructured single crystals
To test this prediction experimentally, we have fabricated TEF devices from asgrown single crystals of PdCoO_{2}, which grow as ultraclean single crystals without the need for any further purification. The synthesis and characterization are described in Supplementary Note 1 and Supplementary Fig. 1, respectively, as well as elsewhere^{29}. The crystals grow as hexagonal platelets (~10–20 µm thick, and several 100 µm in lateral dimensions), with the growth edges of the crystal oriented 90° away from the crystal axes (Fig. 2a).
Critical for the observation of TEF as outlined above is the use of narrow injection nozzles. In twodimensional systems, pointlike contacts can be easily defined lithographically. Here we employ a focused ion beam (FIB)based technique to fabricate narrow nozzles of diameters as small as 250 nm (\(\ll\)λ, L). As a maskless technique capable of etching materials in three dimensions, FIB machining offers a unique way to carve a welldefined microsample out of a larger crystal. The details of FIB micromachining are described in Supplementary Note 2 and Supplementary Fig. 2, and can be found elsewhere^{30}. Given the high degree of control over the material on the subµm scale, this approach may be a viable route towards a more quantitative fermiology based on TEF on even the smallest metallic samples^{27}.
In our typical PdCoO_{2} devices, two sets of nozzles (Fig. 2c, bottom edge: 1–8 and left side: A–D) connect to a central square area and are arranged perpendicular to each other. The orientation of the FS and the Brillouin zone are overlaid in the middle panel. Due to the sixfold rotational symmetry of the FS, these two sets of nozzles will probe TEF directed along different parts of the FS. The bottom panel shows an enlarged view of the nozzles, which are ~250 nm wide and are separated by 1 µm each. Given the lowvoltage levels of the TEF signals, high currents of multiple mA had to be applied, and the resulting high current density in the 10^{6} A cm^{−2} range naturally raises questions about heating and nonlinearities. No such effects were observed, and the signal remains linear in current over the entire studied range (see Supplementary Fig. 3 and Supplementary Note 3). The material can carry such extreme current densities owing to its low inplane resistivity ρ~8 nΩcm. The temperature dependence of the nonlocal voltages (measurement configuration Fig. 2c; data Fig. 2d) observed in our experiment is reproduced quantitatively by solving the Laplace equation numerically in our geometry by using bulk resistivity values^{18}. The quantitative agreement at all temperatures suggests that the FIBfabricated crystalline circuit matches the designed geometry, and the electronic properties over most of the structure are not strongly perturbed by the FIB process.
Directiondependent TEF
For a hexagonal FS, the TEF strongly depends on the crystallographic direction (Fig. 3), because the group velocity of a kstate is always perpendicular to the FS. When the electrons are ejected from a nozzle onto a flat side of the FS hexagon (nozzles A–D), they are collimated into three main directions; hence we term this configuration the “3 beam direction”. If the nozzles are rotated by 90° with respect to the underlying FS (nozzles 1–8), the electrons then travel in a “2 beam direction” configuration. With an applied outofplane magnetic field, the electrons then undergo hexagonal cyclotron motion, yet with different initial conditions. Accordingly, the experimental TEF peaks from nozzle pairs of the same separation occur at different fields (see Supplementary Fig. 8, Supplementary Note 8, and Supplementary Table 2) and differ in shape between the two crystal directions, as seen in Fig. 3b. At negative fields, there is a small, diffusive magnetoresistive component. Since there is only one type of charge carrier (electrons) present in this system, no focusing peaks are observed at negative fields. At small positive fields, where the cyclotron diameter is larger than the separation of the nozzles, a voltage inversion is observed: more electrons reach the large contact (V_{com} in Fig. 2c) than the nozzle. Once the cyclotron diameter equals the distance between the nozzles a voltage peak is detected. As expected for the Blinear period of TEF, the second peak of the 4 µm spaced nozzles coincides with the first peak of the 2 µm spaced nozzles. The Fermi surface is encoded in the peak shape, leading to a significant enhancement of the focusing in the 3beam compared with the 2beam configuration. While in the 3beam direction a large number of electrons are focused into a sharp single peak, the 2beam direction displays a broad shoulder followed by a peak of reduced amplitude. Simple geometric arguments show that for an ideal hexagon, the 3beam direction would exhibit a divergent supergeometric focusing, while the 2beam direction would not focus at all (see Supplementary Figure 6 and sketch in Fig. 3a). The focusing peak in the 2beam configuration as well as the rounding of the peak in the 3beam data arise from the rounded corners of the real Fermi surface as compared with the mathematical hexagon. This is confirmed by Monte Carlo simulations by using a tightbinding approximation of the Fermi surface^{14} based on ARPES data^{31} (see Supplementary Note 11), which reproduce the observed peak shapes and their fine structure (Fig. 3b).
TEF as a probe of the meanfree path
TEF probes semiclassical trajectories of ballistic electrons, and in the absence of scattering would lead to focusing over arbitrary distances. In real crystals, scattering is always present, limiting the range over which focusing can be observed. Accordingly, the height of the first TEF peak shrinks as the nozzle distance is increased (Fig. 3c). It is intuitively clear that the TEF signal will be strongly suppressed for nozzles between which a typical ballistic path would be longer than the meanfree path (see Supplementary Fig. 5 and Supplementary Note 5). A more rigorous calculation, comparing with the observed exponential decay of the focusing signal with nozzle separation, allows us to extract a meanfree path on the order of λ≈15 µm from the data (see Supplementary Fig. 9, Supplementary Note 9, and Supplementary Table 3). This value estimated from TEF is in good agreement with previous studies estimating λ from transport by using a simple Drude model^{19}. As the temperature is increased, the normalized amplitude of the primary peak stays almost constant up to 20 K (Fig. 3d). This is consistent with a roughly constant—and long—momentumrelaxing meanfree path in this temperature region, as reflected by the temperature independent resistivity observed selfconsistently in the devices (Fig. 2d) and in measurements on macroscopic crystals^{14}. From the analysis of a previous flow experiment^{19}, a momentumconserving meanfree path of ~2 µm was deduced for PdCoO_{2} <20 K. The possible effects of this were not included in the models used for the simulations presented in Fig. 3b or Supplementary Fig. S7. Although the simulations clearly capture the main features of our observations very well, there are differences in the details, generally seen in the simulations containing sharper features than the experimental data; it is possible that these differences would be reconciled by including momentumconserving scattering in a more complete analysis. Above 20 K, the focusing peaks gradually decrease until they cease to exist ~70 K, presumably primarily due to the reduction of the momentumrelaxing meanfree path due to Umklapp electron–electron and electron–phonon scattering.
Enhanced apparent specularity due to hexagonal Fermi surface
The first peak B_{1} plays a special role in the TEF geometry as it describes ballistic paths between both nozzles without any interaction with the device boundary. All other peaks at B_{n}, n > 1, correspond to paths that include (n−1) scattering events on the sidewall between the nozzles. This has been exploited to use TEF to perform surface spectroscopy^{24,32}, based on the probabilistic nature of the surface reflection of the quasiparticle back into the channel. A scattering event may be specular, thus conserving its momentum component parallel to the surface, or diffuse, leading to a random continuation of the trajectory after the impact. The specularity of the surface, p, denotes the probability of specular reflection. Conventionally, the amplitude ratios between subsequent peaks, \(q_{\mathrm{n}} = \frac{{A_{{\mathrm{n}} + 1}}}{{A_{\mathrm{n}}}}\), are used to estimate p, as each subsequent peak corresponds to a trajectory that differs by one additional surface impact^{33}. However, even in a simplified analysis adopting the above assumptions, in the supergeometric focusing configuration the pointspread function is so sharply peaked at 2r_{c} (see Fig. 1d) that electrons will be statistically refocused onto the next peak, regardless of the specularity. In agreement, we observe peaks up to n = 8 in the devices, yielding a q value of roughly 0.6 (Fig. 4). Numerical simulations (see Supplementary Fig. 7 and Supplementary Note 7) corroborate this finding, showing that even barely specular surfaces (p = 0.1) are compatible with a qvalue around 0.4 (Fig. 4b).
Discussion
Our experimental data strongly support the hexagonal shape of the semiclassical orbits in PdCoO_{2}, which impacts the TEF signatures. In particular, the directional dependence of the TEF signal as well as the strong distortion of the peak in the 2beam configuration unequivocally evidence noncircular trajectories. The details of the signal are well reproduced by simulations of ballistic transport based on a tightbinding Hamiltonian describing the realistic hexagonal Fermi surface. While the essentially hexagonal shape explains the key features of supergeometric focusing, significant features appear due to the deviation from the ideal hexagon. Most prominently, the 2beam direction is not expected to show any sharp focusing feature in the case of a perfectly hexagonal Fermi surface. Its experimental observation is entirely due to the rounding of the corners, which is well captured by calculations, thus confirming that the present models well describe even subtle details of its electronic system. Besides the trajectories in the bulk, the hexagonal Fermi surface shape will also alter the collimating properties of the nozzles and influence their capability to select certain kstates, which will be the topic of further studies on the TEF dependence on the nozzle width and length. In addition, these results suggest the feasibility of novel electronic devices operating in the ballistic limit exploiting strong deviations from circular Fermi surfaces. This additional avenue of control will enable novel types of functionality. For example, selectively aligning parts of a PdCoO_{2} crystalline circuit along the 3beam or the 2beam direction will completely alter the ballistic response of the device, despite it being chemically and structurally homogeneous. Intriguingly, PdCoO_{2} is an extremely conductive ballistic metal, and therefore automatically incorporates low dissipation, a key prerequisite for highpower and highfrequency applications that is unlikely to be achievable in low carrier density devices based on graphene and 2DES. Promising recent thinfilm results^{34} may indicate a pathway toward largerscale fabrication of such devices.
The almost perfectly hexagonal FS shape of PdCoO_{2} arises from accidental finetuning of hopping parameters in its band dispersion. In general, supergeometric focusing is a generic property of materials with large parallel sections on their FS. Such flat areas on Fermi surfaces can be engineered in 2D materials where arbitrary control over the chemical potential and sometimes even the band structure is possible via gating, as has been shown in bilayer graphene^{26} or in graphenebased moiré superlattices^{27}. Alternatively, flat Fermi surface sections are not rare in bulk crystals such as PdCoO_{2}, and future material science efforts may uncover ballistic behavior in other ultraclean metals. Our FIBbased approach showcases a viable route toward the investigation of ballistic behavior in challenging materials, where the crystal size or chemical composition may impede traditional lithographybased methods to fabricate ballistic devices on the subµm scale.
Data availability
All data needed to evaluate the conclusions in the paper are present in the paper. All raw data underpinning this publication can be accessed in comprehensible ASCII format at DOI: https://doi.org/10.5281/zenodo.3406604. Source code for the simulations can be found at https://github.com/dgglab.
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Acknowledgements
The project was supported by the MaxPlanck Society and has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715730). M.D.B. acknowledges studentship funding from EPSRC under grant no. EP/I007002/1. A.L.S. acknowledges support from a Ford Foundation Predoctoral Fellowship and a National Science Foundation Graduate Research Fellowship. A.L.S. would like to thank Edwin Huang for helpful discussions and Tom Devereaux for letting us use his group cluster. Computational work was performed on the Sherlock cluster at Stanford University and on resources of the National Energy Research Scientific Computing Center, supported by DOE under contract DE_AC0205CH11231. D.G.G.’s and A.W.B.’s work was supported by the U.S. Department of Energy, Office of Science, Basic EnergySciences, Materials Sciences and Engineering Division, under Contract No. DEAC0276SF00515.
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M.D.B., C.P., M.K., and P.J.W.M. fabricated the microstructures, and M.D.B., C.P., and P.J.W.M. performed the measurements. The crystals were grown by S.K. and A.P.M., A.L.S., A.W.B., and D.G.G. performed the ballistic simulations. All authors were involved in the design of the experiment and writing of the paper.
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Bachmann, M.D., Sharpe, A.L., Barnard, A.W. et al. Supergeometric electron focusing on the hexagonal Fermi surface of PdCoO_{2}. Nat Commun 10, 5081 (2019). https://doi.org/10.1038/s41467019130209
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