Disentangling intertwined orders In copper oxide superconductors, several types of order compete for supremacy. In addition to superconductivity, researchers have found periodic patterns in charge density (CDW order), as well as an asymmetry in the electronic density within the unit cell of some cuprates (nematicity). CDW order has been detected in the underdoped regime of all major cuprate families, but the ubiquity of nematicity is less clear. Achkar et al. used resonant x-ray scattering to find that, in the copper oxide planes of three lanthanum-based cuprates, nematicity has a temperature dependence distinct from that of a related structural distortion. This implies that there are additional, electronic mechanisms for nematicity Science, this issue p. 576

Abstract In underdoped cuprate superconductors, a rich competition occurs between superconductivity and charge density wave (CDW) order. Whether rotational symmetry-breaking (nematicity) occurs intrinsically and generically or as a consequence of other orders is under debate. Here, we employ resonant x-ray scattering in stripe-ordered superconductors (La,M) 2 CuO 4 to probe the relationship between electronic nematicity of the Cu 3d orbitals, structure of the (La,M) 2 O 2 layers, and CDW order. We find distinct temperature dependences for the structure of the (La,M) 2 O 2 layers and the electronic nematicity of the CuO 2 planes, with only the latter being enhanced by the onset of CDW order. These results identify electronic nematicity as an order parameter that is distinct from a purely structural order parameter in underdoped striped cuprates.

Key challenges in resolving the cuprate phase diagram are to identify the broken symmetries that generically occur and understand how they are interrelated. Recently, a series of measurements have shown translational symmetry-breaking in the form of charge density wave (CDW) order to be generic to underdoped cuprates (1–9). It has also been proposed that electronic nematicity (intra-unit cell C 4 rotational symmetry-breaking) occurs in the cuprates (10), with experimental evidence for electronic nematicity coming from bulk transport (11–13) and scanning tunneling microscopy (STM) measurements (2, 14). Despite these developments, the role of electronic nematicity, such as whether it is common to the cuprates and how it relates to CDW order or the crystal structure of different cuprate families, has not yet been established.

Relevant in this context are the stripe-ordered La 2– x M x CuO 4 (M is Sr, Ba, Eu, or Nd) cuprates, where it is known that the low-temperature orthorhombic (LTO) phase to low-temperature tetragonal (LTT) structural phase transition (15–18) plays an important role in stabilizing spin and charge stripe order (1, 18). The LTT phase is understood in terms of rotations of the CuO 6 octahedra about axes parallel to the Cu-O bonds, with the rotation axis of the octahedra alternating between the a and b axes (19) of neighboring planes (Fig. 1) (15–18). This induces C 4 symmetry-breaking of the average (or intra–unit cell) electronic structure within an individual CuO 2 plane (referred to here as electronic nematicity) and stabilizes stripes whose orientation rotates by 90° between neighboring planes (1). The LTT distortions are understood to be a structural phenomenon, occurring in order to alleviate a lattice mismatch between the preferred Cu-O bond length of the CuO 2 planes and the (La,M)-O bond length in the (La,M) 2 O 2 layer (16). It is unclear, however, whether there are additional contributions to the electronic nematicity of the CuO 2 planes distinct from that engendered by a purely structural distortion and possibly generic to underdoped cuprates.

Fig. 1 Structure of La-based cuprates. (A) Unit cell of (La,M) 2 CuO 4 in the high-temperature tetragonal phase (HTT). O(1) and O(2) are in-plane and apical oxygen sites, respectively. (B) Octahedral distortions in the LTO and LTT phases. (C) CuO 2 planes showing the octahedral tilt pattern in neighboring layers (z = 0 and 0.5) in the LTT phase. The structural C 4 symmetry-breaking and electronic nematicity alternates between neighboring planes, z = 0 and z = 0.5.

To address this question, we use resonant soft x-ray scattering (RSXS) at different photon energies to probe the relationship between electronic nematicity of the CuO 2 planes, structural distortions, and CDW order. As shown in (20), with the x-ray photon energy tuned to the O K-edge (at an energy that is sensitive to apical oxygen), the (001) Bragg peak can be used to probe the LTO to LTT phase transition. The (001) peak is forbidden in conventional diffraction but is allowed when the photon energy is tuned to an x-ray resonance (21, 22). This occurs because on resonance the scattering cross section develops sensitivity to the orbital symmetry of atoms (21, 22), which differs for atoms in neighboring planes in the LTT phase.

Here, we exploit this orbital and element selective sensitivity and measure the (001) Bragg peak at photon energies corresponding to different elements/orbitals (at the Cu L-, La M-, Eu M-, and O K-edges). This enables one to differentiate the CuO 2 planes from the (La,M) 2 O 2 layer and isolate the orbital symmetry of the Cu 3d states that are most relevant to the low-energy physics of the cuprates. We show that at the Cu L-edge, the (001) peak intensity corresponds to C 4 symmetry-breaking of the Cu 3d states, which we identify as a measure of the degree of electronic nematicity of the CuO 2 planes.

In Fig. 2 we show the (001) peak intensity, , as a function of temperature at absorption edges corresponding to different elements/orbitals: La M, Cu L, Eu M, and O K. At the O K-edge, the in-plane and apical oxygen sites are further distinguished by photon energy, with 528 eV corresponding to in-plane oxygen, O(1), and ~532 eV corresponding to apical oxygen, O(2) (20, 23) (see Fig. 1). We investigated three samples—La 1.475 Nd 0.4 Sr 0.125 CuO 4 (LNSCO), La 1.875 Ba 0.125 CuO 4 (LBCO), and La 1.65 Eu 0.2 Sr 0.15 CuO 4 (LESCO)—having different LTT and CDW transition temperatures, and , respectively. Our principal observation is that measurements of for atoms in the CuO 2 planes [Cu and O(1)] have a distinct temperature dependence from atoms in the (La,M) 2 O 2 layer [La, Eu, and O(2)]. Specifically, the former displays a more gradual temperature dependence than the latter, which exhibits a more pronounced first-order–like phase transition. We ascribe this difference to an additional electronic nematicity in the CuO 2 planes beyond the structural distortions affecting the (La,M) 2 O 2 layer.

Fig. 2 Temperature dependence of the (001) Bragg peak intensity. The intensities are normalized by the corresponding low-temperature values, , with photon energy tuned to the La M-, Eu M-, O K-, and Cu L-edges for (A) LBCO, (B) LESCO, and (C) LNSCO. In all cases, the (001) peak has a more gradual temperature dependence for Cu and O in the CuO 2 planes than for atoms in the (La,M) 2 O 2 layer.

The intensity of the (001) peak at the Cu L-edge directly measures the nematicity of the Cu 3d states (24). Specifically, , where [or by symmetry] and and are diagonal components of the atomic scattering form factor tensor (25) averaged over all Cu sites in neighboring CuO 2 planes at or (Fig. 1). At the Cu L-edge, which probes the Cu 2p to 3d transition, and are directly related to the orbital occupation of the Cu 3d states projected onto the a or b axes, respectively. As such, corresponds to a difference in the symmetry of the Cu 3d states between planes. However, because by symmetry, one can express as , and it thus follows that the (001) peak at the Cu L-edge measures the electronic nematicity of the Cu 3d states within a single CuO 2 plane.

The (001) peak intensity at other absorption edges has similar sensitivity to orbital asymmetry. For the (La,M) 2 O 2 layer, this orbital asymmetry tracks the ionic displacements. The (001) peak intensity measured at a photon energy primarily sensitive to apical O exhibits the same temperature dependence as the (1 0 12) Bragg peak from conventional x-ray scattering, which is sensitive to ionic positions (26). Our measurements of I 001 at energies corresponding to La or apical O also show good agreement with conventional x-ray (26–28) or neutron scattering (1), reinforcing our view of a linear relationship between structural distortions and the (La,M) 2 O 2 layer (001) peaks measured at these photon energies.

Coupling of the symmetry of the electronic structure to ionic positions (e.g., displacements of in-plane oxygen along the c axis in the LTT phase) is also expected for the CuO 2 planes (29). However, the difference in the temperature dependence of the (001) peak between the CuO 2 planes and the (La,M) 2 O 2 layer shows that electronic nematicity of the CuO 2 planes is not simply a trivial consequence of distortions of the (La,M) 2 O 2 layer and that there is an additional, possibly intrinsic, susceptibility to nematic order in the CuO 2 planes.

We next explore the relationship between intra-unit cell ( ) symmetry-breaking probed by the (001) peak and inter–unit cell symmetry-breaking arising from unidirectional CDW order. In Fig. 3, we compare measurements of the temperature dependence of the (001) and CDW maximum peak intensities at the Cu L-edge. An important case is LESCO, where CDW order onsets at temperatures well below (Fig. 3B). Here, the key observation is that the Cu (001) peak is enhanced below ~65 K, coincidently with the onset of CDW order. This suggests that nematicity of the CuO 2 planes and translational symmetry-breaking of the CDW order are intertwined, having a cooperative relationship where CDW order enhances nematicity and vice versa. In contrast, the structural distortion of the (La,M) 2 O 2 layer, measured by the (001) peak at the Eu M, La M or O K (apical) energies, exhibits no such enhancement at (Fig. 2B). Whereas the LTT structural distortion of the (La,M) 2 O 2 layer plays an important role in stabilizing CDW order and nematicity in the CuO 2 planes, it does not appear to be coupled as strongly with the CDW order as the nematicity of the CuO 2 planes is.

Fig. 3 Comparison between nematicity and CDW order. Shown is the temperature dependence of the (001) and CDW Bragg peak maximum intensities (normalized by the low-temperature value) at the Cu L-edge in (A) LBCO, (B) LESCO, and (C) LNSCO. In LESCO, the (001) intensity is enhanced below , indicating a cooperative coupling between nematic and CDW orders. In LBCO, the CDW peak intensity is in good agreement with . CDW peaks were measured at Q = ( 0 1.5), where = –0.238, 0.264, and 0.236 for LBCO, LESCO, and LNSCO respectively.

In LBCO at a doping of , behaves roughly as the fourth power of ( ) (Fig. 3A). However, this power-law relationship is not generic, being seemingly not applicable to LESCO and LNSCO, and may be coincidental. As discussed below, it may also be that 1/8-doped LBCO represents a special case where the order parameters (and/or the coupling parameters) for CDW order, CuO 2 plane nematicity, and structural distortion of the (La,M) 2 O 2 layer are tuned to a common critical point.

In a minimal Landau theory that captures the essential aspects of the relationship between CDW, nematic, and structural orders, we consider two types of order parameters, the electronic nematic order parameter and the CDW order parameter ( ). breaks the point-group symmetry down to . , the complex amplitude of the CDW, also generally breaks translational symmetry. Given this, a suitable Landau free energy, F, is (1)Here, is the structural symmetry-breaking associated with distortions of the (La,M) 2 O 2 layer, representing the three-dimensional (3D) structural phase transition with the octahedral tilting axis rotated by 90° between neighboring planes in the LTT phase. and are associated with a single CuO 2 plane within the 3D unit cell. The parameters a, w, g, ε, r, u, and v are functions of temperature. From experimental observation, acquires a nonzero value through a first-order phase transition at , as described by a supplementary part of the Landau theory (16). We assume that εΦ << 1 because the “direct” coupling of the electronic order parameter to can be expected to be much stronger than the atomic/structural coupling between and the latter CDW terms. As such, we neglect the εΦ term. Also neglected is interlayer coupling of the electronic nematicity beyond that imposed by . Our measurements support this omission, being consistent with the LTT distortion inducing long-range nematic ordering along the c axis. Although limited by the c-axis penetration depth of the x-rays, measurements provide no evidence for a temperature dependent c-axis correlation length to the (001) peak, which would indicate an important role of additional interlayer coupling (24).

Within the context of this model, the observed differences among LBCO, LESCO, and LNSCO involve a material and doping dependence of and also possibly of w, the coupling strength between the LTT structural distortion and electronic nematicity. In contrast, we may conjecture that r and a (the CDW and electronic nematic order inverse susceptibilities) are similar for different cuprate materials at the same hole doping. We identify three distinct cases in the theory and their possible correspondence to materials: Case 1: at . Here, the first-order jump in at is sufficient to induce first-order jumps in and . This case may correspond to LBCO, with lower of x = 0.095 or 0.110 (27), and possibly LNSCO presented here (Fig. 3C). Case 2: at . For , this coefficient ultimately becomes negative. Here, the first-order jump in moves the system to the critical point of the CDW order. This special case requires fine-tuning of parameters but may in fact correspond to LBCO at ~1/8 doping (Fig. 3A). Whether such tuning plays a role in the enhancement of CDW order in 1/8-doped LBCO, where the longest-range CDW order of any cuprate is observed, or in the apparent relationship is presently unclear. Case 3: at . For , this coefficient becomes negative. Here, the CDW ordering temperature is well separated from the LTT phase transition, as in LESCO (Fig. 3B) or LBCO, with (27). Additional work is required to further explore this Landau theory and the relationship of its parameters to material properties (structure, doping, and disorder) and other cuprates.

Further context is obtained by contrasting these results to previous reports of nematicity in YBa 2 Cu 3 O 6+ x (YBCO), Bi 2 Sr 2 CaCu 2 O 8+δ (Bi-2212), and Na x Ca 2– x CuO 2 Cl 2 (NCCOC). Transport measurements of YBCO have shown in-plane anisotropy that is not solely due to crystalline anisotropy (11–13), and x-ray diffraction measurements have shown CDW order to have unidirectional character (30–33). However, the relationship between the orthorhombic structure of YBCO, unidirectional CDW order, and intra–unit cell electronic nematicity is not fully resolved. STM measurements on Bi-2212 and NCCOC arguably show more direct evidence for intra–unit cell electronic nematicity distinct from structure (2, 14, 34). Due to C 4 structural symmetry in these materials, nematicity occurs in Ising-like nanoscale domains. The intra–unit cell electronic nematicity identified here in stripe-ordered La-based cuprates likely represents a long-range ordered analog. This suggests that, like CDW order (1–9), intra–unit cell electronic nematicity may be a generic feature of underdoped cuprates.