Thursday, September 9, 2004: 4 pm at 151 Loomis Laboratory

Phase
diagram showing the destruction of superconductivity: 1) The yellow region represents the ordered phase in which all the electron pairs share the same phase (all arrows pointing up), 2) The elusive Bose metal is in blue in which all the phases are disordered but form a glass, and 3) Beyond the electron pairs fall apart and form an insulator. The vertical axis represents temperature and the in-plane axes any of the tuning parameters that destroy superconductivity such as defects or an external magnetic field. From UIUC News Bureau.

Today’s talk was largely based on Prof. Phillips’ ex-student’s work, Denis Dalidovich on investigating the possibility of a bosonic condensate possessing a finite resistance at absolute zero. The conventional theory results in the two-dimensional localization principle, stating that when noninteracting electrons interact with random scatterers, they always return to their origin. This implies that electrons are not free under such circumstances, but are bound to some potential. Therefore, a metallic state in two dimensions and below is impossible. However, this idea was refuted with the discovery of so-called novel metallic states that contain both strongly correlated electrons as well as disorder sites. Cooper pairs of electrons can behave as bosonic particles and condense to have a well-defined phase. Two well defined eigenstates of such a Bose-Einstein condensate are known, with zero and
infinite conductivity respectively.

The breakup of the condensate can be achieved either by pair-breaking (decreasing the amplitude of the wavefunction) or by phase fluctuations (destroying the coherence over the entire state). The latter may be achieved simply by exploiting the uncertainty relationship between phase and particle number. This duality results in the two extreme cases of insulator, with
well-defined particle number; and semiconductor, with well-defined phase due to entanglement of many number configurations. The insulating state is finitely correlated and hence has a nonvanishing mass associated with the state; in contrast, the superconducting state is completely uncorrelated and hence has a massless mediating particle.

The so-called phase-only model of the superconductor Hamiltonian contains a term which is the product of the charging energy and sum of squares of the conjugate momentum to the phase, and another term that is the product of the Josephson coupling strength and a cosine term describing phase change due to boson hopping. The latter term shows that phase locking maximizes the fluctuation in the energy due to Josephson coupling.

The insulating and superconducting states are diametrically opposed and therefore a phase transition can be observed between them. A particularly neat experimental demonstration is by starting with a perfectly aligned optical lattice created in an intense
laser beam. Decreasing the laser intensity lowers the barrier height and therefore increases the probability of trapped escaping from the electromagnetic well. When the laser beam is sufficiently diffuse, the particles then behave as a superfluid.

The onset of criticality in a bosonic system is seen as a lack of change in some quantity of interest as a function of some tuning parameter, say g. For example, a transition to superconductivity is seen by a sudden drop in resistivity as a function of decreasing grain boundary size as a sign if lower disorder (as well as being a function of temperature). The
standard theory states that below the critical value, g < gc, the resistivity vanishes to give a superconductor. Above the critical value, g > gc, the resistivity is infinite and an insulating state is observed. Also, the resistivity diverges to either 0 or infinity as the temperature tends to zero, except as the critical point g = gc, where the resistivity tends to a finite value of ρc = h/4e2 = 6.5 kOhm. (Note that the charge carriers are electron pairs, hence the charge of 2e.)

Prof. Phillips is of the opinion that the critical resistivity Rc is not a universal quantity, and cites experimental data taken of critical resistivities of Bi(Ge), Al(Ge), Pb(Ge) and Mo(Ge), i.e. various metal layers deposited on top of a germanium substrate. A large scatter was observed. In addition experimental data with magnetic systems also suggest that there are many g values that lead to the resistance leveling off at and below 2K. In fact, there are cross-over points due to
intersection of curves plotted at different temperatures, and also hysteresis effects, both of which are totally forbidden by the standard theory.

The data therefore point to the presence of two critical points between the superconducting and insulating states, with an intermediate phase that is stable to disorder and has intact Cooper pairs. Heuristically, it must also have a finite resistivity is seen to have a power-law dependency on proximity to the superconducting state. This implies the existence of some scattering mechanism, i.e. dissipation, and therefore transport properties near critical points become important. In fact, it could be shown that conductivity must be a monotonic function of frequency divided by temperature.

A toy system of Josephson junctions and superconducting rings was considered. Semiclassical theory gives the conductivity as the product of the number of scattering centers and the mean lifetime between collisions. Quasiparticle excitation of the order parameter regularize the conductivity in this case and cancel out the effect of scattering, but the cancellation is destroyed with a slight perturbation, quickly forming an insulating state. This does not occur because of disorder parameters, describing a random distribution of orientations as well as Josephson coupling magnitudes.

Consideration of disorder leads to the random spin model, which exhibits frustration. This generates a new glassy phase, called a phase glass, that exhibits a non-collinear sum of orientations. This model is characterized by the Edwards-Anderson order parameter, with a correlation function that contains a long-range interaction term in Ï„ -2 describing dissipation (due to the Fourier transform). Such spin glasses have many degenerate states, with many low-lying excitation states that can lead to dissipation. This gives rise to system with superconducting, metallic and insulating phases, and a theory built upon such a model predicts the possibility of finite conductances.

A point of consideration is whether rotating the phase changes the conductivity. It is possible that the stiffness of the phase results in symmetry breaking when shifted, generating massless modes. However this turns out not to be the case, as any phase rotation velocity produces diffusive modes. Stiffness is a transient quantity, with contributions from twisting and limited range of allowable phases. Such stiffness leads to well-confined phases. On the other hand, if all of phase space is allowed, the resistivity vanishes because glasses of this nature have no stiffness.

When an external magnetic field is applied to a superconductor with intrinsic disorder, a vortex glass is produced due to entanglement of magnetic flux lines through the material. The cosine term describing boson hopping then acquires a random gauge term in its argument, giving rise to fluctuations in the Josephson junction coupling strength (by the cosine addition formula). The question is then whether a vortex glass can be a superconductor. If so, a nonlinear IV curve (non-Ohmic behavior) should be obtained below the glass transition temperature, but this is not observed. The reason is due to untrimmed, pointlike defects, as demonstrated by completely Ohmic behavior of yttrium barium cuprates. This suggests that a vortex glass is a manifestation of a Bose metal.

Plotting normal v. critical resistivities shows that they are equal and therefore suggests that the second transition results from the breakup of electron pairs. This follows from the Bardeen relationship, which states that the magnetic field applied associated with breaking up the pairs is proportional to the energy bandgap between the paired and unpaired states.