Amplitude and Phase Fluctuations in Superconductors

The behaviour of classical superconductors is well described by the BCS theory which is a mean-field variational approach with electrons forming pairs and causing a gap in the energy spectrum of single electrons. High temperature superconductors as copper oxides have many features that are not understood in the framework of BCS theory. One of the most important problems is the presence of a region in the normal regime above the critical temperature Tc and below a temperature T* where observable quantities deviate from the free electron gas behaviour. This region is called pseudogap region because it contains effects similar to superconductivity like a partial suppression of electronic density of states.

Schematic experimental phase diagram of cuprate superconductors.

Phase diagram of cuprates
Anomalous effects are measured in the pseudogap region (below T*) of the copper oxides phase diagram. Superconductivity is present only below Tc and some signals persists until T'(Nernst and Hall effects). Eg is the energy scale of the pseudogap. This phase diagram is controversial since the Eg line crosses T' and T*. Usually people present either Eg or T' and T*.

The properties of high temperarture superconductors with low and high superfluid density can be described in term of phase fluctuations (i.e. variartions of the "velocity" of the superfluidity) and amplitude fluctuations (i.e. variation of the "pairing density"): based on a pairing mechanism between electrons, the pseudogap regime is described by taking into account amplitude and phase fluctuations of the pairing field.

Amplitude fluctuations of the pairing field

The superconducting field psi has been simulated in 2 dimensions by using a Ginzburg-Landau action. In this figure, the amplitude or absolute value |psi| of the pairing field is reported. Blue is for a high value of |psi| . Note that fluctuations appear to be large and uncorrelated.

Calculations show good quantitative agreement with specific heat and magnetic susceptibility experiments. We find that the mean-field temperature To has a similar doping dependence as the pseudogap temperature T*, moreover the caracteristic pseudogap energy scale Eg is given by the average amplitude above T_phi :

Comparison between theory (variational method) and experimental specific heat on underdoped YBCO_6.73

The peak at Tc and the wide hump between Tc and T* of the specific heat (thick blue) are the sum of the critical phase part (green) and the amplitude contribution (dashed blue).

Comparison between theory and experimental specific heat on YBCO_{6+x} (average value method)

The specific heat (thick blue) is given by C= C_0 + C_1 , where is the amplitude specific heat C_0 (T) depends only on the average amplitude < |psi| > of the pairing field. C_1 (T) contains the phase part of the Ginzburg-Landau (GL) action. (averages are performed by using cluster Monte Carlo simulations of the GL action)

This fits are obtained by using a Monte Carlo procedure in the parameters space, i.e. by varying parameters randomly until the best fit is obtained. With this method, you can find the absolute best fits.
See the fitting animation for doping x=0.73:

(433 Kb)

Comparison between the measured spin susceptibility (points) of underdoped YBCO_{6.64} and theory

The d-wave (thick blue) fits experiments. The average amplitude is shown in blue and its standard deviation as well.
T_phi is the temperature where the amplitude start to be influence from phases. The green dashed line is the amplitude for uncorrelated phases.

In the underdoped regime especially, the averaged amplitude of the pairing field is of the order of the zero temperature gap up to room temperature. Above some crossover temperature T_phi larger than the critical temperature Tc, the pseudogap region is mainly determined by the amplitude whereas phase fluctuations are only important near Tc up to T_phi .

Extracted phase diagram of cuprates.

Extracted phase diagram from YBCO specific heat
To is the mean-field pairing temperature.

Vo is the phase stiffness and is proportionnal to the charge carrier density.

T_phi is the temperature where correlations disappear.

T_1 is the temperature where the phase dominated specific heat contribution C_1 (T) vanishes.
The hatched region is where the pseudogap starts, i.e. where T* is located.

Inset: Eg is the energy scale of the pseudogap, here defined as the average amplitude < |psi| > at 200K.