Spin-Order Driven Fermi Surface Reconstruction Revealed by Quantum Oscillations in an Underdoped High $T_c$ Superconductor

We use quantum oscillation measurements to distinguish between spin and orbital components of the lowest energy quasiparticle excitations in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.54}$, each of which couple differently to a magnetic field. Our measurements reveal the phase of the observed quantum oscillations to remain uninverted over a wide angular range, indicating that the twofold spin degeneracy of the Landau levels is virtually unaltered by the magnetic field. The inferred suppression of the spin degrees of freedom indicates a spin-density wave is responsible for creation of the small Fermi surface pockets in underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$—further suggesting that excitations of this phase are important contributors to the unconventional superconducting pairing mechanism.

Figure 1

Magnetic quantum oscillations in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.54}$. (a) The solid line shows quantum oscillations in the resonance frequency ($f$) of a tunnel diode oscillator (TDO) to which the sample is coupled inductively, which is proportional to the change in sample in-plane resistivity. For clarity, only the frequency shift ($\Delta f$) is shown. The nonmonotonic field dependence is likely due to a neighboring frequency of identical mass [15, 16, 37]. (b) Fourier analysis, in which the oscillations are multiplied by a Hann window function after background subtraction, yield a single prominent peak at all orientations $\theta$. (c) Magnetic field orientation dependence of the quantum oscillation Fourier frequency (circles and left axis) as determined from a fast Fourier transform (FFT) of the measured oscillations. Also plotted (squares and right axis), is the experimentally determined effective mass obtained by performing fits of the temperature-dependent quantum oscillation amplitude to the Lifshitz-Kosevich expression at different angles in Fig. 2. The $\theta$-dependent frequency and effective mass are further fit by an angular dependence $\propto 1/({cos}^2\theta +\frac{1}{r^2}{sin}^2\theta )$ (green or gray line), yielding an approximately cylindrical $\alpha$ Fermi surface orbit with $m_0^{*}=1.7\pm 0.1m_e$ and $F_{\alpha }\approx 520T$.

Figure 2

Effective mass determination. The data and fits of the temperature-dependent quantum oscillation amplitude to the Lifshitz-Kosevitch expression ($A=A_{0}X/\sinh X$, where $X=2{\pi }^2{m}^{*}{k}_{B}T/\hslash eB$ [6]) are shown together with the fitted effective masses $m_{\theta }^{*}$ and the angles $\theta$ at which each measurement was made.

Figure 3

Magnetic field orientation dependence of the quantum oscillations in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.54}$. (a) A plot of anticipated “spin zero” angles (blue or gray curves) as a function of $\gamma$ determined by setting $\cos (\pi \gamma \frac{m_0^{*}}{m_e}\frac{B}{B_{\perp }})=0$, so that the angle of the $n$th zero is given by Eq. (1). For a Wilson ratio $\gamma =1.1$ [24], given the experimental $m_0^{*}=1.7m_e$, the first spin zero would be expected to occur at $|\theta |\approx 42°$. (b) Quantum oscillations plotted versus the Landau level index $\nu =F_{\theta }/B$ (offset for clarity after polynomial background subtraction) shows that the relative phase of the oscillations is invariant for successive values of $\theta$. The absence of spin zeros indicates that $\gamma$ is vanishingly small (the only finite values of $\gamma$ that cannot be ruled out are in the low range $0.3<\gamma <0.5$). The inset shows the contrasting case of a charge-density wave material $\alpha \mathrm{\text{−}}(\mathrm{BEDT}\mathrm{\text{−}}\mathrm{TTF}{)}_2\mathrm{KHg}(\mathrm{SCN}{)}_4$, where a “spin zero” accompanied by phase inversion of the oscillations is observed experimentally at an angle $\approx 43°$ [30]. (c) Angular dependent amplitude of $F_{\alpha }$ at $B\approx 40\mathrm{T}$, extracted from the fast Fourier transforms of the data. The line shows the anticipated orientation-dependent amplitude according to $A_{\alpha }{R}_{\mathrm{S}}\exp (-\Gamma /B\cos \theta )$ at 40 T for $R_{\mathrm{S}}=1$ (i.e., $\gamma =0$), where the adjustable parameter $\Gamma$ is a damping constant, yielding reasonable agreement with experimental amplitude.

Figure 4

Example of the effect of Zeeman splitting on Fermi surface reconstruction. We consider the simple case of a hole pocket at ($\pi /2$, $\pi /2$) created by commensurate ordering with $\mathbf{Q}=(\pi ,\pi )$ for illustrative purposes. (a) Schematic of a twofold degenerate holelike Fermi surface pocket in the absence of a magnetic field, intersected by a Bragg reflection plane (diagonal line). (b) Schematic showing the Zeeman splitting of the twofold degeneracy (if spin up and spin down are individually well defined so that $\gamma =1$). The red or center shaded area denotes a spin-up Fermi surface while the blue or outer shaded area denotes a spin-down Fermi surface. (c) Schematic showing how the pocket size remains largely invariant to Zeeman splitting if the order parameter couples up and down spins (as in the case of a spin-density wave, so that $\gamma =0$). The purple (shaded) areas show Fermi surfaces comprising spins where the effective moment has been suppressed, potentially due to spin flips at Bragg reflection. While $\mathbf{Q}=(\pi ,\pi )$ is considered for this simple schematic, $\gamma \ll 1$ is expected for more general SDW wave vector scenarios [38].