The origin of high-temperature superconductivity in copper oxides and the nature of the ‘normal’ state above the critical temperature are widely debated1,2,3. In underdoped copper oxides, this normal state hosts a pseudogap and other anomalous features; and in the overdoped materials, the standard Bardeen–Cooper–Schrieffer description fails4, challenging the idea that the normal state is a simple Fermi liquid. To investigate these questions, we have studied the behaviour of single-crystal La2–xSrxCuO4 films through which an electrical current is being passed. Here we report that a spontaneous voltage develops across the sample, transverse (orthogonal) to the electrical current. The dependence of this voltage on probe current, temperature, in-plane device orientation and doping shows that this behaviour is intrinsic, substantial, robust and present over a broad range of temperature and doping. If the current direction is rotated in-plane by an angle ϕ, the transverse voltage oscillates as sin(2ϕ), breaking the four-fold rotational symmetry of the crystal. The amplitude of the oscillations is strongly peaked near the critical temperature for superconductivity and decreases with increasing doping. We find that these phenomena are manifestations of unexpected in-plane anisotropy in the electronic transport. The films are very thin and epitaxially constrained to be tetragonal (that is, with four-fold symmetry), so one expects a constant resistivity and zero transverse voltage, for every ϕ. The origin of this anisotropy is purely electronic—the so-called electronic nematicity. Unusually, the nematic director is not aligned with the crystal axes, unless a substantial orthorhombic distortion is imposed. The fact that this anisotropy occurs in a material that exhibits high-temperature superconductivity may not be a coincidence.
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We acknowledge discussions with Z. Radović, J. Zaanen, S. A. Kivelson, A. Gozar, I. Drozdov, P. N. Armitage and J. C. Davis. The research was done at Brookhaven and was supported by the US Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division. X.H. was supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4410.
The authors declare no competing financial interests.
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Extended data figures and tables
Extended Data Figure 1 Reflection high-energy electron diffraction images taken during growth of an optimally doped (p = 0.16) LSCO film.
a–f, Diffraction patterns obtained from the LSAO substrate just before growth (a), and after the first (b), second (c), third (d), fifth (e) and twentieth (f) LSCO layer. (Each layer was 0.66 nm thick, which is one-half the c axis lattice constant.) Very strong specular reflection indicates that the surface is very flat even on the scale of the electron wavelength (0.1 Å). Long and narrow diffraction streaks indicate high film crystallinity (with some terracing that originates from a slight miscut of the substrate). The LSCO film is epitaxially constrained by the tetragonal LSAO substrate; the in-plane lattice constants are the same, and the Cu–O–Cu direction in LSCO is parallel to the Al–O–Al direction in LSAO, that is, the crystallographic  direction.
Extended Data Figure 2 Inductance data for an optimally doped (p = 0.16) LSCO film show remarkable sample homogeneity.
a, The in-phase (real) component of Vp, the voltage across the pickup coil (proportional to the mutual inductance). It shows diamagnetic screening (the Meissner effect) when the film becomes superconducting. b, The imaginary part of Vp in the same film. c, The same, magnified near Tc. If there were two peaks separated by more than one half-width at half-maximum, these would be clearly resolved. Hence, in this film (of 10 mm × 10 mm) Tc is uniform to within ±0.1 K.
a, The ρT(ϕ) data for three LSCO films of the same doping level (p = 0.02) and taken at the same temperature (T = 295 K). b–d, Atomic force microscope (AFM) images of the three substrates used to grow these films show terraces and steps that originate from the substrate miscut, and are preserved in the films. The scale is the same for all three images, 3 μm × 3 μm. The steps are highlighted using green (in b), black (in c) and blue (in d) dashed lines, with the colours corresponding to those in panel a. The step density and orientation (expressed as the angle relative to the  direction) vary substantially from one film to another, while the nematicity magnitude and direction are similar.
Extended Data Figure 4 Examples of current–voltage characteristics in LSCO films grown by atomic-layer-by-layer molecular beam epitaxy.
a, The dependence of the longitudinal voltage V on the probe current I, at T = 30 K, T = 50 K and T = 295 K, in an LSCO film with p = 0.10. b, The dependence of the transverse voltage VT on the probe current I, measured at the same temperatures. Both I–V and I–VT characteristics are strictly linear in the current range of interest. All other data shown in the paper are taken with a low probe current, I = 2 μA (which corresponds to a current density of 80 A cm−2) and strictly ohmic contacts.
a, The anisotropy in resistivity expressed as the ratio ρb/ρa. Blue solid curve, our data from a heavily underdoped (p = 0.02) LSCO film. Red dashed line, data measured on bulk LSCO crystal with the same doping, reported in ref. 8. b, The same as a, except for p = 0.04. c, The ρT/ρ ratio for several representative doping levels, underdoped (p = 0.10), optimally doped (p = 0.16) and overdoped (p = 0.18). In all cases, there is a substantial, concave T dependence with a fast rise in the region of superconducting fluctuations (not expected for misaligned contacts or, equivalently, for current path meandering due to sample inhomogeneity). The grey bar shows the noise floor. Inset, in heavily overdoped, non-superconducting (p = 0.26) LSCO film, ρT vanishes at 100–150 K. d, Doping dependence of the nematicity magnitude, . Filled red diamonds, data taken at T = 295 K; filled blue circles, at T = Tc(midpoint); filled black circles, at T = 30 K. Open black diamonds, the data from ref. 8, at T = 295 K; open black circles, at T = 30 K. The error bars represent the standard deviation in fitting ρT(ϕ).
Extended Data Figure 6 Systematic doping dependence of nematicity in LSCO films rules out inhomogeneity as the source of transverse voltage.
a, The lithography pattern we used35,37 to fabricate linear combinatorial libraries for measurements of longitudinal and transverse resistances. The central horizontal strip (300 μm wide and 10 mm long) is aligned with a gradient (4% per 10 mm) in Sr doping level (which generates the same gradient in the hole density p) and is connected to 64 gold contact pads. The probe current runs from the contact I+ to I−, while the remaining 62 contacts are used for voltage measurements. This enables simultaneous measurements of ρ in 30 ‘pixels’ and ρT in 31 ‘pixels’, with extremely fine (Δ p = 0.0003) steps in doping between two consecutive pixels. b, The nematicity ratio shows a smooth and monotonic dependence on doping. Open blue circles, the measured data; black dashed line, a linear fit to the data.
Extended Data Figure 7 Spontaneous transverse voltage measured using contacts deposited through a shadow mask.
a, A ring of 36 contact pads (yellow solid circles) deposited on an as-grown p = 0.04 film (light blue square) by evaporating gold through a micromachined shadow mask, without any lithographic steps. The transverse resistance RT is measured using a set of four contacts in a cross; for example, by running the current through contacts 1 and 2 and measuring the voltage at contacts 3 and 4, and analogously for other angles, in steps of 10°. The area of the LSCO film is 10 mm × 10 mm, the diameter of the circle is 5 mm, and the diameter of the contact pads is 0.2 mm. b, The resulting angular dependence of RT measured at T = 295 K in LSCO film doped to p = 0.04. c, The film outside the central circle was etched away and RT(ϕ) was measured again at T = 295 K. The diameter of the remaining circle is 5.5 mm. d, The resulting angular dependence of RT. The differences in the amplitude and the phase of RT(ϕ) oscillations in the two cases indicate that the current flow through the area outside the circle affects the measurements. Moreover, the current flow here is not strictly unidirectional. Thus, the shadow-mask method is less direct than the ‘sunbeam’ lithography method, which requires no modelling. But our key observations of angular oscillations are the same for both methods, ruling out artefacts due to lithographic processing.
Extended Data Figure 8 Rotation of α, the direction of maximum longitudinal resistivity, with temperature.
a, The ρT(T, ϕ = −30°) data for an optimally doped (p = 0.16) LSCO, showing at which temperatures the measurements were made (arrowheads). b, Polar plots of ρT(ϕ) at the temperatures indicated (the same as in a, plus a room temperature measurement). The radial distance measures the magnitude of ρT, with the positive values in blue and negative in red. (For better visibility, the radial extents here are all normalized to the same size.) Here and in the text Tc refers to the midpoint of the resistive transition, which approximately coincides with the maximum of . Apparently, the nematic director of superconducting fluctuations has a different orientation from that in the ‘normal’ state at higher temperatures. c, Dependence of the difference Δα = |α(T = 295 K) − α(T = Tc)| on doping. The angular resolution, and the upper limit on the standard deviation (indicated by the error bars) of α, is ±5°. Note that in all samples α stays essentially constant from room temperature down to the onset of the superconducting transition, suggesting that the observed director ‘rotation’ may in fact result from different preferred (that is, highest conductivity) directions in the normal state and in the region of superconducting fluctuations.
Extended Data Figure 9 Resilience of the transverse resistivity in LSCO to the applied magnetic field.
a, Orange dots, dependence of ρT measured in one LSCO (p = 0.07) device at T = 295 K on the magnetic field applied parallel to the film surface. No statistically significant effect is observed up to B = 1.1 T. Red dots, transverse resistivity measured with the field perpendicular to the film surface, showing a pronounced Hall effect. Note, however, that it has an offset that is not B-dependent, that is, the Hall signal is superposed on top of the ‘nematic’ one. b, A detailed comparison of the ρT(ϕ) angular dependence measured using a sunbeam-patterned LSCO (p = 0.10) film at T = 295 K shows that an in-plane field B = 1.1 T affects neither the nematicity amplitude nor the director orientation. Blue dots, data at B = 0; orange dots, data at B = 1.1 T; red line, fit to .
a, Blue dots, ρT(ϕ) data, measured at T = 295 K, in an underdoped (p = 0.12) LSCO film grown on, and epitaxially anchored to, an orthorhombic NdGaO3 (NGO) substrate. The red lines are fits to . b, The same, measured at Tc-midpoint. c, ρT(ϕ) at T = 295 K in an optimally doped (p = 0.16) LSCO film on an NGO substrate. d, The same, measured at Tc-midpoint. In these films, owing to a substantial (about 1%) orthorhombic distortion, the nematic director is pinned to one of the crystal axes at all temperatures, that is, it does not rotate with p and T. This is also apparent from the ‘clover-leaf’ patterns (polar plots), shown to the right of the corresponding panels.
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Wu, J., Bollinger, A., He, X. et al. Spontaneous breaking of rotational symmetry in copper oxide superconductors. Nature 547, 432–435 (2017). https://doi.org/10.1038/nature23290
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