Magnetic Quantum Oscillations in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathbf{O}}_{6.61}$ and ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathbf{O}}_{6.69}$ in Fields of Up to 85 T: Patching the Hole in the Roof of the Superconducting Dome

We measure magnetic quantum oscillations in the underdoped cuprates ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ with $x=0.61$, 0.69, using fields of up to 85 T. The quantum-oscillation frequencies and effective masses obtained suggest that the Fermi energy in the cuprates has a maximum at hole doping $p\approx 0.11–0.12$. On either side, the effective mass may diverge, possibly due to phase transitions associated with the $T=0$ limit of the metal-insulator crossover (low-$p$ side), and the postulated topological transition from small to large Fermi surface close to optimal doping (high $p$ side).

Figure 1

(a) PDC frequency $f$ versus field $B$ for single crystals of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.69}$ (upper blue trace) and ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.61}$ (lower red trace); $T=1.5\mathrm{K}$ for both. The drop in $f$ corresponds to the irreversibility field [4, 5]. MQOs are visible at high fields for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.61}$. (b) $B$ versus time $t$ profiles for the 60 T Long-pulse and 85 T Multi-shot magnets. (c) SQUID data for the two samples from (a), yielding $T_c$.

Figure 2

(a) PDC resonant frequency $f$ for a ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.61}$ crystal after background subtraction to leave the oscillatory component $\Delta f$; the trace (thick black curve) is an average of three magnet sweeps ($T=1.5\mathrm{K}$). The thinner red line is a fit of Eq. (1) for MQO frequencies 589 and 479 T with $\tau \approx 0.07\mathrm{ps}$. (b) Fourier transform of data in (a) (black thick curve) using a Hann window; the large peak is centered on 570 T. The red curve is a sum of two Gaussians (fine green lines) at 466 and 593 T. (c) Plot of MQO amplitude $A$ divided by $T$ versus $T$; diamonds are from the upsweep of $B$ and dots from the downsweep. The curve is a fit of Eq. (2), giving $m^{*}=1.6\pm 0.1m_e$. (d) Residual [i.e., (data)-(fit)] from (a) versus field (black thicker line). The thinner red curve is a fit of Eq. (1) for a single MQO with $F=270\pm 20\mathrm{T}$.

Figure 3

(a) PDC resonant frequency for a ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.69}$ crystal after background subtraction to leave the oscillatory component $\Delta f$; the trace is a smoothed average of three magnet sweeps ($T=1.5\mathrm{K}$). (b) Oscillation index versus reciprocal magnetic field for the MQOs in (a) (points); dips in $\Delta f$ are indexed by integers and peaks by half integers. The straight line is a fit with a gradient of $550\pm 20\mathrm{T}$.

Figure 4

(a) Summary of MQO frequencies versus $p$ for underdoped cuprates: for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$, ◁ are from Ref. 8, ◇ from Ref. 2, and ○ from this work; $T_c$ and $x$ values are converted to $p$ using Ref. 13. ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$ data from Refs. 3, 4 are squares; the horizontal bar is the spread in $p$ values given for ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$ [3, 4]. Solid symbols (e.g., ●) show the dominant ($F_{\alpha }$) frequency obtained from Fourier analysis; open symbols are from more detailed analyses [e.g., Figs. 2a, 2d or Refs. 2, 19]. (b) Effective mass of the dominant ($\alpha$) MQO frequency $F_{\alpha }$ versus $p$; symbols are the same as in (a) except ◇ are from Ref. 7. (c) Fermi energy from $F_{\alpha }$ and $m^{*}$; for $p$s where several values are given, we take the average. Points are data and the curve is a parabolic fit.