Chapter 7 Summary and Outlook In this thesis the coexistence of Density Waves and singlet Superconductivity was analyzed in some detail. As a first step the gap equations for pure DW order were derived and solved numerically. It was shown that in the high-t c cuprates SDW and CDW order can exist if a suitable interaction is present. However, it was found that DDW order can only exist if the DDW gap is so large that only one of the two DDW bands crosses the Fermi level. Although this may be in agreement with observations made in the pseudo-gap region of the high-t c cuprates [11, 32], clearly further study is required. Especially, calculations going beyond the mean-field level and a detailed analysis of the free energy for a realistic band structure are needed. The set of coupled gap equations determining the SC and DW gaps in the coexistence region was derived. This derivation is based on the assumption that the Density Wave transition temperature is much larger than the superconducting critical temperature, i. e. T DW T c, and that it is therefore allowed to first introduce the DW mean-field and to apply the corresponding Bogoliubov transformation and to introduce the superconducting mean-fields afterwards. It was found that in general, due to the multi-band character of the pure DW phase, there are multiple SC gaps and the pairing has both intra- and inter-band contributions. However, the Cooper pairs consist only of quasi-particles from the same band, since the Fermi surfaces of the different DW bands show a substantial mismatch when the DW gap is not close to zero. It was shown that, at least for T T c the DW gap equation decouples from the SC gap equations. For superconductivity on an SDW background it was found for an arbitrary pairing interaction that both gaps need to change sign when shifted by Q. Therefore, nodeless superconductivity cannot coexist with SDW order. It was noted that the SC gap equations contain the correct limiting case for vanishing SDW gap and therefore one can assume that they are valid for any 122
CHAPTER 7. SUMMARY AND OUTLOOK 123 value of the SDW gap. Considering a momentum independent pairing interaction it was found that, while in the absence of SDW order s-wave superconductivity forms, turning on the SDW order induces lines of nodes on the border of the RBZ and a phase-shift of π between the two SC gaps. Unconventional superconductivity forms. However, since the effective pairing interaction is strongly suppressed by the SDW order, the T c rapidly becomes unobservable small. The situation is different for a pairing interaction in the d-wave channel. In that case the SDW order does not induce additional nodes and no phase-shift between the gaps. However, also in this case superconductivity is destroyed when the SDW order is turned on, but the coexistence region with large T c is much larger than for the momentum independent pairing interaction. The reason for that is that here the effective total pairing interaction is not suppressed by the SDW order, but the effective pairing strength is merely redistributed, weakening the intra-band pairing and enhancing the inter-band pairing strength. Therefore superconductivity can exist until the SDW gap is so large that the first SDW Fermi surface is destroyed. Phasediagrams based on this theory were found to be qualitatively compatible with experimental data of the electron-doped cuprates. In the case of superconductivity on a CDW background, the results are pretty much the opposite to what was found for superconductivity on an SDW background. First of all, here the SC gaps are equal at any pair of momentum points connected by Q for any pairing interaction. Therefore nodeless superconductivity is possible. In general, the gaps must have an even number of nodes between any two momenta connected by Q. Exactly as with the SDW background, the SC gap equations contain the correct limiting case for vanishing CDW gap. For a momentum independent pairing interaction the SC gaps are found to be nodeless, not phase-shifted and almost momentum independent. When the CDW gap is increased the total effective interaction does not change, but the effective pairing strength is merely redistributed between inter- and intraband contributions. Here, the effect of increasing the CDW gap is to weaken the inter- and to enhance the intra-band pairing, effectively decoupling the two bands and thus allowing the possibility of one band superconductivity at large CDW gaps. Therefore, superconductivity is not destroyed when the Fermi surface of the first CDW band vanishes, but exists as long as there is a Fermi surface. Notably, the T c is not necessarily significantly suppressed by the CDW order. If a large portion of the density of states at the Fermi level in the absence of CDW order survives up to large values of the CDW gap, also the T c will remain large. If there is a pairing interaction of the d-wave type the situation is different.
CHAPTER 7. SUMMARY AND OUTLOOK 124 Since there has to be an even number of nodes between momenta connected by Q, additional nodes form at the border of the RBZ and a phase-shift of π is induced between the two SC gaps. Since the total effective pairing strength is rapidly suppressed when the CDW gap is turned on, also the T c quickly becomes unobservable small. If there is a DDW background the situation is different. Here the correct limiting case is not recovered in the limit of a vanishing DDW gap. But since small DDW gaps are not allowed it is not surprising that there are problems in that limit. The SC gaps on the DDW background need to change sign when shifted by Q, i. e. nodeless superconductivity is impossible. If one neglects the fact that there is no solution for the DDW gap such that there are two DDW bands crossing the Fermi level and simply treats the DDW gap as a parameter entering the SC gap equations, one can obtain interesting phaseshifted and not phase-shifted d- and not phase-shifted unconventional s-wave superconductivity in the regime where both DDW band cross the Fermi level. However, as soon as the first DDW band does not cross the Fermi level anymore, all kinds of superconductivity are very strongly suppressed. Therefore, for any allowed value of the DDW gap, there can be no superconductivity with a large T c. This behavior is in contrast with the interpretation of the under-doped half of the SC dome in the hole-doped cuprates as a phase of coexistence of superconductivity and DDW order. The theory described here may be not applicable in the limit of T c T DW. Therefore, it would be interesting to see what results a theory starting from that limit would yield. Even more interesting would be a theory that works for any relation of the two transition temperatures. By now there exists no theory valid for any relation of the transition temperatures and the route to such a theory is not clear yet. The superconducting pairing interaction itself was treated as independent on the DW background and only its dressing by the DW coherence factors was taken into account. If, however, the pairing interaction is purely electronic, it is very likely that the strength and pairing interaction do change as a function of the DW background. In order to investigate such effects it is necessary to choose a specific interaction, which is beyond the scope of this thesis and left for future study. The resonance peak observed by INS in the superconducting state of the high-t c cuprates has been addressed within the spin-exciton scenario. By comparison with the experiments on hole-doped cuprates, it was shown that the description of the resonance mode itself appears to work rather well. However, the theoretical results for the state above T c failed entirely. Here further work is required to understand the behavior above the SC state. But also the question why the description of the SC state, which is apparently
CHAPTER 7. SUMMARY AND OUTLOOK 125 based on an incorrect description of the "normal" state, can work, needs further study. Electron-doped cuprates have been studied within the spin-exciton scenario as well. It was found that the spin-exciton scenario may be applicable, though there remain a few points that require clarification. In particular, further experimental work is required in order to find the relation between the value of the SC gap, especially at the hot spots, and the frequency of the resonance peak. Theoretically, the question whether the strong momentum dependence of the RPA interaction employed in this work is reasonable for the electron-doped cuprates must be addressed. Furthermore, the effect of a magnetic field aligned parallel to the a-b-plane of the cuprates has been analyzed. Here it was found that in a magnetic field of strength H the resonance mode is split into three modes with a frequency distance of 2H. Further experiments are required in order to determine, whether this effect occurs in the cuprates or not. However, even a negative result would not necessarily rule out the spin-exciton. The reason is that a large Γ, describing life-time and resolution effects, appears to be adequate in the electron-doped cuprates. But the splitting is only observable if 2H > Γ. This would with Γ 2meV require H 20T. Such a large magnetic field is difficult to use in INS experiments and also the question of phase transitions induced by the magnetic field becomes important for such large fields. Finally, the evolution of the resonance peak into the Goldstone mode of an SDW state for the case of T c > T N was discussed. It was shown that the resonance mode is over a wide range of temperatures only weakly affected. When the temperature is reduced to values close to T N, however, the resonance frequency at q = Q starts to decrease rapidly towards zero, which is reached at T N. In this process the dispersion of the resonance mode away from Q is changed strongly. While close to T c the resonance mode exhibits almost no dispersion, for temperatures close to T N the resonance mode acquires a V -shaped upwards dispersion. INS experiments on suitable samples are needed. The spin susceptibility of DDW states was calculated. First, for a pure, then for DDW order coexisting with d-wave superconductivity. For the pure DDW state it was found that a resonance peak can form at q = Q. However, the resonance peak becomes rapidly damped away by the continuum of excitations away from that momentum. Therefore, the resonance peak does not have a significant width in momentum like in the superconducting state. With increasing DDW gap the momentum width of the resonance peak becomes somewhat larger, but is still much smaller than in the SC state. Furthermore, the momentum integrated spin susceptibility is found to be almost independent of frequency and the resonance peak has almost no
CHAPTER 7. SUMMARY AND OUTLOOK 126 spectral weight. This behavior is in strong contrast to experiments on the pseudo-gap region of the hole-doped cuprates. In these experiments the momentum integrated spin susceptibility is found to show a strong frequency dependence: It is zero for small frequencies and for larger frequencies the resonance peak emerges. Therefore one can conclude that it is improbable that the pseudo-gap region is actually a DDW phase. Nevertheless, also the spin susceptibility for a phase of DDW coexisting with d-wave superconductivity was calculated. It was found that again a resonance mode forms. For momenta away from Q the resonance mode behaves similar to the pure d-wave result. In the vicinity of Q on the other hand, the DDW order is important. Here, the resonance mode behaves more like in the DDW state, however, with the effective DDW gap enhanced by the SC gap, resulting in a significantly enhanced resonance frequency. In total, the resonance mode looks like in the pure d-wave SC state with a cusp close to Q. It should be stressed that several results presented here have been obtained never before and are completely new. The new results are in particular: First, there is a non-trivial solution of the DDW gap equation only when the α-band Fermi surface has vanished. Second, both the multi-band and multi-gap character of a state with coexistence of Density Waves and superconductivity was studied and the corresponding coupled sets of gap equations, including both intra- and inter-band pairing interactions explicitly, were derived and solved numerically for realistic band structures. Third, the INS experiments on electron-doped cuprates were directly addressed theoretically and found to be, within certain limitations, compatible with the spin-exciton scenario of the resonance peak. Fourth, the quantitative response of the resonance mode to a magnetic field parallel to the a b-plane was calculated and it was found that the mode is split into a triplet. Fifth, numerical results for the continuous evolution of the resonance mode of the SC state into the Goldstone mode of the antiferromagnetic state have been presented for the case when T c is larger than T N. Sixth, the full expression for the spin susceptibility of the DDW state was derived and evaluated numerically for a realistic band structure, showing that a resonance peak forms, but is confined to the direct vicinity of Q. Seventh, the full expression for the spin susceptibility of the combined DDW and d-wave SC state was derived and evaluated numerically for a realistic band structure, showing that a resonance peak forms and is distinguished from the resonance peak in the pure SC state only by a cusp in the dispersion of the resonance frequency in the vicinity of Q.