Quantum Oscillations in the Underdoped Cuprate ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$

We report the observation of quantum oscillations in the underdoped cuprate superconductor ${\mathrm{YBa}}_2{\mathrm{Cu}}_4{\mathrm{O}}_8$ using a tunnel-diode oscillator technique in pulsed magnetic fields up to 85 T. There is a clear signal, periodic in inverse field, with frequency $660\pm 15\mathrm{T}$ and possible evidence for the presence of two components of slightly different frequency. The quasiparticle mass is $m^{*}=3.0\pm 0.3m_e$. In conjunction with the results of Doiron-Leyraud et al. for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.5}$, the present measurements suggest that Fermi surface pockets are a general feature of underdoped copper oxide planes and provide information about the doping dependence of the Fermi surface.

Figure 1

(a) Resonant frequency $f$ of the TDO versus magnetic field $B$ recorded during an 85 T pulse at $T=0.53\mathrm{K}$. Upper inset: ${\mu }_0{H}_{\mathrm{hys}}$ ($\circ$) and ${\mu }_0{H}_m$ (rising—▵, falling—▴) from this work and ${\mu }_0{H}_{\mathrm{irr}}$ (•) from Ref. 16. Lower inset shows expanded view of the ${\mu }_0{H}_m$ data at low $T$. The dashed lines are guides to the eye. (b) Second derivative $d^{2}f/d{B}^2$ of 1.6 K data (solid lines). The dashed line is given by LK theory assuming $\Delta f\propto dM/dB$ [10], and using a suitable scale factor.

Figure 2

Changes in resonant frequency $\Delta f$ of the tunnel-diode oscillator circuit versus $1/B$ recorded during 85 T pulses at various temperatures. A smooth monotonic background has been subtracted [13]. The dotted lines are equally spaced in $1/B$. The oscillatory signal is periodic in $1/B$ with frequency $F=660\pm 15\mathrm{T}$.

Figure 3

(a) Fast Fourier transforms in $1/B$ of $\Delta f(B)$ over the range $60<B<85\mathrm{T}$. The red (black) lines show data for the rising (falling) part of the pulse. A single peak is present in the FFTs with a frequency $F=660\pm 15\mathrm{T}$. An extra, less-reproducible, peak near 200 T has been removed from some of the FFTs by subtracting a slowly varying background. (b) FFT amplitude versus $T$. Open (closed) symbols show rising (falling) field data. The solid line shows the LK damping factor $R_T$ with best-fit value $m^{*}=3.0\pm 0.3m_e$. (c) amplitude of oscillatory function of the form $\sin (2\pi F/B+\varphi )$ with $F=660\mathrm{T}$ fitted to $\Delta f(B)$ in the range $67<B<77\mathrm{T}$. The best-fit $R_T$ curve, shown by a solid line, has $m^{*}=3.1\pm 0.3m_e$. Dashed lines show the LK formula for $m^{*}=2.5m_e$ and $m^{*}=3.5m_e$.

Figure 4

(a) $m^{*}/m_e$ values (⊡) for Y124 (this work), O-II Y123 [1] and ${\mathrm{Ca}}_2{\mathrm{CuO}}_2{\mathrm{Cl}}_2$ (CCOC), the latter from the dispersion of Cu-O orbital states well below the chemical potential measured by ARPES [23]. CCOC is a parent Mott insulator with $p=0$ and no FS. The dashed line is a guide to the eye. (b) Heat capacity anomaly $\delta \gamma$ at $T_c$ for various ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ samples (⊙) and the pseudogap energy $E_G$ (⊡) extracted from the same heat capacity data using a triangular gap model [24]. $\delta \gamma$ at $T_c$ is also shown (▵) for Y124 [26].