Statistical Physics, Second Revised and Enlarged Edition

Superfluid states in Fermi–Dirac systems 171

Temperature

0 TTTc

0

Entropy ntro S

Heat capacitycapa C

Heat capac

ity

C

an

d entropy

S

Normal statetate

S, C =γγTTT

Fig. 15. 1 The normal-to-superconductingphase transition in zero applied magnetic field. Thegraphs show
the dependence on temperature of the electronic heat capacityCand entropyS. Note the increased order
(decreasedS)of the superconductor compared to the normal state at the same temperature. The curves are
related byC=T(dS/dT).

Cooper and Schrieffer (BCS) in 19 5 7) to lower the energyof a coherentground
state. The energy gapis the minimum energy per electron needed to break one of
these pairs. Theexp(−/kkkBT)form ofthethermalproperties atlow temperatures
follows from the basic statistical formulae (actuallyfrom the high energytail of the
FD distribution function) when it is appreciated that there is this energy gap between
thegroundstate (superfluid)andthe excitedstates (normalfluid).
The electrons in the coherentground state have fascinating, somewhat alarming,
properties. They are described by a single wave function (analogous to the boson
groundstate whichdescribes allparticlesintheBEgas). This resultsinthree classes of
remarkableproperty: (i)theelectricalresistanceiszero–comparetheviscosityin^4 He;
(ii) magneticB-fields are excludedfrom the metal,shieldedoutby surface persistent
currents; (iii)there are quantizedstates, correspondingto magneticfluxlinkedin
units ofh/ 2 e,around a loop in the superconductor. In view of the second property, it
is not surprising that the superconducting stateis suppressedbyhighenoughapplied
magneticfields, andthe use ofsuperconductors to producehigh-field, stableandloss-
free magnets is therefore a difficult problem and involves special materials. The third
property gives thebasisfor thewholeideas of‘SQUIDs’ (superconducting quantum
interferencedevices), the most sensitive electro-magnetic measuringinstrumentin