Markus Schmid
Disorder- and Field-Induced Inhomogeneities in Unconventional Superconductors
Supervisor: Prof. Dr. Arno P. Kampf [Theoretical physics III]
Date of oral examination: 02/11/2011
210 pages, english
By this time superconductivity was discovered 100 years ago. A century of research has not only revealed a great variety of fascinating characteristics of superconducting materials, but also paved their way into everyday life. The most prominent example might be the magnetic resonance scanners, but superconductivity is also utilized for superconducting cables, particle accelerators, electric motors and generators etc. Technological applications based on superconductors find their way today in the fields of power supply and industrial production on a larger scale. This thesis is devoted to a better understanding of high-temperature superconductivity (HTSC) and its interplay with antiferromagnetic correlations, which may be crucial for the pairing mechanism and thus for the emergence of HTSC itself. Specific attention is paid to intrinsic disorder which is present in many high-Tc materials. A new era of superconductivity has begun in 1986 when Bednorz and Müller found that the cuprate-perovskite ceramic La2−xBaxCuO4 (LBCO) becomes superconducting at Tc = 35 K. Intense research activity led to the discovery of HTSC in several other compounds of the same material class, e. g. in La1.85Sr0.15CuO4 (Tc = 40 K), YBa2Cu3O7−x (Tc = 93 K), Bi2Sr2Ca1Cu2O8 (Tc = 95 K). To date the highest reproducible transition temperature is found in Hg0.8Tl0.2Ba2Ca2Cu3O8 with Tc = 138 K at ambient pressure. The undoped parent compounds of the cuprates are Mott insulators, but become superconducting with hole doping x, where Tc(x) forms the famous superconducting dome. From the beginning, the high-temperature superconductors revealed unexpected and inscrutable properties. While conventional superconductivity emerges from the instability of a Fermi liquid to an arbitrary small attractive pairing interaction, HTSC does not appear out of a Fermi liquid state by cooling below Tc, except for the heavily overdoped regime. Another mystery was the fact that the conventional phononinduced pairing mechanism cannot explain superconductivity at such high transition temperatures, as phonons rather impede the pairing at large temperatures than the other way round. The absence of the isotope effect in the high-Tc materials substantiated additionally that the attractive pairing interaction is not caused by the exchange of virtual phonons as it is in conventional superconductors. Among other attempts to explain the high-Tc pairing mechanism has been a pairing theory based on the exchange of antiferromagnetic spin fluctuations due to the closeness of the superconducting phase to the antiferromagnetic Mott phase. This ansatz accounts also for the high transition temperatures and explains the superconducting dome. While with increasing doping the concentration of mobile charge carriers is enhanced, the growing spin dilution attenuates the pairing mechanism based on spin-fluctuations. Hence the optimal balance between the two anticorrelated ingredients leads to the appearance of HTSC with a dome-shape of Tc(x). Independent of the determination of the precise pairing mechanism there is an overall agreement that electronic correlations play a crucial role in the high-temperature superconductors and are therefore incorporated in the model presented in this thesis. The existence of a spin-glass phase in La2−xSrxCuO4 (LSCO) and the pinning of stripes in the cuprates by impurities or vortices are only some aspects emphasizing the relevance of antiferromagnetic spin-correlations in HTSCs. Monthoux et al. also showed that a spin-fluctuation mediated pairing mechanism unambiguously imply a dx2−y2 symmetry of the superconducting state. Although the pairing symmetry is not directly accessible by experiments and is still to some degree controversial, much evidence militates in favor of the d-wave symmetry. Therefore we base our theoretical model on the d-wave pairing symmetry and will find further support for it by the comparison of the obtained results with experimental data. In this thesis we focus in large parts on the underdoped regime of the cuprate superconductors manifesting the most unconventional behavior compared to the overdoped regime, which exhibits rather the characteristics of a conventional BCS superconductor. While in conventional superconductors the suppression of superconductivity reveals the normal Fermi-liquid like electronic phase in the absence of superconductivity, the situation is quite different in underdoped cuprates. In the latter case the local suppression of the pair amplitude, e.g. by a vortex leads to the appearance of a competing ordered state stabilized by the proximity to the Mott state. This state is characterized at low temperatures by a static local spin-density wave (SDW) order with an ordering wave vector q near (π, π), an order which is not present in the non-superconducting state above Tc. This was reported first in elastic neutron scattering experiments on LSCO, with a correlation length of several hundred Angstrom, but has been confirmed in other underdoped cuprates as well. An enhancement of incommensurate static order was observed with increasing magnetic field up to 14.5T. Because the signal disappeared above Tc, the magnetism was attributed to the vortices; indeed, scanning tunneling microscopy (STM) measurements on Bi2Sr2CaCu2O8+δ (Bi-2212) were able to directly image unusual charge order near the vortex cores which is almost certainly related to the field-induced SDW detected by neutron scattering. Signatures of antiferromagnetic ordering in the superconducting (SC) state had also been found earlier by nuclear magnetic resonance (NMR) experiments. Subsequently, muon-spin-rotation (μSR) experiments detected magnetic ordering as a wedge-shaped extension of the spin glass phase into the superconducting dome of the temperature vs. doping phase diagram. Lake et al. reported that an incommensurate magnetic order similar to the field-induced state was also observed in zero field. Although it also vanished at Tc, the ordered magnetic moment in zero field had a T dependence which was qualitatively different from the field-induced signal. The zero-field signal was attributed to disorder, but the relation between impurities and magnetic ordering remained unclear. Because strong magnetic fluctuations with similar wavevector are reported at low but nonzero energies in inelastic neutron scattering experiments on these materials, e.g. on optimally doped LSCO samples exhibiting no spin-glass phase in zero field, it is frequently argued that impurities or vortices simply "freeze" this fluctuating order. Describing such a phenomenon theoretically at the microscopic level is difficult due to lack of translational invariance of the inhomogeneous interacting system, but it is important if one wishes to explore situations with strong disorder, where the correlations may no longer reflect the intrinsic spin dynamics of the pure system. Such an approach is proposed here in a model calculation for an inhomogeneous d-wave superconductor with Hubbard-type correlations treated on a mean field level. In this model a single impurity creates, at sufficiently large Hubbard interaction and impurity potential strength, a droplet of staggered magnetization with a size corresponding to the antiferromagnetic correlation length of the hypothetical pure system. When these droplets come close enough to interact, there is a tendency to form incommensurate, phase-coherent Neel domains with a size that is sufficient to explain the observations by Lake et al. in zero field. Such a model explains the empirical observations that both increasing disorder and underdoping enhance the SDW order. STM experiments showed that high-temperature superconductors, in contrast to conventional superconductors, exhibit a spatially inhomogeneous energy gap, which is assumed to arise as well from dopant disorder. We investigate this phenomenon independently of its origin and derive analytic equations which describe the charge redistribution between areas of different pairing interaction. Depending on the density of states (DOS) an area of suppressed superconductivity is charged positively or negatively. In addition, correlations influence the redistribution. We find that the emerging Andreev bound states localized in regions of reduced pairing amplitudes are in remarkable agreement with the experimental results. In large parts of this thesis we explore the role of the impurity- and field-induced quasiparticle bound states in the presence of electronic correlations and compare these results to STM and neutron scattering results. An important task will be the identification of both types of induced local antiferromagnetic order in the superconducting state. Moreover we discuss the origin of the "order by disorder" phenomenon as well as the T evolution of the disordered magnetic state in an applied magnetic field. The presented results reproduce well the qualitative aspects of the experiment by Lake et al.. We find that some features depend on nonuniversal aspects of disorder, in particular the process of domain wall nucleation, and that while disorder- and magnetic-field induced SDW order both add to the ordered moment, the interference of disorder and magnetic-field effects is quantitatively significant. The domain wall formation proves responsible for the distinct temperature dependencies of the field- and the disorder-induced magnetization. The present theory also includes a crossover from magnetic droplets to filamentary stripe-like structures. If the Hubbard repulsion exceeds a critical value, a unidirectional stripe formation of the charges, the magnetization and the superconducting order parameter emerges. In the presence of impurities or vortices these stripes are pinned and are deformed to snake-like paths in the strong disorder limit. The model therefore offers a route to describe the physics of the pinning of stripe correlations in the superconducting state. This insight may prove relevant for many experiments in the underdoped cuprates which have been attributed to stripes.