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

Abstract

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.

Figures (5)

• Figure 1: (a) False-colour scanning electron micrograph of the starting PdCoO2 device S1 for our experiment. Current is injected through gold contacts on the top of a single crystal of PdCoO2 and then flows through meander tracks to become homogeneous through the thickness of the sample before flowing through two conducting channels cut at 30$^{\circ}$ to one another. These are oriented relative to the Fermi surface facets in the so-called 'hard' and 'easy' directions previously identified in studies of directional transport in PdCoO2 in zero applied magnetic field Bachmann2022, and are approximately 63µm wide. The voltage drops within the conducting channels along the easy and hard orientations were measured between contacts $V_{E+}$ and $V_{E-}$, and between $V_{H+}$ and $V_{H-}$ voltage contacts, respectively. (b) The same device after successive FIB narrowing to a channel width of 3.5µm. (c) Magnetoresistance data for five channel widths along the easy and hard directions. The legend shows the average of the channel widths in the two directions, $\bar{w}$.
• Figure 2: Magnetoresistance of PdCoO2 for a range of channel widths. Part (a) illustrates the semiclassical connection between the Fermi surface in reciprocal space and electron trajectories in real space. The hexagonal Fermi surface with average radius $\bar{k}_F$ translates into real-space cyclotron orbits of the same shape but rotated by 90°, with average radius $r_c$. Parts b) and c) show the complete data set for PdCoO2 S1 along the easy and hard directions, respectively. For each channel width, resistivity data are normalised to the value at zero field, with the ratio of the mean free path and channel width $\lambda/w$ indicated by the colour of the trace. Data are plotted against a second dimensionless variable, $w/r_{c}$. The vertical, grey lines indicate the average peak positions in magnetoresistance, at $w/r_{c}=0.78$ (easy) and 0.19 (hard).
• Figure 3: (a) Plotting the magnetoresistance against magnetic field on a semi-log scale reveals the existence of two kinks in the data at characteristic fields $B_{1}$ and $B_{2}$ for both easy and hard direction transport. (b) The exact position of the kinks was determined from local maxima in the second derivative of resistivity with respect to field. (c) Plotting $B_{1}$ and $B_{2}$ for samples with a wide range of widths against $1/w$ reveals that these kinks occur approximately to $w=2r_{c}$ and $w=4r_{c}$, respectively. For $B_{1}$, a small systematic difference between the easy and hard directions is resolved: the hard direction kink occurs at a slightly larger channel width than $2r_{c}$.
• Figure 4: Boltzmann calculations and illustrations of low-field behaviour in the limit of no bulk scattering. We plot the dimensionless ratio $(\rho/\rho_{b})/(\lambda/w)$ where $\rho_{b}$ is the bulk resistivity in zero field. (a) For a circular Fermi surface, the resistivity shows a peak at $w/r_{c}\approx0.55$ as the magnetic field bends electrons towards the sample's edges. (b) For a hexagonal Fermi surface, at low magnetic fields with $0<w/r_{c}<1/2$, the magnetic field enhances edge scattering more strongly in the easy direction than in the hard direction. The easy orientation shows a large field-induced resistivity enhancement, whereas the resistivity in the hard direction is monotonically decreasing. (c) Moving to a realistic Fermi surface for PdCoO2 introduces an initial resistivity increase in the hard orientation at the lowest magnetic fields. This arises from states near the rounded corners of the Fermi surface, which avoid the edges at zero field but are bent towards them by relatively weak fields.
• Figure 5: (a) Illustration of different orbit types. Orbits can be classified by whether they interact with one edge, both, or neither. For $w/r_{c}<2$, the sample supports traversing orbits, making contact with both edges. For $w/r_{c}>2$, the sample supports closed cyclotron orbits, making contact with neither edge. At all fields, the sample supports skipping orbits. Thee orbits start and end at the same edge, and travel in opposite directions at opposite edges. (b) Illustration of the phase space for each orbit type as a function of $w/r_{c}$. The plot indicates the possible positions of the orbit centre (the hypothetical centre point of the closed orbit) relative to the channel's boundaries, marked by dashed lines. (c) Depiction of routes to inter-edge scattering, which is necessary for width-dependent magnetoresistance. For $w/r_{c}<2$, no bulk scattering is required for inter-edge scattering. For $2<w/r_{c}<4$, at least one bulk scattering event is required. For $w/r_{c}>4$, at least two bulk scattering events are required. (d) For a realistic Fermi surface, the condition for the onset of closed cyclotron orbits is not exactly $w/r_{c}=2$ and depends on channel orientation.