3.E. Superconductors

For some metals, the resistance drops to zero when the temperature falls below a critical value T C .  The metal is then said to be in the superconducting state.

 

Naively, one may be tempted to associate zero resistance with an infinite conductivity.  However, the superconducting state cannot be so described, as will be shown below.

 

Substituting the Ohm's law for a metal,

        J=σE                              (3.50)

into the Faraday's law

        ×E= 1 c B t         [Gaussian units]              (3.51)

we get

        B t = c σ ×J

Hence,

        B t =0              for σ

i.e., B is time independent inside a metal of infinite conductivity, which leads to hysteresis ( B depends on sample history) as shown in Fig.3.12. 

 

However, the superconducting state was found experimentally to be a thermodynamic state with perfect diamagnetism ( B=0  regardless of sample history).  This is called the Meissner effect.

 

According to the BCS theory, transition to the superconducting phase is a Bose- Einstein condensation, in momentum space, of Cooper pairs.  Here, a Cooper pair is  a boson representing a bounded state of 2 electrons of opposite spins interacting under an attractive effective potential caused by electron- phonon interactions.

 

Thus, the superconducting state is a macroscopic quantum state.  The onset of superconductivity is an order- disorder phase transition with an effective wave function Ψ as order parameter.  Here, n s = | Ψ | 2  gives the density of superconducting electrons.  The vanishing of electrical resistance is due to the ineffectiveness of the scattering offered by individual impurities against the entire electron condensate.

 

Superconductivity can be destroyed by an applied field H> H C ( T )  (see Fig.3.13).  Thus, H C ( T )  defines the coexistence curve in the H-T plane (see Fig.3.14).  It was found empirically that,

        H C ( T ) H 0 ( 1 T 2 T C 2 )                              (3.53)

where H 0 =H( 0 ) .  Note that

        d H C dT = 2 H 0 T T C 2         Þ                 d H C dT 0  

with d H C dT | T=0 =0 .

 

Using

        dg=sdT 1 4π BdH                               (3.57)

we have, along the coexistence curve,

        s n dT 1 4π B n dH= s s dT 1 4π B s dH                    (3.54)

Þ            ( dH dT ) coex =4π ( s n s s B s B n ) coex =4π ( s n s s ) coex H C ( T )              (3.55)

where we've used B s =0  and B n = H C ( T )  on the coexistence curve.

Eq(3.55) is the Clausius- Clapeyron equation for superconductivity.

 

Using (3.53), eq(3.55) becomes

        2 H 0 T T C 2 =4π ( s n s s ) coex H C ( T )

Þ            ( s n s s ) coex = H 0 T 2π T C 2 H C ( T )= H 0 2 2π T C 2 ( 1 T 2 T C 2 )T

so that

        ( c n c s ) coex = [ T ( s n s s ) T ] coex = H 0 2 2π T C 2 T[ 1 T 2 T C 2 2 T 2 T C 2 ]

                        = H 0 2 2π T C ( T T C 3 T 3 T C 3 )                          (3.56)

For sufficiently low T, the cubic term will be smaller than the linear one so that c n > c s  in the coexistence region.  At T= T C , we have

        ( c n c s ) T= T C = H 0 2 π T C

Note also that

        ( s n s s ) T=0 =0

in agreement with the 3rd law.

 

Integrating (3.57) at fixed T gives

        g( T,H )=g( T,0 ) 1 4π 0 H BdH               (3.58)

Using B n =H  and B s =0 , we have

        g n ( T,H )= g n ( T,0 ) 1 8π H 2          (3.59)

        g s ( T,H )= g s ( T,0 )         for H H C              (3.60)

Finally, with the help that, on the coexistence curve,

        g n ( T, H C )= g s ( T, H C )                           (3.61)

we can combine (3.59-60) into

        g s ( T,0 )= g s ( T, H C )= g n ( T, H C )

                        = g n ( T,0 ) 1 8π H C 2          (3.62)

Thus, the condensation energy is

        g n ( T,0 ) g s ( T,0 )= 1 8π [ H C ( T ) ] 2

with

        g n ( T C ,0 ) g s ( T C ,0 )=0