Quantum Oscillations from Nodal Bilayer Magnetic Breakdown in the Underdoped High Temperature Superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$

We report quantum oscillations in underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.56}$ over a significantly large range in magnetic field extending from $\approx 24$ to 101 T, enabling three well-spaced low frequencies at $\approx 440\pm 10$, $532\pm 2$, and $620\pm 10\mathrm{T}$ to be clearly resolved. We show that a small nodal bilayer coupling that splits a nodal pocket into bonding and antibonding orbits yields a sequence of frequencies, $F_0-\Delta F$, $F_0$, and $F_0+\Delta F$ and accompanying beat pattern similar to that observed experimentally, on invoking magnetic breakdown tunneling at the nodes. The relative amplitudes of the multiple frequencies observed experimentally in quantum oscillation measurements are shown to be reproduced using a value of nodal bilayer gap quantitatively consistent with that measured in photoemission experiments in the underdoped regime.

Figure 1

(a) Magnetic quantum oscillations (solid line) measured in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.56}$ obtained using the contactless resistivity technique at $T=1.6(1)\mathrm{K}$ consisting primarily of $\approx 440\pm 10$, $532\pm 2$ and $620\pm 10\mathrm{T}$ frequencies [see (b)]. A slowly varying background has been subtracted, the data have been multiplied by the monotonic function $\exp [75\mathrm{T}/B]$ for visual clarity. The upper (red) dotted curve is the simulated waveform for a nodal pocket split into bonding and antibonding orbits of frequency $F_0\pm \Delta F$ (using $F_0=530$ and $\Delta F=90\mathrm{T}$ from the experimental data) with $n=2$ magnetic breakdown gaps [see Fig. 3b] and a characteristic breakdown field $B_{\mathrm{MB}}\approx 14\mathrm{T}$. The lower (blue) dotted curve is the simulated waveform for a nodal pocket split into bonding and antibonding orbits of frequency $F_0\pm 2\Delta F$ with $n=4$ magnetic breakdown gaps (see Fig. 4) and a characteristic breakdown field $B_{\mathrm{MB}}\approx 4\mathrm{T}$. Simulations are made using Eq. (2), in which the product $R_s{R}_D{R}_T$ [26] is a monotonic function of $B$, and a negative prefactor is used for the $n=2$ case. (b) Fourier transform (using a Blackman window) of the measured quantum oscillation data compared with simulated data. The three experimental low frequencies are seen to be reproduced by the simulation with the right ratio of amplitudes.

Figure 2

(a) A schematic of two ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ unit cells where the ${\mathrm{CuO}}_2$ bilayers are shaded. Hopping matrix elements within ($t_{\perp }$) and between ($t_c$) bilayers is shown. (b) A contour plot of $t_{\perp }(\mathbf{k})$ within the ${\mathrm{CuO}}_2$ planes for the simplified model expressed as Eq. (1), in which case the bilayer coupling vanishes at the nodes along which $|k_x|=|k_y|$ (shown by dotted gray lines).

Figure 3

A schematic nodal hole pocket before (a) and after (b) bilayer splitting, showing each of the allowed orbits with frequencies $F_0\pm \Delta F$, and the additional orbit allowed by magnetic breakdown tunnelling (dotted line) with frequency $F_0$. The location of the nodal magnetic breakdown gap ${\epsilon }_g$ is shown by arrows.

Figure 4

A schematic of a diamond-shaped nodal pocket [6, 24] split by bilayer coupling, yielding 4 magnetic breakdown junctions with each of the possible orbits shown. (a) shows the two orbits (in this case $F_0-2\Delta F$ and $F_0+2\Delta F$) yielded by bilayer splitting of the original pocket; (b), (c), and (d) show additional allowed orbits due to nodal magnetic breakdown tunneling, yielding frequencies $F_0-2\Delta F$, $F_0-\Delta F$, $F_0$, $F_0+\Delta F$, and $F_0+2\Delta F$ (the small breakdown field in the simulations of Fig. 1 causes $F_0\pm 2\Delta F$ to be weak in amplitude).