3.5 Fermi liquid-like behavior in the underdoped cuprates
3.5.5 Strong Coupling Formalism
In this section, the Migdal-Eliashberg formalism presented in section 1.3.2 (or strong coupling formalism) is used to fit the optical spectra of HgBa2CuO4+x UD67. The main goal of this analysis is to clarify the different correlation mechanisms present in the cuprates by extracting the bosonic spectral (glue) functionΠ(ω) from the optical spectra. The first part presents the fit of the data using the histogram representation presented in section 3.5.5.1. Next, the Fermi-liquid model is fitted to the dc-transport conductivity. Finally the improved model that combines both approaches is introduced and shows that the Fermi-liquid electron-electron coupling and the other bosonic coupling are distinguishable but also partly overlapping in energy which causes a complex interplay in this energy scale.
In previous works, the bosonic spectral function fit procedure was developed by van Heumen et al. and was based on the fit of the Bi2Sr2CuO6+x and optimally doped HgBa2CuO4+x. The results published in Ref.[35, 154]had led to the conclusion that among different models the simple histogram representation was an efficient way of extracting the bosonic spectral function compared to the model constrained marginal Fermi-liquid (MFL)[60]and the MMP[61]. The histogram model was representingΠ(ω)by six amplitude and energy dependentblocks(or histograms). In this study, the procedure of van Heumenet al. was extended with more blocks of fixed width, which indeed increases the number of free parameters. The extended model was initially developed using the RefFIT program, but then it was realized that a more efficient algorithm was crucial far an increased number of fitting parameters. The fitting procedure was then adapted to a multicore and GPU-enabled MATLABR fitting algorithm, which resulted in a nearly instantaneous fitting of the data. In addition the combined Fermi-liquid and histogram model was further refined in the Ph.D. thesis of Mirzaei[131]to which the reader may refer for further details of the computation. The different models may have a slightly different output, so that the mean squared error〈χ2〉is a good indication of fitting quality.
3.5.5.1 Histogram model
The results of the histogram model are summarized in Figure 3.27. The fitting procedure was done simultaneously on the real and imaginary parts of the memory function but also on a weighted point at zero energy of the dc transport. The model is built on 40 blocks ranging from 0+ to 150 meV which corresponds to an energy resolution of 3.75 meV. The model has the advantage of showing the best fit but it also has the higher number of free parameters. One can expect that with such a high resolution, the phonon peaks may be also fitted which could either
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Figure 3.27: Results of the histogram model fit to the memory function of UD67. (left)Comparison between the model (circles) and the data (plain lines) for the real and imaginary part at 80 K and 380 K. (middle)Resulting bosonic spectral function at selected temperatures; the data is stacked up for clarity. (right)Temperature dependent coupling constantλand mean squared error〈χ2〉.
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Figure 3.28: (left)Results of the Fermi-liquid model fit (blue dots) and the dc resistivity of UD67 (plain line). The dashed line represents the extrapolation of the high temperature linear regime.
The inset shows the resulting bosonic spectral function with a cutoff at about 118 meV. (right)Real and imaginary part of the memory function obtained from the fit.
suppress or increase any block intensity depending on the width of the phonon. This model allows anyways disentangling the different features appearing in the optical data: Below 200 K and roughly 15 meV the bosonic spectral function has a very small but non-zero amplitude which corresponds to the Fermi-liquid interactions. Above this energy, no correlations are seen up to about a peak at about 50 meV. This peak is a common feature of the cuprates[59,154,177]and can directly be linked to the saturation of the real part of the memory function. The sharpening of this feature upon increasing the temperature is comparable to the underdoped Bi2Sr2CuO6+x, but the strong increase of its amplitude seems to be sample dependent. The broad feature at 80 meV is quickly suppressed above 100 K, which suggests that it could be related to the superconducting fluctuations. Note that in the analysis of van Heumenet al. this feature is persistent up to room temperature.
3.5.5.2 Fermi-liquid model
In the Fermi-liquid theory, the number of particle-hole excitations is proportional to the energy.
The bosonic glue functionglue function is then modeled by a linear function of energy and has a cutoff atωc representing either theoretically the bandwidth or simply the boundary of the model. The slope and the cutoff energy were adjusted in order to match the dc transport
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Figure 3.29: Results of the combined model fit to the memory function of UD67. The inset of the top left figure shows the value of the cutoff energy used for the low energy Fermi-liquid model, the shaded area represents the region where the temperature is aboveT∗∗that is seen by a strong fluctuation of the parameters.
data. Shown in Figure 3.28, the match is excellent up to 220 K, after which a deviation toward a lower linear but lower resistivity is seen. The subtle deviation from the Fermi-liquid above T∗∗is very important: First, it was shown that a cutoff at 118 meV corresponds to a change from a quadratic to a linear behavior at pπkBT ≈218 K which is exactlyT∗∗with p=2. With p=1.5 it should correspond to 88.5 meV. Then, if the resistivity changes in both case to a linear behavior aboveT∗∗, the slope at high temperature but also the way it change betweenT∗∗
and T∗is completely different, which indicates that this model is oversimplified to accurately describe the Fermi-liquid to non-Fermi-liquid transition in the cuprates. In addition, this model moves the maximum of M1 to 118 meV which is not representing the energy dependent data anymore. It is then clear that the 50 meV to 80 meV features obtained in the histogram model are necessary to describe the entire spectra and that the 118 meV cutoff needs to be lowered in accordance with these changes.
Finally, the two models with the common features seen in the two previous models were combined together[131]. It was shown that either using a free cutoff energy or a free slope parameter for the Fermi-liquid model gave almost the same mean squared error. The first procedure was preferred because it was closer to the bare histogram representation and also because a change of slope for the Fermi-liquid model would have corresponded to a change of
density of states The results shown in Figure 3.29 indicate an excellent description of both the dc transport measurement and the optical spectra. Compared to the previous model the cutoff energy of the Fermi-liquid model is reduced from 118 meV to about 27 meV which is very close to the 33 meV found as boundary to the Fermi-liquid in section 3.5.4. From 70 K to 210 K the cutoff energy is decreasing with increasing the temperature while the 50 meV peak is slightly increasing.
3.5.5.3 Discussion
This analysis points out three different components: at low energy the Fermi liquid is responsible for a coupling proportional to the energy up to about 30 meV, followed at higher energy by a sharp temperature dependent mode at 50 meV and a broad feature around 80 meV. The precise interplay of the different components is still confusing since they are close in energy and may probably overlap. In addition, the phonon energy range is right in the 10 meV to 100 meV which may also have a large coupling with the electrons resulting in an asymmetric shape that would certainly play a role in an artificial increase (or decrease) of the bosonic glue function[178]. As an example, the effect of a peak in the bosonic spectral function is a bending down of the real part memory function; indeed in[154]the bosonic spectral function was never zero so that it is possible that the strong phonon dip seen in real part of the memory function would have: 1) forced a higher amplitude of the bosonic spectral function 2) decreased the memory function over an energy span and 3) suppressedΠ(ω)in the same energy range. Such effect could typically cause local minima which in principle should be less prevailing in the van Heumen 6-block model.
This analysis was done using the UD67 memory function. If the phonons are problematic for this sample, they should be even more for the lower dopings. However, it would indeed be tempting to remove the phonon features prior to the fit, which may be done in a later study.