Extended Drude analysis

The extended Drude model is used to describe the single component low energy excitations through an energy dependent optical scattering rate 1/τ_opt and a dynamical mass renormal-ization factorm^∗/m. This single component model does not intend to describe the interband transitions so that they need to be removed prior to the conversion. It was shown that the high-energy oscillators fitted using the Drude-Lorentz model had a constant dielectric function (ε_b) below about 1 eV so that the natural way was to subtract them from the total dielectric constant following Equation 1.67b. The phonons are also sharp features that could in principle be removed from the electronic background. They were kept in order to avoid the residual features shown in Figure 3.13 (b) and (c) an increased number of parameters, which would lead to higher uncertainty of the procedure. Additionally, even if very small, there is an indication of certain asymmetry of the phonon modes that may introduce important bias in the data analysis if not removed properly.

Using the plasma frequency, the memory function was calculated from the interband free dielectric constant Equation 1.64; the real and imaginary part of the memory function are displayed at selected temperatures in Figure 3.19. The curves belowT_c are plotted in grayscale and are not of physical relevance since the Drude model is not able to describe superconductivity.

At high temperature, the real part of the memory function of UD45 and UD55 is negative which indicates that for these temperatures, the samples have an incoherent transport and that the single component approach can not be used anymore. This trend is increasingly observed by lowering the doping which indicates that the cuprates also become increasingly incoherent by losing their coherent spectral weight (see next section 3.4.7). Putting the phonon peaks aside, the common trend observed for all dopings and for temperaturesT_c<T ¯T^∗∗, is the linear slope of M₁ extrapolating to ω = 0. This behavior is exactly the expected from the Fermi liquid theory (see Equation 1.124); the slope of this function corresponds tom^∗/m−1 and it is not constant over the whole temperature range. The sharp structure at about 0.1 eV in M₁ appears only below T_c for the optimally doped sample while it is much broader and appears at higher temperatures for the underdoped samples. For UD67, this feature disappears at around 300 K which is somewhat lower than T^∗≈350 K. In the case of UD55 and UD45 the maximum becomes less noticeable, probably due to the considerable broadening. The dynamical relaxation rate M₂(ħhω,T) =ħh/τ(ħhω,T)has an energy dependence exhibiting an upward curvature for all temperatures and for all dopings. Plotting the optical scattering rate as the squared photon energies in Figure 3.20 points out that the ω² dependence is clearly

0 0.1

0.5 OpD97 (van Heumenet al.)

0.1

Figure 3.19: Real and imaginary part of the HgBa₂CuO_4+x memory function at selected tempera-tures. The temperatures of the legend are shown as thick lines in the graphs.

0 0.01 0.02 0.03 (ħhω)² (eV²)

0 0.1 0.2 0.3 0.4 0.5 0.6

ħhM2(eV)

390 K 300 K 200 K 100 K 70 K

Figure 3.20: Optical scattering rate of HgBa₂CuO_4+x UD67 plotted as a function ofω²for temper-atures aboveT_c. The thick lines represent the temperature shown in the legend.

observed for UD67. On the other hand, it is totally linear for OpD97 and remains questionable mostly due to suppression of the coherent spectral weight and the strong phonon background for the two strongly underdoped samples. The maximum seen in M₁ is associated with an inflection point in M₂ and shifts from roughly 200 K to 100 K when the energy is raised from 10 meV to 50 meV. The interesting feature appears atT_c with a suppression of scattering below about 0.1 eV which seems to be transferred above that energy, producing a plateau-like structure with an onset evolving from 115 meV for the optimally doped sample to 135, 160 and 145 meV for the underdoped. Very recently the expected superconductingd-wave gap of HgBa₂CuO_4+x was reliably measured[166]. Its maximum amplitude was found at 39 meV and a quasiparticle peak was found in the nodal dispersion at approximately 51 meV below the Fermi level which is different from the other cuprates with similarT_c. It would, therefore, be tempting to look at the same energies in optics and following Ref.[22]with the new values of the gap[166] the onset of the gap in optics would happen at≈130(35)meV. In optics if the onset is defined where ħh/τis the steepest, it correspond where the M₁ reaches its maximum near 100 meV which agrees very well for all doping.

Figure 3.21 shows the temperature dependence of the dynamical effective mass and the optical scattering rate and for the three samples. The T² dependence of the optical scattering rate is seen at low energy and temperature for all three underdoped samples. For these samples, it is also possible to guess the onset of the linear regime at about T^∗. On the other hand, the optimally doped sample clearly shows a linear temperature trend right aboveT_c.

The temperature dependence for selected energies ofm^∗/mis shown in Figure 3.21. The black round dot is the zero energy and zero temperature effective massm^∗/m(ω=0,T =0). The 10 meV and 20 meV curves of all dopings point out two characteristic behaviors: at low

0 200 400

Figure 3.21: Temperature dependence of(left)the dynamical mass renormalization factor and (right)the optical scattering rate for the four different dopings and selected photon energies. The thick black dots represent the mass renormalization factor extrapolated from the data. The 10 meV data for sample UD45 and UD55 lies outside the experimental range or is too noisy. The gray area represents the data underT_c which are kept but are out of validity of the extended Drude model.

The gray dashed points represent the expected scenario of an effective mass varying as T⁻¹ for T≥T^∗∗.

temperature but above T_c, the effective mass is almost flat with a small hump like structure for UD67 and UD55. Whereas at a higher temperature it decreases monotonously for the underdoped sample but not for the optimally doped sample. By plotting the maximum of effective mass at point T_max as a function of the selected energy, it is interesting to see that for UD67 it decreases and extrapolates to 212 K ≈ T^∗∗ for ω → 0, which is another way of identifying T^∗∗. Furthermore in the Fermi-liquid model the energy scale is given by ξ² = (ħhω)²+ (pπk_BT)² so that if the Fermi-liquid disappears at about 212 K in the dc limit it should disappear at lower temperatures once the energy is finite and increases. This observation is an indirect confirmation that UD67 possesses Fermi-liquid like properties. Above 60 meV the plateau goes below T_cso that the maximum is very roughly extrapolated to about 80 K a value not so far from pπk_BT =88 K if the value of the scaling parameter p=1.5 is taken (see section 3.5). A similar trend is observed in UD55 and the plateau gets to higher temperatures accordingly to the higher T^∗∗. However, it is much more difficult to confirm this trend with UD45 since this sample has indeed a very weak coherent carrier peak so that it would necessitate to go much lower in energy to precisely see the extension of the plateau. For UD67 the effective mass indeed increases from about 3 at 390 K to 5 at T^∗∗, taken together with the strong temperature dependence ofM₁near its maximum at 0.1 eV, indicates that the charge carriers become increasingly renormalized when the temperature decreases. The effective mass at zero temperature is respectively 7.1, 6.2, 4.2 and 2.85 starting from UD45. In Figure 3.21, the gray circles represent a fit tom^∗/m_b=a/T for the lowest energy data and T≥T^∗∗. A recent study by Bariši´cet al. is pointing to a new interpretation of the phase diagram which reveals, in fact, the hidden Fermi-liquid behavior throughout the entire doping range[132]. This scenario is based on the assumption that the resistivity that is seen proportional to the temperature appears as a result of a temperature-dependent carrier densityn. They propose that the real archetypal Fermi-liquid behavior has to be seen in a quadratic scattering rate and that below T^∗, the resistivityρ =m^∗/(ne²τ)(T ≤ T^∗∗) is obtained from a constant carrier density and effective mass. But the difference comes aboveT^∗where the ratiom^∗/nvary with temperature as 1/T. This scenario implies that the lowest energies presented of m^∗/m_b follow the gray circles, which is the case for UD67 and UD55 but not for the optimally doped sample which show a rather linear optical scattering rate above T_c and a constant effective mass.