Compensated electron and hole pockets in an underdoped high-$T_{\text{c}}$ superconductor

We report quantum oscillations in the underdoped high-temperature superconductor ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ over a wide range in magnetic field $28\le {\mu }_{0}H\le 85\text{T}$ corresponding to $\approx 12$ oscillations, enabling the Fermi surface topology to be mapped to high resolution. As earlier reported by Sebastian et al. [Nature (London) 454, 200 (2008)], we find a Fermi surface comprising multiple pockets, as revealed by the additional distinct quantum oscillation frequencies and harmonics reported in this work. We find the originally reported broad low-frequency Fourier peak at $\approx 535\text{T}$ to be clearly resolved into three separate peaks at $\approx 460$, $\approx 532$, and $\approx 602\text{T}$, in reasonable agreement with the reported frequencies of Audouard et al. [Phys. Rev. Lett. 103, 157003 (2009)]. However, our increased resolution and angle-resolved measurements identify these frequencies to originate from two similarly sized pockets with greatly contrasting degrees of interlayer corrugation. The spectrally dominant frequency originates from a pocket (denoted $\alpha$) that is almost ideally two-dimensional in form (exhibiting negligible interlayer corrugation). In contrast, the newly resolved weaker adjacent spectral features originate from a deeply corrugated pocket (denoted $\gamma$). On comparison with band structure, the $d$-wave symmetry of the interlayer dispersion locates the minimally corrugated $\alpha$ pocket at the “nodal” point ${\mathbf{k}}_{\text{nodal}}=\left(\pi /2,\pi /2\right)$, and the significantly corrugated $\gamma$ pocket at the “antinodal” point ${\mathbf{k}}_{\text{antinodal}}=\left(\pi ,0\right)$ within the Brillouin zone. The differently corrugated pockets at different locations indicate creation by translational symmetry breaking—a spin-density wave has been suggested from the suppression of Zeeman splitting for the spectrally dominant pocket. In a broken-translational symmetry scenario, symmetry points to the nodal $\left(\alpha \right)$ pocket corresponding to holes, with the weaker antinodal $\left(\gamma \right)$ pocket corresponding to electrons—likely responsible for the negative Hall coefficient reported by LeBoeuf et al. [Nature (London) 450, 533 (2007)]. Given the similarity in $\alpha$ and $\gamma$ pocket volumes, their opposite carrier type and the previous report of a diverging effective mass in Sebastian et al. [Proc. Nat. Am. Soc. 107, 6175 (2010)], we discuss the possibility of a secondary Fermi surface instability at low dopings of the excitonic insulator type, associated with the metal-insulator quantum critical point. Its potential involvement in the enhancement of superconducting transition temperatures is also discussed.

Figure 1

(Color online) (a) A schematic of the small hole (blue) pocket $\alpha$ and small electron (red) pocket $\gamma$, illustrating greatly contrasting degrees of corrugation and approximate locations within the sector of the Brillouin zone bounded by (0,0), $\left(\pi /2,\pi /2\right)$, and $\left(\pi ,0\right)$, shown for $-2\pi <k_z<2\pi$. Also shown in semitransparent fashion is the larger hole orbit $\beta$ [corresponding to a separate Fermi surface section or potentially a magnetic breakdown orbit (Ref. 12)]. The degree of corrugation is determined from angle-dependent Fourier transforms shown in Fig. 3 and frequencies shown in Fig. 4a. On using the Onsager relation, the three pockets collectively yield an effective hole doping of $p=11.7\pm 0.2\mathrm{\%}$. (b) Estimates of the ratio of the exciton binding energy to kinetic energy ${\text{Ry}}^{\ast }/E_{\text{ke}}$ and effective exciton separation $r_{\text{s}}$ as a function of oxygen concentration $x$, using the expressions in Appendix and effective masses from Refs. 16, 17, with arrows indicating the appropriate axes. Also shown are the corresponding values of $T_{\text{c}}$ (renormalized) from Ref. 7 and different shadings representing the insulating and metallic regimes at low $T$.

Figure 2

(Color online) (a) Quantum oscillations made using the contactless conductivity method on a sample of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6.54}$ over a broad range of magnetic fields $28<{\mu }_{0}H<85\text{T}$ at $\approx 1.5\text{K}$, corresponding to $\approx 12$ oscillations of the prominent $\alpha$ frequency. The experimental data below and above 65 T are from measurements made using the same sample on the same probe in two different pulsed magnets. (b) A Fourier transform of the quantum oscillations both without (thick black curve) and with (thin red curve) a Hann window applied. The inset shows a comparison of the separation in frequency $F_{\gamma ,\text{neck}}-F_{\alpha }$ or $F_{\alpha }-F_{\gamma ,\text{belly}}$ (shown by the horizontal dotted line) with the frequency resolution limit $\Delta F_{\text{lim}}\approx \left[1/\Delta \left(1/{\mu }_{0}H\right)\right]$ (left axis), as a function of the number of $\alpha$ oscillation periods $n_{\alpha }$ (bottom axis). The frequency separation exceeds the frequency resolution for $n_{\alpha }\gtrsim 8$, enabling all three frequencies to be clearly resolved in Fourier transforms and fits of the quantum oscillations. All three frequencies are found to be present in a Fourier transform only at high enough resolution (i.e. when the number of oscillations exceeds eight), and irrespective of the choice of window function (e.g. Boxcar, Hann, Blackman), confirming that they are genuine spectral features.

Figure 3

(Color online) (a) Fourier transform before (thin purple) and after (thick magenta) subtracting the dominant $\alpha$ component of an unconstrained fit to Eq. (1) (dot-dashed green line in the upper inset) to the quantum oscillations (black line in the upper inset) measured in a sample of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ with $x=0.56$ at $T\approx 1.5\text{K}$ using the contactless conductivity technique (see Appendix ). The weaker $F_{\gamma ,\text{neck}}$ and $F_{\gamma ,\text{belly}}$ frequency peaks in the Fourier transform are clearly visible after the dominant oscillatory component of the oscillations has been subtracted (unlike expected behavior of sidelobes, that would vanish along with the subtraction of the central dominant frequency). A polynomial is used to fit the background and the waveform is modulated by a Hann window prior to Fourier transformation. The prominent oscillations and harmonics are indicated. Extremal Fermi surface cross sections given by $A_k=\left(2\pi e/\hslash \right)F$ are obtained from the observed frequencies $F_{\alpha }$, $F_{\gamma ,\text{neck}}$, and $F_{\gamma ,\text{belly}}$ and harmonics $2F_{\alpha }$, $2F_{\gamma ,\text{neck}}$, and $2F_{\gamma ,\text{belly}}$ as indicated; here $H=\left|\mathbf{H}\right|$. The red line (solid gray online) in the upper inset is an unconstrained fit to Eq. (1), where the dot-dashed green line and the dashed blue lines are the $\alpha$ and $\gamma$ components, respectively. The purple curve (upper inset, lower panel) shows the result of subtracting the $\alpha$ component (dot-dashed green line) of the unconstrained fit to Eq. (1) from the raw data, revealing a distinctive beat pattern in the lower panel of the inset. The vertical lines represent the fields at which nodes occur for $n=2,3,4$ from Eq. (3) using the fit value of $\Delta F_{\gamma ,0}$. The lower inset shows the temperature dependences of the $F_{\alpha }$, $F_{\gamma ,\text{neck}}$, and $F_{\gamma ,\text{belly}}$ amplitudes (the latter two renormalized to equalize the amplitudes at the highest temperature), indicating similar effective masses $m^{\ast }$ for the $\alpha$ and $\gamma$ pockets. A fit to $\Delta f=a_0{R}_T$ (Ref. 21) yields $m_{\alpha }^{\ast }=1.56\pm 0.05m_{\text{e}}$, $m_{\gamma ,\text{neck}}^{\ast }=m_{\gamma ,\text{belly}}^{\ast }=1.6\pm 0.2m_{\text{e}}$ (where $m_{\text{e}}$ is the free electron mass). [(b) and (c)] Examples of Fourier transforms of oscillations measured in a sample with $x=0.54$ at $T\approx 1.5\text{K}$ and different angles $\theta$ [data in Figs. 4b, 4c] before and after subtracting the fitted $\alpha$ component [see Fig. 4b] as described in the text (expanded to show the finer high frequency structure)—further Fourier transforms before and after subtraction are shown in the Appendix . Note that the higher frequency $F_{\beta }$ (when observed) is $3.14\pm 0.05$ times larger than $F_{\alpha }$, inconsistent with a harmonic of $F_{\alpha }$ (Refs. 12, 16).

Figure 4

(Color online) (a) The product $F\text{cos}\theta$ (for both fundamental and harmonic—solid symbols for unsubtracted data and hollow symbols for subtracted data) obtained from the Fourier analysis shown in Figs. 3b, 3c and Appendix plotted versus $\theta$. $F_{\gamma ,\text{neck}}$ and $F_{\gamma ,\text{belly}}$ are obtained by Fourier analysis after subtracting the fitted $\alpha$ component [using Eq. (1)]. For clarity, the data have been symmetrized with respect to $\pm \theta$: both the swept $\theta$ data [shown in Fig. 6a] and measurements made over extensive range of $\theta$ in Ref. 14 indicate the oscillations to be symmetric in $\pm \theta$, as expected for a detwinned single-phase orthorhombic crystal. Lines indicate the expected $\theta$ dependences according to Eqs. (2, 4), using the fitted parameters in Fig. 3a. Consistency with Eq. (4) indicates a deeply corrugated $\gamma$ section of Fermi surface (Refs. 22, 23). The harmonics are obtained directly from peaks in Fourier transforms of the raw data—consistency between fundamentals and harmonics confirms our finding of similarly sized but distinctly different $\alpha$ and $\gamma$ Fermi surface sections. (b) Examples of quantum oscillations measured in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ with $x=0.54$ at different angles $\theta$, as indicated (where $\phi \approx 0$) together with the fitted $\alpha$ component [green lines, using Eq. (1)] as described in the text. (c) The quantum oscillations measured in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ with $x=0.54$ at different angles $\theta$ after subtracting the green fits in (b), yielding residuals dominated by the $\gamma$ pocket oscillations. Curves are offset according to $\theta$ (right axis). Also plotted (red lines) are the predicted values of $H_n$ from Eqs. (3, 4) using the fit value of $\Delta F_{\gamma ,0}$.

Figure 5

(Color online) (a) Schematic unreconstructed Fermi surface depicting the translational vectors ${\mathbf{Q}}_{\text{s},\pm }$, the line thickness (obtained by a perturbation in $r$ as described in the text) represents the interlayer dispersion $k_z$ responsible for corrugation and is seen to be greater in the antinodal $\left(\pm \pi ,0\right)$, $\left(0,\pm \pi \right)$ compared to the nodal $\left(\pm \pi /2,\pm \pi /2\right)$ region. (b) Representative reconstructed Fermi surface according to models used in Refs. 12, 27, 28, 29 (described in the text and Appendixes ). Blue and red lines correspond to the nodal hole pockets and antinodal electron pocket, respectively. The line thickness (obtained by a perturbation in $r$) represents the interlayer dispersion $k_z$ and is significantly bigger for the antinodal compared to the nodal pocket. (c) Reconstructed Fermi surface consisting of only a single type of pocket $\left(\beta \right)$ after increasing the coupling ${\Delta }_{\text{s}}$ (neglecting ${\Delta }_{\text{c}}$). Antiferromagnetic bilayer coupling (Ref. 19) (shown in Fig. 10 of Appendix ) would yield two slightly different variants of the Fermi surfaces in (b) and (c), but with very similar pocket areas. (d) Schematic dispersion prior to Fermi surface reconstruction depicted in one dimension. (e) Schematic dispersion of the three pockets. (f) Schematic showing the opening of a gap between the $\alpha$ and $\gamma$ pockets, as expected to result from an excitonic insulator instability. Here the charge potential ${\Delta }_{\text{c}}$ is not included; its effect on the $\beta$ orbit (or remnants thereof, shown as a dotted line) will depend on its strength and on the specifics of the superlattice (see Appendix ). The effects of bilayer splitting are not shown here for clarity, but are discussed in Appendix .

Figure 6

(Color online) (a) Examples of angle-swept measurements made from positive to negative values of $\theta$ on a sample of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ with $x=0.56$ at fixed field ${\mu }_{0}H=45\text{T}$, for different in-plane orientations $\left(\varphi \right)$ of the in-plane component of the magnetic field ${\mathbf{H}}_{\parallel }=\left(H\text{sin}\theta \text{cos}\varphi ,H\text{sin}\theta \text{sin}\varphi ,0\right)$. Curves shown correspond to $-54\le \varphi \le 156°$ (bottom to top) in steps of $15°$. The gap in angular data around $\theta =0$ is due to a correction made for a small misalignment of the crystal, explained in Appendix . (b) The $\varphi$ dependence of the caliper radius $k_{\parallel }$ of the $\gamma$ pocket determined from fits [see Appendix , Eq. (E1)] of superimposed $\alpha$ and $\gamma$ oscillation waveforms to the data in (a)—using values of $\Delta F_{\gamma ,0}$ and ${\Gamma }_{\alpha }$ obtained from fits to the field swept data. Fit frequencies are given in Appendix . The full $360°$ $\varphi$ rotation is inferred from performing a rotation over $180°$ in $\varphi$ combined with fits to $\pm \theta$, with only $k_{\parallel }$ allowed to vary. The magenta line is the $\varphi$ dependence of $k_{\parallel }$ expected for the single ${\mathbf{Q}}_{\text{s},\pm }$ model shown in Fig. 5b. (c) An example fit (red line—solid gray online) to $\varphi =51°$ (equivalent to $231°$) data (black line), with further fits shown in Appendix .

Figure 7

(Color online) (a) Quantum oscillations measured as a function of the angle of inclination to the crystalline $c$ axis ($\theta$) on a sample of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ ($x$=0.56) at a constant field ${\mu }_{0}H$=45 T and a fixed in-plane angle $\varphi$ at $T$=1.5 K using the contactless conductivity technique. Periodicity in $1/B\cos \theta$ of both the fundamental (dotted pink lines) and harmonic (solid green lines) oscillations (dominated by the $\alpha$ frequency) are revealed. The gray vertical lines label the expected peak positions for fundamental $\alpha$ quantum oscillations periodic in $1/B\cos \theta$ with periodicity $F_{\alpha }$, and the dotted vertical cyan lines label the expected peak positions for the corresponding $2F_{\alpha }$ harmonic. The harmonic oscillations are extracted by subtracting the fundamental oscillations fit using Eq. (E1) from the measured oscillations as a function of $\theta$ (fit and measurements shown in Figs. 6, 9). A residual background has been further subtracted for visual clarity.

Figure 8

(Color online) (a)–(d) Fourier transforms of the oscillations measured in ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ with $x=0.54$ at different angles $\theta$ before (thin mauve line) and after (thick magenta line with gray shading) subtracting the $\alpha$ component of a fit to Eq. (1) of the main text—a subset of which are shown in Figs. 3b, 3c of the main text. The resolved $\gamma$ frequencies are used to construct Fig. 4a of the main text. Frequency peaks corresponding to 2F harmonic oscillations are observed in the Fourier transform at all measured angles.

Figure 9

(Color online) (a) Swept $\theta$ oscillations measured in a sample of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ with $x=0.56$ at constant field ${\mu }_{0}H=45\text{T}$ and different fixed values of $\varphi$ at $T\approx 1.5\text{K}$ fit to Eq. (E1) (dashed black lines) as described in Fig. 6. The gaps in the data result from a correction for a sample misalignment of $\approx 7°$ (see experimental details). (b) Fourier transforms in $1/{\mu }_{0}H\text{cos}\theta$ of the oscillations shown in (a), which are dominated by the $\alpha$ pocket. As expected for a nearly ideal two-dimensional Fermi surface section, the dominant $\alpha$ frequency obtained is the same as that from fixed $\theta$ swept magnetic field experiments [i.e., Fig. 6a] at all angles of rotation. Periodicity of the oscillations in $1/B\text{cos}\theta$ is shown in Appendix . The curves correspond from bottom to top to $-54\le \varphi \le 156°$ in $15°$ steps.

Figure 10

(Color online) Calculated Fermi surfaces in which the effects of bilayer splitting are included as a perturbation in $r$ (here $r$ is the hopping “range” parameter discussed in Sec. 2B, see Ref. 19). The effects of corrugation are not shown for clarity, but are the same as in Fig. 5. The difference in area between different bilayer-split variants of the Fermi surface is represented by the thickness of the line, where the colors (and “$+$” and “$-$” labels) indicate the sign of this difference. (a) shows the translation of the unreconstructed Fermi surface where the coupling between bonding and antibonding bands is represented by different signs for the difference in area (black for “$-$” and gray for “$+$”) between ${\epsilon }_{\mathbf{k}}$ and ${\epsilon }_{\mathbf{k}+\mathbf{Q}}$. (b) shows the reconstructed Fermi surface where the difference in area (red for “$+$” and pink for “$-$” for the electron pocket, and blue for “$+$” and cyan for “$-$” for the hole pocket), can be seen to cancel for each pocket.

Figure 11

(Color online) Error analysis of unconstrained oscillation fit to two Fermi surface sections. (a) Least squares error of the unconstrained fit of Eq. (1) to the measured oscillations of the sample with oxygen concentration $x=0.56$ as a function of different ratios between the corrugation of the $\alpha$ and $\gamma$ pockets is shown on the left-hand axis. The corresponding difference between the fit corrugations of the $\alpha$ and $\gamma$ pockets is shown on the right-hand axis. A clear minimum is observed for $\Delta F_{\gamma ,0}/\Delta F_{\alpha ,0}\approx 6.2$, yielding the best fit parameters—while the error is seen to rapidly increase if the corrugations of the two pockets are similar. (b) Example best fit (red line) to the measured oscillations for the scenario where $\Delta F_{\gamma ,0}\approx \Delta F_{\alpha ,0}$. The fit does not yield good agreement, demonstrating the need for two low frequency Fermi surface sections of significantly different warpings to explain the measured oscillations.

Figure 12

(Color online) Swept $\theta$ oscillations measured by contactless conductivity on a sample of ${\text{YBa}}_2{\text{Cu}}_3{\text{O}}_{6+x}$ with $x=0.54$ at constant field $H=45\text{T}$ and different fixed values of $\varphi$ at $T\approx 1.5\text{K}$.

Figure 13

The left-hand panel shows the Fermi surface calculated for a helical spin-density wave model. Using a hole doping $p=11.7\mathrm{\%}$ and $\delta \approx \frac{1}{16}\approx 0.06$ as constraints, adjustment of a single parameter ${\Delta }_{\text{s}}$ yields a Fermi surface with frequencies comparable to those observed—i.e., $F_{\alpha ,0}=575\text{T}$, $F_{\beta ,0}=1583\text{T}$, and $F_{\gamma ,0}=517\text{T}$. The largest difference from measured area for the case of the $\beta$ pocket corresponds to 0.2% of the paramagnetic Brillouin zone area. The helical spin-density wave model used is described in Ref. 12. On including an ortho-II potential (right-hand panel), using the formalism described in Ref. 29, pockets with frequencies—$F_{\alpha ,0}=517\text{T}$, $F_{\beta ,0}=1641\text{T}$, and $F_{\gamma ,0}=515\text{T}$—closer to those observed experimentally are obtained in the right-hand panel. The largest difference from measured area for any one pocket now corresponds to 0.07% of the Brillouin zone.

Figure 14

(Color online) The actual Brillouin zone for Fermi surface reconstruction by ${\mathbf{Q}}_{\text{s}}=\left(\pi \left[1\pm 2\delta \right],\pi \right)$, assuming $\delta =\frac{1}{16}$, illustrating the possible superlattice modulation vectors ${\mathbf{Q}}_{\text{c}}$ and ${\mathbf{Q}}_{\perp }$ accompanying an excitonic insulator instability.

Figure 15

(a) Experimentally measured angular dependence of the dHvA frequencies in ${\text{LaRhIn}}_5$. (b) Band structure calculations of the Fermi surface topology of the $\alpha$ pockets in ${\text{LaRhIn}}_5$ (taken from Ref. 71).

Figure 16

(a) In-plane Fermi surface cross section of $\beta -{\left(\text{BEDT}-\text{TTF}\right)}_2{\text{IBr}}_2$ from Ref. 22. (b) $\theta$ dependence of the difference frequency $\Delta F_{1,3}=F_{\text{belly}}-F_{\text{neck}}$ (also from Ref. 22) for two different $\varphi$ angles, which are $90°$ apart according to (a).