Quantum oscillations and magnetic reconstruction in the delafossite ${\mathrm{PdCrO}}_2$

We report quantum oscillation data on the metallic triangular antiferromagnet ${\mathrm{PdCrO}}_2$. We find that, to very high accuracy, the observed frequencies of ${\mathrm{PdCrO}}_2$ can be reproduced by reconstruction of the (nonmagnetic) ${\mathrm{PdCoO}}_2$ Fermi surface into a reduced zone. The reduced zone corresponds to a magnetic cell containing six chromium sites, giving a $\sqrt{3}\times \sqrt{3}$ in-plane reconstruction, and $\times 2$ interplane reconstruction. The interplane ordering represents a reduction in lattice symmetry, possibly to monoclinic, and an associated lattice distortion is expected. In addition, we report a magnetic transition under an applied in-plane field that is probably equivalent to the spin-flop transition reported for ${\mathrm{CuCrO}}_2$, and present data on its field-angle dependence. We also report measurements of the resistivity of ${\mathrm{PdCrO}}_2$ up to 500 K.

Figure 1

Magnetic oscillations in sample #1 at three field angles. The $y$ scale is the resistance of the piezoresistive sense element on the cantilever.

Figure 2

The observed oscillation frequencies of ${\mathrm{PdCrO}}_2$, multiplied by $cos\left(\theta \right)$, against field angle $\theta$. The field rotation axes are shown at bottom right, referenced to the nonmagnetic zone. The frequencies are from Fourier transformation of the data over the range 4.2 to 15 T. Frequencies that are clearly sum or difference frequencies are not shown. The indicated amplitudes of the oscillations are peak-to-peak amplitudes of the resistance of the sense element on the cantilever. The ${\mathrm{PdCoO}}_2$ frequencies are taken from the model in Ref. [10], and have been scaled by 93.3%.

Figure 3

A two-dimensional (2D) model of the reconstruction. Top: the ${\mathrm{PdCoO}}_2$ Fermi surface in the 2D nonmagnetic zone, and magnetic zone arising from a $\sqrt{3}\times \sqrt{3}$ reconstruction. Bottom: reconstruction into the magnetic zone. The fully reconstructed orbits $\alpha$ and $\gamma$ are illustrated in the left-hand panel, and the breakdown orbits $\beta$ and $\delta$ in the right-hand panel.

Figure 4

Left: a possible magnetic cell for ${\mathrm{PdCrO}}_2$, that contains 6 Cr sites. The Cr sites are colored black, gray, and white to indicate the three spin directions of each layer. Right: the nonmagnetic first Brillouin zone (black), together with the ${\mathrm{PdCoO}}_2$ Fermi surface and the six-site zone, into which the nonmagnetic Fermi surface is reconstructed.

Figure 5

Red circles: calculated frequencies based on reconstruction of the ${\mathrm{PdCoO}}_2$ Fermi surface into the magnetic zone shown in Fig. 4. The upper and lower panels show the $\gamma$ and $\alpha$ frequencies, respectively. In the lower panel, the diameter of the symbols is proportional to the logarithm of the expected oscillation amplitude, based solely on the curvature of the Fermi surfaces. Black circles: the frequencies observed in sample #1, with the diameter of the symbols proportional to the logarithm of the amplitude.

Figure 6

Perpendicular magnetization against applied field at 0.7 K, for sample #1, where the field was rotated in a $\langle 0001\rangle -\langle 1\overline{1}00\rangle$ plane. Each curve is offset by the field angle $\theta$; the offsets are indicated by the thin horizontal lines at the left and the angle scale is on the right. The solid lines show downward field sweeps, the dashed lines upward. The curves are smoothed, to exclude quantum oscillations.

Figure 7

An illustration of the spin flop. In zero field, the spins lie in $\langle 0001\rangle -\langle 1\overline{1}00\rangle$ planes (left panel). When a field is applied in a $\langle 1\overline{1}00\rangle$ direction, the spin plane rotates by ${90}^{\circ }$ (right panel). Note that in ${\mathrm{PdCrO}}_2$ the spins are not perfectly coplanar; see Ref. [1] for details. Also, there may be domains of different $\langle 0001\rangle -\langle 1\overline{1}00\rangle$ spin planes; the evolution of each possible domain with applied in-plane field is discussed in Ref. [21].

Figure 8

The resistivity of ${\mathrm{PdCrO}}_2$ against temperature, for two samples. Sample A had two pairs of voltage contacts. The point at 295 K, with error bars, indicates the room-temperature resistivity: $8.2\pm 1.0\mu \mathrm{\Omega }\mathrm{cm}$. Data on a ${\mathrm{PdCoO}}_2$ sample are also shown, scaled to the room-temperature resistivity determined in Ref. [10]: $2.6\pm 0.2\mu \mathrm{\Omega }\mathrm{cm}$. Bottom panel: $d\rho /dT$ for sample A.

Figure 9

Lifshitz-Kosevich fits to the amplitudes determined over the field range 8.5–9.5 T.

Figure 10

Fourier transforms of the oscillations below (thin lines) and above (thick lines) the magnetic transition, which at the angles in this figure occurs at $B\sim 10$ T.

Figure 11

Oscillation frequency against field angle for two further possibilities for the magnetic reconstruction, discussed in the text.