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# Carrier concentration

The density of itinerant carriers depends on doping, but not in a simple way. In the low doping $x$ regime, the carrier concentration $n$ approximately follows the rule $n\approx x$, instead of $n = 1 + x$ that would be expected from non-interacting electron band theory.

One way that is used to determine effective value of carrier density is from optical conductivity. This method relies on the application of the Drude model of conductivity, which for isotropic impurity scattering with rate $1/\tau$ gives $\sigma(\omega) = n e^2\tau /(1 + i\omega\tau)$. Within this model, integration of the real part of optical conductivity gives a measure of mobile carrier density.
For hole-doped cuprates, however, $\sigma(\omega)$ does not have a simple Lorentzian form; instead, typically there is a sharp peak below 100 meV, followed by a broad hump up to 1 eV. The result of frequency integration, therefore, depends on the choice of the frequency integration window. For any integration window up to 1 eV the carrier density extracted this way increases with doping.

Another way to extract the carrier density is based on the Hall effect measurement. Within the simplifying assumptions of a single scattering rate, the Hall resistivity is independent of this rate inversely proportional to the carrier density, $R_H = 1/(ne c)$. Experimentally, it has been found that the carrier density $n$ extracted this way is strongly temperature dependent. In the case of L$_{2-x}$Sr$_x$CuO$_4$, the temperature independent part is consistent with Sr doping, $n \approx x$, and the rest appears to be activated with the doping-dependent gap of about 0.1 eV that vanishes near optimal doping [1].

The results for the density obtained by these two methods are qualitatively consistent, see Fig. 1. [2]