3.4 Optical measurements
3.4.2 Normal incidence reflectivity
184.108.40.206 In-plane measurement
The FIR to NIR in-plane reflectivity at near-normal incidence of these extremely small samples was measured using the Bruker IFS 113v spectrometer coupled to the ultra-high vacuum and high stability home-made cryostat presented in section 2.2.3. The regular working vacuum of the order 10−8mbar was necessary to avoid as much as possible condensation of water and other residual vapors on the sample surface. For the UD55 sample, a high-energy upgraded Bruker IFS 66v/S extended this reflectivity range to 3 eV; for which another cryostat was used to fit the apparatus. In both setups, the gold and silver evaporators allowed an in situ deposition of a thin reference layers on the sample surface. This technique produced reproducible absolute reflectivity for these mm-sized samples. Several detectors and beam-splitters were used; Table 3.2 summarizes the couples of detectors for sample UD67. The spectral resolution is 0.1 meV in the FIR. Due to the small sizes of UD55 and UD45, the measurement used slightly different energy ranges.
Due to the high stability of the cryostat, the measurements were conducted from 9 K to 390 K at a cooling speed of about 1 K min−1. This also corresponds to one spectrum per kelvin. In order to increase the signal to noise ratio, the data were averaged over 10 K intervals. The long-term drift of the light sources and detectors was calibrated using the high-stability flipping-mirror placed in front of the cryostat window. The mirror was flipped up and down every 20 minutes, providing a continuous monitoring of the light intensity drift. The remaining uncorrected drift was analyzed during the warm-up process, which leads to a thermal hysteresis effect below the noise level in the FIR and below 0.5 % at higher energies. The final absolute and calibrated reflectivity data are shown in Figure 3.8 for the three samples at selected temperatures, together with the data of van Heumenet al..
Between each measurement, the evaporated metallic layer was removed using scotch tape.
It was common that some part of gold was remaining on the sample surface, which in these cases required re-polishing the surface. The good overlap of the data along the different spectral ranges confirms that the bulk properties are not affected by the procedure. Prior transferring the sample to the cryostat, the sample surface quality was verified using an optical microscope.
The reflectivity data of the three dopings is shown in Figure 3.8 at selected temperatures.
The data of Ref.  ranging from 20 K to 290 K is shown as a comparison. Most of the temperature dependence is happening at low energy, this part is magnified in the right graphs.
The metal-like reflectivity shown in Figure 3.8 is dominated by the excitations in the CuO2 plane which are strongly influenced by the sample doping. The relative accuracy of the low-energy reflectivity as a function of temperature can be assessed by fitting the low-energy reflectivity to the Hagen-Rubens equation presented in section 220.127.116.11, which is accurate in the ω 1/τregime. This assumption is at least fulfilled above Tc for UD45 but the range
0 0.2 0.4 0.6 0.8 1
Figure 3.8: In-plane reflectivity of the HgBa2CuO4+x samples at selected temperatures. From top to bottom: OpD97, UD67, UD55, UD45. (left)Full energy span. The dashed gray rectangles indicate the magnified FIR range(right). In-plane and residualc-axis phonons are shown by the black arrows in the last graph.
Table 3.2: Combination of sources, beam splitters and detectors used for the measurement of the underdoped HgBa2CuO4+x samples. The low cutoff energy was respectively 11, 7 and 8 meV respectively starting from UD45.
High energy cutoff (meV)
UD45 - 72 145, 372 471 1240
-UD55 9 74 149 471 1054 3100
UD67 11 80 130 366 1240
-Source Hg Hg Globar Globar Tungsten Tungsten
Beam splitter Si 10µm Ge/Mylar 6µm Ge/KBr Ge/KBr Si/CaF Si/CaF
Detector Bolo 1.6K Bolo 4K Bolo MIR MCT MCT Diode
Polarizer PE Gold KrS-5 KrS-5/MIR MIR/NIR Visible
of the fit is reduced to lower energies for UD67 since the scattering rate is about 15 meV at 70 K. The comparison of the infrared data to the dc transport shown in Figure 3.7 confirms that HgBa2CuO4+x exhibits the lowest residual resistance among the cuprates. The transition from a quadratic to a linear temperature dependence of the resistivity at aboutT∗∗is also confirmed such as the linear temperature dependence atT∗.
The FIR reflectivity increases by lowering the temperature. This is directly due to the lowering of the carrier scattering rate. At about 0.5 eV, the broad bump-like structure is gradually phased out by going underdoped; for UD45, the reflectivity is almost linearly decreasing in that range. This effect can be ascribed to the decrease of metallicity and has a direct impact on the optical phonons that are less screened and thus more apparent at low doping. For instance, the in-plane oxygen vibration mode at about 42 meV is much stronger in the reflectivity of UD45. In contrast, The weak doping dependence is seen as a slight decrease of the reflectivity from 11 % for UD45 to about 9 % for OpD97. At the same time, the reflectivity minimum is almost at the same energy. In Figure 3.9 the reflectivity is shown for the entire energy range, including the ellipsometric measurements (see section 3.4.3). Up to around 2 eV, the reflectivity shows a featureless region, which is an illustration of the charge transfer gap between the O2p orbital and the Cu 3dupper Hubbard band (see section 3.1.2). Comparing with Figure 3.10 of the next section, the arrows shown on the last graph of Figure 3.8 point three residualc-axis phonons (19.2, 52.7 and 80.6 meV). This observation emphasizes that the use ofs-polarized light didn’t suppress all the phonons so that a small inhomogeneity in the samples is probable. The highest mode shouldn’t be confused with a strong in-plane phonon mode close to 80 meV. It is also important to remember that the phonon dip is the tail of the phonon mode so that in the real part of the conductivity, the energy of the phonon mode is a bit lower as can be seen in Figure 3.10.
The reflectivity is a useful way of analyzing strong and sharp modes such as phonons.
Nevertheless, the subtle change of the low-energy reflectivity slope due to the low energy
0 1 2 3 4 5 Photon energy (eV)
0 0.2 0.4 0.6 0.8 1
390 K 300 K 200 K 100 K 10 K
Figure 3.9: Reflectivity of HgBa2CuO4+x UD67 at selected temperatures. The high energy data were converted from the dielectric function of the ellipsometric measurements.
correlation is difficult to interpret. Such analyze is better done by using the optical conductivity, which is shown in section 3.4.6.
18.104.22.168 c-axis measurement
Using a Bruker 70v coupled to a IR-microscope thec-axis reflectivity of the HgBa2CuO4+x UD67 and UD45 reflectivityR(ω)was also measured at room temperature using the polished edge of the samples. The data from OpD97 was measured on the ac-plane of a different sample from 3.7 meV to 2.5 eV by C.C. Homes. A thin gold layer was sputtered on one side of the sample while the other part was kept protected with a thin aluminum foil. Thec-axis reflectivity was measured by taking spectra on either side of the interface with the portion covered with gold. The measurement was repeated several times and the reproducibility was within 1 % at 0.1 eV. Compared to the optimally doped sample which was measured using a large surface and a conventional technique, the diffraction limits of the IR-microscope limited the low-energy data to about 50 meV; the highest energy was 1.5 eV. By virtue of an almost flat reflectivity at high energy, the reflectivity was fitted using the Drude-Lorentz model and extrapolated as a constant to high energy. The reflectivity curves are shown at room temperature in Figure 3.10;
the four expected phonons are the mercury and barium vibration modes at about 11 meV and 19 meVand also the vibration of the in-plane oxygen at 44 meV and 74 meV. The highest mode is the apical oxygen vibration mode, which is probably split due to disorder. in chapter 4 other possible scenarios for the phonon splitting are introduced for Sr2RuO4. Thec-axis optical conductivity of the cuprates is known to be dramatically different from the in-plane which emphasize the strong anisotropy of quasiparticles around the Fermi surface[134, 135].
0 0.02 0.04 0.06 0.08 0.1 0.12 Photon energy (eV)
0 0.2 0.4 0.6 0.8 1
OpD97 UD67 UD45 c-axis
Figure 3.10: c-axis reflectivity of HgBa2CuO4+x at room temperature. The data of UD67 and UD45 was measured on a Bruker 70v attached to a IR-microscope thus limiting the beam size to about 50µm. The small sample size and the diffraction limit restricted the measurement to about 40 meV.
The OpD97 data are reproduced from Ref..
The ellipsometric measurements complet and extend the data up to 5 eV for the UD67 and 3.1 eV for the UD45. Due to a technical issue, no ellipsometric data were taken for the UD55 but the spectral range was obtained the mean of reflectivity measurement. The temperature resolution is 2 K. The angle of incidence of light was respectively 61◦and 62◦. As explained in section 22.214.171.124, the dielectric function is a mix of theab-plane andc-axis response. In the limit described in section 126.96.36.199, the trueab-plane dielectric functionε(ω) =ε1(ω) +iε2(ω)was extracted from ˜ε(ω)by inverting the Fresnel equations. In order to do so, the complexc-axis dielectric functionεc(ω)was obtained from thec-axis reflectivity measurement. This procedure was performed using the RefFIT software[16, 22]. Measurements were also performed for different angles of incidence, 65◦, 70◦and 80◦ with no notable difference on the final dielectric function. The resulting dielectric function is shown in Figure 3.11 for the UD67 sample and for few selected temperatures. Converting the ellipsometric data using Equation 1.27 in the 0.8 eV to 1.2 eV range shows a good agreement with FTIR measurements.