Fermi surface reconstruction from bilayer charge ordering in the underdoped high temperature superconductor YBa₂Cu₃O_6+x

Our level of understanding of metals typically depends on the extent to which Fermi surface cross-sections observed in quantum oscillation experiments can be explained by comparison with band structure calculations, and the degree to which Fermi liquid behavior prevails [1–4]. Additional factors, such as a periodic modulation of the electronic states—caused, for example, by a spin- or charge-density wave—can introduce additional gaps, in which case the Fermi surface needs to be understood in terms of the original band structure translated by ordering wavevectors [4–7].

In the case of high temperature superconductors, a determination of the electronic structure is essential for understanding the origin of pairing. Yet, the small Fermi surface pocket observed in the underdoped cuprates YBa₂Cu₃O_6+x [8] and YBa₂Cu₄O₈ [9] cannot readily be explained in terms of the unmodified band structure [9–17]. The momentum-space cross-section of the experimentally observed pocket is roughly thirty times smaller than the size of the large hole Fermi surface sections expected for the CuO₂ planes from band structure calculations [18–20], with the cyclotron motion having apparently the opposite sense of rotation to that expected [10, 12]. A small pocket has been proposed to originate from band structure involving the mixing of the chain states with BaO orbitals in YBa₂Cu₃O_6+x [19, 20], but the relevant band is reported to lie significantly (≈0.6 eV) below the Fermi energy in photoemission experiments [11].

The emergence of long range charge order reported in underdoped YBa₂Cu₃O_6+x by nuclear magnetic resonance (NMR) [21] and x-ray scattering [22–24] may contribute to the observed difference between the small Fermi surface pocket measured and the large Fermi surface expected from band structure. A direct link, however, has yet to be established between the structure of the measured order and reconstruction of the large Fermi surface to yield the observed small pocket. Furthermore, some differences are reported in the ordering wavevectors seen in different experiments [21–24], and it remains to be settled whether the observed charge order in YBa₂Cu₃O_6+x is stripe-like [25–27] as opposed to biaxial [17, 28] in nature.

A satisfactory electronic structure model of underdoped YBa₂Cu₃O_6+x would be expected to explain the ordering wavevectors found in both NMR and x-ray experiments, while accounting for the measured distribution of bilayer-split frequencies in quantum oscillation experiments [16, 29], as well as other experimental observations relating to the electronic structure [10–12, 17, 30, 32]. A single layer scheme of Fermi surface reconstruction by biaxial charge order has previously been shown to yield nodal electron pockets [15, 17, 28, 34] consistent with experimental observations of the small size of the electronic heat capacity at high magnetic fields [30]⁴, the negative electrical transport coefficients [10, 12] and the predominantly nodal Fermi surface location inferred from photoemission, Fermi velocity and chemical potential oscillation measurements [11, 17, 32]. In this paper, we show that the introduction of bilayer coupling [33] to the single layer scheme [15, 17, 28, 34] leads to the possibility of a modulation period and Fermi surface pocket of different size corresponding to each of the bonding and antibonding bands, consistent both with experimental measurements of the charge ordering [21–24] and quantum oscillations [16, 29].

NMR experiments provide evidence for long range charge ordering in a magnetic field B⪆15 T [21] in underdoped YBa₂Cu₃O_6+x, while recent x-ray scattering experiments reveal ordering vectors that are two-dimensional (2D) [22–24]. The x-ray results are significant for two reasons. Firstly, they reveal the scattering intensity and correlation lengths to be equivalent along the a and b lattice directions of the orthorhombic unit cell [18], with the orthorhombicity and CuO chain oxygen ordering having no apparent effect in causing stripe domains to form preferentially along either a or b. Secondly, they reveal the charge ordering to be incommensurate with wavevectors $\textbf {Q}_a=(\frac {2\pi }{\lambda _a a},0,Q_z)$ and $\textbf {Q}_b=(0,\frac {2\pi }{\lambda _b b},Q_z)$ (of in-plane period λ_a = λ_b ≈ 3.2 lattice constants) consistent with a wavevector nesting the bilayer-split bonding band [22–24] (where Q_z refers to the c-axis component of the ordering vector). The periods reported in x-ray scattering experiments, however, differ from that (λ_a = 4.0) reported in NMR studies [21] of similar YBa₂Cu₃O_6+x samples and that (λ ≈ 4.2) [35] typically found by x-ray diffraction in the same region of the phase diagram in single layer spin-stripe ordered cuprates [36]. Furthermore, no obvious relationship is found between the charge ordering wave vectors found in x-ray experiments on YBa₂Cu₃O_6+x and low energy spin excitation wave vectors [37].

Given the unambiguous planar origin of the 2D charge ordering found in x-ray scattering experiments in YBa₂Cu₃O_6+x [24], we focus here on a model dispersion solely for the CuO₂ bilayers. While photoemission experiments have yet to identify signatures of band folding consistent with charge ordering, the wave vectors found using x-rays in YBa₂Cu₃O_6+x [22, 23], as well as those found earlier in doping-dependent scanning tunneling microscopy measurements in other cuprates [38, 39], suggest Fermi surface nesting at the antinodes [34, 38]. The opening of an antinodal gap (shaded light grey in figure 1) that is expected to accompany ordering at such wave vectors is consistent with a reconstructed Fermi surface that involves relative translations of the remaining arc-like segments of Fermi surface (represented by thick black lines in figure 1 [17, 28, 34]). These translated segments of Fermi surface would occupy the same regions in momentum space where 'Fermi arcs' are observed in photoemission experiments in underdoped cuprates [11, 39–41].

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The single layer unreconstructed Fermi surface produced using ε(k) = ε₀ + 2t₁₀[cos ak_x + cos bk_y] + 2t₁₁[cos(ak_x + bk_y) + cos(ak_x − bk_y)] + 2t₂₀[cos 2ak_x + cos 2bk_y] has a single nesting wavevector spanning the antinodal regions (i.e. |Q_a| = |Q_b| in figure 2(a)). On introducing bilayer splitting, two nesting vectors now span the bonding and antibonding bands such that |Q^B_a| = |Q^B_b| and |Q^AB_a| = |Q^AB_b| are inequivalent (shown in figure 2(b)). For the single-layer dispersion⁵, we choose t₁₁/t₁₀ = −0.32 and t₂₀/t₁₀ = 0.16 after Andersen et al [18] and Millis and Norman [25]. The bonding and antibonding bands are obtained by diagonalizing

$$\begin{equation} {\bf H}_{\rm bi}=\left(\begin{array}{@{}cc@{}}\varepsilon({\bf k}) & -t_\perp({\bf k})-t_{{\rm c}}\,{\rm e}^{+{\rm i}ck_z}\\ -t_\perp({\bf k})-t_{{\rm c}}\,{\rm e}^{-{\rm i}ck_z} & \varepsilon({\bf k})\\ \end{array} \right) \end{equation} \noindent \tag{ 1 }$$

after Garcia-Aldea and Chakravarty [33], where

$$\begin{eqnarray*} &&t_\perp({\bf k})=\frac{t_{\perp0}}{4}[\cos(ak_x)-\cos(bk_y)]^2 \end{eqnarray*}$$

and t_c represent intrabilayer and interbilayer couplings, respectively. On evaluating equation (1) one obtains

$$\begin{equation} \varepsilon({\bf k})^{\rm B,AB}=\varepsilon({\bf k})\mp\sqrt{t_\perp({\bf k})^2+t_{{\rm c}}^2+2t_\perp t_{{\rm c}}\cos ck_z}, \end{equation} \tag{ 2 }$$

for the bonding and antibonding bands (shown for cos ck_z = 0 in figure 2(b), where c refers to the c-axis lattice parameter. Setting t_⊥0/t₁₀ ≈ 0.5 and t_c/t_⊥0 = 0.25 produces a bilayer splitting at the antinodes ($\textbf {k}_{\mathrm { antinode}}=(\pm \frac {\pi }{a},0,k_z)$ and $(0,\pm \frac {\pi }{a},k_z)$) resembling that in bandstructure calculations [19, 20] in YBa₂Cu₃O_6+x, but with a reduced nodal gap (ε_g ≈ 2t_c) that falls within the experimental upper bound from photoemission [41] and magnetic quantum oscillation measurements [16].

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The corrugation of the unreconstructed bilayer Fermi surface with respect to the interlayer momentum k_z (shown in figure 3) caused by t_⊥0 and t_c implies that Fermi surface nesting is optimized by acquiring a Q_z = π component to the wave vector. Such a component produces a reconstructed Fermi surface in which the fundamental component of the corrugation is suppressed, leaving a small residual second harmonic corrugation of order t²_c/t_⊥0 (originating from the square root in equation (2)). Hard x-ray scattering experiments find direct evidence for an interlayer component to the ordering vector [12]. Furthermore, quantum oscillation experiments performed over a broad range of magnetic field and angles are found to be consistent with a bilayer-split Fermi surface that exhibits a comparatively weak corrugation [15, 16]. The strong suppression of the interlayer optical conductivity within the underdoped regime [42] could be a further signature of Fermi surface corrugation suppressed by nesting. The small size of the residual corrugation term (which we estimate later in the manuscript) compared to the other energy scales enables us to consider a simplified 2D approximation to the Fermi surface, which we obtain by neglecting the k_z-dependence in equation (2).

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To obtain the full set of reconstructed bands, the relevant couplings (V^B,AB_a,b) between bands with relative translations by ±Q^B_a, ±Q^B_b, ±Q^AB_a and ±Q^AB_b need to be considered (including both the bonding and antibonding bands given by equation (2)), requiring the construction of a Hamiltonian in the form of a matrix (see e.g., [25, 33]). Orthogonality between the Bloch wavefunctions of the bonding and antibonding bands ensures that charge-density modulations form independently for those two bands, with the coupling between them being a second order effect⁶. The strongest coupling V_a,b(k') is therefore expected to occur between states within each of the bands translated by wavevectors intrinsic to that band—i.e. between ε(k)^B and ε(k ± Q^B_a,b)^B and between ε(k)^AB and ε(k ± Q^AB_a,b)^AB.

While the complete set of bands for strictly incommensurate order (in which λ^B and λ^AB are irrational) requires an infinite number of translations and a matrix of infinite rank, which cannot be diagonalized, it was shown in  [34] that one can approximate incommensurate behavior by considering rational values of λ (in which λ is the ratio of two integers)—yielding large but manageable matrices. Provided 0 ≪ |V_a,b(k')| < |t₁₀|, the Fermi surface consists of multiple repetitions of the same electron pocket throughout the Brillouin zone [34]. Here we go a step further in simplification by considering only the minimal number of terms necessary to produce a single instance of the electron pocket for each band (e.g. those illustrated in figure 1). Hence,

$$\begin{eqnarray} \fl H^{\rm B,AB}= \left( \begin{array}{@{}cccc@{}} \varepsilon({\bf k})^{\rm B,AB} & V_a({\bf k})^{\rm B,AB} & V_b({\bf k})^{\rm B,AB} & 0\\ V_a({\bf k})^{\rm B,AB} & \varepsilon({\bf k}+{\bf Q}_a^{\rm B,AB})^{\rm B,AB} & 0 & V_b({\bf k}+{\bf Q}_a^{\rm B,AB})^{\rm B,AB}\\ V_b({\bf k})^{\rm B,AB} & 0 & \varepsilon({\bf k}+{\bf Q}_b^{\rm B,AB})^{\rm B,AB} & V_a({\bf k}+{\bf Q}_b^{\rm B,AB})^{\rm B,AB}\\ 0 & V_b({\bf k}+{\bf Q}_a^{\rm B,AB})^{\rm B,AB} & V_a({\bf k}+{\bf Q}_b^{\rm B,AB})^{\rm B,AB} & \varepsilon({\bf k}+{\bf Q}_a^{\rm B,AB}+{\bf Q}_b^{\rm B,AB})^{\rm B,AB}\end{array} \right),\nonumber\\ \end{eqnarray} \noindent \tag{ 3 }$$

which must be diagonalized for both bonding and antibonding bands. With these simplifications, we can estimate the size of the pockets, their effective masses and contributions to the electronic DOS for arbitrary values of λ^B or λ^AB—bearing in mind that the complete Fermi surface will have these same pockets repeated by ±Q^B_a, ±Q^B_b, ±Q^AB_a and ±Q^AB_b and combinations thereof throughout the Brillouin zone.

We proceed to obtain the Fermi surface for a given strength (V₀) of the potential by determining the minimum in the total electronic DOS for the bonding and antibonding bands as a function of Q^B_a,b and Q^AB_a,b. We utilize the fact that the DOS is further minimized by considering momentum-dependent couplings of the form [34]

$$\begin{eqnarray} &&\begin{array}{l} V_a({\bf k})^{\rm B,AB}=V_0\textstyle \frac{1}{1-r^{\rm B,AB}}(1-r^{\rm B,AB}\cos bk_y),\\ V_b({\bf k})^{\rm B,AB}=V_0\textstyle \frac{1}{1-r^{\rm B,AB}}(1-r^{\rm B,AB}\cos ak_x). \end{array} \end{eqnarray} \tag{ 4 }$$

In real-space, r corresponds to a modulation of bond strength, for which there is evidence in recent experiments [24]. The value of r is tuned so as to suppress the energetically unfavourable coupling between translated states whose group velocities point in the same direction. By adjusting V₀, t_⊥0, r^B and r^AB (where B and AB refer to the bonding and antibonding bands, respectively), we are then able to tune the locations in λ^B and λ^AB of the minima in the DOS.

We find that the wavevectors Q^B_a,b and Q^AB_a,b at which the electronic DOS is minimized when V₀/t₁₀ = 0.7, t_⊥0/t₁₀ = 0.54, r^B ≈ 1.65 and r^AB ≈ 1.35 correspond closely to the wavevectors reported in x-ray scattering and NMR experiments. These same wavevectors also produce nodal pockets (see figure 4) with areas consistent with those found in recent quantum oscillation experiments uncovering a bilayer-split Fermi surface (see figure 5(a)). Figure 5(b) shows the electronic DOS (solid lines) for V₀/t₁₀ = 0.7 plotted as a function of 1/λ for both the bonding and antibonding bands in units of the effective mass. The deep minima (where the free energy is minimized) are located at λ^B_a,b = 3.30 and λ^AB_a,b = 4.05, which are very similar to the values λ_a,b = 3.28 and λ_a = 4.00 reported in x-ray scattering [22–24] and NMR [21] experiments, respectively, performed on underdoped YBa₂Cu₃O_6+x.⁷

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Using Onsager's relation, F = (ℏ/2πe)A_k (where A_k is the Fermi surface orbit cross-section in momentum space), we obtain F^B ≈ 337 T and F^AB ≈ 709 T quantum oscillation frequencies for the model Fermi surface pockets in figure 4 when V₀/t₁₀ = 0.7 (see figure 5). The size of the closed orbits is not found to be significantly impacted by the inclusion of higher order terms in the Hamiltonian in the limit |V_a,b(k')| < t₁₀, justifying our neglect of such terms for estimating the orbit areas and their effective masses. The observed frequencies would be further modified due to magnetic breakdown tunnelling, which is expected due to the suppressed gap ε_gap ≈ 10 meV between bonding and antibonding bands at the nodes in underdoped YBa₂Cu₃O_6+x, as seen in photoemission experiments [41]. The low effective mass of m₀ ≈ 0.34 m_e for the prominent frequency F₀ from band structure calculations [18, 25] due to the hopping parameter t₁₀ = 380 meV implies a renormalization of the bands (and subsequently V₀) by a factor of ≈4.7 in the vicinity of the Fermi energy for consistency with the experimentally measured value m* ≈ 1.6 m_e. Such a value of the renormalization yields for the residual corrugation (neglected in the 2D Fermi surface approximation) (t²_c/t_⊥0)/4.7 ≈ 2.5 meV, which is comparable in magnitude to the small interlayer coherence temperature (≈27 K) reported in interlayer transport experiments [43]. Finally, we obtain V₀ = 0.7 × 380/4.7 ≈ 57 meV for the strength of the charge order parameter, which, on taking T₀ ≈ 135 K [23] for the ordering temperature, yields 2V₀/k_BT₀ ≈ 10. While this ratio is large, it lies within the range of values, 6 <2V₀/k_BT₀ < 17, found experimentally in typical charge-density wave systems [44–48]. V₀ is also of sufficient strength to overcome incoherence of the bands reported at the anodes [52].

The magnetic breakdown resulting from the small gap ε_gap between bonding and antibonding bands at the nodes (see figure 6) is expected to occur at four points around each pocket (in the repeated Brillouin zone), giving rise to a sequence of five frequencies F₀ + mΔF (where m = −2, −1, 0, 1 and 2) in which the original bilayer-split frequencies correspond to F^B = F₀ − 2ΔF and F^AB = F₀ + 2ΔF [16] (see figure 4). A small magnetic breakdown gap (ε_gap) would further cause the central frequency F₀ (depicted in figure 6) to dominate [16] with weaker adjacent features at F₀ − ΔF and F₀ + ΔF [16], while the spectral features at F₀ ± 2ΔF are expected to be further suppressed in amplitude [16]. The calculated values F₀ ≈ 519 T and ΔF ≈ 95 T compare favorably with those F₀ ≈ 532 T and ΔF ≈ 90 T found experimentally [16].

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We turn to the question of why the wavevectors for the bonding and antibonding bands might be detected in different experiments [21, 22] and consider whether details of the Fermi surface reconstruction could provide potential clues. X-ray scattering experiments are expected to be sensitive to the momentum-dependent susceptibility [22–24], which is largest for the sections of the Fermi surface that are better nested and most completely gapped once Fermi surface reconstruction takes place. We find the electronic DOS to be lower for the bonding band (reflecting its flatter topology), which could potentially account for the observation of λ^B in x-ray scattering experiments [22–24]. Further experiments are required to determine whether a secondary feature can be seen with x-rays corresponding to the antibonding band at λ^AB_a,b ≈ 4; possibly induced by the application of a magnetic field at low temperatures [49]. NMR, by contrast, is expected to yield a splitting when the charge modulations become commensurate with (or locks-in to) the underlying crystalline lattice [21]; owing to its proximity to an integer value, such a possibility is more likely for λ^AB.

We also note that whereas the formation of a CuO₂ plane-centered charge modulation involving the bonding band [23] is mostly decoupled from oxygen ordering within the chains, as indicated by recent x-ray scattering experiments [24], the opposite may be true for a charge modulation involving the antibonding band owing to a greater overlap of its wave function with the chains (the chains themselves being prone to different forms of order [50, 51]). Local interactions with the oxygen order could cause the latter to become more susceptible to a lock-in transition or to differences in the strength of the modulation along the a and b lattice directions, potentially revealed in NMR experiments [21].

In summary, we have shown that the different wavevectors reported in x-ray scattering and NMR experiments and the frequencies observed in quantum oscillation experiments [16] can be consistently explained by biaxial charge ordering involving different nesting vectors for the bonding and antibonding bands in the bilayer cuprate YBa₂Cu₃O_6+x [16, 18, 33]. We note that the negative electrical transport coefficients, small size of the high magnetic field electronic heat capacity and chemical potential oscillations are similar to those expected in previously considered biaxial charge ordering models in which the effects of bilayer coupling were neglected [15, 28, 34]. The high purity of YBa₂Cu₃O_6+x samples together with their suitability for both quantum oscillation and x-ray scattering experiments make these materials a model system for understanding the interplay between charge ordering and superconductivity in the cuprates.

This work is supported by the Royal Society, King's College (Cambridge University), US Department of Energy BES 'Science at 100 T,' the National Science Foundation and the State of Florida.