High T $_\textbf{\small c}$ superconductivity

The high temperature superconductors are highly unconventional in that the normal state is rather abnormal, characterized by a pseudogap (i.e., strong depletion of density of states around the Fermi level) in the single particle excitation spectrum in the underdoped regime. The seminal BCS theory of superconductivity predicts that the superconducting order parameter (magnitude) and the excitation gap are identical, and therefore, the single-particle excitation gap vanishes at and above the transition temperature $T_c$. However, it evidently fails to explain the existence of the pseudogap. This normal-state gap persists up to a crossover temperature $T^* (> T_c)$, and evolves continuously to the excitation gap below $T_c$. The origin and the nature of the pseudogap have been highly controversial and still remain the hottest topic in high $T_c$ superconductivity research.

My Ph.D research work with K. Levin, I. Kosztin, and B. Jankó sets up a pairing fluctuation theory, which extends the mean-field BCS theory to short coherence length superconductors in a natural way so that finite-momentum pairing fluctuations play a progressively more important role as the pairing strength increases. The cuprate (and organic) superconductors belong to a large class of short coherence length superconductors, different from normal-metal superconductors. Therefore, the fluctuation effects are expected to be strong and important. Based on a $T$-matrix approximation, our theory treats incoherent pair excitations (pairons) in a self-consistent fashion so that the order parameter $\Delta_{sc}$ and the excitation gap $\Delta$ are distinct in general. While the order parameter vanishes at $T_c$, the excitation gap persists up to a higher temperature $T^*$ at which (metastable) pairs start to form. This distinction, characterized by the pseudogap $\Delta_{pg}$, is accounted for by the incoherent pair excitations, and essentially vanishes in the weak coupling BCS limit. In our picture, the pseudogap $\Delta_{pg}$ shares the same origin as the order parameter and can be regarded as a precursor to superconductivity.

We have shown that finite momentum pairing leads naturally to a pseudogap, providing a most intuitive explanation for the origin of the pseudogap. The Thouless criterion for superconducting pairing instability can be generalized to the BEC condition for pairons. In addition to the fermionic quasiparticles of BCS theory, pairons are a new type of excitations in our theory.

We are among the first who have studied the pseudogap behavior below $T_c$ systematically. we have shown that the pseudogap persists down to $T=0$, with the total gap given by $\Delta = \sqrt{\Delta_{sc}^2 + \Delta_{pg}^2}$. In particular, we have obtained a cuprate phase diagram, in (semi-) quantitative agreement with experiment. Pair excitations play an important role in destroying superfluidity and thus provide a natural explanation for the otherwise troublesome quasi-universal behavior of the temperature dependence of the superfluid density for different doping concentrations.

We have predicted, at a generalized mean-field level, that pair excitations lead to new low-temperature power laws in physical quantities such as the penetration depth and the specific heat. This provides so far the only theoretical explanation for the $T^{3/2}$ power law of the penetration depth in the BEDT family of the organic superconductors observed by R. Giannetta et al. and recently by others. We have also found other preliminary experimental support.

We have also studied the signatures of phase coherence in phase-insensitive experiments such as specific heat, and the collective modes and gauge invariance in the presence of strong pairing fluctuations, and the magnetic field effects on the pseudogap phenomena. Our results have been quite satisfying.

My recent work with J. R. Schrieffer has extended the pairing fluctuation theory to superconductors with nonmagnetic impurities. This is very important since impurities are always present, even in the cleanest samples, and are essential for understanding the ac conductivity, among other physical quantities. Such a theory is crucial in understanding the impurity effects on the pseudogap phenomena. To this end, one needs to go far beyond the standard so-called self-consistent (impurity) $T$-matrix approximations at the BCS level, and include the pseudogap itself as an intrisic part of the theory. Due to the complexity and technical difficulties of this problem, there has been virtually no other work on this issue. In our work, both the particle-particle scattering and the impurity scattering $T$-matrices are treated self-consistently. In this context, we have studied the effects of impurity scattering on $T_c$ and the pseudogap at arbitrary scattering strength (from the Born to unitary limit) and different hole doping concentrations. We have found that both $T_c$ and the order parameter decrease with the impurity concentration, whereas the pseudogap is relatively insensitive. This reveals that finite momentum pair excitations are more robust against impurity scattering than the condensate. This finding is in some sense similar to effects of magnetic field on the pseudogap. We have also shown that the chemical potential adjusts itself with the impurity level, and the density of states at the Fermi level is always finite at finite impurity concentrations, in good agreement with experiment (but different from some other predictions).

My recent work with Z. Tešanovic concerns the superconductor-insulator transition at the lower critical doping and the transport properties in the pseudogap region of the cuprate phase diagram. Since the cuprate superconductors are quasi-two dimensional layered materials and are close to Mott insulators at half filling, phase fluctuations of the pairing field are expected to be very strong. In the deep underdoped regime, vortex proliferation provides another natural scenario for destroying the superconducting ordered state. Experiment has shown that at very low doping, the dc resistivity increases rapidly with decreasing temperature at very low $T$. Using a particle-vortex duality transformation generalized to the case of finite temperatures, we have shown that below the lower critical doping, the system behaves like an insulator at low temperatures, consistent with experimental observations.

My work addresses the central issues in high $T_c$ superconductivity, and has been directed towards understanding experiment. I have kept in close contact with a variety of different experimental groups.