Eschrig - Theory Of Superconductivity, A Primer

The integral on

righ

hand s de is the total hange of the phase of the wavefun tion (7) around

the

ontour, whithe must

be an

integer multiple of 2 sin e the wavefun tion itself must be unique.

Hen e,

I

A + js

~

(16)

dl = q 2 n:

The left hand integral has been namedCthe uxoid by F. London. In the situation of our ring we nd

=

~2 n:

(17)

By dire tly measuring the ux

q

value of the super ondu ting harge was

the absolute

:

measured:

quantumj j = 20e;

=

h

(18)

f the super urren j

along the ontour C is non-zero, then the

is not quantized any more,

(The sign of the ux quantum ma

be de ned

0

2

is the proton harge.)

the uxoid (16), however, is always

tized.

sample

hi h rotates with the

I

order to

termine

sign ofquan

onsiderarbitrarily;

s

super ondu tinguxha ge density qn is

angular velo ity !. Sin e thesample is neutral,

normal urrentdensity jqni side thesample) yieldits now

Ampere's law (in the absenneutralizede of

by the

ha ge

y

of

remainder of the material.

B

B = 0

j qnBv ;

B

r

r

r

onsidering

r

is the super urrent with respe t to the

where v = ! r is the lo l velo ity of the sample, and j

s

rest oordinates. Taking again the url and

leads to

v =

! r

= !

!

r = 3! ! = 2!

2 B = 0

js 2 0qnB

!:

We de ne the London eld

r2

2

r

! =

2mB

!

(19)

BL

2 L 0qnB

q

and onsider the se ond London equation (11) to obtain

2

B B

L

:

(20)

r2 B =

2

Deep i

side a

L

al to the homogeneous

super ondu tor the magneti eld is not zero but q

London eld.

Independenrotatingmeasurements of the ux quantum and the London eld result in

q = 2e;

m

= 2m

:

(21)

B

e

The bosoni eld

is omposed of pairs of ele trons.

11

3

T

THE MODYNAMICS OF THE

PHASE TRANSITION1

y

f

bosoni

. From

Up to here w

super ondu tivity as a

ofexperimenthetra-

w k

ow, that

onsideredered

phenom na are

propertup

o the riti

temp ratur

T

sition,as

rises. The paramet rs of the theory, n

and

, areondensexpeto b

ted temperature

depe

from the super o du ting state, indexedpresenb

,

to the norm

al ondu ting

state,

indexed by

dent:

ust vanish at T .

vi inity

B

L

Intemperaturethis nd the next hapters we onsider the

of the phase transition, T T T .

3.1

B

The Free Energy

t giv

temp

T , pr ssure p, and

Experiments are normally done

eld B. Sin e

to

rst London equa ion

there is no stationary state at E magneti=6 0, w must keep E = 0

a ordingthermodynami equilibrium state. Hen e, wrature

the

(Helmholtz) Free Energy

4)

F

(10)T; V; B);

F (T; V; B);

F

s

F

onsider

= V m;

(25)

= S;

= p;

F

T

V

B

First, the dependen e of Fs on B is

wh re S is the entropy, and m is the

density.

determined from the fa t that in the bulk of

super ondu tor

B

extmagnetization

0

m = 0

(26)

+ B

= B +

as it follows from the se ond London equation (11). Hen e,

V B2

F

V B

Bs

= +

=) Fs(B = Fs(0) +

2

:

(27)

The magneti sus eptibility of a normal (non-magneti ) metal is

0

hen e it may be negle ted here:

0

j 1 = j m;sj;

8)

j

F (B) F

n

(0):

(29)

Eq. (27) implies ( f. (25))

m;n

V B2

Fs

Fs(T; V; 0) +

0

;

(30)

p(T; V; B) =

p(T; V; 0) 2

:

2

The pressure a

exerts on its

B

0

Fsurroundings= B2

redu es in an external eld B: The eld B

implies a for e superareaondu tor

(31)

on the surfa e of the super ondu tor with normal .2 0

1L. D. Landau and E. M. Lifshits, Ele trodynami s of

Continuous Media, Chap. VI, Pergamon, Oxford, 1960.

13

3.2

The Free Enthalpy

y (Gibbs Free Energy) G at B = 0

The relations between the Free Energy F and the Free En

and

B 6= 0 read

Fs(T; V; 0) = Gs(T; p(T; V; 0); 0)thalp(T; V; 0)V

Fs(T; V; 0) +

V B2

Fs( V; B) =

2 0

B); B) p(T; V; B)V =

V B2

= Gs(T; p(T; V; 0)

B2

:

These relations ombine toG

2 0 ; B) p(T; V; 0)V +

2 0

(T; p(T; V; 0); 0) = G

(T; p(T; V; 0)

B2 ; B);

or

s

s

B2

2 0

(32)

In a ord with (31), the e e t of

G (T; p; B) = Gs(T; p + 2

; 0):

external magneti eld B0

n the F Enthalpy is a redu tion of

the pressure exerted on the surrouan

by B2=2

. In the normal statree, from (29),

G

0

(T; p; 0):

(33)

(T; p; B) = G

n

dings,

The riti al temperature T (p; B) is given byB2

; 0) = Gn(T ; p; 0):

(34a)

Likewise B (T; p) from

Gs(T

; p +

2 0

2

Gs(T; p +

B

; 0) = Gn(T; p; 0):

(34b)

B

2 0

B

T

B (T )

T (B)

Figure 6: The

temperature as

fun tion of the applied magneti T eld and the thermodynami

riti al eld as ritifunaltion of temperature.

14

3.3

riti al

states is u ually

, so that

Theat

Free Enthalp di eren e between the normal and

<The(B = 0)

eld

B (T ) for whi h

holds is also small. Taylor

left hand side of

yields

expansion of

2

G (T; p) = G

B

2

G

=

B

(35)

(T;(34)p + riti al

Gsuper(T; )ondu+ tingV(34)T; p; B = 0):

s

2

0

p

is

2

0

Experiment shows that at B = 0

order,

Hen e,

G (T;phase) G (T; p) = a T (p) T

2

:

6)

thermodynami

transition

se ond

B (

(37)

where a is a onstant, and b = p2

p) = b T (p) T

;

a=V .T;

(p) is meant for B = 0.

B (T )

normal state

B

B

B

= 0

P

0

0M

= V Bm 0 G

T B (T )

b(T (p) T )

m = 1

e e tM issner

T (p) T

FIG. 8: The magn tization urve

f a super ondu tor.

FIG. 7: The thermodynami riti al eld.

We onsider all thermodynami parameters T; p; B at the

transition point P of Fig. 7. From

(32),

S

G

B2

; 0);

Ts

= Ss(T;ph+ase 0

G

2

Vs

(T; p; B)

=

= Vs(T; p +

B

; 0):

(38)

Di erentiating (34b) with

ps

2

to T yields, with (38),

0

respeG (tT; p +

2

(T; p)

T

B

; 0) =

Gn

(T; p; 0 or B );

2

T

0

2

Ss(T; p; B ) +

Vs(T; p; B )

2 0

T B (T; p) = Sn(T; p; B );

(39)

S(T; p; B

) = S

(T; p; B

) S

n

(T; p; B

)

=

Vs(T; p; B )B

(T; p) B ( ; p):

s

15

0

T

A ording to (37) this di eren e is non-zero for B

6= 0 (T < T

(p)): For B 6= 0 the phase transition

is rst order with a laten

heat

Q

p; B ):

(40)

= T S(

For T ! 0, Nernst's theorem demands S

s

= S

= 0, and hen e

nB (

T;p)

= 0:

(41)

3.4

Heat

y jump

lim

T

2

T !0

= T

2

yields

For B 0, T T ( ) we an use (35). Applying T

T V (T; p)

2

2

C

apa it

= T

2

=

(42)

= Cp;s Cp;n

T

2

Gs(T; p) G (T; p)

2

T 2 B (T; p):

The thermal expansion V

2

gives a small

tribution

!2

h has been0negle ted. With

B

B

2

B

2

= 2B

on

= 2

whi

2

B

+ 2B

2

we nd

T

T

T

T

= T

!

#

TV

"

B

2B

Cp =

2

(43)

For T ! T

(p), B

0

T

+ B

T 2

! 0 the jump in the

heat is

T V

B

!2

T

V

Cp

spe i

=

b2

:

(44)

=

0

T

0

It is given by the slope of B (T ) at T (p).

Cp