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The in-plane properties of layered metals can sometimes be obtained from measurements of out-of-plane quantities. For example, angular magnetoresistance oscillations (AMRO), which are angular variations in the interlayer resistivity induced by rotating a magnetic field H in a polar plane relative to the conducting layers, can provide detailed information on the shape of the in-plane Fermi surface (FS) in layered metals. Here we resolve for the first time, the momentum (k) and energy (ω or T) dependence of the in-plane transport lifetime τ in an overdoped cuprate Tl2Ba2(Ca0)Cu1O6+δ (Tl2201) through advances, both experimental and theoretical, in the AMRO technique. Experimentally, we extend the temperature range of previous AMRO measurements on overdoped Tl220113 (with a superconducting transition temperature Tc=15 K) by more than one order of magnitude. Theoretically, we derive a new general analytical expression for the interlayer conductivity in a tilted H that incorporates basal-plane anisotropy. For T>4 K, the AMRO can only be explained by inclusion of an anisotropic scattering rate 1/τ whose anisotropy grows with T. Significantly, the anisotropy in 1/τ and its T dependence up to 55 K can quantitatively account for both the robust linear-in-T component to the in-plane resistivity ρab and the T-dependent Hall coefficient RH over the same temperature range14,15. These anomalous behaviours are not characteristic of a simple Fermi liquid, which is often the starting point for modelling overdoped cuprates. We discuss the consequences of these findings for our understanding of the normal-state transport in cuprates.

As described in the Supplementary Information, detailed azimuthal and polar-angle-dependent AMRO data were taken at 4.2 K and 45 T and fitted to the Shockley–Chambers tube integral form of the Boltzmann transport equation, modified for a quasi-two-dimensional (quasi-2D) metal16 (and assuming an isotropic mean-free-path ℓ), to generate a full 3D parameterization of the FS wavevector kF(ϕ,θ) consistent with previous measurements13 (here ϕ refers to in-plane angles and θ to polar angles out of the plane). Before studying the T dependence of the scattering rate, a self-consistency check was carried out on the fitting procedure by varying H at a fixed temperature. The solid lines in Fig. 1a represent polar-angle-dependent changes in the interlayer resistivity at 4.2 K (normalized to the zero-field resistivity) for various fields 20 T≤μ0H≤45 T at a fixed azimuthal orientation of the inclined sample φ=29∘ (relative to the Cu–O–Cu bond direction) where all AMRO features are visible (here μ0 is the permittivity of free space). The magnetoresistance is determined by the magnitude of ωcτ, the product of the cyclotron frequency and the transport lifetime. The dashed lines are simulated curves produced simply by scaling ωcτ (=0.4(μ0H/45)) whilst keeping all other parameters fixed at their 45 T values. The data scale very well, implying that the isotropic formalism16 remains valid with decreasing H and that no additional angular dependence appears owing to the presence of inhomogeneous superconducting regions (with different Tc values) or anomalous vortex-liquid phases17.

Figure 1: Field and temperature dependencies of the polar AMRO in overdoped Tl2201 (Tc=15 K) at a fixed azimuthal direction φ=29°.
figure 1

a, Solid lines: normalized data for different field strengths. Dashed lines: simulated AMRO fits using the same kmn coefficients as given in Supplementary Information, Fig. S1 and ωcτ values scaled simply by the field scale (that is, ωcτ=0.4(μ0H/45)). b, Solid lines: normalized data at 45 T for different temperatures between 4.2 K and 55 K. Dashed lines: best least-squares fits using the same kmn coefficients as given in Supplementary Information, Fig. S1 and assuming an isotropic ωcτ. c, As b but with an anisotropic ωcτ=ω0τ0/[1+αcos 4ϕ].

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Figure 1b shows the temperature dependence of up to 55 K (μ0H=45 T). Remarkably, AMRO features remain discernible at all temperatures, in particular the kink around θ=30∘. Comparison of the data in Fig. 1a and b reveals that the AMRO evolve differently depending on whether ωcτ is reduced by decreasing H or by increasing T. In the former case, both the peak at H∥c and the peak at intermediate angles diminish at approximately the same rate, whereas in the latter, the intermediate peak is found to survive up to much higher temperatures. The dashed lines in Fig. 1b show the best least-squares fits to the data assuming all parameters except the product ωcτ remain constant up to 55 K. These fits are clearly inferior to those in Fig. 1a.

To proceed, we relax the constraint that ωcτ remains isotropic at all temperatures and generalize the expression16 for to incorporate basal-plane anisotropy in the relevant parameters. We first parameterize the Fermi velocity as vF(ϕ)=vF0(1+βcos 4ϕ), where β is the anisotropy in vF, and the variation of ωc around the FS as

The generalized expression for then becomes

where is the c-axis reciprocal lattice vector, is the interlayer velocity, ħ is the reduced Planck’s constant, d is the interlayer spacing (=1.16 nm for Tl2201), P=G(2π,0) is the probability that an electron makes a complete orbit of the FS without being scattered and

This formalism holds irrespective of whether hopping is coherent or weakly incoherent (that is, when , the interlayer hopping energy, and is ill-defined)18. In the latter case, AMRO arise from differences in Aharonov–Bohm phases acquired in hopping between layers for positions ϕ1 and ϕ2 on the FS19.

Consistent with the tetragonal symmetry of Tl2201, we write 1/τ(ϕ)=(1+αcos 4ϕ)/τ0. Although no unique and independent determination of the various anisotropy parameters can be made from fits of theoretical curves to AMRO data alone, certain features of the data tightly constrain the parametrization, in particular the FS parameters defining kF(ϕ,θ) (ref. 13). Furthermore, as there is no experimental evidence to suggest changes in the FS topography with temperature, we fix these parameters to their values at 4.2 K. Similarly, β, the anisotropy in vF, is assumed to be constant. Finally, to minimize the number of fitting parameters, we assume that ωc is isotropic (=ω0) within the basal plane. Thus, we can provisionally ascribe the evolution of the AMRO uniquely to changes in 1/ω0τ(ϕ) and extract 1/ω0τ0(T) and α(T) from fits to the data at different temperatures. The best least-square fits are shown in Fig. 1c. The quality of the fits at all T is clearly much improved with just the inclusion of α(T), the anisotropy in the scattering rate. Moreover, the subsequent fitting to the in-plane transport data is sufficiently good (see below) that the introduction of additional parameter(s), for example, to account for any possible T dependence in β, seems to be unnecessary (for more details see the Supplementary Information).

The consequences of the above analysis are examined in Fig. 2. To aid our discussion, we show schematically in Fig. 2a the in-plane geometry of various relevant entities with respect to the 2D projection of the FS of overdoped Tl2201 (red curve in Fig. 2a). The purple line represents the d-wave superconducting gap, whereas the blue solid line shows our deduced geometry of 1/ωcτ(ϕ) (as governed by the sign of α), its maximum being at ϕ=0∘. Note that the scattering anisotropy and the superconducting order parameter have the same symmetry. This is consistent with earlier azimuthal AMRO data20 but contrasts with recent angle-resolved photoemission spectroscopy (ARPES) measurements21. However, we note that the ARPES-derived scattering rate is one order of magnitude larger, suggesting that the two probes are not measuring the same quantity.

Figure 2: Determination of the in-plane transport coefficients from 45 T polar AMRO.
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a, Red curve: schematic 2D projection of the FS of overdoped Tl2201. Purple curve: schematic representation of the d-wave superconducting gap. Blue curve: geometry of (ωcτ)−1(ϕ). Black dashed line: isotropic part of (ωcτ)−1(ϕ). b, T dependence of (1−α)/ω0τ0, that is, the isotropic component of (ωcτ)−1(ϕ) and sole contribution along the ‘nodal’ region indicated by the green arrow in a. The green dashed curve is a fit to A+B T2. c, T dependence of 2α/ω0τ0, that is, the anisotropic component of ωcτ−1(ϕ) and the additional contribution that is maximal along the ‘anti-nodal’ direction indicated by the orange arrow in a. The orange dashed curve is a fit to C+D T. d, Black circles: ρab(T) data for overdoped Tl2201 (Tc=15 K) extracted from ref. 14. Purple dashed curve: simulation of ρab(T) from parameters extracted from our AMRO analysis. To aid comparison, 1.9 μΩ cm have been subtracted from the simulated data. (It is not unreasonable to expect different crystals to have different residual resistivities.) e, Black circles: RH(T) data for the same crystal14. Purple dashed curve: simulation of RH(T) from parameters extracted from our AMRO analysis. In this case, no adjustments have been made. ρab(T) and RH(T) were calculated using the Jones–Zener expansion of the linearized Boltzmann transport equation for a quasi-2D FS (ref. 7). Note that using (1), we can re-express the expressions in ref. 7 solely in terms of parameters extracted from our analysis. The error bars in b,c are the covariance matrix of the multidimensional least-squares fitting routine.

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To give our anisotropic function for ωcτ more physical meaning, we re-express (1+αcos 4ϕ)/ω0τ0 as (1−α)/ω0τ0+(2α/ω0τ0)cos22ϕ. The isotropic part (1−α)/ω0τ0 (black dashed line in Fig. 2a) is the sole contribution along the diagonal ‘nodal’ direction (indicated by the green arrow) where the pairing gap vanishes. The T dependence of (1−α)/ω0τ0 is plotted in Fig. 2b and as shown by the dashed line, follows a simple quadratic law (A+BT2). In contrast, the anisotropic component 2α/ω0τ0, maximal in the direction given by the orange arrow in Fig. 2a and plotted in Fig. 2c, is seen to grow approximately linearly with temperature, this linearity extending at least down to 4.2 K.

To our knowledge, this is the first quantitative determination of the momentum and temperature dependence of the in-plane mean-free-path in cuprates. Together with the complete FS topology, this is all we need in principle to calculate the various coefficients of the in-plane conductivity tensor. Figure 2d shows ρab(T) as determined from our analysis, superimposed on published data for overdoped Tl2201 at the same doping level (with the superconductivity suppressed by a large magnetic field)14. The form of ρab(T), in particular the strong T-linear component below 10 K and the development of supra-linear behaviour above this temperature, is extremely well reproduced by the model. The corresponding RH(T) is shown in Fig. 2e. Significantly, the absolute change in anisotropy in (ω0τ)−1(ϕ,T) can account fully for the rise in RH(T), at least up to 40 K. Above 40 K, the simulation has a slightly weaker T dependence, possibly due to the increased disorder in the AMRO sample, known to weaken the overall T dependence of RH(T) in cuprates2, and/or the emergence of vertex corrections that manifest themselves only in the in-plane transport22. Overall however, the same parametrization of 1/ω0τ(ϕ,T) described in Fig. 2b,c gives an excellent account, not only of the evolution of the AMRO signal (Fig. 1c), but also of the ‘anomalous’ transport behaviour. Given the gradual evolution of the transport properties in Tl2201 with doping23, we believe these findings will be relevant to crystals with higher Tc values.

We now discuss the implications of our results for existing theories of transport in high-Tc cuprates. Several contrasting approaches dominate much current thinking: Anderson’s resonant-valence-bond picture24, marginal Fermi-liquid phenomenology6 and models based on fermionic quasiparticles that invoke specific (anisotropic) scattering mechanisms within the basal plane owing either to anisotropic electron–electron (possibly Umklapp) scattering7 or coupling to a singular bosonic mode, be that of spin8,9, charge10 or superconducting fluctuations11. Our analysis clearly supports those models in which anisotropy in the inelastic part of ℓ(k) is responsible for the anomalous RH(T). Empirically, both RH(T) and the T-linear component of ρab(T) are derived from a T-linear anisotropic scattering term that is maximal along the Cu–O–Cu bond direction. The magnitude of the anisotropy is large, even at such an elevated doping level. At T=55 K, for example, ℓ(k) varies by a factor of two around the in-plane FS. Significantly, in non-superconducting cuprates, ρab(T)∝T2 at low temperatures with no evidence of a T-linear term12,23. This implies that the development of superconductivity (from the overdoped side) is closely correlated with the appearance of the T-linear resistivity and anisotropic inelastic scattering, a correlation that will be explored further in future studies. (Recall that 1/τ(ϕ) also has the same angular dependence as the superconducting gap.)

Our analysis implies the presence of (at least) two inelastic scattering channels in the current response of superconducting cuprates. Recent ARPES measurements25 on Bi2Sr2CaCu2O8−δ also found evidence for two contributions to the quasiparticle (single-particle) scattering rate; one quadratic in ω and one linear in ω that develops a kink below Tc. A scattering process that is quadratic in both temperature and frequency is characteristic of electron–electron scattering. Given that the Hall conductivity is dominated by those regions (in this case, the nodal regions) where scattering is weakest, we thereby ascribe the T2 dependence of the inverse Hall angle cot θH in cuprates to such scattering. Recall that cotθH does not vary markedly across the cuprate phase diagram8,23 and so the strength of electron–electron scattering seems largely doping independent.

The second term (seen by ARPES) has been attributed to scattering off a bosonic mode, though its origin and its relevance to high-Tc superconductivity remain subjects of intense debate26. Possible candidates include phonons, d-wave pairing fluctuations, spin and charge fluctuations but as all, bar phonons, seem to vanish in heavily overdoped non-superconducting cuprates11,27,28, it is difficult to single one out at this stage. For a bosonic mode to be the source of anisotropic scattering revealed by AMRO however, the continuation of its linear T-dependence to very low temperatures is highly constraining, requiring as it does the presence of an extremely low energy scale. An alternative origin for this k-space anisotropic scattering is real-space (correlated) inhomogeneity. Indeed in underdoped cuprates, intense anisotropic scattering exists alongside gross inhomogeneity and checkerboard charge order29. As the Mott insulator is approached, the degree of inhomogeneity grows and the simple anisotropic metal evolves into a more exotic ‘nodal’ metallic state in which the FS is reduced to a series of Fermi arcs in those (nodal) regions where scattering is weakest30. We stress again though that both the anisotropy and the T dependence of the scattering need to be accounted for. Nevertheless, the stronger T-linear behaviour seen in ρab as maximum Tc is approached points to an increase in the anomalous term with lower doping, and the connection between the anisotropy in the under- and overdoped regimes is clearly an important avenue for future research.

Finally, this work demonstrates that AMRO can be an extremely powerful probe of intralayer anisotropies in layered metals, beyond mere determination of the FS. The formalism and procedure we have used here could be applied to a host of other layered correlated metals, for example, molecular superconductors and ruthenates, to establish whether anisotropic scattering also plays an important role in the unconventional behaviour observed in these systems.