Zero-field resistivity<\/p>\n
Combinatorial LCCO thin films were grown epitaxially on SrTiO3 substrates from targets with x\u2009=\u20090.10 and 0.19, facilitating measurements over a large region of the LCCO phase diagram in minute steps in x. A comprehensive description of the deposition and patterning processes can be found elsewhere (ref. 10<\/a>). Two films, S1 (0.116\u2009\u2264\u2009x\u2009\u2264\u20090.159) and S2 (0.159\u2009\u2264\u2009x\u2009\u2264\u20090.188), were patterned into microbridges and their in-plane resistivity measured using a standard four-point a.c. lock-in method. Seven channels could be measured simultaneously on a single film and in total, 13 were measured on S1 and 7 on S2 with one common doping level at x\u2009=\u20090.159. As shown in Extended Data Fig. 1a<\/a>, the two \u03c1(T) curves at x\u2009=\u20090.159 coincide below T\u2009=\u200970\u2009K provided that the data for S2 are scaled (reduced) by a factor of 1.5. This same scaling factor was then applied to all channels in S2. Note that the data for film S1 (Extended Data Fig. 1a<\/a>) and Supplementary Fig. 1<\/a> are only discrete points (the zero-field values for each field sweep) rather than a continuous curve. To obtain a full, uninterrupted \u03c1(0,T) curve between 2\u2009K and 290\u2009K inside the Bitter magnet would have required at least 8\u2009h of continuous measurement, time that was not available at the high-field facility when measuring S1. Fortunately, for S2, the cryostat was available for an extended period following the measurement, making it possible to perform a warming run in zero-field over 3\u2009days.<\/p>\n Extended Data Fig. 1b,c<\/a> shows the zero-field in-plane resistivity \u03c1(0,T) and the corresponding derivatives d\u03c1\/dT for a selection of channels on S2, respectively. As clear from Extended Data Fig. 1e<\/a>, \u03c1(0,T)\u2009\u221d\u2009A1T\u2009+\u2009A2T2 at low T (below 60\u2009K), with A2 constant and A1 decreasing with increasing x (and decreasing Tc (ref. 10<\/a>)). This is the main reason why the coupling strength parameter g(x) was assigned uniquely to the T-linear term in equation (1<\/a>). Consistent with previous reports45<\/a>,54<\/a>, \u03c1(0,T) develops a stronger T dependence at higher T (>70\u2009K). In-plane MR measurements were carried out in a 35-T Bitter magnet at HFML-FELIX in Nijmegen with the magnetic field applied perpendicular or parallel to the CuO2 planes. The sample stage was glued with GE varnish to maintain its orientation.<\/p>\n Longitudinal MR<\/p>\n In a quasi-two-dimensional (2D) metal hosting Landau quasiparticles, a magnetic field oriented parallel to the direction of the injected current will generate a negligible Lorentz force, and as a result, the orbital MR is expected to be close to zero. Extended Data Fig. 2<\/a> compares the transverse and longitudinal MR for three overdoped concentrations\u2014x\u2009=\u20090.159, 0.174 and 0.188\u2014at T\u2009=\u200975\u2009K. Clearly, the longitudinal MR is negligible at this temperature, implying that the MR of LCCO is entirely orbitally driven (at lower temperatures, a non-zero MR appears that we attribute to the emergence of superconducting fluctuations). This anisotropy in the MR in electron-doped cuprates has been observed previously22<\/a>,49<\/a>,54<\/a>,55<\/a>, in addition to an isotropic negative component in the longitudinal MR in PCCO for x\u2009\u2264\u2009xFSR that was attributed to AFM order55<\/a>. In the p-doped cuprates Bi2Sr2CuO6+\u03b4 and Tl2Ba2CuO6+\u03b4, by contrast, the longitudinal MR is found to have a similar magnitude as the transverse MR18<\/a>, behaviour that is inconsistent with conventional (Boltzmann) transport theory.<\/p>\n Boltzmann formalism in electron-doped cuprates<\/p>\n The purely orbital nature of the MR in overdoped LCCO motivates us to examine its T and H dependencies within a standard Boltzmann treatment, although modified to take into account the presence of a marked in-plane anisotropy of the scattering rate. We use the SCTIF applied to a 2D metal in the time basis to avoid divergences that can occur as B\u2009\u2192\u20090. The SCTIF is the general solution of the Boltzmann transport theory equation within the relaxation time approximation and is used to determine the response of quasiparticles to an external electromagnetic field at arbitrary field strengths. The expression for conductivity within the time basis, after neglecting the temperature broadening of the FS, is56<\/a>,57<\/a><\/p>\n $${\\upsigma }_{{ij}}=\\frac{1}{4{{\\uppi }}^{3}}{\\int }_{\\rm{FS}}{{\\rm{d}}}^{2}k\\frac{1}{{\\hslash }{v}_{\\rm{F}}}q{v}_{i}{\\int }_{0}^{{\\infty }}q{v}_{j}\\left(-t\\right){P_{{\\upphi}}}\\left(t\\right){\\rm{d}}t,$$<\/p>\n \n (2)\n <\/p>\n where vi,j are the velocity components with {i,j}\u2009=\u2009{x,y}. P\u03d5(t) is the probability of a quasiparticle starting at angle \u03d5 to survive for time t, defined as<\/p>\n $${P}_{{\\rm{\\phi }}}\\left(t\\right){\\rm{:= }}\\exp \\left[-{\\int }_{0}^{t}\\frac{d{t}^{{\\prime} }}{{{\\tau }}\\left({t}^{{\\prime} }\\right)}\\right],$$<\/p>\n \n (3)\n <\/p>\n where \u03c4(t) is the velocity relaxation time, and all other parameters take their usual values. The role of the magnetic field can be made explicit within the SCITF formalism by making a change of variables from t to \u03c6, the (azimuthal) angle swept around the FS under the influence of the field H\u2225c. Writing \\(\\varphi ={\\omega }_{\\rm{c}}t\\), where \\({\\omega }_{\\rm{c}}\\left(\\varphi \\right)={eB}{\\vec{k}}_{\\rm{F}}\\left(\\varphi \\right).{\\vec{v}}_{\\rm{F}}\\left(\\varphi \\right)\/\\hslash {k}_{\\rm{F}}^{2}\\left(\\varphi \\right)\\), we can re-express the SCTIF formalism as<\/p>\n $${{\\rm{\\sigma }}}_{{ij}}=\\frac{1}{4{{{\\uppi }}}^{3}}{\\int }_{\\rm{FS}}{\\mathrm{d}}^{2}k\\frac{1}{{{\\hslash }}{v}_{\\rm{F}}}q{v}_{i}{\\int }_{0}^{2\\uppi }{\\mathrm{d}}\\varphi \\frac{q{v}_{j}\\left(\\varphi \\right)}{{\\omega }_{\\rm{c}}\\left(\\varphi \\right)}{\\int }_{0}^{{{\\infty }}}{\\mathrm{d}}{\\varphi }^{{\\prime} }\\exp \\left({\\int }_{\\varphi }^{\\varphi {\\prime}}{\\mathrm{d}}\\varphi {\\prime} {\\prime} \\frac{-\\varphi {\\prime} {\\prime} }{{\\omega }_{\\rm{c}}\\left(\\varphi {\\prime} {\\prime} \\right)\\tau \\left(\\varphi {\\prime} {\\prime} \\right)}\\right)$$<\/p>\n \n (4)\n <\/p>\n The key parameter that determines the H dependence of the MR is the mean free path \u2113 (vF\u03c4), particularly its anisotropy. The FS of LCCO does not exhibit sharp corners and is not in close proximity to a van Hove singularity. Thus, the k dependence of vF (which incorporates the variation in the density of states around the FS) is anticipated to be limited. Indeed, for x\u2009=\u20090.188, where the FS curvature is most pronounced, vF varies by \u226410% (ref. 26<\/a>). It is, thus, the anisotropy of \u03c4 that is expected to dominate the MR. A standard tensor inversion is then used to convert the magnetoconductivity to MR.<\/p>\n Given the strong magnetic order observed below xFSR, spin correlations probably manifest at all x even when their obvious signatures in the MR vanish. Such spin correlations may necessitate going beyond the relaxation time approximation by, for example, including the effects of CVCs, as highlighted in the main text. These CVCs have been predicted to give rise to a non-Fermi-liquid T dependence of the resistivity and have a crucial impact on the Hall effect30<\/a>, as discussed later.<\/p>\n Tight-binding parameterization<\/p>\n The general tight-binding model for 2D systems with tetragonal symmetry is given by<\/p>\n $$\\begin{array}{l}E(k)={\\varepsilon }_{0}-2t(\\cos ({k}_{x}a)+\\,\\cos ({k}_{y}a))-4t{\\prime} (\\cos ({k}_{x}a)\\cos ({k}_{y}a))\\\\\\qquad\\quad-2t{\\prime} {\\prime} (\\cos (2{k}_{x}a)+\\,\\cos (2{k}_{x}a))\\end{array}$$<\/p>\n \n (5)\n <\/p>\n where t, t\u2032 and t\u2033 are the nearest, next-nearest and next-next-nearest-neighbour hopping parameters, and \u03b50 is a measure of the chemical potential. We use the conditions given in ref. 26<\/a> in which t\u2009=\u20090.3 and t\u2033\/t\u2032\u2009=\u2009\u22120.5 remain constant at all x, and assume that the Luttinger count \\(x=\\frac{2{A}_{\\text{FS}}}{{A}_{\\text{BZ}}}-1\\) is satisfied to fix \u03b50 and t\u2032\/t. Here AFS(BZ) is the area of the FS (BZ), respectively. The TB parameters for LCCO are listed in Extended Data Table 1<\/a>. LCCO is tetragonal with an in-plane lattice parameter a\u2009=\u20094.01\u2009\u00c5. The \\({{\\hslash }}\\vec{v}={\\nabla }_{k}E\\) relation is used to obtain the components of the velocity in equation (2<\/a>), leaving \u03c4 as the only parameter that needs to be explicitly defined.<\/p>\n Simulations<\/p>\n Although the form of the scattering rate \u03c4\u22121\\(\\left(\\varphi ,T,x\\right)\\) (redefined as \\(\\varGamma \\left(\\varphi ,T,x\\right)\\) hereafter) expressed in equation (1<\/a>) is a result of extensive modelling, it contains only those components required to capture all the key features of the experimental resistivity and MR data (outside of the FSR regime) as a function of field strength, temperature and doping with a single doping-dependent variable parameter g(x). In this section, we justify the inclusion of each component in \\(\\varGamma \\left(\\varphi ,T,x\\right)\\) and present alternative scenarios that fail to capture all of the key features of the MR.<\/p>\n As discussed in the main text, there are two primary candidates for the dominant, anisotropic inelastic scattering rate in overdoped LCCO: spin fluctuations and short-range charge correlations. According to a recent angle-resolved photoemission spectroscopy (ARPES) study, the nodal regions of the FS in LCCO exhibit non-Fermi-liquid behaviour58<\/a>, probably due to their close proximity to the AFM zone boundary. There are no known wavevectors connecting two nodal points, however, and no experimental evidence so far for hotspot scattering at the nodes. Hence, we do not consider this scenario here.<\/p>\n Within the spin fluctuation scenario, the core element of \\(\\varGamma \\left(\\varphi ,T,x\\right)\\) is a four-fold symmetric, double-peaked component that gives rise to eight broad peaks, each located at the intersection of the underlying FS and the AFM BZ boundary (Fig. 3<\/a>) and reproduced in Extended Data Fig. 3b1<\/a> (note that the loci of these hotspots shift in accordance with the doping level). For simplicity, both the T-linear and T2 components of \\(\\varGamma \\left(\\varphi ,T,x\\right)\\) are assigned this anisotropic form, though only the T-linear term incorporates g(x), as motivated by the derivatives of the zero-field resistivity presented in Extended Data Fig. 1c<\/a>. Combined with an isotropic impurity scattering rate \u03930 of appropriate magnitude, this T-dependent scattering rate (labelled hereafter \u0393T) captures the most striking feature of the MR data, namely, the growth of \u2206\u03c1\/\u03c1(0) with increasing T (Extended Data Fig. 3a3<\/a>). Recall that Kohler\u2019s rule would predict an opposite T dependence for \u2206\u03c1\/\u03c1(0). The H dependence of the MR is also reasonably well captured by \u03930\u2009+\u2009\u0393T, except two important details. First, at low fields, the derivatives fan out from H\u2009=\u20090 (Extended Data Fig. 3b4<\/a>), whereas in the data (Extended Data Fig. 3a4<\/a>), the initial slopes are essentially T independent. Second, the field dependence tends towards saturation as the anisotropy in \u0393T becomes washed out by ever-increasing cyclotron motion at low T, and at T\u2009=\u200950\u2009K, the MR is purely quadratic. In reality, the MR of overdoped LCCO remains H linear up to fields as high as 65\u2009T (ref. 24<\/a>). This observation justifies the inclusion in our simulation of loci (\u2018hotspots\u2019) on the FS that impede cyclotron motion19<\/a> at all temperatures in this range.<\/p>\n Extended Data Fig. 3c,d<\/a> considers a combination of spin and charge fluctuation contributions to \\(\\varGamma \\left(\\varphi ,T,x\\right)\\) to illustrate the fact that adding extra complexity does not necessarily improve the simulations. In Extended Data Fig. 3c<\/a>, for example, we add small hotspots at the antinodal regions to mimic additional scattering off charge fluctuations between the antinodes (Fig. 3<\/a>, grey double-headed arrows)50<\/a>,51<\/a>.<\/p>\n The resulting simulations show an MR that is quadratic at low fields but which rapidly transitions towards saturation, as shown in the 20K MR curve (Extended Data Fig. 3c3<\/a>) and its derivative (Extended Data Fig. 3c4<\/a>). This tendency is due to the fact that the SCTIF concerns the average scattering rate around the FS; although this scenario contains substantive and complex anisotropy, such a combination of two anisotropic components generates a total scattering rate that is, in fact, on a coarse level, more isotropic, thereby explaining the tendency towards MR saturation.<\/p>\n In Extended Data Fig. 3d<\/a>, we present the converse situation with dominant scattering from charge fluctuations, combined with subdominant hotspots at the AFM BZ boundary. In this case, the anisotropic inelastic component is a broad peak centred around the antinodes. This simulation matches more closely with the experimental curves; the initial slopes are essentially independent of temperature in this scenario (Extended Data Fig. 3d4<\/a>), although once again, the simulated MR shows a tendency towards saturation at higher field strengths, particularly at lower T.<\/p>\n Finally, to convince the reader that hotspots alone are insufficient to reproduce the experimental data, Extended Data Fig. 3e<\/a> shows the results of a simulation in which the anisotropy in \u0393(\u03c6) is contained solely in the sharp hotspots positioned at the AFM zone boundary. Although the T dependence of \u03c1(0,T) is reasonably well-matched in this scenario, (Extended Data Fig. 3e2<\/a>), the T dependence of the MR is inverted (Extended Data Fig. 3e3<\/a>). This demonstrates the clear need for the inclusion of \u0393T in the full expression for \\(\\varGamma \\left(\\varphi ,T,x\\right)\\).<\/p>\n The scattering rate parameters that provide the best overall fits to the x\u2009=\u20090.159 data based on a pure spin or charge fluctuation scenario (Fig. 4<\/a>) are listed in Extended Data Table 2<\/a>. The hotspot scattering rate \u03c4HS\u22121 is constructed as a narrow Gaussian peak whose height and width are also given. These hotspot peaks have a similar magnitude as the inelastic scattering rate, resulting in an overall \u0393 that is either peaked at the AFM zone boundary or at the antinodes, both scenarios providing the anisotropy that is required to reproduce the MR at low T. This parameterization is also consistent with the magnitude of the residual resistivity. Since \\({\\tau }_{{\\rm{HS}}}^{-1}\\) peaks sharply around the intersections of the FS and the AFM BZ, the dominant contribution to the residual resistivity is \\({\\tau }_{{\\rm{imp}}}^{-1}\\). Ignoring the slight in-plane anisotropy in kF and vF, we can rewrite \\({\\rho }_{0}=\\left(2\\uppi \\hslash c\/{e}^{2}{k}_{\\rm{F}}{v}_{\\rm{F}}\\right){\\tau }_{{\\rm{imp}}}^{-1}\\). Inserting the values for kF and vF from Fig. 4a(i)<\/a> and for \\({\\tau }_{{\\rm{imp}}}^{-1}\\) from Extended Data Table 2<\/a>, we arrive at \u03c10\u2009\u2248\u200915\u2009\u03bc\u03a9\u2009cm. Using the hotspot parameters quoted in Extended Data Table 2<\/a>, we estimate that the angle-averaged \\({\\tau }_{{\\rm{HS}}}^{-1}\\)\u2009\u2248\u20094\u2009ps\u22121, giving a total \u03c10\u2009\u2248\u200917.2\u2009\u03bc\u03a9\u2009cm. From Extended Data Fig. 1<\/a>, we estimate that the actual value of \u03c10\u2009\u2248\u200917.5\u2009\u03bc\u03a9\u2009cm.<\/p>\n Incorporation of CVCs into the Boltzmann formalism<\/p>\n In a 2D metal, the Hall conductivity \u03c3xy is determined by the \u2018Stokes\u2019 area swept out by the \\(\\vec{{\\ell}}(\\varphi )\\) vector as \u03c6 varies around the FS29<\/a>. The FS of LCCO (Figs. 3<\/a> and 4<\/a>) has only hole-like curvature, meaning that \\({\\vec{v}}_{{\\rm{F}}}\\) has the same circulation at all \u03c6. In this circumstance, even the inclusion of strong (orders-of-magnitude) anisotropy in \u03c4\u22121(\u03c6) cannot cause \u03c3xy, and therefore RH, to become negative. In LCCO, for x\u2009>\u2009xFSR, however, RH(T) drops monotonically from close to its isotropic (Drude) value towards zero and even becomes negative at elevated temperatures10<\/a>,27<\/a> (Extended Data Fig. 4<\/a>). Although a two-band model is able to account for the behaviour of the Hall resistance, it neither respects the FS measured in ARPES nor can it simultaneously account for the MR59<\/a>. In the absence of any additional FSR, for which there is no reported evidence, the only effective way of explaining the sign change in RH(T) within a single fluid picture is a breakdown of the relaxation time approximation and the incorporation of CVCs into the calculation for \u03c3xy (ref. 28<\/a>).<\/p>\n In this section, we develop a semiquantitative picture of the evolution of RH(T) by incorporating the effect of CVCs into the analysis of the Hall response. For these simulations, we use the Jones\u2013Zener expansion of the Boltzmann transport equation. Although this form of the Boltzmann equation is only applicable at low magnetic fields, RH itself has only been measured so far at relatively low field strengths. Moreover, this form allows the use of analytical expressions for most of the key quantities.We start by introducing analytical expressions for kF(\u03c6) and vF(\u03c6) that simplify the TB parameterizations above and preserving their four-fold symmetry:<\/p>\n $${k}_{{F}}(\\varphi )={k}_{F0}(1+\\alpha\\;{\\sin }^{2}(2\\varphi ))$$<\/p>\n \n (6)\n <\/p>\n $${v}_{{F}}(\\varphi )={v}_{F0}(1+\\beta\\;{\\sin }^{2}(2\\varphi ))$$<\/p>\n \n (7)\n <\/p>\n Here kF0\u2009=\u20095.424\u2009\u00d7\u2009109\u2009m\u22121, \u03b1\u2009=\u20090.11, vF0\u2009=\u20094.946\u2009\u00d7\u2009105\u2009m\u2009s\u22121 and \u03b2\u2009=\u20090.03. Note that in our simulations of the Hall response, the variation in vF(\u03c6) at the antinodes has been flattened slightly relative to equation (7<\/a>) to approximate the TB parameterization more closely. This has no impact on \u03c1(H,T). vx(\u03c6) and vy(\u03c6) are then obtained from vF(\u03c6) using the following analytical expressions:<\/p>\n $$\\begin{array}{l}{v}_{x}(\\varphi )={v}_{{F}}(\\varphi ){\\cos }(\\varphi -\\gamma )\\\\ {v}_{y}(\\varphi )={v}_{{F}}(\\varphi ){\\sin }(\\varphi -\\gamma )\\end{array}$$<\/p>\n \n (8)\n <\/p>\n where60<\/a><\/p>\n $$\\gamma (\\varphi )={\\tan }^{-1}\\left(\\partial \\left[\\mathrm{ln}\\left[{k}_{\\rm{F}}(\\varphi )\\right]\\right]\/\\partial \\varphi \\right)$$<\/p>\n \n (9)\n <\/p>\n In electron-doped cuprates close to commensurate AFM order, the dominant scattering vector is \\(\\vec{Q}\\)\u2009=\u2009(\u03c0, \u03c0). When CVCs are introduced, momenta linked through \\(\\vec{Q}\\) are heavily mixed through strong scattering, as a result of which the total current Jk becomes approximately parallel to vk\u2009+\u2009vk+Q and is, thus, no longer perpendicular to the FS. Previously, Kontani used the linear response theory to derive the following expression to describe how \\({\\vec{J}}_{k}\\) transforms in proximity to the AFM hotspot30<\/a>:<\/p>\n $${\\vec{J}}_{k}=\\frac{1}{1-{{{\\epsilon }}}_{k}^{2}}\\left({\\vec{\\nu }}_{k}+{{{\\epsilon }}}_{k}{\\vec{\\nu }}_{k\\pm Q}\\right)$$<\/p>\n \n (10)\n <\/p>\n where \\({{{\\epsilon}}}_{k}\\approx \\left(1-c{{\\rm{\\xi }}}_{\\text{AFM}}^{2}\\right) < 1,{{\\rm{\\xi }}}_{\\text{AFM}}\\) is the AFM correlation length and c is a constant. To incorporate CVCs into our Boltzmann framework, we first substitute this expression for \\(\\vec{{J}_{k}}({\\vec{v}}_{k})\\) into the kinetic equation and recalculate vx and vy:<\/p>\n $$\\begin{array}{l}{v}_{x}^{\\ast }(\\varphi )=\\frac{1}{1-\\epsilon {(\\varphi )}^{2}}\\left({v}_{x}(\\varphi )+\\epsilon (\\varphi ){v}_{x}\\left({\\varphi }^{{{{\\prime} }}}\\right)\\right),\\\\ {v}_{y}^{\\ast }(\\varphi )=\\frac{1}{1-\\epsilon {(\\varphi )}^{2}}\\left({v}_{y}(\\varphi )+\\epsilon (\\varphi ){v}_{y}\\left({\\varphi }^{{{{\\prime} }}}\\right)\\right),\\end{array}$$<\/p>\n \n (11)\n <\/p>\n where \u03c6\u2032\u2009=\u2009\u03c6\u2009\u2013\u2009(\u03c0, \u03c0). This process effectively links the hotspots through the AFM Q vector. For each value of x, the angles \u03c6n associated with the hotspots are determined by the locations at which the FS crosses the AFM zone boundary (for example, Extended Data Fig. 5c,d<\/a>). For \u03f5(\u03c6), we assume a Gaussian with a fixed breadth and height (but, for simplicity, ignore the prefactor 1\/(1\u2009\u2013\u2009\u03f52)), using the following expression:<\/p>\n $${{\\epsilon }}\\left({{\\varphi }}\\right)=\\exp \\frac{-{\\left(\\varphi -{\\varphi }_{n}\\right)}^{2}}{2{\\sigma }^{2}}$$<\/p>\n \n (12)\n <\/p>\n with \u03c3\u2009=\u20090.07 (chosen to approximate the anisotropy expressed in equation (1<\/a>)). We then substitute these expressions for \\({\\nu }_{x}^{* }\\) and \\({\\nu }_{y}^{* }\\) into the Jones\u2013Zener expansions for \u03c3xx and \u03c3xy:<\/p>\n $${{\\rm{\\sigma }}}_{{xx}}=\\frac{{e}^{2}}{2{{{\\uppi }}}^{2}{{\\hslash }}c}{\\int }_{0}^{2{{\\uppi }}}\\frac{{k}_{\\rm{F}}\\left({{\\varphi }}\\right){v}_{x}^{* }\\left({{\\varphi }}\\right)\\cos \\left({{\\varphi }}-{\\rm{\\gamma }}\\right)}{\\varGamma \\left({{\\varphi }}\\right)\\cos {\\rm{\\gamma }}}\\,{\\mathrm{d}}{{\\varphi }}$$<\/p>\n \n (13)\n <\/p>\n $${{\\rm{\\sigma }}}_{{xy}}=\\frac{-{e}^{3}}{2{{{\\uppi }}}^{2}{{{\\hslash }}}^{2}c}{\\int }_{0}^{2{{\\uppi }}}\\frac{{v}_{x}^{* }\\left({{\\varphi }}\\right)}{\\varGamma \\left({{\\varphi }}\\right)}\\frac{\\partial \\left[{v}_{y}^{* }\\left({{\\varphi }}\\right)\/\\varGamma \\left({{\\varphi }}\\right)\\right]}{\\partial {{\\varphi }}}\\,{\\mathrm{d}}{{\\varphi }}$$<\/p>\n \n (14)\n <\/p>\n where c\u2009=\u20096.2\u2009\u00c5 is the c-axis lattice spacing. Finally, for \\(\\Gamma \\left(\\varphi ,T,x\\right)\\), we adopt the same form as in equation (1<\/a>), which is reproduced here combined with the (isotropic) impurity scattering rate \u03930:<\/p>\n $$\\varGamma \\left(\\varphi ,T,x\\right)={\\varGamma }_{0}+{\\varGamma }_{\\text{HS}}({{\\varphi }})+\\left(g\\left(x\\right){{\\rm{\\alpha }}}_{1}T+{{\\rm{\\alpha }}}_{2}{T}^{2}\\right){\\sin }^{12}\\left[2\\left({{\\varphi }}\\pm \\left(\\frac{{{\\uppi }}}{4}-{\\rm{HS}}\\right)\\right)\\right]$$<\/p>\n \n (15)\n <\/p>\n where \u0393HS is the additional sharp Gaussian contribution at each hotspot whose parameters are given in Extended Data Table 2<\/a>.<\/p>\n Anisotropic scattering term in LCCO<\/p>\n Extended Data Fig. 5a,b<\/a> shows vx(\u03c6) and vy(\u03c6), respectively, together with their CVC-modified counterparts \\({v}_{x}^{* }\\left(\\varphi \\right)\\) and \\({v}_{y}^{* }\\left(\\varphi \\right)\\) for the parameterizations defined in equation (11<\/a>). The expression for the Hall conductivity \u03c3xy of a 2D metal29<\/a> (equation (14<\/a>) and reproduced in the box at the top of the figure) is set by the product of \u2113x(\u03c6) and \u2202\u2113y(\u03c6)\/\u2202\u03c6. In the absence of anisotropy in \u2113(\u03c6) or any CVCs, \u2113x \u00d7 \u2202\u2113y is always positive. Introducing the CVCs, however, creates a window in \u03c6, where \\({v}_{x}^{* }\\left(\\varphi \\right)\\) and \\(\\partial {v}_{y}^{* }\/\\partial {\\rm{\\varphi }}\\) have opposite sign, as indicated by the vertical shaded bars in both panels. Within the Ong construction, this sign change arises due to a change in circulation of the \\(\\vec{{\\ell}}\\) vector around the FS. These regions of opposite circulation are highlighted in Extended Data Fig. 5c,d<\/a> (thick solid arrows). Correspondingly, these regions can contribute a component to \u03c3xy of opposite sign, even when the scattering time \u03c4 is isotropic. The width of these regions is set by the width of the \u2018influence\u2019 of the CVCs. Incorporating anisotropy into \u03c4(\u03c6) (or equivalently \u0393(\u03c6)) will act to enhance or reduce this effect, depending on the form of \u0393(\u03c6) itself.<\/p>\n For RH(T) to drop in magnitude with increasing T relative to its isotropic, low-T value, we require that \u2113 in those regions in which the circulation of \u2113(\u03c6) is opposite to that of \u03c6 to become longer\u2014a scenario we label \u0393N in Extended Data Fig. 5c<\/a>. Conversely, if \u2113 is shorter at the antinodes, the opposite effect occurs and \u03c3xy (and hence RH) will become larger as the anisotropy in the total scattering rate grows. This scenario, labelled \u0393AN, is shown in Extended Data Fig. 5d<\/a>. The former situation (\u0393N) is relevant for the case in which spin fluctuation scattering is dominant because the nodes are notably closer to the AFM hotspots. \u0393AN(\u03c6), on the other hand (Extended Data Fig. 5d<\/a>), is more likely to be associated with small-\\(\\vec{Q}\\) charge fluctuations or strong electron\u2013phonon scattering, as suggested by recent ARPES measurements on LCCO that report the presence of a ubiquitous antinodal kink at a characteristic (doping-independent) energy scale (~45\u2009meV). Its ubiquity points towards coupling to phonons61<\/a>. Although this phonon-derived mode may be manifest in the single-particle response, its impact on the transport properties has not yet been verified.<\/p>\n An inspection of Extended Data Fig. 5c,d<\/a> provides a qualitative understanding of which form of \u0393(\u03c6) will act to reduce RH with increasing T. As explained in the figure caption, for \u0393N(\u03c6), the counter-circulation of \\(\\vec{{\\ell}}(\\varphi )\\) occurs in the regions in which \u2113 is the longest. Hence, as anisotropy in \u0393N(\u03c6) grows with increasing T, the negative contributions to \u03c3xy also grow, causing RH(T) to drop. Conversely, were \u0393AN to be the appropriate form of the T-dependent anisotropic component, RH(T) would increase with increasing T.<\/p>\n This analysis is put on a more quantitative footing in Extended Data Fig. 6<\/a>, where RH is calculated within the same Jones\u2013Zener formalism for different forms of \u0393(\u03c6). For all panels, we use a parameterization for kF(\u03c6) and vF(\u03c6) that is consistent with those used for x\u2009=\u20090.159 in Fig. 4<\/a>. Extended Data Fig. 6a<\/a> shows the isotropic case: \u0393(\u03c6)\u2009=\u2009\u03930. In this instance, we obtain RH\u2009=\u2009+8.9\u2009\u00d7\u200910\u221210\u2009m3\u2009C\u22121, about 20% higher than the Drude estimate for a (hole) carrier density of 1\u2009\u2013\u20090.159\u2009=\u20090.841 per unit cell and in excellent agreement with the data shown in Extended Data Fig. 4a<\/a> for x\u2009=\u20090.16 at low T, even with the inclusion of CVCs. For \u0393(\u03c6)\u2009=\u2009\u03930\u2009+\u2009\u0393HS (Extended Data Fig. 6b<\/a>), we find RH\u2009=\u2009+5.6\u2009\u00d7\u200910\u221210\u2009m3\u2009C\u22121, showing that the addition of hotspot scattering leads to a slight reduction in RH.<\/p>\n As argued above, a larger RH is found when the scattering rate (between the hotspots) is larger in the antinodes. Extended Data Fig. 6c<\/a> depicts a form of \u0393(\u03c6)\u2009=\u2009\u03930\u2009+\u2009\u0393HS\u2009+\u2009\u0393AN. This form of scattering could reflect a scenario in which strong charge fluctuations coexist with the AFM hotspots. In this case, RH\u2009=\u2009+1.03\u2009\u00d7\u200910\u22129\u2009m3\u2009C\u22121, confirming that additional scattering at the antinodes leads to a further increase in RH in the presence of CVCs. Hence, as anisotropy in \u0393(\u03c6) grows with increasing T, so will RH. This is opposite to what is observed experimentally10<\/a>,27<\/a>. Clearly, to simulate a vanishing RH, the anisotropic component of \u0393(\u03c6) needs to be inverted.<\/p>\n This is done in Extended Data Fig. 6d<\/a>, where now \u0393(\u03c6)\u2009=\u2009\u03930\u2009+\u2009\u0393HS\u2009+\u2009\u0393N. Using the same parameters that were used to model the MR in the main text (in the absence of CVCs), we find RH\u2009=\u2009+4.3\u2009\u00d7\u200910\u221210\u2009m3\u2009C\u22121, which is about half the value found for \u03930 only.<\/p>\n Finally, Extended Data Fig. 6e<\/a> shows that removing the AFM hotspots and considering only CVCs and anisotropic scattering created by short-Q charge fluctuations leads to a marked enhancement in RH. This is due to the fact that at the loci on the FS where the mixing of states occurs (that is, where the CVCs are finite), (i) the mean free path is strongly suppressed and (ii) the FS is essentially flat with very little curvature. Indeed, despite extensive analysis, we were unable to find a simulation based purely on dominant short-Q charge fluctuation scattering that could reproduce both the growth in the MR and the concomitant drop in RH(T) with increasing temperature. We, thus, do not consider this scenario further in this article. Finally, in Supplementary Fig. 5<\/a>, we show that the incorporation of modified velocities (from spin fluctuation scattering) into the SCTIF has a negligible effect on the modelling of the MR.<\/p>\n In summary, we find that in addition to CVCs due to AFM spin fluctuations, the inclusion of an associated \u0393N with a magnitude that leads to a doubling of the MR (Fig. 4<\/a>) also causes RH to halve. This form of \\(\\varGamma \\left(\\varphi ,T,x\\right)\\), thus, enables the two key T dependencies in the magnetotransport of LCCO beyond x\u2009>\u2009xFSR to be simulated within the same parameterization. This result also echoes an earlier study of the frequency (\u03c9) dependence of the Hall angle in overdoped PCCO where a sign change of RH(T,\u03c9) in an FS with only a singular curvature was also interpreted as evidence for AFM CVCs28<\/a>. This analysis also suggests that it should be possible to reproduce both RH(T,x) and the MR from the ARPES-derived FS of LCCO using a single set of parameters. Given the large number of variables (width, height of hotspot, precise form of the anisotropic component and so on), however, here we simply present this semiquantitative analysis to justify the phenomenological form of \u0393(\u03c6) and close this section with some preliminary thoughts on its possible physical origin.<\/p>\n The phenomenological form of \u0393(\u03c6)\u2014necessitated primarily by the unusual growth in \u2206\u03c1\/\u03c1(0) in LCCO over a temperature range in which \u2113 itself is diminishing\u2014has three principal features: T linearity; large, smooth anisotropy; and maxima at the AFM BZ boundary. The general approach to account for such a combination of features is to invoke the presence of low-lying bosonic excitations. The most natural candidates for these excitations are magnon modes in the vicinity of \\(\\vec{Q}\\)\u2009=\u2009(\u03c0, \u03c0). As shown in Fig. 3<\/a>, the node is three times closer to the AFM zone boundary than the antinode. Thus, any spectral broadening of the (\u03c0, \u03c0) mode (due to magnon lifetime effects) or any thermal occupation of modes up to about 10\u2009meV at 50\u2009K will disproportionately widen the hotspot towards the node. Both effects have been observed in electron-doped cuprates46<\/a>,61<\/a>,62<\/a> and, when combined with the analysis presented above, offer a viable explanation as to why RH(T) decreases with increasing T but the MR increases (at least up to 70\u2009K). This also may explain the T linearity of \\(\\varGamma \\left(\\varphi ,T,x\\right)\\).<\/p>\n A final key question to address, though, is how scattering off these magnon modes can induce such a large anisotropy in \u0393. Indeed, recent calculations based on the dynamical cluster approximation applied to overdoped cuprates show that scattering off low-lying spin excitations generates only mild anisotropy in the scattering rate63<\/a>. One possible source for the enhancement seen in the experiment is small-angle scattering. Cuprates are known to exhibit primarily out-of-plane disorder in the charge-transfer layers rather than in CuO2 planes, resulting in small-angle scattering on the electrons64<\/a>,65<\/a>. Small-angle scattering can be understood as Brownian motion or as an effective diffusion process across the FS, and the expected time taken to diffuse into the nearby hotspots scales as the sector size squared (where the hotspots act as \u2018sinks\u2019 similar to the necks in elemental Cu)66<\/a>,67<\/a>. If the hotspot in LCCO indeed widens predominantly towards the node with increasing temperature, then the shrinking nodal sector can cause a marked increase in the efficacy of small-angle scattering to relax the velocity, thereby enhancing the anisotropy between the nodal and antinodal regions. Thus, one possible interpretation of the increasing anisotropy in LCCO with temperature through the g(x) term is that small-angle scattering exacerbates the anisotropic scattering. Although modelling this scenario is beyond the present work, we suggest that hotspot broadening combined with small-angle scattering may explain the origin of the unusual T-dependent anisotropy in \u0393 obtained here.<\/p>\n Before closing, we highlight two aspects of the magnetotransport in LCCO that are not yet understood. First among these is the T2 resistivity that extends to high T (refs. 45<\/a>,59<\/a>). As shown in Fig. 1<\/a>, this behaviour contrasts markedly with the T linearity found in p-type cuprates at high T. The fact that \u03c1ab(T) in LCCO extends beyond the estimated Mott\u2013Ioffe\u2013Regel limit suggests that this superlinear behaviour may arise from a gradual loss of carriers or a transfer of low-energy spectral weight with increasing T. The latter could be revealed, for example, in optical conductivity measurements at elevated temperatures66<\/a>.<\/p>\n The other outstanding issue is the persistence of H-linear MR to the lowest T (ref. 24<\/a>). Within the model presented above, the H-linear MR is a result of impeded cyclotron motion due to the presence of hotspots. Of course, there may be a physical reason for this, for example, non-local (k-dependent) impurity potential scattering or the presence of a quantum critical phase and associated residual scattering off quantum fluctuations associated with the AFM order, but for the latter, it is not immediately obvious why it should persist down to low T. An alternative explanation is that this residual, small H-linear MR arises due to the compositional variation along the film created by the combinatorial synthesis itself. Longitudinal variation in the Hall voltage is a long-standing explanation for H-linear MR in metals68<\/a> and, indeed, has recently been proposed in the context of cuprates69<\/a>. We note, however, that H-linear MR is also observed at low T in LCCO films that are not grown combinatorially24<\/a>. With this in mind, it is tempting to attribute the persistence of a sizable 1\/\u03c4HS down to low temperatures to the sublinear behaviour of \\(1\/\\tau (\\propto \\sqrt{T})\\) predicted near the hotspots70<\/a> within spin fluctuation theory, though this is clearly an outstanding issue that awaits explanation based on future microscopic theories.<\/p>\n Evolution of the high-field slope over the full (T,x) range<\/p>\n As a further, final hint to the origin of the anisotropic scattering rate in LCCO, Extended Data Fig. 7<\/a> shows the high-field slope \u03b31, defined as the value of d\u03c1\/d\u00b50H at 33\u2009T, with respect to T and x. At all doping levels, irrespective of the AFM order, the slope of MR falls as T\u2009<\u200975\u2009K. Within the AFM regime, the slope becomes negative below T\u2009\u2248\u200920\u2009K without discontinuities, suggesting that the origin of its T dependence across the entire T range is closely linked to the evolution of AFM correlations, given the negative slope is associated with the AFM order8<\/a>. When x\u2009>\u2009xFSR, the slope is positive at all T, but its magnitude also falls below 75\u2009K before levelling off at 20\u2009K.<\/p>\n The smooth variation in the MR and the distinct similarity across the AFM boundary strongly links the unusual T dependence of the MR below 75\u2009K to AFM spin correlations that persist across the doping series. Note that the T dependence diminishes as x increases, reflecting the weakening spin fluctuations implied by the decrease in the scaling parameter g(x). Alternatively, the monotonic x dependence is also evident from the gradual variation in d\u03c1\/d\u00b50H (plotted versus H\/T in Fig. 2h\u2013l<\/a>) with increasing x. The low-T negative component seen at low x also evolves continuously, becoming positive with no discontinuity near xFSR.<\/p>\n","protected":false},"excerpt":{"rendered":"Zero-field resistivity Combinatorial LCCO thin films were grown epitaxially on SrTiO3 substrates from targets with x\u2009=\u20090.10 and 0.19,…\n","protected":false},"author":2,"featured_media":940083,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[24],"tags":[8878,49,48,8877,8882,8881,3881,3673,8876,8879,8880,314,66,3884,8875],"class_list":["post-940082","post","type-post","status-publish","format-standard","has-post-thumbnail","category-physics","tag-atomic","tag-ca","tag-canada","tag-classical-and-continuum-physics","tag-complex-systems","tag-condensed-matter-physics","tag-electronic-properties-and-materials","tag-general","tag-mathematical-and-computational-physics","tag-molecular","tag-optical-and-plasma-physics","tag-physics","tag-science","tag-superconducting-properties-and-materials","tag-theoretical"],"_links":{"self":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/940082","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/comments?post=940082"}],"version-history":[{"count":0,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/posts\/940082\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media\/940083"}],"wp:attachment":[{"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/media?parent=940082"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/categories?post=940082"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.newsbeep.com\/ca\/wp-json\/wp\/v2\/tags?post=940082"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}