Choice of doping dependences of distribution parameters
This protocol is extracted from research article:
Unusual behavior of cuprates explained by heterogeneous charge localization
Sci Adv, Jan 25, 2019; DOI: 10.1126/sciadv.aau4538

The simplest assumption of a linear decrease of the mean energy and doping-independent distribution width may be relaxed by introducing additional parameters and assumptions. While these assumptions are purely phenomenological (and detract somewhat from the main message of the simplest possible calculation), an eventual microscopic theory should be capable of providing the true doping dependences, at least in principle. To test the robustness of our calculation, we introduced a curvature into the dependence of the mean gap on doping, using the function of the formEmbedded Image(7)

This function still crosses zero at p = pc (and contains no additional parameters) but has an upward curvature at higher doping. A similar function can be used for the dependence of the distribution width on doping, but with a more general formEmbedded Image(8)where β is a numerical constant. The more constrained form with β = 1/tanh(1) cannot be used for δ, since it would lead to a zero distribution width at pc and nonphysical divergences in the calculations. We chose β ~ 0.4, but again, it turned out that the exact value is not very important. The main features of the phase diagram were unchanged, but introducing these nontrivial doping dependences of Δp and δ somewhat broadened the region of the phase diagram where the resistivity is linear in temperature on the overdoped side (fig. S2). To introduce curvature in Δp(p) at low doping, in line with some experiments (Fig. 3 in the main text), we further modified Eq. 7 and cast it in the formEmbedded Image(9)without introducing additional free parameters. This form gives better agreement between calculated and measured Hall coefficients for strongly underdoped LSCO (see the Supplementary Materials), as well as a better match between the modeled mean energy (dashed line in Fig. 3) and the characteristic high-energy scale from experiments. Moreover, it has the physically appealing feature that Δ0 is approximately 1 eV, the charge-transfer gap at zero doping as determined from Hall-effect measurements (24). Yet again, this introduces no considerable changes to the overall picture.

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