Directional ballistic magnetotransport in the delafossite metals PdCoO$_2$ and PtCoO$_2$

Studies of electronic transport in width-restricted channels of PdCoO$_2$ have recently revealed a novel `directional ballistic’ regime, in which ballistic propagation of electrons on an anisotropic Fermi surface breaks the symmetries of bulk transport. Here we introduce a magnetic field to this regime, in channels of PdCoO$_2$ and PtCoO$_2$ along two crystallographically distinct directions and over a wide range of widths. We observe magnetoresistance distinct from that in the bulk, with features strongly dependent on channel orientation and becoming more pronounced the narrower the channel. Comparison to semi-classical theory establishes that magnetoresistance arises from field-induced modification of boundary scattering, and helps connect features in the data with specific electronic trajectories. However, the role of bulk scattering in our measurements is yet to be fully understood. Our results demonstrate that finite-size magnetotransport is sensitive to the anisotropy of Fermi surface properties, motivating future work to fully understand and exploit this sensitivity.

💡 Research Summary

This paper investigates magnetotransport in width‑restricted channels of the delafossite metals PdCoO₂ and PtCoO₂, focusing on the so‑called “directional ballistic” regime that emerges when electrons travel ballistically on a highly anisotropic Fermi surface. The authors fabricate a single PdCoO₂ crystal into two parallel channels oriented along the crystallographic a‑axis (“easy” direction) and at 30° to it (“hard” direction). Starting from a width of ≈63 µm, each channel is progressively narrowed by focused‑ion‑beam milling down to ≈0.75 µm, yielding a dense data set across a wide span of widths. Magnetoresistance is measured at low temperature (≈4 K) in a perpendicular magnetic field up to 9 T for each width and orientation.

Key observations are:

1. Zero‑field size effect – As the channel narrows, the zero‑field resistivity rises due to increased boundary scattering, with the hard direction showing a larger baseline resistivity because electrons are already aligned toward the walls.

2. Low‑field magnetoresistance peak – For w/r_c < 1 (where w is channel width and r_c the cyclotron radius), the easy direction exhibits a pronounced resistivity peak that grows with the ratio λ/w (λ being the bulk mean free path). This peak reflects magnetic‑field‑induced bending of electron trajectories toward the sidewalls, enhancing boundary collisions. In contrast, the hard direction shows a monotonic decrease or a much weaker peak because the electron flow is already parallel to the walls.

3. High‑field kinks – At larger fields (w/r_c > 2) the resistivity decreases overall but displays two distinct kinks, labeled B₁ and B₂. Systematic analysis shows B₁ occurs when w ≈ 2 r_c and B₂ when w ≈ 4 r_c, independent of material. These kinks correspond to geometric transitions in the set of possible electron orbits: at w ≈ 2 r_c the first class of skipping orbits that graze the walls disappears, while at w ≈ 4 r_c a second class of longer‑period orbits is eliminated. The kink positions are essentially set by geometry (w/r_c) whereas their amplitudes scale with λ/w, i.e., the relative importance of bulk versus boundary scattering.

4. Fermi‑surface anisotropy – The authors compare three theoretical treatments: (i) a circular (isotropic) Fermi surface, (ii) an ideal hexagonal Fermi surface, and (iii) a realistic PdCoO₂ Fermi surface obtained from ARPES and dHvA data, which includes rounded corners. Boltzmann‑equation calculations (and complementary Monte‑Carlo simulations) show that only the realistic hexagonal model reproduces the directional differences observed experimentally. In the easy direction the magnetic field strongly enhances edge scattering, while in the hard direction the field initially bends trajectories from the rounded corners toward the walls, producing a modest low‑field rise that later turns into a decrease.

5. Role of bulk scattering – The semi‑classical model assumes a single mean free path λ to capture bulk scattering. While this parameter successfully accounts for the scaling of feature amplitudes, the detailed shape of the high‑field tails and the exact magnitude of the kinks remain only qualitatively described. The authors acknowledge that a full theory incorporating momentum‑dependent scattering, electron‑phonon coupling, and possible surface potential variations is still lacking.

Overall, the work demonstrates that in quasi‑two‑dimensional metals with a faceted Fermi surface, finite‑size magnetotransport is governed by two dimensionless ratios: w/r_c, which determines the magnetic‑field‑induced geometric restructuring of electron orbits, and λ/w, which sets how strongly those geometric changes affect the measured resistance. The directional dependence—absent in isotropic metals—highlights the sensitivity of boundary‑dominated transport to Fermi‑surface shape.

The authors suggest that these findings open avenues for exploiting anisotropic ballistic transport in device concepts such as high‑resolution magnetic sensors, direction‑selective interconnects, or low‑loss microwave waveguides where the interplay of geometry and magnetic field can be tuned to control scattering. Future work is proposed to extend the study to other anisotropic conductors, to explore temperature and current‑density dependence, and to develop a comprehensive microscopic theory that fully captures bulk scattering mechanisms alongside boundary effects.