Angle-dependent magnetoresistance as a probe of Fermi surface warping in ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$

We develop a model for the angle-dependent magnetoresistance of ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$ in the underdoped regime where the Fermi surface is thought to be reconstructed by an ordered state such as a charge-density wave. We show that such measurements can be employed to unambiguously distinguish the form of the Fermi surface's interlayer warping, placing severe constraints on the symmetry and nature of the reconstructing order. We describe experimentally accessible conditions in which our calculations can be put to the test.

Figure 1

A quasi-two-dimensional Fermi surface with (a) simple cosine interlayer warping and (b) staggered twofold interlayer warping. In both cases, the in-plane Fermi surface is circular before interlayer warping is imposed, and two periods of the interlayer warping are shown for clarity. The arrows extending from each figure indicate the direction of the Fermi velocity for those points on the Fermi surface.

Figure 2

The various in-plane Fermi surface shapes considered in this paper. The left column shows ellipses with increasing eccentricity $\left(\epsilon \right)$ from top to bottom. The right column shows diamond-shaped pockets with increasing concavity from top to bottom; $\alpha$ is the angle that must be subtended on a circle to create each arc that makes up the pocket. Note that the area enclosed by all of these shapes is the same; we constrain the cross-sectional area to be consistent with quantum oscillation measurements, as described in the text.

Figure 3

The position of a quasiparticle, represented in this figure by an open circle, can be specified by the central height of the orbit it is tracing $\left(k_z^0\right)$ and its azimuthal position $\left(\phi \right)$. Note that $\phi$ is defined with respect to $k_x$, as is $\varphi$, the azimuthal angle of the applied magnetic field. The polar angle of the applied field, $\theta$, is defined with respect to $k_z$.

Figure 4

An illustration of the effect of Bragg diffraction on quasiparticle motion, including (a) a top-down and (b) an angled view. In this example, the original Fermi surface is cylindrical while the reconstructed Brillouin zone is rectangular. The dotted lines indicate quasiparticle orbits around the unreconstructed Fermi surface, while the solid arcs show a new quasiparticle orbit about the reconstructed Fermi surface. The solid gray line indicates the Bragg diffraction vector. As illustrated by the figure above, the value of $k_z$ will be the same before and after Bragg diffraction; however, the value of $k_z^0$ will change by $\mathrm{\Delta }{k}_z$ as the quasiparticle moves to a different orbit about the Fermi surface. The value of $\mathrm{\Delta }{k}_z$ depends on the geometry of the Fermi surface and the Brillouin zone, as well as the quasiparticle's azimuthal position on the Fermi surface when it undergoes Bragg diffraction.

Figure 5

Concave diamond-shaped pockets at the corners of a Brillouin zone, as predicted for a biaxial CDW reconstruction. The dotted line marks the Brillouin zone boundary. Panel (a) shows the path of a quasiparticle that undergoes Bragg diffraction, while panel (b) shows the path of a quasiparticle that undergoes magnetic breakdown. The shaded pocket in each panel is meant to emphasize the fact that even when considering magnetic breakdown effects, the area we fix to match quantum oscillation measurements is that of the diamond-shaped pockets.

Figure 6

Calculated ADMR of Hg1201 with ${\omega }_0\tau =1$ and $\varphi =0$ for various combinations of in-plane shape and interlayer warping. The top panels show ADMR for a simple cosine interlayer warping with (a) elliptical and (b) diamond-shaped in-plane Fermi surfaces. The bottom panels show ADMR for a staggered twofold interlayer warping with (c) elliptical and (d) diamond-shaped in-plane Fermi surfaces. While the different in-plane Fermi surfaces cause some variation, there is a clear qualitative distinction between the two interlayer warpings regardless of the in-plane shape.

Figure 7

Calculated ADMR of Hg1201 for select in-plane Fermi surface shapes, with $\varphi =0$ and with (a,d) ${\omega }_0\tau =0.1$, (b,e) ${\omega }_0\tau =1$, and (c,f) ${\omega }_0\tau =10$. Plots in the top row show ADMR calculated with a simple cosine interlayer warping, while those in the bottom row show ADMR calculated with a staggered twofold interlayer warping.

Figure 8

Calculated ADMR of Hg1201 for diamond-shaped in-plane Fermi surfaces with $\varphi =0$ and ${\omega }_0\tau =1$ and with various degrees of magnetic breakdown: (a,d) $B_0/B=0.0001$, (b,e) $B_0/B=1$, and (c,f) $B_0/B=10000$. Plots in the top row show ADMR calculated with a simple cosine interlayer warping, while those in the bottom row show ADMR calculated with a staggered twofold interlayer warping.

Figure 9

A schematic diagram of where the ADMR measurements described in this paper would be feasible, based on the data and assumptions described in the text. The dome on the left labeled “SC” is the superconducting region, based on an estimate of $H_{c2}$ with the magnetic field in the interlayer direction. Outside the superconducting region, the color gradient represents the value of ${\omega }_0\tau$, with a darker color denoting a larger value. The dashed line indicates an estimated cutoff for the criterion ${\omega }_0\tau \ge 0.1$; the region below the dashed line is where the distinction between interlayer warpings could most clearly be distinguished by ADMR measurements.

Figure 10

Calculated ADMR of Hg1201 with a simple cosine interlayer warping (top row) or a staggered twofold interlayer warping (bottom row) for eight different values of $\varphi$ as indicated in the legend. Results are shown for four different in-plane Fermi surfaces: (a,e) circular, (b,f) elliptical, (c,g) square diamond, and (d,h) concave diamond.

Figure 11

Calculated ADMR of Hg1201 with two different interlayer lattice parameters. Results are shown for a circular in-plane Fermi surface with a simple cosine interlayer warping (top row) or a staggered twofold interlayer warping (bottom row). (a,c) ${\omega }_0\tau =1$, (b,d) ${\omega }_0\tau =10$.

Figure 12

Calculated ADMR of Hg1201 with various scattering rates, where $g\left(\phi \right)$ parametrizes the variation of the scattering rate, $\tau$, about the Fermi surface. Results are shown for Fermi surfaces with a simple cosine interlayer warping (top row) or a staggered twofold interlayer warping (bottom row). Results are shown for two different in-plane Fermi surfaces: (a,c) elliptical with $\epsilon =0.7$ and (b,d) square diamond with $\alpha =0$.

Figure 13

Calculated ADMR of Hg1201 for an elliptical Fermi surface with various orientations. The in-plane Fermi surface is an ellipse with $\epsilon =0.9$. The variable ${\phi }_{\text{tilt}}$ indicates the angle between $k_x$ and the major axis of the ellipse. The magnetic field is fixed at $\varphi =0$ for all calculations. (a) Results for a simple cosine interlayer warping. (b) Results for a staggered twofold interlayer warping.