3.5 Fermi liquid-like behavior in the underdoped cuprates
3.5.4 Fermi-liquid optical conductivity
In this section the underdoped HgBa2CuO4+x are assumed to be Fermi-liquid-like materials which are governed by the experimentally-defined scaling parameter p=1.5. The Fermi-liquid Equation 1.119 developed in Ref.[38]and shown in section 1.4.1 is used to characterize the
three temperature-independent parameters: T0 the Fermi-liquid scale, ZΦ(0) the coherent weight and ZΓ the impurity scattering rate. Note that the impurity scattering rate of the model is linked to the experimental value through 1/τ(ω = 0,T = 0) = 2ZΓimp/(ħhZe). In order to unveil the incoherent MIR intraband conductivity, asumption have to be made on the temperature dependence of this conductivity[65, 172]. Without having enough arguments in hand for deciding between the scenarios, it was chosen to keep the two scenarios and use two fitting procedures, keeping the Fermi-liquid parameters either free or fixed upon increasing the temperature. The intraband response was extracted and becomes prominent when the coherent transport disappear at very low doping.
In a log-log plot, the Fermi-liquid optical conductivity exhibits a characteristic non-Drude feature at the thermal energyħhω≈kBT, in fact exactly where it gets its scaling form. Shown in Figure 1.11, this hump-like structure can easily be misleading and either is interpreted as a finite energy mode. Compared to a single Drude peak, the fit of the low-energy data using the complete Equation 1.119 (not an approximation) is the correct way of describing the low-energy optical properties of the HgBa2CuO4+x samples, but extra care has to be taken since the incoherent MIR response is close in energy and may strongly influence the results[173, 174]. Due to the complexity of Equation 1.119, the fitting procedure is performed using a Fortran routine[43]that uses the standard Nelder-Mead (Simplex) algorithm for unconstrained mini-mization[175, 176]. The input data was either the reflectivity or directly the complex optical conductivity. It was found that using the latter was more convenient but the fit of the reflectivity could also be advantageous as a better extrapolation to the low energy Hagen-Rubens regime (see Figure A.1 in the appendix for an example). For all samples the real and imaginary parts of the optical conductivity data were simultaneously fitted at temperatures aboveT0; the routine was kept fitting up to the higher temperature since the deviations aboveT∗are meaningful for the discussion. Except for UD45, the data were fitted from the lowest photon energies. For all samples and at low temperature, the data strongly deviate from the model at approximately 33(3)meV. In order to test for a possible sensitivity to the fit boundaries, several limits were tested. It was found that changing the energy from 30 meV to 80 meV by steps of 10 meV was not affecting the onset of the deviation whereas the low energy conductivity was overshoot by pushing the limit above 40 meV. Consequently, this indicated that 33 meV is about the energy where the MIR incoherent response becomes dominant over the coherent free carriers.
Subsequently the maximum energy used for the fit was fixed at 33 meV for all the samples The first procedure (procedure A) consisted in fitting all temperatures with free parameters.
It was found that the impurity scatteringZΓ was usually zero or fluctuating at very small values so that it was simply set to zero. The results are presented for UD67 in Figure 3.25. In the pure Fermi-liquid model, the temperature dependence is directly imposed in the model of the self-energy Equation 1.117 and in the Bose and Fermi functions so that in principle in this regime the fitting should in principle give constant parameters. Keeping the parameters free
101 102 Photon energy (meV)
100 101
σ1(kScm−1 )
100 K 150 K 200 K 300 K 390 K (a) Hg1201 UD67
100 200 300
Photon energy (meV) 0
0.5 1
∆σ1(kScm−1 ) (b)
100 200 300
Temperature (K)
100 400 700 0T(K) (c)
1 1.5
ZΦ(eV)
0 50 100 ZΓ(meV) fixed
100 200 300
Temperature (K)
0 0.5 1 1∆σ(kScm) −1
10 meV 20 30
40 50 100
Figure 3.25: Fermi-liquid fit to the low energy optical conductivity of HgBa2CuO4+x UD67 . The Fermi-liquid parameters T0 andZΦ(0)are allowed to vary for as a function of temperature while ZΓ is fixed at zero. (a)Experimental conductivity (thin lines). The Fermi-liquid model (dashed lines+thick lines). The thick line represent the fitting regionħhω≤33 meV. (b)Subtraction of the model to experimental data as a function of energy (left) or as a function of temperature (right).
The black arrows indicatesT∗∗andT∗. (c)Temperature variation of the Fermi-liquid parameters;
fixed parameters are labeled as is.
allowed to see the deviations from constant values: while the coherent quasiparticle weight is about remaining constant, the Fermi-liquid scale is strongly reduced already above T∗. It is interesting to note thatZΦ(0)is nearly constant belowT∗∗ and starts to increase linearly at a higher temperature. This phenomenon is seen for all underdoped samples. The figures for the other samples are presented in the appendix, see Figures A.2, A.4 and A.6. The optimally doped sample was also fitted as a comparison; no kink is observed in the 150 K to 250 K regions so that the kink present for the underdoped sample is intrinsic. At 100 K, the energy onset of the thermal regime is predicted to be 40 meV, a slight change of slope is already seen at 25 K.
At 150 K, it is already corresponding to 61 meV which is in the incoherent region. Compared to the strongly underdoped samples, UD67 is the only one clearly showing the Fermi-liquid thermal regime. In section 1.4.1, the Fermi-liquid thermal regime was predicted to occur for temperature much lower than T0. If it is indeed the case for UD67 with T0 ≈ 700 K, it becomes questionable for UD55 and UD45 whereT0 is respectively≈300 K and 240 K for the low temperature fit. In contrast, it will be shown that for Sr2RuO4 the Fermi-liquid regime is seen at temperatures one ordre of magnitude below the Fermi scaleT0. At very low doping, it is possible that the Fermi-liquid regime survives but is strongly weakened by the increasing lack of coherent spectral weight.
Figure 3.25 (b) shows the data subtracted from the fitted conductivity. The residual conduc-tivity is attributed to the intraband MIR peak which is described in the two component model of Ref.[96, 149, 172]. The difference here is that the Fermi-liquid model is used in place of a simple Drude component
σ(ω,T) =σFL(ω,T) +σMIR(ω,T) (3.5) Illustrated in Figure 3.15 at 10 K, where the superconducting gap suppresses most of the coherent peak the Fermi-liquid, keeping onlyσMIR fit of the data allows in the same fashion to unveil the temperature dependence of the MIR which turns out to indicate a suppression of the low energy part upon increasing the temperature. This scenario is found in good accordance with the prediction of Mileyet al. in[65]. Their study states that the increase of temperature is setting more free carriers in the MIR peak which in turn induces a spectral weight transfer to the Fermi-liquid Drude. Whereas this scenario could, at first sight, weaken the Fermi-liquid picture by contributing extrinsically to the spectral weight. It could explain the deviations with the expected scaling parameterp=2.
The second procedure (procedure B) shown in Figure 3.26 was used to find the best Fermi-liquid parameters for the data between 90 K and 100 K. At higher temperature,ZΦ(0)was kept free. While in a pure Fermi-liquid regime the parameters would remain constant as it is the case for Sr2RuO4 (see next chapter), the coherent spectral weight is decreasing linearly as a function of temperature. Moreover, the temperature trace indicates that at T∗∗, the deviations
101 102 Photon energy (meV)
100 101
σ1(kScm−1 )
100 K 150 K 200 K 300 K 390 K (a) Hg1201 UD67
100 200 300
Photon energy (meV) 0
0.5 1
∆σ1(kScm−1 ) (b)
100 200 300
Temperature (K)
700 800 0T(K) fixed (c)
0.8 1 ZΦ(eV)
0 50 100
ZΓ(meV)
fixed
100 200 300
Temperature (K)
0 0.5 1
∆σ1(kScm −1)
10 meV 20 30
40 50 100
Figure 3.26: Fermi-liquid fit to the low energy optical conductivity of HgBa2CuO4+x UD67. The Fermi-liquid parameterT0and ZΓ are fitted below 110 K and fixed at higher temperature,ZΦ(0)is kept free for all temperatures.
occur already at zero energy, which is expected in the scenario when ξ2max = 88.5 meV = (ħhω)2+ (pπkBT)2 has to be conserved.