Electronic diffusion in a normal state of high-Tc cuprate YBa2Cu3O6+x

The bad metallic phase with resistivity above the Mott–Ioffe–Regel (MIR) limit, which appears also in cuprate superconductors, was recently understood by cold atom and computer simulations of the Hubbard model via charge susceptibility and charge diffusion constant. However, since reliable simulations can be typically done only at temperatures above the experimental temperatures, the question for cuprate superconductors is still open. This paper addresses this question by resorting to heat transport, which allows for the estimate of electronic diffusion and it further combines it with the resistivity to estimate the charge susceptibility. The doping and temperature dependencies of diffusion constant and charge susceptibilities are shown and discussed for two samples of YBa2Cu3O6+x. Results indicate strongly incoherent transport, mean free path corresponding to the MIR limit for the underdoped sample at temperatures above ~200 K and significant effect of the charge susceptibility on the resistivity.

conductivity (8,9).For the insulating parent compound, the magnonic  was determined as the difference between the measured in-plane and out-of-plane conductivities allowing the extraction (10) of very long magnonic mean free paths (longer than 100 unit cells).Introductions of impurities (9,10), defects (11) and magnetic fields (10) were used to reveal the conduction and scattering mechanisms.Furthermore, directly measured heat diffusion constant was discussed in terms of Planckian dissipation and a soup like mixture of electrons and phonons (12).

Results and Discussion
Heat transport is special in a sense that all quantities in its Nernst-Einstein equation  =  %  can be independently measured -even the heat diffusion constant  % (12).The challenge however is the separation of phononic  &' and electronic  #$ contributions to the total conductivity  =  &' +  #$ .Takenaka et al. (9) carefully analyzed the O and Zn doping dependence of  for YBa2Cu3O6+x (YBCO) and obtained good estimates for  #$ (Fig. 1A).We use these data, together with the data for  #$ (Fig. 1B) from Loram et al. (4) to estimate the electronic heat diffusion constant  %,#$ via the Nernst-Einstein relation  %,#$ =  #$ / #$ .The resulting  %,#$ for two measured compounds with  !∼ 60 K (x ∼ 0.68) and  !∼ 90 K (x ∼ 0.93) in the non-superconducting regime is shown in Fig. 1C. %,#$ decreases with increasing T as expected, since it is related to the mean free path  via  %,#$ = /2 and  normally decreases with increasing T due to increasing scattering.Here  is a mean quasiparticle velocity.By using its estimate  = 2.15 × 10 ) m/s (12) and setting  to minimal value ( ∼  with  being a lattice constant) one obtains a MIR limit for  %,#$ (Fig. 1C).Remarkably,  %,#$ for the  !∼ 60 K compound decreases to such value already at relatively low  ∼ 200 K and shows signs of saturation close to MIR value at higher .This reveals strongly incoherent behavior in such regime.From value of  %,#$ and  one can estimate  (right y axis in Fig. 1C).This average value of  is small and reaches  ∼ 6 at lowest  ∼  ! . dependence of  %,#$ can be related to the  dependence of scattering time  via  %,#$ =  * /2.In Fig. 1D we show that for  !∼ 60 K compound the behavior at lowest  is most consistent with Fermi liquid (FL) like scattering 1/ ∝  * , advocated, e.g., in Ref. 13, while the  !∼ 90 K compound shows more 1/ ∝  like behavior.We show in Fig. 1D also the comparison to the suggested ( 14) lower bound on diffusion  &.,.= ℏ * /(2 -) obtained from a Planckian timescale  = ℏ/( -). &.,. is shown for a numerical prefactor  of the order unity set to  = 1. %,#$ is much smaller and the consistency with  &.,.for a  !∼ 90 K compound requires  ~ 0.2, while smaller  ~ 0.05 is needed for consistency with  !∼ 60 K compound.By having values for  %,#$ , we can make a step towards estimating  ! . ! and  %,#$ typically do not behave the same, but they are related.It was found that within a dynamical mean field theory (15) they are at low  related by  !=  %,#$ , with a factor  = 1/ * and  being a quasiparticle weight.By assuming a constant factor  in regime of measured data and setting it to an approximate value  = ./.0 !, we obtain a sensible approximation for  ! .This can be used together with the data on resistivity (9) (Fig. 2A) to calculate  !via the Nernst-Einstein relation  != 1/( ! ).The resulting  ! is shown in Fig. 2B.It is compared to the theoretical estimate  !≈  / *  / , with  / being a non-interacting density of states. ! is larger than theoretical estimate and in addition shows some T dependence.A clear cusp is seen for  !∼ 60 K compound at  ∼ 120 K presumably corresponding to the charge density-wave (CDW) phase transition (16).More relevant for our discussion is its T and doping dependence above such T.While both compounds show decreasing  ! with increasing T, the decrease for  !∼ 60 K compound is stronger.From 200 K to 300 K its  !decreases by about 15%, which is comparable to the decrease of  ! for about 20% (together they give an increase of  for about 50%).This is similar to the behavior observed in the Hubbard model (1).T dependence of  !therefore plays an important role for  in YBCO and may turn out to be even greater at higher T as  ! is expected to be saturating while  increases further.This agrees well with the picture from the optical conductivity (17), in which the increase of resistivity with increasing T in the bad metallic phase is due to the transfer of low frequency spectral weight to higher frequencies, while the mean free path is saturating.It is interesting to note, that with decreasing doping,  !( ∼ 150 K) is increasing, which is consistent with the phase separation tendency, for which  !diverges at some point in the extended phase diagram (18).We also note that the Nernst-Einstein relation is applicable also to the pseudogap phase as it relies (2,14) only on the normal behavior with some scattering or current relaxation mechanism leading to the finite diffusion constant.One should keep in mind, that the analysis here is based on experimental data, which have some degree of uncertainty and that the properties are averaged over all carriers with no differentiation, e.g., nodal vs. anti-nodal Fermi surface parts.However, the main outcomes like strongly incoherent electronic transport close to expected MIR saturation and the first estimate of  !suggesting notable T and doping dependence could withstand future tests.The presented behavior and insight also hint on possible better understanding and calls for further experimental efforts to pin down at least one additional quantity, either charge diffusion constant  ! or charge susceptibility  ! .

Materials and Methods
Some details on calculations can be found in the SI Appendix.
All the data are taken from references and results calculated as explained in the text.

Figures
Figure 1.A) Electronic heat conductivity  #$ for two YBCO samples ( !∼ 60 K and  !∼ 90 K) from Ref. 9. Transition temperatures  ! are indicated.B) Electronic specific heat as measured for similar samples from Ref. 4. C) Calculated electronic heat diffusion constant  %,#$ for two samples showing decrease with T and for  !∼ 60 K sample also values and tendency for saturation close to the MIR limit at higher T. Due to some uncertainty in definition of MIR limit, it is shown with 30% range from  = .Right y-axis shows estimated mean free path.D)  %,#$ in logarithmic scale compared with 1/ * , 1/ and suggested Planckian lower bound for a prefactor  = 1.A B