Drude-Lorentz analysis

Using the RefFIT software[16, 28], the reflectivity and ellipsometer data were fitted simul-taneously using the Drude-Lorentz model for all temperatures. The UD55 sample has only reflectivity data up to 3.1 eV. The room temperature Drude-Lorentz parameters are summarized in Table 3.3 for sample UD67. The other doping analysis obtained comparable results with the exception that UD45 required an extra secondary Drude peak; this can be understood from the fact that this sample almost loses entirely its coherent spectral weight at room temperature, while the MIR component is much less dependent on temperature[65, 96, 149]. As shown in Figure 3.12 (a), the model and Kramers-Kronig transformation of the data are almost indistin-guishable which indicate the quality of the fitting procedures. The lowest interband transition starts at about 1.4 eV that allows to adequately set the limit between the free and bound charges response at 1 eV. All the modes below this energy can be either the coherent carrier response (Drude peak), the incoherent free carrier response (σ_MIR) or any mode such as phonons and magnons. The free carrier plasma frequency is formed by the geometric sum of the spectral weight of intraband contributions except the phonons and is not very sensitive to doping nor temperature: 2.000(24), 2.008(21), 2.053(20)and 2.046 eV starting from UD45. And forε_b, respectively 4.59(5), 5.13(5), 4.10(2)and 3.66, the singularε_bmeasurement observed in UD55 might come from the different experimental technique used at high energy (reflectivity instead of ellipsometry). Comparing to the expected value of 3.57 calculated with the Clausius-Mossotti relation in section 1.2.3.1 the experimental values are higher and anti-correlate with doping.

This is an indication that the oxygen cation is responsible for an important part of the high energy response although other contributions are also required to describe the entire picture, notably the fact thatε_bincreases at low doping which is seen by a higher reflectivity in both our data and in La_2−xSr_xCuO₄ [66].

In theħhω <1 eV energy range which is the region of interest for this study, the interband transitions are represented through the bound-charge dielectric function defined using Equa-tion 1.60. As shown in Figure 3.14, this funcEqua-tion is practically constant in that range and does only contribute as a constant parameter, which is combined withε_∞ as the total high energy response. The confirmation of the energy independence of this parameter on the low energy part of the spectra is shown in Figure 4.19 Using the fit parameters, the data was extrapolated to zero energy and to much higher energies following the procedure explained in section 1.2.4.

The complex part of the reflectivity (or its phase) was completed through Kramers-Kronig

0 1 2 3 4

0 0.5 1

1.5 (a)

0 1 2 3 4

Photon energy (eV) 0

0.2 0.4 0.6

T ≈T_c (b)

intraband interband

0 20 40 60 80 100

1 1.3 1.6 1.9 2.2 1σ(kScm) −1

(c)

0 20 40 60 80 100

Photon energy (meV)

0 0.1 0.2 1∆σ(kScm) −1

(d) σ1(kScm−1 )

Figure 3.13: (a)Real part of the optical conductivity at room temperature for the Drude-Lorentz fit to the data by Equation 1.56 and Table 3.3. The thick orange line represents the total conductivity of the fitted model; the first oscillator in black is the Drude peak. The solid line is the conductivity obtained from the reflectivity and ellipsometry data using Kramers-Kronig relations.(b)intraband and interband conductivities from the model.(c)Comparison of the fit (dashed line) and Kramers-Kronig transformed data at 390 K, with and without phonon subtraction. (d)Difference between the Kramers-Kronig conductivity and the Drude-Lorentz fit with and without the phonon subtraction.

The phonons at 15, 28, 30, 42, 69 and 78 meV were fitted using standard Drude-Lorentz line shape.

The feature at 82 meV is due to the small reflectivity mismatch of two measurement ranges and was not fitted. Some parts of the phonons are still noticeable which indicates that the phonons are not completely fitted and might be better described by adding electron-phonon coupling.

Table 3.3: Room temperature Drude-Lorentz model parameters of the in-plane optical response of HgBa₂CuO_4+x UD67. All numbers are given in eV. The high-energy dielectric constant isε_∞=2.2.

The intraband plasma frequency obtained from the intraband response isħhω_p=2.053(20)eV.

intraband interband

j=0 j=1 j=2 j=3 j=4 j=5 j=6 j=7 ħhω_0,j 0 0.116 0.542 0.908 1.443 2.430 2.939 5.204 ħhω_p,j 0.857 1.461 0.969 0.638 0.444 0.947 0.510 4.390 ħhγ_j 0.046 0.337 0.747 0.805 0.598 1.126 0.636 2.723

0 0.2 0.4 0.6 0.8 1 Photon energy (eV)

−10

−5 0 5 10

Dielectricfunction

Re(ε−ε_b) Im(ε−ε_b) Reε_b

Imε_b

Figure 3.14: Free-charge and bound-charge contributions to the dielectric function of HgBa₂CuO_4+x at 290 K. The negligible energy dependence of the bound-charge response indicates a clear separation between the intraband and interband responses below 1 eV.

relations and the resulting data were then transformed to the complex dielectric function using Equation 1.27. The consistency of the procedure was checked by comparing the output real and imaginary parts of the dielectric function to the ellipsometric data but also by comparing optical conductivity obtained by the Kramers-Kronig transformation of the reflectivity to the model (see Figure 3.13). The resultingε(ω)is shown in Figure 3.11 along with the pseudo-dielectric function at room temperature.

3.4.6 Optical conductivity

The optical conductivity is obtained from the dielectric function using Equation 1.15 and shown in Figure 3.15 at selected temperatures (30 K steps). For clarity the curves under T_care plotted in gray. The small but sharp low energy features at 15, 28, 30, 42, 69 and 78 meV are the optical phonons also seen in Figure 3.13 (b,c). Below 140 meV and belowT_ca strong gap-like suppression of conductivity is observed for both OpD97 and UD67; this effect is caused by the transfer of spectral weight into the zero-energy condensate peak[149, 150]. However, only the coherent spectral weight is removed by the superconducting transition, keeping the MIR conductivity (σ_MIR) unaffected. Additionally the suppression remains visible up to about 250 K which is much higher than the temperature of the superconducting fluctuation scenario (T⁰≈84 K) for UD67[104]) and thus is a clear optical signature of the pseudogap.

A direct comparison of the different samples at room temperature is shown in Figure 3.16.

The Drude peak due to the free coherent carrier response progressively narrows upon lowering the temperature. If the coherent weight is strong and indicates a Drude response for the optimally doped sample, it clearly weakens by decreasing the doping and the response becomes

0 0.2 0.4 Photon energy (eV)

0 1

2 OpD97

0 1

2 UD67

0 1

2 UD55

0 0.2 0.4

Photon energy (eV) 0

1

2 UD45

0 0.2 0.4 0.6

Photon energy (eV)

0 1 2

0 1 2 10 K 100 K 300 K 390 K

0 1 2

0 0.2 0.4 0.6

Photon energy (eV)

0 1 2

σ1(kScm−1 ) σ2(kScm −1)

Figure 3.15: Real and imaginary parts of the in-plane optical conductivity of four different dopings of HgBa₂CuO_4+x at selected temperatures. From top to bottom: OpD97 from[22], UD67, UD55, UD45. The gray curves represent the data for T ≤ T_c and the dashed lies are the low energy extrapolation.

0 0.1 0.2 0.3 0.4 0.5 Photon energy (eV)

0 0.5 1 1.5 2 2.5 3

σ1(kScm−1 )

OpD97 UD67 UD55 UD45 290 K

Figure 3.16:Comparison of the real part of the optical conductivity of the underdoped HgBa₂CuO_4+x samples at room temperature. The data of OpD97 is reproduced from Ref.[22]

increasingly incoherent which is typically the case at high temperature for the UD45. Seen as a negativeσ₂, the absence of coherent peak makes the extended Drude analysis meaningless for this sample at high temperature.

For UD55 and UD45 the hallmark of the superconducting transition in the optical conductivity data is less clear; this can be understood by calculating the superfluid density for T T_cwhich is directly proportional to the strengthω_p,sof the superconducting condensate peak[59]. This can either be done by fitting the low energy reflectivity to a two fluid model consisting of a standard Drude plus a zero scattering orδ peak, or by assuming that the condensate peak gives rise to aε₁(ω)≈ −ω_p,s/ω² contribution, so that it can be estimated by extrapolating the data to zero energy. Using the later estimation, the superfluid densityρ_sis much smaller for the underdoped samples which is directly seen in the superfluid strengthħhω_p,s= 0.61(6), 0.69(7), 1.03(11)and 1.19(5)eV starting from UD45. This strong suppression is also seen in other studies[59, 139, 151]and the linear scaling with T_cwas first observed by Uemuraet al.

in Ref.[152]. Later Homeset al. expanded the relation by takingσ_dc(T =T_c)as an additional parameter into account [153]. The reported experimental scaling form follows the scaling relation ρ_s ≡ω²_p,s = (120±25)σ_dcT_c. Using the calibrated dc conductivity or an quadratic extrapolation these measurement for the samples that have filamentary superconductivity (UD45 and UD55), the product with σ_dcT_c is respectively 31.8(9), 29.6(7), 44.2(10) and 39.2(8)kScm⁻¹K. The new data is introduced in the Homes scaling shown in Figure 3.17. Even using the uncalibrated conductivity given in Ref.[81]the data for the HgBa₂CuO_4+x samples seems to have a much higher conductivity at T_c which moves all the points toward the right part of the graph. The scaling is also observed but the slope is smaller. There is presently no explanation to this mismatch.

10¹ 10² 10³ 10⁴ 10⁵ 10⁶ σ_dcT_c (Scm⁻¹K)

10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸

ρs=ω

2 p,s

(cm−2 )

YBa₂Cu₃O_6+x Pr-YBa₂Cu₃O_7−x YBa₂Cu₄O₈ Bi₂Ca₂Cu₄O_8+x

Y/Pb-Bi₂Ca₂SrCu₂O_8+x Tl₂Ba₂CuO_6+x

Nd_2−xCe_xCuO₄ La_2−xSr_xCuO₄ HgBa₂CaCuO_4+x

sheet conductance

Figure 3.17: Homes scaling plot with the additional values of this study (dark yellow circle). The reference to the literature can be found in Ref.[153]