Marder M.P. Condensed Matter Physics

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Phenomenology of Superconductivity 843

Taking

the

curl

Eq. (27.20)

ensure that

the

final results

are

gauge-invariant

gives

A£

VxVxß

= 0,

(27.21)

recovering Eq. (27.5).

Equation (27.21)

is not

quite

general

Eq. (27.5) because

the

first involves

B while

the

second involves

B/ß. A

magnetic permeability different from unity

can

introduced

if one

takes

ß to be the sum of

two terms,

one of the

form

(27.14)

and the

second

the form (27.18).

The

physical interpretation

adding

these two terms together

that some electrons

in a

solid, such

as the

valence elec-

trons,

are

confined

individual atoms

and

create

magnetic response

ß.

Another

population

electrons creates superconductivity. Adding together these

two re-

sponses continues

produce

superconductor,

but the

penetration depth changes

and

given

Eq. (27.6).

27.2.2 Thermodynamics

Superconductors

Suppose

one

takes

long thin cylinder

superconducting material

and

places

external magnetic field

strength

that points along

the

cylinder axis.

The

long cylindrical shape

desirable because its demagnetizing factor is zero,

and the

normal component

the magnetic field automatically vanishes along most

of its

surface. Experimentally, Meissner

and

Ochsenfeld (1933) found that

at a

critical

applied field

the

superconductivity

destroyed

and

magnetic flux penetrates

the sample.

The

transition

completely reversible,

right

this critical field

the

superconducting state

and

normal state

the material must be

equilibrium.

The

free energy

per

volume

the metal

in the

normal state must be just

^normal

+ Ö ^c

•

(27.22)

ÖTTß

Because

the

superconducting state expels

the

magnetic induction,

its

free energy

JUSt Jsuperconducting-

There

now

appears

to be a

problem.

The

free energy

of the

normal state

greater than that

the superconducting one,

and the

magnetic field energy

also

positive.

Yet

when

the

critical field

applied, (27.22)

supposed

equal

^superconducting- What

has

gone wrong?

The

answer

that thermodynamics

is be-

ing used improperly.

The

reversible experiments

are

carried

out by

slowly varying

external currents,

not by

controlling

directly.

In the

instant that the normal metal

expels

B and

becomes superconducting,

host

magnetic flux lines blasts

off to

infinity. The work done during this instant must be accounted for; equivalently,

one

must choose

work with thermodynamic variables that really

vary smoothly

during

the

change from normal metal

superconductor.

The

correct thermody-

namic potential,

as in Eq.

(24.27),

= ?-B-% = ?

nonnal

]

-B

(27.23)

844

Chapter 27. Superconductivity

Therefore the difference in free energy per volume at any given temperature be-

tween superconducting and normal metal must be exactly

A 5 = Sn,

normal -^superconducting

87T//

(27.24)

Because \i

for superconducting metals, Eq. (27.24) is often written

AJ=#.

8TT

(27.25)

One should remember, as mentioned in Section

24.2.3,

that the magnetic field H

inside a long thin cylinder with axis along the field is always spatially uniform and

equal to the externally applied field. It is magnetic induction B that is expelled

from superconducting samples, not the magnetic field H.

By differentiating Eq. (27.25) with respect to temperature, one finds immedi-

ately that the excess entropy of normal metal relative to the superconductor must

AS=—

AJ:

4^W

(27.26)

Meissner and Ochsenfeld determined that as temperature rises towards the critical

temperature T

where superconductivity vanishes in zero magnetic field, the critical

field H

also goes to zero. It follows immediately from Eq. (27.26) that the latent

heat of transformation is zero, and therefore the transition is second order.

27.2.3 Landau-Ginzburg Free Energy

The most productive phenomenological description of superconductivity combines

ideas from the free energy (27.10) with the idea that the superconductivity resides

in a population of superconducting electrons whose density vanishes at a critical

temperature T

. Landau and Ginzburg (1950) described the superconducting state

through a macroscopic wave function

that plays a role for superconductivity

identical to the role played by the wave function

<I>

employed in Eq. (15.98) for su-

perfluid helium. If the supercurrent density n = \^\

vanishes at T

, then it should

be possible to expand G

ß (see Eq. (27.10)) in powers of l^l

. Landau and Ginz-

burg guessed the free energy to be

ßi

2m*

i c

*(r)

(27.27)

The first two terms in Eq. (27.27) are procured directly from Landau's general

theory of second-order phase transformations, discussed in Section

24.6.1.

The

remaining terms provide a particular realization of Eq. (27.10) and were inspired

by the idea that the superconducting electron wave function might interact with

the vector potential like a single macroscopic particle. The effective charge e* =

Phenomenology of Superconductivity

845

2e multiplying A is due to the fact that pairs of electrons are responsible for the

superconductivity. Minimizing Eq. (27.27) with respect to A leads to

Vxß= —/, (27.28)

with

2eh - - 4<?

j(r) =

[**V*

- #V#*1 A***. (27.29a)

2im*

m*c

Minimizing with respect to

\I>*

leads to

i(»vA-)

Integrate so that all spatial derivatives

xjj act on

SP,

then take functional iy-i 9QM

derivative with respect to \P*. ^ ' '

Equations (27.29) are the Landau-Ginzburg equations.

Boundary Condition. To solve problems in particular geometries, it is always

necessary to specify the behavior of * at the boundary of a sample. When a super-

conductor contacts vacuum no current can flow out of the boundary, so

/H^ 2e ~\

•

—

A I ty = 0

Here

" is normal to the boundary; see Lan- (27 30)

\i c ' ' dau and Ginzburg (1950), de Gennes

(

1992),

'Via. 2e

->\

W =

177, or Wertheim (1969), p. 327.

Landau and Ginzburg observe that "it is natural to demand that the wave function

at the boundary of

the

metal should vanish" but such an additional boundary condi-

tion on the superconducting wave function would eliminate solutions in thin films

except for quantized values of the film thickness. Because this phenomenon is not

observed, the seemingly natural boundary condition \I/ = 0 must be rejected.

27.2.4 Type I and Type II Superconductors

The Landau-Ginzburg equations depend upon two lengths. The first is a coherence

length £ that sets the scale for spatial variations of the order parameter ^/. This

length is given by

2m*\a\

The second length is the London penetration depth. In terms of Landau and Ginz-

burg's parameters it is given by

= T5^Ü- C"2>

47r|a|(2e)

The coherence length and penetration depth, as well as critical temperature and

lower critical field are tabulated in Table 27.1.

846

Chapter 27. Superconductivity

Table 27.1. Properties of superconductors

Compound

As (P =14 GPa)

Ba (P =20 GPa)

Bi (P =8 GPa)

Ce (P =5 GPa)

Cs(P=13GPa)

P(P=17GPa)

Se (P =13 GPa)

Si (P =12 GPa)

Te (P = 8 GPa)

Y (P =17 GPa)

YBa

HgBa

(K)

1.18

0.5

5.3

0.02

8.55

0.52

1.7

1.6

1.09

0.02

3.95

3.41

0.10

6.0

0.1

0.0005

0.92

9.3

5.8

7.20

0.49

6.9

7.1

3.7

4.46

7.8

4.3

1.37

0.42

2.4

1.8

0.02

2.7

0.85

0.53

18.5

135

(G)

105

58.9

340

289

20.1

1096

1980

803

308

831

1410

162

180

1.07

500

(Â)

000-16000

7600

2400-3 500

380

510-960

1000-3 000

4200

4-8

(Â)

160-500

1 100

380-450

390-640

390

390-630

340-750

1600

900-8 000

Critical temperature T

, lower critical magnetic field H

where flux first

penetrates sample at zero temperature, coherence length £ and London

penetration depth \

for selected superconductors. Measurements are at

atmospheric pressure unless otherwise indicated. Source: Grigoriev and

Meilkhov (1997), pp. 566-567 and Plakida (1995).

Phenomenology of Superconductivity

847

Calculation of Coherence Length £. These lengths emerge from some simple

calculations based upon Eq. (27.29b). First, consider a case in which externally

imposed magnetic fields vanish. One sees immediately from Eq. (27.29b) that the

spatially uniform solutions are

|*|

= I P

(27.33)

The constant ß must always be positive or else

is minimized by sending

oo.

The nonzero case in Eq. (27.33) is only possible if a < 0, and it corresponds to a

spatially uniform superconducting state. The free energy per volume of this state,

from Eq. (27.27), is

v—v

and comparing with Eq. (27.25) shows that

«? - *f. (2735,

When superconductivity is present, it makes sense to scale \& by this uniform

value \I/o and define

= l

, (27.36)

in terms of which Eq. (27.29b) becomes

-£

V>-V> + V#|

= 0,

WhenA = 0.

(27.37)

£ being given by Eq. (27.31). Thus £ is the characteristic scale on which tp varies

in the absence of magnetic fields.

One-Dimensional Interface. The physical significance of £ is illustrated by ex-

amining the scale over which the wave function ip varies. Take tp to be real, and

consider a one-dimensional interface. In one dimension one has

-ÊV-V' + ^

=0- (27.38)

Multiplying Eq. (27.38) by tp' and integrating, one has the first integral

-Ç

(ip')

-ip

+ -ip

= Const (27.39)

Equation (27.39) is only consistent with an asymptotic value of

if one takes

the right hand side constant to be

— 1

/2.

In this case, one can then write

T// = -L-(1-V

) (27.40)

which has a solution

?/; = tanh-^. (27.41)

v2£

This solution describes a transition between two spatially uniform superconducting

regions where the phase of

the

wave function changes sign across the interface. The

coherence length £ sets the spatial scale for this transition.

848

Chapter 27. Superconductivity

Penetration Depth.

The

second length,

the

penetration depth, arises when

one

considers

superconductor occupying

the

region, say,

x < 0, in an

exceedingly

weak field. When

the

field

zero,

one has the

solution

* =

\E»o

for x < 0. In

the presence

of a

very weak field,

will change

linear order

in the

field,

but

because

it is

nonzero without

field,

leading order

one can

continue

to use

\I>

*o- Looking

Eq. (27.29a) and keeping only the leading terms gives

- - 4e

-,-

j=— Vxß = -mlA.

(27.42)

47T m*c

Taking the cross product

Eq. (27.42) gives

VxVxß= *gß

-Ar

ß. (27.43)

m*c

Using Eq. (27.33) immediately gives Eq. (27.32)

for

the penetration depth.

The ratio

the penetration depth and coherence length

*LlS

= -r\iz

(27-44)

fact the only free parameter

the Landau-Ginzburg theory, because all other

constants can

scaled out,

rescaling ip, distance, and

specific, define

(27.45)

C\j2nï

\a\

and measure

all

distances

units

£. Then Eq. (27.29b) becomes

ip-ip\tp\

-(-iV

ä/2)

= 0

(27.46)

while Eq. (27.29a) becomes

-f V x V x a =

(^*

V^ - ^Vf) - |Vfa. (27.47)

s '

Surface Energies. A particularly important result of Landau and Ginzburg's

theory is the calculation of

the

surface energy between normal and superconducting

metals. For K < \j\fl the energy is positive; however, if

> l/\/2, the surface

energy is negative. Once one has a negative surface energy, it becomes favorable

for magnetic flux to penetrate a superconductor and form interlocking regions of

normal and superconducting metal. Superconductors for which this happens are

called type II; those for which it does not are called type I. The way in which the

flux penetrates was worked out by Abrikosov (1957).

In order to discover why the magnitude of K plays such an important role,

consider a superconductor in a magnetic field larger than H

, so that the super-

conductivity is destroyed and ^ = 0. Now begin to lower the field, and examine

Phenomenology of Superconductivity

849

the linear stability of the state \I> = 0 by expanding everything to first order in *.

From Eq. (27.29a), one sees that the current j vanishes to this order, and therefore

one can take A simply to be the vector potential associated with a uniform field H,

pointing, say, along the z axis. Writing Eq. (27.29b) to linear order then gives

' -jfiV +

—)

-Q*.

(27.48)

2m*

This equation is the same as the eigenvalue equation for an electron in a constant

external magnetic field, and so one can immediately write down the solution by

comparison with Eq. (25.48). Define

to be the largest magnetic field permitting

a nonzero solution of Eq. (27.48). The lowest-energy eigenvalue is hco

/2, where

in this case

"c = ^, (27.49)

m*c

because in Eq. (27.48) the charge carrier has charge — 2e. The larger H becomes,

the larger a would need to be in order for a solution to be possible. For a given a,

the largest H at which Eq. (27.48) has nonzero solution is therefore

—a = \a\ = -. (27.50)

m*c

Using Eq. (27.35) to express the critical field H

in terms of a and ß, along with

Eq. (27.44) to define

gives

-^ = V2K. (27.51)

Now one can see why

= \/\/2 divides the two types of superconductors. If

> l/\/2, then

lies above H

. For applied magnetic fields between H

and

, it is energetically unfavorable to expel all magnetic flux from the system, but

favorable to form at least one superconducting vortex, corresponding to solutions

of Eq. (27.48). Equation (27.48) predicts that a vortex will form, but is unable to

describe its strength or final form; these can only be obtained from more elaborate

calculations taking into account the nonlinear terms of Eq. (27.47). As the mag-

netic field descends below

toward H

, more and more vortices fill the system,

until at H

they coalesce and expel all the magnetic flux. By contrast, if

< l/\/2,

forming a vortex does not become possible until the magnetic field has dropped

below H

. It is reasonable to guess, and possible to verify, that the system does

not choose to form vortices when it has the option of expelling all the magnetic

field and to become superconducting instead. The picture that led to Eq. (27.48),

in which a normal metal in a magnetic field develops a superconducting wave func-

tion to order

\&,

is no longer valid. It is more favorable for

to grow to size —a/ß

and take over the whole sample.

Interface Energies. Problem 2 describes the free energies of superconducting-

metal interfaces in two limits. For

= 0,

hf^l

(27

850

Chapter 27. Superconductivity

and forming the interface costs a finite positive energy per unit area

For K

—>

oo,

Ç H

7 = -^ (27.53)

87T

because this energy is negative, the system will respond by trying to form as much

interface as it can. The result is the Abrikosov lattice of flux lines penetrating the

sample depicted in Figure 27.2.

Figure 27.2. A Type II superconductor is unstable to the formation of flux tubes that

penetrate the sample trying to generate a maximal area where superconductor and metal are

in contact. The lowest free energy configuration is often a triangular lattice. (A) Electron

hologram from the side of magnetic flux entering a lead film [Source: Tonomura et al.

(1986),

p. 93.] (B) Top view of an Abrikosov lattice of flux tubes in NbSe2, taken with

scanning tunneling microscopy. [Courtesy of S. Pan and A. de Lozanne, University of

Texas.1

27.2.5 Flux Quantization

The Landau-Ginzburg equations have an effective charge — 2e seated in the spot

one would expect to be occupied by the electron charge —e. This effective charge

has a convoluted history. Ginzburg had guessed that the superconducting charge

carriers might have an effective charge e*. Landau objected that effective charges

violate gauge invariance, and their paper takes the position that "e is a charge,

which there is no reason to consider as different from the electronic charge." In

fact, arguments based upon gauge invariance require only that charges be integral

multiples of the electron charge; in two-dimensional space these arguments permit

effective particles with arbitrary charge, as shown in Figure 25.11 and discussed by

Wilczek (1990). Despite the fact that the microscopic theory of superconductivity

involved pairing of electrons, the correct form of the Landau-Ginzburg equations

was not clear, and the effective charge —2e was a surprise when discovered exper-

imentally by Deaver and Fairbank (1961) and Doll and Näbauer (1961) in studies

of quantized magnetic fluxes.

The Landau-Ginzburg equations predict that the magnetic flux trapped within

a superconductor is quantized (Figure 27.3). Suppose, however, that the effective

Phenomenology of Superconductivity 851

charge — e* and mass m* of the superconducting particles were not known. The

superconducting current associated with \I> would be

Letting

gives

e*H - - e*

lim*

m*c

M?)

\&(f) = fy

P(r>

Take

<j>

to be real.

-v<t>=-

e*\

A +

e*H\7(f>

m -f e ->

e*^>l

(27.54)

(27.55)

(27.56)

(27.57)

Simply demanding that the phase

</>

be differentiable is enough to prove that the

magnetic flux is quantized, just as in Eq. (15.106). Because \I/ must return to itself

when

<j>

changes through 27r, one obtains

ds- V(j> = 2nl.

l is an

integer.

As a result,

ds-

m -> e ->

e*^l c

2vr/.

(27.58)

(27.59)

Choosing a contour of integration that lies within the cylinder at a depth greater

than the skin depth (Figure 27.3), j is negligible, and

r B

= $

ds-A = 2-nl (27.60)

2irlhc e

= / — $

. See definition of $

in Eq. (25.51). (27.61)

e* e*

Figure 27.3. Magnetic flux that pierces a su-

perconducting ring is quantized in units of

$o/2.

The integration contour in Eq. (27.59)

is taken on the dotted line, which lies deep in

the superconductor where j vanishes.

So the integrated magnetic flux $ that penetrates a hole through a supercon-

ductor is quantized in units of (e/e*)&o, and can be used to measure the effective

charge e*. The measurements in Figure 27.4 show that e* — 2e, leading to the con-

ventional form of Eq. (27.27). The fact that the only phase changes observed in

Figure 25.5 are 0 and ir is another experimental demonstration of this same fact.

852

Chapter

27.

Superconductivity

0.1 0.2 0.3 0.4

Magnetic field H (G)

0.5

Figure 27.4. Trapped magnetic flux in a superconducting cylinder as a function of applied

field. The dimensions of the cylinder are on the order of 10 /im, and it is made of tin. The

flux is measured by moving the cylinder up and down at 100 Hz toward and away from a

conducting coil and measuring the electromotive force induced in the coil. [Source: Deaver

and Fairbank

(1961

p. 44; a nearly identical experiment was published simultaneously by

Doll and Näbauer (1961). Theoretical discussions by Byers and Yang and Onsager are

sandwiched between these two experimental papers.]

27.2.6 The Josephson Effect

It takes a certain degree of courage to adopt a phenomenological function like \l/,

treat it as a physical entity, and predict new phenomena. Josephson (1962) pre-

dicted that the wave function could be induced to interfere with itself and oscil-

late if two superconductors were separated by a small strip of nonsuperconducting

material. A qualitative derivation of Josephson's equations is presented at first;

Section 27.2.9 obtains the basic results in a more careful way.

Consider two superconductors separated by a thin insulating layer. On the

opposite sides of this weak

link,

the two superconductors interact only because of

their exponentially small residues as they decay across the barrier into each others

territory. The change in energy of two nearly independent systems due to tunneling

is given in general by

I drU{r) (^(r)*

(r) + ^

(r)^(r))

(27.62)

(27.63)

where the wave function without the argument r refers to its value at some reference

point inside the bulk, and e is some very small quantity with dimensions of energy.

Writing down a Schrodinger equation for two wave functions coupled by such an

interaction leads to

0*1

[£i*i+e#

]

0^7 -i

-2.

[£A

*,,

(27.64a)

(27.64b)