Angle-dependent magnetoresistance oscillations of cuprate superconductors in a model with Fermi surface reconstruction and magnetic breakdown

We calculate angle-dependent magnetoresistance oscillations (AMRO) for interlayer transport of cuprate superconductors in the presence of ($\pi ,\pi$) order. The order reconstructs the Fermi surface, creating magnetic breakdown junctions; we show how such magnetic breakdown effects can be incorporated into calculations of interlayer conductivity for this system. We successfully fit experimental data from an overdoped cuprate using our model, showing that behavior previously attributed to anisotropic scattering in this material may in fact be due to ($\pi ,\pi$) ordering. This work paves the way for the use of AMRO as a tool to distinguish ordered states that have different ordering wave vectors.

Figure 1

Fermi surface reconstruction of a Q2D material under ($\pi , \pi$) antiferromagnetic order, as viewed along $k_z$. (a) The Fermi surface (FS) and first Brillouin zone (BZ) of a Q2D tetragonal material; (b) the reconstructed BZ and (c) reconstructed FS of the material following the onset of ($\pi ,\pi$) antiferromagnetic order; (d) the repeated-zone view of the reconstructed FS, illustrating the small cross sections of FS that have replaced the unreconstructed cylindrical FS. The gray line illustrates Bragg diffraction between two magnetic breakdown junctions at the BZ boundary. The angle $\xi$ is also defined here; it will be used in our conductivity calculations.

Figure 2

When a magnetic field is applied to a Q2D material, quasiparticles will trace out orbits on the cylindrical Fermi surface that are perpendicular to the applied field. On the lowest orbit in this figure, we define the parameters $k_z^0$ and $k_B$. On the upper two orbits, we illustrate the fact that a quasiparticle undergoing Bragg diffraction moves to a different cross section of the Fermi surface. The dashed lines represent the Brillouin zone boundaries for the reconstructed Fermi surface. Note that the azimuthal angle $\varphi$ is defined with respect to $k_x$ of the unreconstructed system.

Figure 3

Calculated dimensionless resistivity of ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$ as a function of the orientation of applied magnetic field for different values of $B_0/B$ and ${\omega }_0\tau$. We plot resistivity in units of ${\pi }^4{\hslash }^4{\omega }_0/m_0^{*}c{\left(ae{t}_{\perp }{k}_{21}\right)}^2$ so that it is dimensionless. The resistivity in plots (a, b) and (e, f) is scaled up by a factor of 5 for visual clarity. (a, c, e, g) Interlayer resistivity as a function of both $\theta$ and $\varphi$. (b, d, f, h) Interlayer resistivity as a function of $\theta$ for select values of $\varphi$; these azimuthal angles were chosen to facilitate comparison to Ref. [21].

Figure 4

Temperature dependence of the interlayer AMRO of overdoped ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$ ($T_c=$ 15 K) at a fixed field of 45 T and a fixed azimuthal direction of $\varphi =7^{\circ }$. The solid lines are $c$-axis magnetoresistivity data, taken from French et al. [38]. These data have been normalized to the zero-field resistivity of the sample at each temperature. The dashed lines are simulations of AMRO for ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$ under antiferromagnetic order, calculated as described in the text. The simulations have been scaled to match the data at $\theta =0^{\circ }$ for each temperature. When producing these simulations, all the parameters related to Fermi surface geometry were fixed and the only parameters allowed to vary with temperature were ${\omega }_0\tau$ and $B_0$.

Figure 5

Temperature dependence of (a) $1/{\omega }_0\tau$ and (b) $B_0$ extracted from fits to data. Error bars are standard errors extracted from the covariance matrix of the least-squares fitting at each temperature (see Appendix pp4).

Figure 6

The angle $\xi$ is defined with respect to the reconstructed Fermi surface of ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$. The Fermi surface is shown in the repeated zone scheme, with the reconstructed Brillouin zone overlaid.

Figure 7

Position of the eight magnetic breakdown junctions on the Fermi surface of ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$ under $(\pi ,\pi )$ order. The reciprocal space axes shown correspond to the reconstructed Brillouin zone.

Figure 8

Three small Fermi surface pockets are formed when the Fermi surface of ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$ is reconstructed under $(\pi ,\pi )$ order.

Figure 9

Simulations of in-plane transport for ${\mathrm{Tl}}_2{\mathrm{Ba}}_2{\mathrm{CuO}}_{6+\delta }$ compared to experimental data taken from Ref. [54]. (a) Calculated values for the in-plane resistivity and for the cotangent of the Hall angle in a 45-T field plotted versus temperature (inset: data at 7 T). (b) Calculated values of the Hall coefficient as a function of temperature in a 45-T field (inset: data at unspecified field).