# Is Nematicity in Cuprates Real?

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}-

_{x}Sr

_{x}CuO

_{4}(LSCO), a prototype high-temperature superconductor (HTS) cuprate, a nonzero transverse voltage is observed in zero magnetic fields. This is important since it points to the breaking of the rotational symmetry in the electron fluid, the so-called electronic nematicity, presumably intrinsic to LSCO (and other cuprates). An alternative explanation is that it arises from extrinsic factors such as the film’s inhomogeneity or some experimental artifacts. We confront this hypothesis with published and new experimental data, focusing on the most direct and sensitive probe—the angle-resolved measurements of transverse resistivity (ARTR). The aggregate experimental evidence overwhelmingly refutes the extrinsic scenarios and points to an exciting new effect—intrinsic electronic nematicity.

## 1. Introduction

#### 1.1. Electronic Nematicity

#### 1.2. Nematicity in LSCO

_{c}superconductor, La

_{2-x}Sr

_{x}CuO

_{4}(LSCO). Note, however, that most of our discussion and arguments are more general and apply to other alleged nematic materials.

_{orth}, the Cu-O octahedra tilt and the unit cell become orthorhombic. T

_{orth}depends on doping, decreasing as x increases and vanishing at x

_{c}≈ 0.22. Relevant to the present discussion, note that even bulk LSCO crystals are tetragonal in a large part of the phase diagram. On the other hand, the samples under study here are ultrathin (10–20 nm thick) single-crystal LSCO films grown by atomic-layer-by-layer molecular-beam epitaxy (ALL-MBE) on LaSrAlO

_{4}(LSAO) substrates polished perpendicular to the [001] crystallographic direction. LSAO is tetragonal, and so are all these LSCO films at any temperature and doping. This has been verified by X-ray diffraction with a resolution high enough so that a tiny ($\le 0.08\%$) orthorhombic distortion was detected in thicker LSCO films, while being basically negligible in the films discussed here.

**C**

_{4v}(in the Schoenfliss notation) or 4 mm (in the International Hermann–Mauguin notation).

**C**

_{4v}=

**C**

_{4}+ $\widehat{\sigma}$

_{v}

**C**

_{4}, where

**C**

_{4}= {E, Ĉ

_{4}, Ĉ

_{2}, Ĉ

_{4}

^{−1}}, Ĉ

_{4}and Ĉ

_{2}are rotations by π/2 and π, respectively, around the z-axis, and $\widehat{\sigma}$

_{v}= $\widehat{\sigma}$

_{y}is the ‘vertical’ mirror reflection in the xz plane. By “electronic nematicity in LSCO”, we will refer to the situation in which the electron transport properties show that

**C**

_{4}symmetry is broken. We will not differentiate between this originating from a bona fide spontaneous long-range order or from a large “nematic susceptibility” triggered by a small external perturbation, since both of these classify as “real”, reflecting an intrinsic property of the material. However, we differentiate this from any effect entirely caused by extrinsic factors that could be eliminated by improving samples or measurements.

#### 1.3. Transverse Voltage Due to Anisotropic Resistivity

**C**

_{4v}symmetry, one would expect the longitudinal resistivity r to be isotropic in the xy plane (i.e., a scalar). In general, i.e., if the symmetry is orthorhombic (

**C**

_{2v}) or lower, ρ is a rank-2 tensor with the highest resistivity ρ

_{a}in some direction that is generally not aligned with any of the high-symmetry crystal axes. Restricting ourselves to in-plane properties, in an orthorhombic material, this tensor has the components:

#### 1.4. The Angle-Resolved Transverse Resistivity Measurements

_{T}≡ (V

_{34}+ V

_{56}+ V

_{78})/3. Other than this averaging, all the data are presented as measured. To factor out the current and the geometry, we introduce the transverse resistivity $\rho $

_{T}≡ (V

_{T}/I)d, where I ≡ I

_{12}is the probe current and d is the film thickness. Using this 36-beam pattern, we can measure $\rho $ in 72 devices and ${\rho}_{T}$ in 108 devices, with the angular resolution of $\pm 5\xb0$. We will refer to this as angle-resolved resistivity (ARR) and angle-resolved transverse resistivity (ARTR) measurements, respectively.

_{4}-symmetric) samples must be zero by symmetry. Note that most of the techniques mentioned in Section 1.1 above (with the notable exceptions of the cross-polarized Raman spectroscopy, second-harmonic generation, and THz dichroism) also probe diagonal elements and are thus subject to large backgrounds.

**C**symmetry being broken by any external factors. For this reason, a single measurement on a single sample is generally not sufficient; one needs a systematic examination of a large set of samples and parameters, and then the systematic trends in the dependences on T, B, and x can indeed differentiate unambiguously, as will be shown below.

_{4}#### 1.5. Key Experimental Observations: Unexpected Transverse Voltage

_{c}by assuming a putative parabolic relation, as we discussed in [29] and as it is customary; we use it to facilitate the comparison with the literature, but none of our conclusions depend on this convention.) The data agree with the previous results in the literature and show a superconducting transition at T

_{c}≈ 30 K. The same device, already at room temperature, shows a nonzero transverse voltage even without any magnetic field applied. This means that the corresponding transverse resistivity ${\rho}_{T}\left(T\right)$ is nonzero, as shown in Figure 2b, and hence, the fourfold rotational symmetry (

**C**) is broken.

_{4}**Temperature dependence of**${\rho}_{T}$. Figure 2b shows that as T is lowered, ${\rho}_{T}\left(T\right)$ decreases gradually and changes its slope for T < 70 K. A prominent and sharp peak in ${\rho}_{T}\left(T\right)$ appears in the vicinity of the superconducting transition in the temperature range where various techniques observe superconducting fluctuations. The ratio ${\rho}_{T}/\rho $, which is a measure of transport anisotropy, in fact decreases with temperature, as one would expect; nevertheless, at this doping level (p = 0.10), it remains finite and measurable even at room temperature.

**Angular dependence of**${\rho}_{T}$

**.**As an example, in Figure 3, we show $\rho \left(\varphi \right)$ and ${\rho}_{T}\left(\varphi \right)$ data measured at 36 angles in an LSCO (p = 0.04) film patterned into a sunbeam at T = 30 K. The raw experimental ${\rho}_{T}\left(\varphi \right)$ data (blue dots in the upper panel) fit very well to $\Delta \rho \mathrm{sin}\left[2\left(\varphi -\alpha \right)\right]$ with $\Delta \rho =424\mathsf{\mu}\mathsf{\Omega}\mathrm{cm}$ and $\alpha =60\xb0$ (the solid red curve). The lower panel shows the experimental $\rho \left(\varphi \right)$ data (solid black diamonds). The dashed red line is not an independent fit; it was generated by just translating the solid red line up by $\overline{\rho}=2\mathrm{m}\mathsf{\Omega}\mathrm{cm}$ and to the left by $45\xb0$. The agreement is reasonable but not nearly as good as for the ${\rho}_{T}\left(\varphi \right)$ data because of the on-chip device-to-device variations in the “background” longitudinal resistance. This clearly illustrates the advantages of the background-free ARTR technique.

**Rotation of the nematic director with the temperature.**In Figure 4a, we show the ${\rho}_{T}\left(T\right)$ data for an optimally doped (p = 0.16) LSCO film at one fixed azimuth angle. In Figure 4b, we show the fully angle-resolved data for the same sample at six fixed temperatures. In these polar plots, the radial distance measures the magnitude of ${\rho}_{T}$ as a function of the azimuth angle $\varphi $. The red color is used for positive and blue for negative values. Comparing the six panels, one can see that as the temperature is lowered, the principal axes of the resistivity tensor change their orientation with respect to the [100] crystal direction. The apparent rotation of the nematic director is strong in the relatively narrow temperature region near T

_{c}.

**Doping dependence of**${\rho}_{T}$. In LSCO, the ratio ${\rho}_{T}/\rho $ varies strongly and systematically as a function of the level of chemical doping x from more than 50% for x = 0.02% to less than 0.5% for x = 0.22 (see Figure 5).

**Time and length scales.**In Figure 1b, the width of one Hall-bar device is 100 µm, and the distance between two contacts for the longitudinal voltage measurements is 300 µm. The entire sunbeam circle diameter is 5 mm. The observed $\mathrm{cos}\left(2\varphi \right)$ dependence for the 36 Hall bars of one sunbeam pattern implies that

**C**symmetry is broken on the 5 mm scale. The time it takes us to complete one set of T-dependent ARTR measurements is typically on the scale of several days to weeks. Again, the observed $\mathrm{cos}\left(2\varphi \right)$ dependence means that the anisotropy map is stable over at least that time scale. We remeasured a few sunbeam devices after storing them for extended periods (years), and these did not show any changes.

_{4}**Temperature cycling.**The sunbeam pattern (Figure 1a) has 253 gold contacts. The number of lead wires in our cryogenic transport measurement setups is limited to 48 to keep the thermal load reasonably low and to be able to reach down to T = 0.3 K. This means that we cannot measure all the Hall bars simultaneously. Rather, we wire-bond to subsets of contacts and make the T-dependent measurements on the corresponding devices. Then, we warm the system up, take the film out, and remove these wires. Then, we rewire another segment and perform another set of T-dependent measurements, etc., until the whole pattern is covered. Thus, to map one complete set of $\rho (\varphi ,T$) and ${\rho}_{T}\left(\varphi ,T\right)$ curves, it takes several room-temperature-to-low-temperature (4 K or 0.3 K) cycles. Again, the observed $\mathrm{cos}\left(2\varphi \right)$ dependence means that nematicity’s overall orientation and amplitude are robust against thermal cycling between room temperature and low temperature. A possibility remains that this could change if the film is heated to some higher temperature. Still, we consider it unlikely since it was shown that several samples with the same doping—all of which were synthesized at $T>600\xb0\mathrm{C}$—have very similar nematicity amplitude and director orientation.

**The organization of this paper.**The above examples illustrate our typical experimental observations of nonzero transverse resistivity in LSCO films. Since the crystal structure of these films is essentially tetragonal, one is tempted to attribute this observation to a new and exciting physical phenomenon, electronic nematicity. However, before jumping to that conclusion, one should carefully consider, and rule out by firm experimental evidence, the possibility of any extrinsic factors, such as experimental artifacts or the film’s inhomogeneity, that might generate a nonzero transverse voltage. In this review, we ponder all such artifacts that we could think of and discuss in detail how we have ruled them out in our experiments. The aggregate evidence seems to disqualify this scenario overwhelmingly.

## 2. Reproducibility and Statistics

## 3. Artifacts

#### 3.1. Contact Misalignment

**The observed effect is way too large.**In the lithography pattern we used (Figure 1), the width of the Hall bar is W = 100 µm, and the distance between the two neighboring voltage contacts is L = 300 µm. Assuming, conservatively, the upper limit for the misalignment, $\Delta $l = 0.5 µm, this would give ${V}_{3,4}=\left(0.5/300\right)RI$ and ${\rho}_{T}/\rho =\left(\Delta l/W\right)=0.5\%$. However, this is much smaller than the experimental value in Figure 2c, where ${\rho}_{T}/\rho \approx 2.5\%$ at room temperature, while near T_{c}it surges to 30%. To produce such a big effect, the contact misalignment would have to be 30 µm, and that would be visible under an optical microscope or even to the naked eye.- ${\rho}_{T}\left(T\right)$
**does not scale with**$\rho \left(T\right)$. From the above formula, the ratio ${\rho}_{T}/\rho $ is determined solely by the value of $\Delta l/W$, so it should be independent of the temperature. However, this contradicts our experimental observations; as illustrated in Figure 2, the ratio ${\rho}_{T}/\rho $ varies significantly with T, increasing very steeply near T_{c}. - ${\rho}_{T}\left(\varphi \right)$
**does not scale with**$\rho \left(\varphi \right)$. We demonstrated that both ${\rho}_{T}$ and $\rho $ vary strongly and systematically with the azimuth angle $\varphi $—they oscillate with the same amplitude and period ($180\xb0$) while they are phase-shifted by $45\xb0$ from one another, as shown in Figure 3. The misalignment of transverse voltage contacts certainly cannot explain the observed angular dependence of the longitudinal resistivity, $\rho =\overline{\rho}+\Delta \rho \mathrm{cos}\left[2\left(\varphi -\alpha \right)\right]$. - ${\rho}_{T}\left(x\right)$
**does not scale with**$\rho \left(x\right)$**.**We studied the dependence of ${\rho}_{T}$ and $\rho $ in LSCO as a function of the level of chemical doping x and found that the ratio ${\rho}_{T}/\rho $ varies strongly and systematically with x, from more than 50% for x = 0.02 to less than 0.5% for x = 0.22 (see Figure 5). Since we used the same lithography mask to fabricate all these devices, the contact misalignment should not change with x.

#### 3.2. Orthorhombic Distortion

_{orth}, which decreases with doping and vanishes at x ≈ 0.22. While this orthorhombic distortion breaks the

**D**symmetry in principle, it cannot account for the observed nematicity in LSCO thin films.

_{4h}**The observed effect is way too large.**The orthorhombic distortions in our films are suppressed because the films are very thin (20-unit-cells thick) and epitaxially anchored to the tetragonal LaSrAlO_{4}(LSAO) substrates. In consequence, the crystal structure of our LSCO films is almost exactly tetragonal at all doping levels. To be quantitative, X-ray diffraction data in twice-thicker LSCO films grown on LSAO substrates by ALL-MBE show the in-plane orthorhombic distortion of just 0.08% in insulating, 0.04% in optimally doped, and 0.01% in overdoped metallic LSCO films, respectively [31]. Hence, even at the lowest doping levels, the orthorhombic distortion in films is quite small—at least a factor of 20 smaller than in the corresponding bulk samples—and the distortion energy must be reduced even more. Consider a charge-density wave (CDW) in the simple Peierls’ model for a rough estimate. The total CDW condensation energy E(Q) should scale as ${Q}^{2}\mathrm{ln}Q$, where Q is the distortion amplitude. Using this rule of thumb, the estimated E(Q) in our thin films should be about three orders of magnitude smaller than in the bulk crystals, i.e., basically negligible.- ${\rho}_{T}$
**does not depend on substrate.**As a control experiment, we deposited LSCO on orthorhombic NdGaO_{3}(NGO) substrates. In these substrates and the thin LSCO films grown on top, the orthorhombic distortion is substantial, with $\left(a-b\right)/\left(a+b\right)\approx 0.5\%$, where a and b are the in-plane lattice constants. Nevertheless, we found that the amplitude of nematicity, $N=\left({\rho}_{a}-{\rho}_{b}\right)/\left({\rho}_{a}+{\rho}_{b}\right),$ was nearly the same in these orthorhombic LSCO films as in the tetragonal LSCO films on LSAO substrates. The only difference was that in the films on NGO, the nematic director was pinned to the crystallographic [100] direction. In contrast, in the LSCO films on LSAO, the angle was generally different and dependent on doping and temperature. - ${\rho}_{T}$
**does not depend on uniaxial pressure.**In the second control experiment, we used a custom-built mechanical clamping device to apply uniaxial pressure on LSCO films grown on LSAO substrates. We could achieve up to 0.9% uniaxial strain and orthorhombic distortion. However, we found no effect on the amplitude of nematicity [32], consistent with paragraph 2 above. Note that this distortion is larger than the one spontaneously occurring in free-standing LSCO crystals (0.8% in undoped La_{2}CuO_{4}and decreasing with Sr doping). **The temperature dependence of**${\rho}_{T}$**does not match.**In free-standing bulk LSCO crystals, orthorhombic distortion develops in a mean-field manner and becomes nearly constant below about 0.5 ${T}_{orth}$. In stark contrast, ${\rho}_{T}\left(T\right)$ shows a strong peak near T_{c}, rising by a factor of 5–10 in a narrow temperature interval above T_{c}.

#### 3.3. Strain

_{T}. How this could actually work is unclear to us. Nevertheless, the same arguments as in Section 3.2 above also rule out this scenario.

#### 3.4. Substrate Miscut

_{T}. However, we can rule out this scenario based on the following arguments.

- ${\rho}_{T}$
**does not depend on the miscut angle and orientation.**We have investigated in detail whether there is a correlation between the step orientation and the uniaxial anisotropy in the electronic transport, i.e., the direction in which the maximum of ${\rho}_{T}\left(\varphi \right)$ occurs. Since the miscut angle varies randomly from one substrate to another, the same is true of the density and orientation of the surface steps. However, the nematicity amplitude and director orientation are reproducible in the films with the same doping level, while they vary systematically with doping and temperature. This is illustrated in Figure 8, where we present AFM images of three LSAO substrates, each showing clear atomic steps. The postgrowth AFM scans of the LSCO films deposited on these LSAO substrates confirmed that the steps were carried over into the films to the surface. The densities and orientations of atomic steps vary significantly from one to another of these three films (Figure 8b–d). Yet, both the amplitude and the phase of the angular oscillations of ${\rho}_{T}\left(\varphi \right)$ remain nearly the same (Figure 8a). - ${\rho}_{T}$ is the same in films and bulk. We also observed nematicity in bulk single crystals of LSCO, with the amplitude essentially the same as in LSCO films doped to the same level. Our data agree with those obtained independently by other groups [6,7]. These millimeter-thick bulk LSCO crystals have neither misfit nor antiphase dislocations, and the atomic steps at the surface have an unmeasurably small effect on the transport properties. Altogether, the ${V}_{T}$ observed in both LSCO films and bulk crystals cannot originate from surface steps.

#### 3.5. Thermal Gradient

^{2}. This corresponds to the probe current I

_{x}= 2 µA, while I-V relations are linear up to at least 10 µA for both V and V

_{T}contacts.

_{T}that we measure is odd.

#### 3.6. Thickness Gradient

- Our ALL-MBE system is equipped with a scanning quartz-crystal oscillator monitor, which enables the measurement of the local deposition rates across the substrate [9]. If we use a single thermal effusion cell, the deposition rate varies linearly across the substrate, and the gradient is 4% per 1 cm. However, we compensate for this gradient by using a pair of identical sources placed symmetrically (at azimuth angles that differ by 180°). The resulting gradient is thus much less than 1% over 1 cm, so the variations across a single 100 µm wide Hall bar are negligible.
- We digitally control the film thickness by monitoring the intensity of the Reflection High-Energy Electron Diffraction (RHEED). By scanning the RHEED beam across the film surface, we can verify that the thickness is uniform. Moreover, we can also see that the RHEED pattern and the atomic termination of the film surface are the same across the entire wafer.
- The geometry of the synthesis system determines the variations in film thickness. These should reproduce from one film to another, while we see that ${\rho}_{T}$ has a strong and systematic doping dependence.
- Moreover, ${\rho}_{T}$ strongly depends on temperature, and so does the orientation of the nematic director. That rules out any purely geometric origin.
- Within this scenario, the measured ${\rho}_{T}$ would always be proportional to the measured $\rho $, at every temperature, the azimuth angle $\varphi $, doping, etc., which is at variance with the experimental facts.

#### 3.7. Composition Gradient

- Since the geometry of our MBE system fixes our source positions and orientations with respect to the substrate, such gradients would not change from one film to another if the targeted film composition is kept the same.
- This would change with the doping level, but the doping dependence would be the opposite of what we see. The effect (if any) of such a gradient would increase with doping (it would scale with x). We see that ${\rho}_{T}$ decreases fast with doping, by a few orders of magnitude, until, on the extremely overdoped side, we hit the noise floor (see Figure 5). To throw in some numbers, if, for example, the residual gradient was 1% over 1 cm, for x = 0.20, this would amount to $\Delta x=2\times {10}^{-3}$, while for $x=0.02$, it would be 10 times smaller, i.e., $\Delta x=2\times {10}^{-4}$. In contrast, the observed nematicity amplitude is, in fact, more than an order of magnitude larger in LSCO, with $x=0.02$ rather than $x=0.20$.
- The nematicity amplitude varies strongly and systematically with the temperature, while the gradient in doping level would not depend on T.
- Moreover, had this been the culprit, the observed nematic director would always point in the same direction, while its orientation, in fact, changes substantially with the doping level and temperature.

_{2-x}Sr

_{x}CuO

_{4}(LSCO). We measure x using quartz-crystal microbalance (QCM), Rutherford back-scattering, X-ray reflectance fringes, etc., to the aggregate absolute accuracy of a few percent. On the other hand, using our COMBE technique, we can generate a 4%/cm linear gradient of x in one direction. This is just a geometrical effect, so it can be calculated very accurately. We verified this by direct measurements using our scanning QCM as well [30]. We pattern the strip into 32 pixels; then, the difference in x between two neighboring pixels is Dx = 0.04/32 x = 0.00125 x. Near optimal doping, this gives Dx = 0.0002. So, while we do not know the absolute value of x to better than a few percent, we know the relative pixel-to-pixel change with two-orders-of-magnitude-better precision. Thus, we can look at the doping dependence in exquisitely fine steps. This is very valuable if, e.g., one is studying scaling and critical exponents near a doping-controlled quantum phase transition [36].

#### 3.8. Schottky Barrier Contacts

## 4. Sample Inhomogeneity

#### 4.1. Phase Separation

**Our LSCO films are not granular.**The X-ray diffraction and atomic force microscopy (AFM) studies indicate that the crystalline structure and morphology of our LSCO films are very uniform both microscopically and macroscopically. According to RHEED, high-resolution XRD, AFM, scanning cross-section transmission electron microscopy (STEM), etc., our LSCO films synthesized by ALL-MBE are single crystals, atomically smooth, and nearly perfect. This results from 30 years of sustained and focused development of the ALL-MBE technique and technology [36,37,38,39,40,41]. No static ordered phase (CDW, SDW, ferro- and antiferromagnetism) is observed in the entire doping range, while some nematicity is always present. While CDW and AF fluctuations may be present, these would average to zero with time and space and would not be macroscopically aligned.**Our LSCO films are ‘clean’.**One figure of merit, frequently used in the literature to benchmark the film quality, is the residual resistivity ${\rho}_{0}=\rho \left(T\to 0\right)$ or, more tellingly, the residual resistivity ratio (RRR), defined as $\rho \left(T=300K\right)/{\rho}_{0}$. In Figure 11, we show the $\rho \left(T\right)$ dependence in a representative LSCO film that we synthesized using ALL-MBE, while the measurements were carried by an independent group (at NHMFL) [41]. It shows that ${\rho}_{0}\approx 0$, and RRR > 200. By this standard, this (single-crystal) LSCO film is more perfect and less disordered, by far, than any bulk LSCO single-crystal reported in the literature. Suppose one uses the standard Drude formula and inserts the measured and broadly accepted values for the Fermi velocity, effective mass, and carrier density; the inferred mean free path would be longer than 1.7 µm, i.e., more than 4000 lattice constants. This should be considered exceptionally clean within the accepted standards and terminology [42]. An alternative is to postulate a breakdown of the canonical description of metals and superconductors. Nevertheless, this film shows a substantial nonzero transverse resistivity already at room temperature and down to T_{c}. It also features a pronounced peak near ${T}_{c}$, while no such peak exists in the longitudinal resistivity. Moreover, ${\rho}_{T}$ changes the sign four times as the in-plane angle varies [1]. Therefore, ${\rho}_{T}$ is proportional neither to $\rho \left(T\right)$ nor to the derivative $\mathrm{d}\rho \left(T\right)/\mathrm{d}T.$- ${N}_{s}\left(T\right)$
**is linear in T.**In a clean unconventional superconductor with a V-shaped d-wave gap, the dependence of the superfluid density on temperature ${N}_{s}\left(T\right)$ is expected to be linear. Scattering on any disorder, impurities, and other defects would cause the breaking of near-nodal pairs, filling in the gap and rounding it near the node, and leading to the ${T}^{2}$ dependence of ${N}_{s}\left(T\right)$ below some characteristic temperature T*. We have shown that in LSCO, ${N}_{s}\left(T\right)$ is linear for all doping levels, in some films down to 1 K. This is only expected in a very clean d-wave superconductor [29]. - $T,\varphi ,B,$
**and**$x$**dependences are systematic.**The strong, smooth, and reproducible dependences of the nematicity amplitude and orientation on $\varphi ,T,B,$ and $x$, impose very tight constraints on any model. It is difficult to account for all of these within any inhomogeneity scenario. Handwaving is not sufficient—one ought to present a concrete model of inhomogeneity that can at least qualitatively reproduce the $\mathrm{sin}\left(2\varphi \right)$ dependence of $\rho $ coincident with the $\mathrm{cos}\left(2\varphi \right)$ dependence of ${\rho}_{T}$ at fixed T and doping; the decrease in ${\rho}_{T}$ with x; the rotation of the nematic director with T, B, and x; the disappearance of the peak in ${\rho}_{T}\left(T\right)$ with B [43], etc.

#### 4.2. Guided Vortex Motion

_{c}and attributed this to a peculiar motion of superconducting vortices [44,45,46]. In cuprates, thermally generated superconducting vortices and antivortices abound at temperatures near T

_{c}as low-energy excitations. Suppose that, for whatever reason, the current direction deviates locally from the Hall-bar direction. In that case, the vortex motion will have a component along the current path, generating a voltage between the transverse contacts [44,45,46]. The same could happen even if the electric current is strictly parallel to the Hall-bar direction if the vortex motion is constrained and guided, e.g., by spatially organized pinning or barriers. However, given the following experimental evidence, we can rule out this scenario.

- In one device, and for one pair of voltage contacts, a scratch, a microcrack, or local inhomogeneity in oxygen distribution could redirect the current locally, thus generating a transverse voltage. It is almost impossible to believe, though, that this could, just by chance, happen at all 108 positions measured and in such a way to create a $\mathrm{cos}\left(2\varphi \right)$ curve in every film and for every doping.
- Even if the guided vortex motion was relevant for explaining the pronounced peak in ${\rho}_{T}\left(T\right)$ near T
_{c}, such as in Figure 2b, it could not be at the origin of the electronic nematicity we observe at room temperature, where no superconducting vortices are present. - The hypothesis of guided vortices is also incompatible with our angle-resolved transverse magnetoresistance (ARTMR) data [43]. If the moving vortices are preferentially driven in a particular direction—by a gradient in the distribution of dopant ions or some other defects, step edges and antiphase dislocations due to substrate miscut, etc.—one could expect some anisotropic transverse voltage because of broken
**C**symmetry, and this could peak near T_{4}_{c}. However, as more vortices are generated, this effect should grow stronger with the magnetic field. Moreover, one would also expect to see the same effect in the longitudinal resistivity. This is the opposite of what we observe. First, the peak near T_{c}in ${\rho}_{T}\left(T\right)$ decreases by an order of magnitude in the field B = 6 T. Second, there is no peak (and almost no MR whatsoever) in $\rho \left(T\right)$ at the same temperature and field. Third, if the sample were effectively orthorhombic due to some compositional or structural reasons, one would expect this to change neither with the magnetic field nor with the temperature, while the nematicity magnitude and director orientation show a strong and systematic T- and B-dependence [43].

#### 4.3. Randomness of Sr Doping

_{c}and the superfluid density with doping, may be attributed to a peculiar distribution of the Sr dopant atoms. This scenario may warrant a separate section because of a semantic ambiguity—should this be called extrinsic or intrinsic. As far as it is known, Sr dopants distribute randomly; LSCO is a solid solution. Hence, this randomness is unavoidable for entropy reasons and, in this sense, is intrinsic to the thermodynamically stable form of LSCO. Strictly speaking, this disorder breaks down all spatial symmetries, both translational and rotational. However, XRD in LSCO shows sharp Bragg peaks [31]. Moreover, at least in overdoped LSCO, angle-resolved photoemission spectroscopy (ARPES) shows sharp quasiparticles and a well-defined Fermi surface, also confirmed by the study of angle-resolved magnetoresistance oscillations (AMRO). Hence, LSCO behaves quite like a crystal and is generally modeled assuming long-range order and crystallographic-space-group symmetry, which should manifest itself in all macroscopic physical properties. However, our ARTR experiments show that the rotational symmetry is broken in electrical transport, while the detailed findings do not seem consistent with any Sr-distribution-related scenario.

**This randomness averages out on the relevant (µm, mm) length scales.**The average Sr–Sr distance is smaller than 1 nm, and since the distribution of Sr atoms is random, local variations are expected on a very small (nm) scale. However, this randomness should be averaged out already on the scale of, say, 100 nm. It should be thoroughly washed out on the scale of the width of a single Hall bar, 100 µm, and even more so on the scale of the sunbeam device illustrated in Figure 1a, which is larger than 0.5 cm.**Sr clustering is not seen by structural probes.**In one proposed model, the density of Sr atoms is assumed to vary locally, forming domains or clusters. However, this hypothesis can be refuted based on the arguments listed in Section 4.1, which rule out phase separation and granularity of whatever origin. Sr clusters have been observed neither by nanoscale synchrotron-based techniques such as X-ray fluorescence nor by TEM-based high-resolution electron-energy-loss spectroscopy (HR-EELS).**Sr clustering is inconsistent with the transport and susceptibility data**. In Figure 12, we show the two-coil mutual inductance data for an optimally doped (x = 0.16) LSCO film. The in-phase signal measures the reactive response and shows the Meissner effect with a sharp onset at T_{c}= 40.8 K. The half-width-at-half-maximum of the peak in the out-of-phase (dissipative) response puts an upper limit on the variations of T_{c}in this film, $\Delta {T}_{C}<0.1K$, over a large area of 10 × 10 mm^{2}. Using the empirically known relations that link $\rho ,{T}_{c}$, and x, one can estimate the corresponding upper limit on the gross local variations in $\rho $. For $\Delta {T}_{c}/{T}_{c}\approx 0.1/40=0.25\%$, and assuming the worst-case scenario when the variation is in the form of a gradient along one line, this would result in measured $\left({\rho}_{a}-{\rho}_{b}\right)/\left({\rho}_{a}+{\rho}_{b}\right)<0.1\%$—at least two orders of magnitude smaller than the anisotropy we measure near T_{c}. This rules out attribution to a substantial inhomogeneity in the film of the observed phenomena, including, in particular, the peak in V_{T}near T_{c}, which is typically 1–2 orders of magnitude broader.**Sr clustering is inconsistent with the**${\rho}_{T}\left(\varphi ,x\right)$**dependence**. The probability that sufficiently large domains, with a large enough difference in stoichiometry, could be present in our films is quite small. However, let us grossly overestimate it as P = 10% for one device. What, then, is the probability that this happens in each of the 108 devices we fabricate out of one film, and that this happens in such a way as to generate the $\mathrm{cos}(2\varphi )$ dependence on the azimuth angle, with the $\pi /4$ shift between ${\rho}_{T}\left(\varphi \right)$ and $\rho \left(\varphi \right)$, as shown in Figure 3. Essentially, it is null. Moreover, inhomogeneity would have to be organized on a macroscopic scale in such a way that (a) the film shows a preferred transport direction, (b) when the film is ‘sliced’ into 108 small devices, this remains the same in each device, in both magnitude and orientation, (c) this reproduces in other films of the same doping, and (d) as we change the doping level, this “organized inhomogeneity” would need to have a smooth and monotonic doping dependence. More is different here; our extensive statistics allow for definitive statements and rule out the dominant role of such random agents.**Sr clustering is inconsistent with the**${\rho}_{T}\left(\varphi ,T\right)$**dependence**. The systematic temperature dependence, particularly the director rotation, is the ultimate challenge to the Sr-clustering model and any other scenario based on random or organized chemical or structural disorder. The Sr distribution does not change with the temperature.

#### 4.4. T_{c} Distribution

_{c}shows some spatial variations. They found that this model can account for a sizeable transverse resistivity ${\rho}_{T}$ within the temperature interval $\Delta {T}_{c}={T}_{c2}-{T}_{c1}$, where T

_{c1}is the lowest and T

_{c2}the highest value. In this case, ${\rho}_{T}$ scales with the temperature derivative of the longitudinal resistivity $\rho $. This proportionality, ${\rho}_{T}\left(T\right)\propto \Delta {T}_{c}\times \mathrm{d}\rho \left(T\right)/\mathrm{d}T$, is the fingerprint of the distribution of T

_{c}in the sample.

_{c}. However, a more detailed analysis clearly shows that this cannot be the key contributing factor for several reasons.

**Not at room temperature.**This mechanism only works near T_{c}, where R(T) drops sharply. It does not work at room T, where R(T) evolves gradually. Even if the observed broadening of the resistive transition originated entirely from the disorder and inhomogeneity—which, as we argue in detail in (vi) below, is decidedly not the case—it is unclear how this could account for the observed nematicity at room temperature.**Not for x**≲**0.07.**As shown in Figure 13, in LSCO with x = 0.063, the $\rho \left(T\right)$ dependence is not monotonous. As T decreases, $\rho \left(T\right)$ decreases (like in metal) but then flattens and starts to increase (like in a semiconductor) until it starts dropping fast again in the region near T_{c}where superconducting fluctuations take over. Hence, the derivative $\mathrm{d}\rho /\mathrm{d}T$ changes sign twice with temperature. In contrast, ${\rho}_{T}\left(T\right)$ always stays positive. For $x\lesssim 0.06$, LSCO is neither superconducting nor metallic; $\rho \left(T\right)$ is semiconductor-like, and $d\rho /dT$ is negative in the entire range $0T300\mathrm{K}$. In contrast, ${\rho}_{T}\left(T\right)$ is always positive in this temperature interval.**Not for**$\varphi =\alpha $**.**As we mentioned in Section 1.5 (ii) above, ${\rho}_{T}=\Delta \rho \mathrm{sin}\left[2\left(\varphi -\alpha \right)\right]$, and, hence, ${\rho}_{T}=0$ for $\varphi =\alpha \mathrm{mod}\left(\pi /2\right)$. On the other hand, $\rho =\overline{\rho}+\Delta \mathrm{cos}\left[2\left(\varphi -\alpha \right)\right]$, so for $\varphi =\alpha $, $\rho =\overline{\rho}+\Delta \rho ={\rho}_{a}$, while for $\varphi =\alpha \pm \mathsf{\pi}/2$, $\rho =\overline{\rho}-\Delta \rho ={\rho}_{b}$. Since neither ${\rho}_{a}\left(T\right)$ nor ${\rho}_{b}\left(T\right)$ is zero anywhere above T_{c}, clearly, ${\rho}_{T}\ne \mathrm{d}\rho /\mathrm{d}T$.**Not random; systematic x-dependence.**Here, inhomogeneity means the randomness of some structural and electronic features at different locations in the film. If this was the dominant source of V_{T}and if we systematically measure ${\rho}_{T}$ at various locations along a strip that is patterned in the film, the recorded ${\rho}_{T}$ should either vary randomly with the position, if the domains are relatively large, or be constant (i.e., independent on the position) if the domains are much smaller than the strip width, and thus, averaged out. To test this possibility, we synthesized an LSCO film with a built-in continuous gradient in doping level x by means of the combinatorial molecular beam epitaxy (COMBE) [30,36]. Then, we used lithography to pattern this film, as shown in Figure 10a. This pattern, with 2 current contacts and 31 pairs of voltage contacts, allows us to measure $\rho \left(x\right)$ and ${\rho}_{T}\left(x\right)$ at 30 positions in the LSCO film with a continuous doping gradient. We built the corresponding electronics that allow us to measure all 30 channels simultaneously, thus reducing the scatter due to possible variations in temperature, etc. As seen in Figure 10b, the measured ${\rho}_{T}/\rho $ ratio shows a linear dependence on p; clearly, it is not random at different locations in the film.**Broadening of the resistive transition in cuprates does not arise from disorder.**In Section 4.3 (iv) above, we showed that in our LSCO films, the spread in T_{c}, as measured by the mutual inductance technique, can be as small as ±0.1 K over the 10 × 10 mm^{2}area. On the other hand, in LSCO (and all other superconducting cuprates), the resistive transitions are quite broad [42], typically, DT_{c}≈ 5–10 K. The actual value is somewhat vague since various definitions of T_{c}are possible and indeed have been used in the literature, e.g., the transition onset T^{O}; the temperature at which the resistivity drops to 90%, 50%, or 10% of the normal-state value just above the onset, etc. However, all of these suffer from ambiguity in the choice of the normal-state resistivity. It is less subjective to use the temperature T_{M}at which the slope of $\rho \left(T\right)$ is the largest; this can be determined unambiguously as the temperature at which the derivative curve, $\mathrm{d}\rho \left(T\right)/\mathrm{d}T$, reaches the maximum. The most conservative choice, however, and the one we always adhere to, is to use the temperature at which the sample resistivity vanishes, i.e., T_{c}≡ T(R = 0). We found (with extensive statistics) that this is also the temperature at which the Meissner effect onsets. Using this definition, we find that T_{c}(as determined by high-precision mutual inductance measurements) is quite sharp. In Figure 14, we show a set of $\rho \left(T\right)$ characteristics measured in one LSCO film patterned into a linear combinatorial library of devices [30]. For each device, we indicated the transition onset T^{O}, the temperature T_{1}at which the resistivity drops to 90%, T_{2}where it drops to 50%, T_{3}at which it drops to 10%, and T_{M}at which the derivative $\mathrm{d}\rho \left(T\right)/\mathrm{d}T$ has the maximum. It is apparent from Figure 14b that, as long as we consistently use any of these characteristic temperatures, it stays essentially constant, on the scale of 0.1 K, in all devices across the entire film.**Broadening of the transition in cuprates arises from superconducting phase fluctuations.**The resolution of this apparent ‘paradox’—the simultaneous occurrence of a very sharp Meissner signal and a broad resistive transition—is well-known [42,48,49,50,51,52].- Cuprates are extreme type-II superconductors with the ratio ${\lambda}_{L}/{\xi}_{0}>100$, where ${\lambda}_{L}$ is the London penetration depth and ${\xi}_{0}$ the coherence length. Therefore, one indeed expects, from the Ginzburg–Levanyuk criterion, that there should be a few Kelvin-wide temperature regions in which superconducting fluctuations dominate the transport. As the temperature is increased, vortex–antivortex pairs are generated and thermally dissociated. The probe current causes free vortices to move, and vortex flow causes dissipation and finite resistance [42].
- In LSCO, at every doping level, the superfluid density is very low, i.e., a few orders of magnitude lower than in conventional superconductors such as Nb or Pb. Expressed in the units of Kelvin, the phase stiffness ${\rho}_{s0}$ is nearly equal to T
_{c}[29]. This fact implies that thermal phase fluctuations essentially determine the superconducting transition by destroying the global phase coherence [52]. - Cuprates are quasi-two-dimensional—they behave like vertical stacks of intrinsic Josephson junctions. A single LSCO layer can host HTS with T
_{c}, with the superfluid density equal to that in the bulk samples [29]. Hence, in cuprates, one should expect to see Berezinskii–Kosterlitz–Thouless (BKT)-like physics. Indeed, dynamic BKT transition has been observed by MHz susceptibility measurements [48] and microwave [49] and THz spectroscopies [50,51]. - Almost every type of experiment that could detect thermal phase fluctuations in principle has shown that in cuprates, they persist in a vast temperature region. Both ARPES and scanning tunneling microscopy (STM) data show that the superconducting gap does not close at T
_{c}but persists well above, up to as much as 1.5 T_{c}. Microwave and THz spectroscopies show superconducting phase fluctuations up to 20–30 K above T_{c}[49,50,51]. Magnetoresistance data show superconducting fluctuations well above the critical field that causes nonzero resistance. In underdoped LSCO, evidence for superconducting vortices at temperatures as high as 250 K has been found in the Nernst effect, diamagnetism, torque magnetometry, etc.

## 5. Control Experiments Using Ti Films

_{2}RuO

_{4}(SRO) films [26]. ARTR measurements were made using the same experimental setup. To compare all these films on the same footing, we normalized the measured ${\rho}_{T}\left(\varphi \right)$ by the corresponding average longitudinal resistivity $\overline{\rho}$. In Figure 15, we compare the ARTR data for Ti and SRO. Apparently, in sharp contrast to large and oscillating ${\rho}_{T}\left(\varphi \right)$ in SRO and LSCO, ${\rho}_{T}$ is zero within the noise in the Ti film.

## 6. Nematicity beyond ARTR in LSCO

#### 6.1. Other Cuprates

_{2}Cu

_{3}O

_{7-d}is orthorhombic, and it contains layers of Cu-O chains that run preferably in one direction. This causes anisotropy in electron transport that originates from the crystal structure; this must be carefully discerned and subtracted before asserting the presence of additional electronic nematicity. Moreover, the chain layers are typically disordered, which also needs to be considered.

_{2}Sr

_{2}CaCu

_{2}O

_{8}(BSCCO) is also orthorhombic for a different reason: the BiO layers are buckled along the diagonal, forming a superstructure modulation with a period of about five lattice constants. While the in-plane lattice constants are not very different, ${a}_{0}\approx {b}_{0}$; a significant transport anisotropy again originates from the crystal structure. Substituting Pb at Bi sites has been shown to reduce the amplitude of this superstructure modulation. Hence, studying BSCCO with and without Pb substitution may be a way to discern between the anisotropy that originates from the lattice and the purely electronic contribution.

#### 6.2. Bulk Crystals

#### 6.3. Other Techniques

- Some techniques, such as scanning tunneling microscopy (STM) and angle-resolved photoemission spectroscopy (ARPES), are very surface-sensitive. Then, one needs to worry whether all the structural and electronic properties of the surface layer (that is probed) are the same as the bulk (that is not probed by ARPES or STM but is probed by transport or magnetic measurements to determine T
_{c}). In fact, in most ionic crystals, it is guaranteed that the topmost surface layer will not have the bulk structure because of the long-range (Madelung) origin of the cohesion [31,54]. Here, BSCCO may be an exception because of the weak Van der Waals coupling between the layers (which also makes it cleavable). - Most of these other techniques have background-related problems since one is looking for small angle-dependent deviations from some relatively large average value. Then, it is the question of how big these deviations are compared to the error bars and sensitivity of the measurement for one orientation.

## 7. Conclusions

**C**rotational symmetry in the electron fluid, i.e., the electric nematicity in copper oxide superconductors. At least for now, this appears to be the most plausible scenario. We feel that the burden of the proof is now with the nay-sayers. Handwaving is not sufficient—one ought to present a concrete model of something extrinsic that reproduces the $\mathrm{sin}\left(2\varphi \right)$ dependence of $\rho $ and the $\mathrm{cos}\left(2\varphi \right)$ dependence of ${\rho}_{T}$ at fixed T and doping, the decrease in ${\rho}_{T}$ with x, the rotation of the nematic director with T, B, and x, the disappearance of the peak in ${\rho}_{T}\left(T\right)$ with B, etc. These are very tight constraints on any model.

_{4}## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The 36-Hall-bar (“sunbeam”) pattern, with the bars spaced in the steps of 10 degrees. To measure the Hall effect, we apply a magnetic field

**B**perpendicular to the film surface. (

**b**) One Hall-bar device with 8 contacts, allowing 2 measurements of the longitudinal resistivity $\rho $ and 3 of the transverse resistivity ${\rho}_{T}$. This entire sunbeam pattern allows for angle-resolved measurement of $\rho \left(\varphi \right)$ in 72 devices (’pixels’) and transverse resistivity ${\rho}_{T}$ in 108 devices. Adapted from [1].

**Figure 2.**The temperature dependences of (

**a**) the longitudinal resistivity $\rho $, (

**b**) the transverse resistivity ${\rho}_{T}$, and (

**c**) their ratio ${\rho}_{T}/\rho $, in LSCO (p = 0.10). Adapted from [1].

**Figure 3.**Relation of angular dependences of ${\rho}_{T}$ and $\rho $ in an underdoped (p = 0.04) LSCO film. Upper panel: blue dots, the measured values of ${\rho}_{T}\left(\varphi \right)$ at T = 30 K; solid red line, the fit to $\Delta \rho \mathrm{sin}\left[2\left(\varphi -\alpha \right)\right]$. Bottom panel: black diamonds, the measured values of $\rho \left(\varphi \right)$; the dashed red line is obtained by shifting the solid red line upward by $\overline{\rho}=2\mathrm{m}\mathsf{\Omega}\mathrm{cm}$ and to the left by $45\xb0$. Adapted from [1].

**Figure 4.**(

**a**) ${\rho}_{T}\left(T\right)$ in an optimally doped (p = 0.16) LSCO film at one fixed azimuth angle. (

**b**) ${\rho}_{T}\left(\varphi \right)$ in the same sample, at six selected temperatures. (The lower five are as indicated by arrows in panel a). In the polar plot, the radial distance indicates the magnitude, and the in-plane angle is measured from the [100] direction. The red color denotes positive and blue negative values. Compared to room temperature, near T

_{c}, the nematic director rotates by about 60°. (

**c**) Dependence of the difference $\Delta \alpha =\left|\alpha \left(T=295\mathrm{K}\right)-\alpha \left(T=Tc\right)\right|$ on doping. The angular resolution, and the upper limit on the standard deviation (indicated by the error bars) of α, is ± 5°. Adapted from [1].

**Figure 5.**(

**a**) The doping dependence of ${\rho}_{T}\left(x\right)$ and $\rho \left(x\right)$, measured in LSCO thin films. Open black squares, longitudinal resistivity $\overline{\rho}\left(x\right)$ measured at T = 295 K; solid blue circles, ${\rho}_{T}^{0}\left(x\right)$ at T = 295 K; solid orange triangles, ${\rho}_{T}^{0}\left(x\right)$ at $T={T}_{C}$ (midpoint); solid green diamonds, ${\rho}_{T}^{0}$ at T = 30 K are shown (in lieu of $T={T}_{C}$) for the samples with x = 0.02 and x = 0.04 that are non-superconducting. (

**b**) The doping and temperature dependence of the nematicity magnitude, $N\equiv \Delta \rho /\rho $. In the grey areas, the signal is below our noise floor. Adapted from [1].

**Figure 6.**The 32-pixel pattern, with 64 Au contacts spaced in 300 µm steps. This pattern allows for simultaneous measurement of the longitudinal resistivity or magneto-resistivity (using, for example, pads A and C) in 30 devices (‘pixels’) and the transverse resistivity and Hall effect (using pads A and B) in 31 devices. Adapted from [30].

**Figure 7.**The schematic drawing of a Hall bar with the longitudinal voltage contacts spaced at the distance L and the transverse voltage contacts slightly misaligned by $\Delta l$. The transverse voltage that originates from this contact misalignment should be proportional to the displacement, ${V}_{T}=\left(\Delta l/L\right)RI$, where R is the longitudinal resistance and I the probe current. Adapted from [2].

**Figure 8.**(

**a**) ${\rho}_{T}\left(\varphi \right)$ for three LSCO (p = 0.02) films synthesized by ALL-MBE under the same growth conditions but on three different substrates. (

**b**–

**d**) The atomic force microscopy (AFM) images were taken on the surfaces of these three substrates. The dashed lines mark the atomic steps. In plot (

**a**), the green diamonds 1 correspond to Figure 8b, the black disks 2 to 8c, and the blue triangles 3 to 8d. Adapted from [2].

**Figure 9.**‘Nematicity’ in LSCO films and bulk crystals. (

**a**) The anisotropy in resistivity, expressed as the ratio ${\rho}_{b}/{\rho}_{a}$. Blue solid curve: our data from a heavily underdoped (x = 0.02) LSCO film. Red dashed line: data measured on bulk LSCO crystal (after Ref. [34]). (

**b**) The same as in (

**a**), but for x = 0.04.

**Figure 10.**(

**a**) The lithography pattern used to fabricate a COMBE LSCO library. The central Hall bar is aligned along the direction of the Sr doping gradient. One pair of contacts, e.g., {1, 3}, measure $\rho $ at one sector (one “pixel”) of the Hall bar, and 30 pixels in total comprise one library of doping-dependent $\rho \left(x\right)$. Opposed pairs, e.g., {1, 2}, are used to measure the transverse ${\rho}_{T}\left(x\right)$. (

**b**) Open circles: ${\rho}_{T}\left(x\right)/\rho \left(x\right)$ measured using this technique in an LSCO thin film. Black dashed line: a linear fit to the data. The measured ${\rho}_{T}/\rho $ ratio is approximately linear in x. Adapted from [1].

**Figure 11.**The resistivity of a representative LSCO film synthesized by ALL-MBE. The dashed black line indicates a linear-fit extrapolation of the resistivity to temperatures below the superconducting transition, $\rho ={\rho}_{0}+\alpha T$, where ${\rho}_{0}\approx 1.5\left(\pm 1.5\right)\mathsf{\mu}\mathsf{\Omega}\mathrm{cm}$ and $\alpha \approx 1.02\left(\pm 0.01\right)\mathsf{\mu}\mathsf{\Omega}\mathrm{cm}/\mathrm{K}$. Adapted from [41].

**Figure 12.**(

**a**) Mutual inductance, the real part, showing Meissner effect. (

**b**) Imaginary part, showing the dissipation peak. (

**c**) The same as in panel (

**b**) but magnified near T

_{c}. One can take the HWHM of this peak ($\approx 0.1\mathrm{K}$) as an upper limit on the variations of T

_{c}in the film. NB: If the film contained two areas with ${T}_{c1}\ne {T}_{c2}$, and $\left|{T}_{c1}-{T}_{c2}\right|<0.1\mathrm{K}$, we would have resolved two peaks in ImM. If there were a (quasi)continuous distribution of domains with a spread $\Delta {T}_{c}>0.1\mathrm{K}$, we would have observed one broad peak with a width larger than $\Delta {T}_{c}$. Adapted from [29].

**Figure 13.**(

**a**), The temperature dependence of longitudinal resistivity $\rho \left(T\right)$ in LSCO (x = 0.063). (

**b**) The temperature dependence of the derivative $\mathrm{d}\rho \left(T\right)/\mathrm{d}T$. (

**c**) The temperature dependence of transverse resistivity ${\rho}_{T}\left(T\right)$ in LSCO (x = 0.063). The derivative $\mathrm{d}\rho /\mathrm{d}T$ changes sign twice as T is lowered, while in contrast, ${\rho}_{T}$ is always positive, so apparently, ${\rho}_{T}\left(T\right)$ does not scale with $\mathrm{d}\rho \left(T\right)/\mathrm{d}T$.

**Figure 14.**(

**a**) $\rho \left(T\right)$ dependence is measured in several devices fabricated in an LSCO thin film. The superconducting transitions are relatively broad and thus can be characterized in various ways: by the temperature T

^{O}at which the superconductivity onsets, T

^{M}at which $\rho \left(T\right)$ has the largest slope, and the temperatures at which the resistance drops to 90% (T

_{1}), 50% (T

_{2}), or 10 % (T

_{3}), etc. (

**b**) However, each of these characteristic temperatures, T

^{o}, T

_{1}, T

_{2}, T

_{3}, and T

^{M}, shows very little variation device-to-device. Adapted from [30].

**Figure 15.**Comparison of SRO, an electronic nematic, with the Ti control sample. In the Ti film, ${\rho}_{T}$ is zero at T = 295 K (purple dots) and T = 4 K (magenta dots). This is in stark contrast to SRO, where ${\rho}_{T}\left(\varphi \right)$ is substantial at T = 295 K (light blue dots) and very large at T = 4 K. (dark blue dots). The solid lines are the fits to ${\rho}_{T}\left(\varphi \right)={\rho}_{T0}\mathrm{sin}\left[2\left(\varphi -\alpha \right)\right]$ for experimental data in matching colors. The solid black line stands for ${\rho}_{T}=0$. Adapted from [26].

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**MDPI and ACS Style**

Božović, I.; He, X.; Bollinger, A.T.; Caruso, R.
Is Nematicity in Cuprates Real? *Condens. Matter* **2023**, *8*, 7.
https://doi.org/10.3390/condmat8010007

**AMA Style**

Božović I, He X, Bollinger AT, Caruso R.
Is Nematicity in Cuprates Real? *Condensed Matter*. 2023; 8(1):7.
https://doi.org/10.3390/condmat8010007

**Chicago/Turabian Style**

Božović, Ivan, Xi He, Anthony T. Bollinger, and Roberta Caruso.
2023. "Is Nematicity in Cuprates Real?" *Condensed Matter* 8, no. 1: 7.
https://doi.org/10.3390/condmat8010007