High-temperature and other unconventional superconductors
High-temperature and other unconventional superconductors |
|
Theoretical problems in cuprates such as the strong correlations between electrons, a microscopic explanation of high-Tc superconductivity, and the competition of superconductivity with other structural phases are studied in the weak- and strong coupling limits. The obtained results are compared with photoemission, inelastic neutron and light scattering experiments.
The most important structural element of high-Tc superconductors are electron- or hole-doped CuO2 planes. The dynamics of electrons in these planes may be described by a Hubbard model or its strong-coupling descendent, the t-J model. In the weak-coupling case the calculations are based on the fluctuation-enhanced exchange (FLEX) approximation, whereas in the strong-coupling case the leading terms of a 1/N expansion (N is the number of internal degrees of freedom for the electrons consisting of a spin and an additionally introduced flavor index) are considered.
Elementary excitations in superconductors are of central interest in order to learn more about the correlations and the mechanism for Cooper-pairing. In conventional strong-coupling superconductors like lead (Pb) the understanding of the elementary excitations in tunneling spectroscopy played the most important role in accepting the picture of electron-phonon mediated superconductivity. In the high-Tc cuprates it is important to understand the spectral density, i.e. the local density of states, of the quasiparticles measured by angle-resolved photoemission spectroscopy (ARPES) and their interdependence with spin excitations observed by inelastic neutron scattering (INS) experiments.
 |
Figure 1: Calculated spectral density N(k,ω) in the normal state for temperature T=100K along the nodal (0,0)->(π,π) direction (from left to right) as a function of frequency in the first Brillouin zone. The peak positions (connected by the solid line to guide the eye) refer to the renormalized energy dispersion ω(k) that is calculated self-consistently. A kink structure at an energy approximately ωkink=75±15meV occurs that results from coupling of the quasiparticles to spin fluctuations. |
Employing the FLEX approximation to the one-band Hubbard model our calculations show a abrupt change (kink) of the Fermi velocity of holes within the CuO2-plane at same characteristic energy if they interact with magnetic excitations. This is shown in Figure 1 where we analyze the spectral density N(k,ω) in the nodal direction, (0,0)->(π,π), in the normal state. Furthermore, below Tc, in regions where the superconducting gap is large, we find a strong renormalization of the spectral function. Both effects are fingerprints of the pairing interaction. We compare the results of our electronic theory with available ARPES data and find fair agreement.
 |
Figure 2: Calculated feedback of superconductivity on the spin susceptibility Im χ(q,ω) for cuprates at optimal doping (x=0.15) for wavevector q=Q. The solid curve refers to the normal state (T=1.5Tc), while the dashed curve denotes the renormalized spin susceptibility in the superconducting state at T=0.7Tc. |
If Cooper-pairing in the high-Tc cuprates is mediated due to antiferromagnetic spin fluctuations, it is important to study their interdependence with elementary excitations. In analogy to conventional superconductors in which the tunneling density of states was analyzed, we calculate the feedback effect of the superconducting gap function Δ(k,ω) on the spin susceptibility for high-Tc cuprates. If the gap is large enough, the feedback effect yields a resonance peak at some characteristic energy ~2Δ0 shown in Figure 2. We compare our results with INS data (B. Keimer's group) and find fair agreement.
Various experiments show the existence of a local gap with d-wave symmetry in underdoped cuprates at temperatures above Tc. The nature of this pseudogap is presently unclear and quite different proposals for its explanation have been put forward. Because small coherence lengths, which are typical for underdoped cuprates in contrast to conventional superconductors, are associated with large fluctuation effects of the superconducting order parameter the phase coherence may be lost above Tc whereas the absolute value of the superconducting order parameter remains finite. Our calculations show that without long-range Coulomb interactions classical thermodynamic fluctuations dominate at the relevant temperatures which cause, for instance in the spectral function of electrons, only rather small broadenings in momentum space. Coulomb forces, on the other hand, yield also important quantum fluctuations above Tc which could, at least partially, be responsible for the observed strong increase of the incoherent part in the spectral function and of the scattering rates in the pseudogap phase.
 |
Figure 3: Phase diagram in the J=0.3 and different Coulomb interaction strengths, characterized by the nearest-neighbor constant Vnn. N, SC, FL, and ICDW denote the normal, superconducting, flux, and incommensurate charge density wave phases, respectively. |
Alternative explanations of the pseudogap are based on the competition of superconductivity with structural phases such as antiferromagnetism (AF) or charge-density waves (CDW). FLEX calculations, appropriate for the weak-coupling case of the Hubbard model, suggest such a competition of AF and a CDW with d-wave symmetry. In the strong-coupling case a 1/N expansion (1/N is used as an expansion parameter to extrapolate to the physical case N=2) yields the phase diagram shown in Figure 3. All energies are given in units of the nearest-neighbor hopping t.
For δ>0.14 (δ denotes hole doping away from half-filling) the normal state N at high temperatures is unstable with respect to d-wave superconductivity SC with a Tc which increases with decreasing doping. At δ~0.12 a new phase boundary between SC or N and a d-CDW (often also called staggered flux phase and denoted by FL) appears which competes with SC and suppresses Tc with decreasing doping. It turns out that the FL phase is unstable against phase separation but that it can be stabilized by a screened Coulomb interaction characterized by the nearest-neighbor interaction Vnn. The Coulomb interaction induces incommensurable CDW at very low δ's but does not change much the FL boundary. In this model the pseudogap phase originates from the d-CDW, has d-wave symmetry like the superconducting order parameter and coexists with it below Tc. Calculations have been carried out to identify the collective modes of these phases, to predict characteristic consequences of this model for electronic Raman scattering and c-axis tunneling, and to confront the results with the experimental data.
It is now widely believed that the superconductivity in MgB2 is caused by a strong electron-phonon coupling in this material. One effect of this coupling can be seen in the Raman spectrum shown in Figure 4. Besides of the phonon line at 600cm-1 there is a sharp peak in the experimental curve (solid line) near the superconducting gap due to σ electrons at around 110cm-1. According to theory the scattering of electronic excitations across the gap by the strong electron-phonon coupling leads to a bound state inside the gap which manifests itself as the sharp peak seen in the theoretical curve (dashed line). The theory explains the observed selection rules and the different behavior of σ and π electrons. However, there is one quantitative problem: An explanation of the strong Fano effect seen experimentally near the phonon line would require a much larger coupling constant λ which would be incompatible with the small energy of the bound state.
 |
Figure 4: Experimental (solid line) and theoretical (dashed line) E2g Raman susceptibility. |
Similar bound states as in MgB2 have not been found in the cuprates suggesting a weak electron-phonon coupling and a mainly electronic mechanism for superconductivity in layered high-Tc superconductors. Various lattice effects such as superconductivity-induced phonon shifts and broadenings, isotope effects in Tc, the penetration depth, infrared and photoemission spectra, however, have been observed in the cuprates. The recent detection of a kink in the electron dispersion of several cuprates at energies typical for phonons has been explained in terms of phonons using a bare λ of about 0.7 and the 1/N expansion to include electronic correlation effects. New Monte Carlo calculations, suggesting a surprising increase of λ with U at large U's, persuaded us to carry out new analytic calculations on the influence of electronic correlations on λ using instead of the 1/N expansion the Kotliar-Ruckenstein method. At high temperatures, where the Monte Carlo have been performed, the obtained results look similar to the Monte Carlo data, predict, however, also an entropy driven transition to a phase-separated state which seems to disagree with the numerical simulations. At low temperatures the obtained results disagree on a quantitative level with those from the 1/N expansion suggesting a substantially larger suppression of λ by correlation effects. More work seems to be necessary to reach a final assessment of the role played by phonons in high-Tc superconductors.
The discovery of high-Tc superconductivity in the cuprates led to extensive searches for other superconducting transition metal oxides. One important example is the novel superconductor strontium ruthenate (Sr2RuO4) which was discovered by Y. Maeno and co-workers in 1994. Its crystal structure is iso-structural to that of (La,Sr)2CuO4, but has a Tc~1.5K and is a triplet superconductor. Ruthenium has four remaining electrons within the 4d-shell and crystal field effects split the 4d-states into t2g and eg sub-shells. The negative charge of O2- causes the t2g states to lie lower in energy and the corresponding xz-, yz-, and xy-orbitals form the Fermi surface yielding three bands, the so-called α-, β-, and γ-band.
 |
Figure 5: Calculated normal state temperature dependence of the nuclear spin-lattice relaxation rate T1-1 of 17O in the RuO2-plane for the external magnetic field applied along the c-axis (dashed curve) and along the ab-plane (solid curve). The corresponding susceptibilities include spin-orbit coupling. Down- and up-triangles are experimental points taken from Ishida et al. |
Using a three-band Hubbard Hamiltonian and including the important spin-orbit coupling (SOC) we calculate various physical properties and describe Cooper-pairing by the exchange of spin excitations. We find p-wave triplet superconductivity due to large ferromagnetic excitations from the γ-band accompanied by a high electronic density of states. However, the α- and β-band reveal strong antiferromagnetic excitations at wavevector Q=(2/3,2/3) due to their nesting properties. Both effects are coupled via SOC and result in a strong magnetic anisotropy in the normal state, χzz(Q)>χ+-(Q), that is shown in Figure 5. Taking the parameters from LDA band-structure calculations, we compare the results of our electronic theory with NMR data and find good agreement. Furthermore, the magnetic anisotropy is responsible for (a) lifting the degeneracy of the three possible states of the chiral A-phase yielding a superconducting d-vector d(k)~Δ0(sin kx + i sin ky)z and (b) for nodes of the superconducting gap function between the RuO2-planes.
fkf.mpg.de)fkf.mpg.de)fkf.mpg.de)fkf.mpg.de)fkf.mpg.de)