Extraction of the pseudogap temperature and magnitude

In the EDC’s of Fig. 6.4, it is visible that a gap opens. For example, this can be seen in EDC’s of sample x0.32 when looking first at the spectra on high temperatures. These spectra at T=175 K and 150 K rotate due to the Fermi function around one point, whereas the spectra at low temperature are shifted away from this point.

In the following, I will derive quantitative results for the dependence of the gap on tempera-ture. Notwithstanding my suggestion above, in the following I must quantify a gap of unknown origin. To do this, I must choose a method of data processing which does not rely on a the-ory. In a case where the spectral function is known, one would be able to fit the spectra.

Figure 6.8: Directive of the leading edge midpoint: First determine the average in-tensity of the constant background on the occupied side and the height of the first maximum near E_F. Then the energetic position of half of the intensity between the background and the maximum must be chosen.

Aside from a fitting routine, there are typically two methods to quantify the gap: One is by theleading edge method and the other is by thesymmetrization method [246]. Both methods must be used because each has its own problems. For the examination of the temper-ature behavior, the leading edge method is adequate.

After an introduction of the method, I will discuss some specialities in the temperature behavior. As the lead-ing edge method gives the lowest boundary but not the real gap, the symmetrization method will be used for an estimation of the real gap magnitude. The results of both methods will then be compared and shown in the section thereafter.

Pseudogap temperature and magnitude accord-ing to the leadaccord-ing edge method

The leading edge method can be seen as the air lever of gap determination in ARPES. It is simple and compara-ble but does not give the real gap magnitude but rather the shift of the spectrums leading edge. The used mea-surement directive of the leading edge midpoint (LEM) is illustrated in Fig. 6.8: The LEM is obtained by first determining the average intensity of the constant back-ground on the occupied side and the height of the first maximum near E_F. The energetic position of half of the intensity between background and maximum must then be chosen. This position is found by a linefit using the points in an interval ±3 meV around the position of half the intensity.

A good discussion about the quality and problems of the leading edge method is provided by Kordyuk et al. [247]. Let me note one problem of the leading edge midpoint. For this, I assume

Figure 6.9: The leading edge midpoint(LEM) in dependence of temperature

6.1 T^∗ and ∆^∗ measured by ARPES

a non-gapped spectral function written in simple form as A(~k, ω, T)' Σ⁰⁰

(ω−E(~k))²+ Σ⁰⁰² . (6.5)

I now must assume a self-energy. One which fits well with the temperature dependency is the empirical spectral function of Kordyuk et al. [247]. There, the imaginary part of the self energy is given as Σ⁰⁰ = ^p(αω)²+ (βT)². In principal, this spectral-function is nothing more than a simplified one of a marginal Fermi-liquid. However, basically the excitation, which will be measured by the leading edge, can be considered as a Lorentzian which broadens with temperature. But when this broadening occurs, the midpoint changes. Thus, the LEM not only gives the gap but is also sensitive to a change in width of the probed spectral function. As described in [248, 247, 249], the energy distribution curves (EDC) can be integrated over a finite momentum range around k_F to ensure that the leading edge midpoint of a non-gaped spectrum stays at the Fermi level. For the following analysis it was integrated over ± 0.06 Å⁻¹. This area is sufficiently large which might also be concluded from Fig. 6.5 as the area includes all significant changes in the maximum intensity due to temperature. It should be stated that this routine does not prevent the shifting of the LEM in a gapped spectrum due to change in the self-energy. Therefore, relative to the temperature, the LEM measures an opening of a gap but also a change in excitation width.

The reader may want to verify some of the following interpretations. Therefore, the resulting LEM’s relative to the temperature are shown in Fig. 6.9.

Figure 6.10: The temperature range where the LEM gap saturates ∆T^∗ vs doping and La concentration.

Also included is the position of the 3rd depression and a cubic spline as guide to the eye.

In all measurements, the expected opening of the pseudogap can be seen by the shift of the leading edge midpoint. We see the LEM start-ing to move away from E_F and saturate at low temperatures. The saturation sometimes needs a large temperature interval as in sam-ple x0.66 or, as in x0.32, an abrupt jump of the LEM instead of a saturation. For the quan-titative analysis, the pseudogap-temperature T^∗ and a maximum LEM-pseudogap ∆^∗_mid must be defined. The pseudogap-temperature is taken as the temperature where the LEM changes significantly. The LEM-gap ∆^∗_midwas chosen as the difference of the Fermi energy E_F and a ’saturated’ LEM position shortly above the superconducting transition temper-ature T_C. In some measurements - see x058-2 - this might not be a perfect description. I will discuss the qualitative results later. Here, it is enough to know how T^∗ and ∆^∗_mid can be determined.

What can also be observed by the figures is

6.1 T^∗ and ∆^∗ measured by ARPES

that there is no large contribution of a superconducting gap in respect to the limited temperature range. For BCS theory, the gap is proportional to T_C; When using the BCS-formula for Bi2201 we would expect a zero-temperature superconducting-gap of up to about 4 meV. Thus, I cannot state that there is no superconducting gap. What I can state is that, in respect to the resolution and the achievable temperature, in the antinodal direction there is no large contribution from the superconducting gap, but instead from the pseudogap.

When looking at x0.32, at T^∗, a bounce in the LEM might also be observed. Let me note here that such behavior of the LEM-gap was recently found in Bi2212 [248]. From the previous discussion, we know that an increase or decrease in the leading edge midpoint does not necessarily mean that the gap changes. It could also be a broadening or narrowing of the excitation.

Although more examination is necessary, I attribute a possible change in linewidth at T^∗ to an onset of ordering. For this ordering, I will now refer back to observations that I am able to quantify by my conducted measurements, e.g. that the LEM sometimes jumps and sometimes saturates. Let me refer back to the simple model described above; The model depicts that an ordered state exists at a depression and disorder occurs away from a depression. Let us further consider the onset of electronic ordering as a kind of crystallization process; An ordering from a high-temperature disordered state can be more easily achieved in a system with holes equal to the filling factor of a depression as in an electronic structure with (intrinsic) defects. In my opinion, exactly this behavior can be observed in the temperature range the LEM gap saturates.

In the following, this temperature range is noted as ∆T^∗. In the case where ∆T^∗ is small, the system is nearly clean, whereas if ∆T^∗ is large, many defects are present within the system. One can say that ∆T^∗probes the population of antinodal defects which can be determined from Fig.

6.9 by a 10%-90% criterium as it is used for ∆T_C in AC-susceptibility (see Appendix A.2). This refined ∆T^∗ vs doping and La concentration is shown in Fig. 6.10. It can clearly be stated that, around the depression, ∆T^∗ is small. Note that the sample x0.32 is near the next depression.

In the case where the temperature range that the LEM needs to saturate is proportional to the number of defective holes, it can therefore be stated that the ground state at a depression is the cleanest pseudogap state.

The symmetrization method

For the final comparison of the gap magnitude, I will explain the symmetrization method [246]

which I additionally use to quantify.

Figure 6.11: Directive of the gap estimation for the symmetrization method: Mirror the spectrum at EF. The dotted lines are the spectra and its mirror image.

The straight line is the addive of both. The gap is then given by the position of the peak maximum which is next nearest to zero binding energy.

The symmetrization directive is depicted in Fig. 6.11. By mirroring the spectrum at E_F, the contribution of the Fermi-function is basi-cally removed. The gap is then given by the position of the next peak maximum nearest to E_F. The problem with the symmetrization method is its sensitivity to the correct deter-mination of E_F. From a comparison of a fit of the gold spectra and the nodal direction for sample x.032, the uncertainty in E_F can be es-timated as about ±3 meV. This seems like a small value but let me point out that the sym-metrization method is essentially a deconvo-lution where a small shift can produce a high error. From the directive of Fig. 6.11, the difference between the leading edge midpoint and the symmetrization gap can be seen; The leading edge gives a much smaller gap than the symmetrization.

The symmetrization gap was taken at the low-est temperature. For the sample x0.61, no result was obtained by the symmetrization method as the spectrum was very broad due to the enhanced scattering explained before.

It can only be stated that a gap < 100 meV exists for that sample.