quasiparticles is not conserved.) Although the terms in mk mk 0 in Eq. (7.5.2)

represent interactions among the fermions, the de¬nition of the quasiparticle

energy adopted in Eq. (7.5.3) allows us to treat them as independent. We can

then obtain an approximate solution of Eq. (7.5.1) by replacing mk 0 and mÀk 0

by their thermal averages. If we abbreviate the Fermi function of Eq. (7.5.4)

by f °Ek Þ then we ¬nd that to satisfy Eq. (7.5.1) we must put

X

^

2E k uk vk À °u2 À v2 Þ Vkk 0 uk 0 vk 0 ½1 À 2f °Ek 0 Þ ¼ 0:

k k

0

k

249

7.5 The transition temperature

The only di¬erence between this equation and Eq. (7.3.8) lies in the extra

factor of 1 À 2f °Ek 0 Þ that multiplies the matrix element Vkk 0 . Consequently if

we again make the substitution (7.3.9), but this time replace the de¬nition of

Ák given in Eq. (7.3.11) by the de¬nition

X

Ák °TÞ ¼ Vkk 0 °1 À x2 0 Þ1=2 ½1 À 2f °Ek 0 Þ °7:5:5Þ

k

4

0

k

we regain Eq. (7.3.12). The temperature-dependent gap parameter Á°TÞ is

then found by resubstituting Eq. (7.3.12) in Eq. (7.5.5). One has

1X Ák 0 °TÞ

Ák °TÞ ¼ ½1 À 2f °Ek 0 Þ: °7:5:6Þ

Vkk 0 2

^

2 k0 ½E k 0 þ Ák 0 °TÞ 1=2

2

This equation still contains the excitation energy Ek which we evaluate by

taking the thermal average of Eq. (7.5.3). We ¬nd

X

^k

Ek ¼ E k °u2 À v2 Þ þ 2uk vk Vkk 0 uk 0 vk 0 ½1 À 2f °Ek 0 Þ

k

k0

^2

¼ ½E k þ Á2 °TÞ1=2 ; °7:5:7Þ

k

which is identical with our previous expression (7.4.5) except that Á is now a

function of temperature. On substituting for Ek 0 and f °Ek 0 Þ in Eq. (7.5.6) we

¬nd

^2

1X Ák 0 °E k 0 þ Á2 0 Þ1=2

Ák ¼ : °7:5:8Þ

k

tanh

Vkk 0 2

^

°E k 0 þ Ák 0 Þ

2 1=2

2 k0 2kT

In the simple model de¬ned by Eq. (7.3.15) this reduces to

° 0!D ^

tanh½°E 2 þ Á2 Þ1=2 =2kT ^

d E ¼ 1: °7:5:9Þ

VD°Þ

^

°E 2 þ Á2 Þ1=2

0

At zero temperature this equation for Á reduces to our previous solution

(7.3.16). When the temperature is raised above zero the numerator of the

integrand is reduced, and so in order for Eq. (7.5.9) to be satis¬ed the

denominator must also decrease. This implies that Á is a monotonically

decreasing function of T; in fact it has the form shown qualitatively in

Fig. 7.5.1. The initial decrease is exponentially slow until kT becomes of the

order of Á°0Þ and the quasiparticle excitations become plentiful; Á°TÞ then

250 Superconductivity

Figure 7.5.1. The energy gap parameter Á decreases as the temperature is raised

from zero, and vanishes at the transition temperature Tc .

begins to drop more rapidly until at the transition temperature Tc it vanishes.

The magnitude of Tc in the BCS model is found from Eq. (7.5.9) by putting

Á°Tc Þ ¼ 0. We then have

° 0!D =2kTc

xÀ1 tanh x dx ¼ 1

VD°Þ

0

or

° 0!D =2kTc

1

0! =2kTc

½ln x tanh x0 D À sech2 x ln x dx ¼ :

VD°Þ

0

For weak-coupling superconductors we can replace tanh °0!D =2kTc Þ by

unity and extend the upper limit of the integral to in¬nity to ¬nd

°1

0!D 1

À sech2 x ln x dx ¼ :

ln

2kTc VD°Þ

0

The integral is more easily looked up than evaluated, but either way is equal

to ln 0:44, from which

kTc ¼ 1:140!D eÀ1=VD°Þ : °7:5:10Þ

Comparison with Eq. (7.3.17) shows that in this model

2Á°0Þ

¼ 3:50;

kTc

251

7.5 The transition temperature

a result in adequate agreement with the experimentally observed values of

this parameter, which for most elements lie between two and ¬ve.

The existence of an isotope e¬ect is an obvious consequence of Eq. (7.5.10).

The simplest form of isotope e¬ect occurs when VD°Þ is independent of the

ionic mass A, as then Tc depends on A only through the Debye energy 0!D ,

which is proportional to AÀ1=2 . The fact that the electron“phonon enhance-

ment of D°Þ is independent of A has already been considered in Problem 6.1,

and the demonstration that V should also have this property follows similar

lines. One should, however, note that Tc is very sensitive to changes in the

density of states D°Þ as a consequence of the fact that VD°Þ ( 1. Thus

if VD°Þ ¼ 1 then from Eq. (7.5.10) one sees that a one percent decrease in

8

D°Þ will cause an eight percent decrease in Tc . This makes it not very

surprising that inclusion of the Coulomb repulsion in V or the use of other

more complicated models can lead to di¬erent kinds of isotope e¬ect.

The electronic speci¬c heat C may now also be calculated for the BCS

model. The energy E°TÞ of the superconductor at temperature T will be

the average expectation value of the Hamiltonian (7.5.2). We ¬nd

X

^2 ^

2 2

E°TÞ ¼ ½2E k v k þ °u k À vk Þ2fk E k À °1 À 2fk Þuk vk Á

k

X

Á2

^

¼ E k À Ek À °1 À 2fk Þ

2Ek

k

from which

dE°TÞ

C¼

dT

X

Á2 df 1 2 d°1=EÞ

¼ 2 EÀ þ °1 À 2f ÞÁ

2E dT 2 dT

k

X

1 À 2f

df 2d

¼ þÁ

2E

2E

dT dT

k

X dÁ2 2E 2 @f

¼ À : °7:5:11Þ

@E

dT T

k

Reasonably good agreement with experiment is usually obtained with this

formula. The observed discontinuity in C at the transition temperature arises

from the term dÁ2 =dT, which is zero for T > Tc but ¬nite for T < Tc . The

prediction of Eq. (7.5.11) is that C increases by a factor of 2.43 as the sample is

cooled through Tc ; observed increases are within a factor of four of this value.

252 Superconductivity

7.6 Ultrasonic attenuation

In Fig. 7.1.1 the damping coe¬cient for low-frequency sound waves in a

superconductor was shown as a function of temperature. The rapid decrease of

as the sample is cooled below Tc suggests that the attenuation of an ultrasonic

wave depends on the presence of quasiparticle excitations in the material. We

can see why this should be so when we recall that the frequencies used in such

ultrasonic experiments are typically less than 100 MHz, so that each phonon

has an energy of less than 10À6 eV. Unless T is very close to Tc this phonon

energy will be much less than 2Á, which is the minimum energy required to

create a pair of quasiparticles. The phonons can then only be absorbed by

scattering already existing quasiparticles from one k-state to another.

The canonical transformation that we adopted in Section 6.5 allowed us to

ignore the phonon system in calculating the BCS ground state; we had removed

the ¬rst-order e¬ects of the electron“phonon interaction and we assumed the

transformed phonon system just to form a passive background that did not

a¬ect our calculations. In the theory of ultrasonic attenuation, however, the

occupation number nq of the applied sound wave must be allowed to change as

the wave is damped, and so we must exclude this particular phonon mode q

from the canonical transformation that leads to Eq. (6.5.4). We are con-

sequently left with a Hamiltonian that still contains the set of terms Hq and

HÀq where

X y

Mq c kþq;s ck;s °ay þ aq Þ:

Hq ¼ Àq

k;s

The annihilation operator aq will reduce the occupation number of the phonon

mode q by unity, and thus e¬ect the damping of the applied sound wave. The

e¬ect of this process on the electron system is seen by transforming from the

electron operators c to the quasiparticle operators

by means of Eqs. (7.3.3)

and (7.3.4). In the notation in which k k" and Àk Àk# this gives us

Xy

Hq ¼ Mq °ay þ aq Þ °c kþq ck þ cy cÀ°kþqÞ Þ

Àq Àk

k

X y

¼ Mq °ay þ aq Þ ½°ukþq

kþq þ vkþq

À°kþqÞ Þ°uk

k þ vk

y Þ

Àq Àk

k

y

þ °uk

y À vk

k Þ°ukþq

À°kþqÞ À vkþq

kþq Þ

Àk

X y

y

½°ukþq uk À vkþq vk Þ°

kþq

k þ

y

À°kþqÞ Þ

¼ Mq °aÀq þ aq Þ Àk

k

y

þ °ukþq vk þ uk vkþq Þ°

kþq

y þ

À°kþqÞ

k Þ: °7:6:1Þ

Àk

253

7.6 Ultrasonic attenuation

y

The terms in

kþq

y and

À°kþqÞ

k lead to the simultaneous creation or

Àk

destruction of two quasiparticles and, as we have seen, cannot represent

energy-conserving processes when 0!q is negligible. We are thus left

y

with the terms

kþq

k and

y

À°kþqÞ , which can lead to energy-conserving

Àk

processes if either Ekþq ¼ Ek þ 0!q or EÀk ¼ EÀ°kþqÞ þ 0!q . For such pro-

cesses to be allowed, the quasiparticle states k and À°k þ qÞ must initially

be ¬lled and the ¬nal states °k þ qÞ and Àk must be empty. We then ¬nd

the total probability Pa of a phonon of wavenumber q being absorbed to be

given by

X

Pa / °ukþq uk À vkþq vk Þ2 ½ fk °1 À fkþq Þ °Ekþq À Ek À 0!q Þ

k

þ fÀ°kþqÞ °1 À fÀk Þ °EÀk À EÀ°kþqÞ À 0!q Þ:

The set of terms HÀq is similarly calculated to give a certain probability Pe of

a phonon of wavenumber q being emitted when a quasiparticle is scattered

from k to k À q. We assume this phonon mode to be macroscopically occu-

pied, so that we can neglect the di¬erence between nq and nq þ 1. We then

¬nd the net probability P that a phonon is absorbed to be

P ¼ Pa À Pe

X2

/ °uk À v2 Þ2 ½° fk À fkþq Þ °Ekþq À Ek À 0!q Þ

k

k

þ ° fÀ°kþqÞ À fÀk Þ °EÀk À EÀ°kþqÞ À 0!q Þ; °7:6:2Þ

where °ukþq uk À vkþq vk Þ2 has been replaced by °u2 À v2 Þ2 , the justi¬cation

k k

being that q is small while in the present context uk and vk are slowly varying

functions. We similarly make the approximation

@f

fk À fkþq ¼ Àq Á

@k

dE dE @f

¼ Àq Á

dk dE @E

^

E @f

¼ Àq0vk cos ;

E @E

with the angle between q and k. On changing the sum in Eq. (7.6.2)

to an integral over energy and solid angle and substituting for °u2 À v2 Þ2

k k

254 Superconductivity

we ¬nd

° ^ 2

^ ^

E E @f E ^

P / D°Þ q0vk cos q0vk cos À 0!q d°cos Þ d E

E @E

E E

° ^ 2

0!q E

E @f ^

¼ D°Þ j cos j cos À d°cos Þ d E

^

@E q0vk E

E

°1

!q @f

™ À2D°Þ dE

Á qvk @E

/ f °ÁÞ:

The ratio of the damping s in the superconducting state to that in the

normal state is thus approximately

s f °ÁÞ

¼

n f °0Þ

2

¼ ;

exp °Á=kTÞ þ 1

in good agreement with experiment.

7.7 The Meissner effect

In the Meissner e¬ect the vanishing of the magnetic ¬eld inside a bulk super-

conductor must be attributed to the existence of an electric current ¬‚owing in

the surface of the sample; the magnetic ¬eld due to this current must exactly

cancel the applied ¬eld H0 . We investigate this phenomenon within the BCS

model by applying a weak magnetic ¬eld to a superconductor and calculating

the resulting current density j to ¬rst order in the applied ¬eld. We avoid the

di¬culties of handling surface e¬ects by applying a spatially varying mag-

netic ¬eld de¬ned by a vector potential A ¼ Aq eiq Á r and examining the

response j ¼ jq eiq Á r in the limit that q becomes very small.

It had been realized for many years before the BCS theory that any ¬rst-

order response of j to the vector potential A would lead to a Meissner e¬ect.

If we de¬ne the constant of proportionality between j and A so that

c

j¼À °7:7:1Þ

A;

42

255

7.7 The Meissner effect

with c the speed of light, and then take the curl of this relation we obtain the

London equation,

c

r‚j¼À B:

42

The use of the Maxwell equations

4

r‚H¼ rÁB ¼ 0

j;

c

and the fact that B ¼ H when all magnetization is attributed to the current j

gives us the equation

r2 B ¼ À2 B:

The solutions of this equation show a magnetic ¬eld that decays exponen-

tially with a characteristic length , which can thus be identi¬ed with the

penetration depth discussed in Section 7.1. In a normal metal would be

in¬nite, and so from Eq. (7.7.1) we should expect to ¬nd no ¬rst-order term

in an expansion of j in powers of A.

We now proceed to perform the calculation of jq in the BCS model. As we

saw in Section 3.10, a magnetic ¬eld enters the Hamiltonian of a single

electron as a perturbation term

e e2

H1 ¼ A À pÁA À AÁp ;

2mc c

relativistic e¬ects and the spin of the electron being neglected. We drop the

term in A2 and write H1 in the notation of second quantization as

eX y

hkj À p Á A À A Á pjk 0 ick;s ck 0 ;s

H1 ¼

2mc k;k 0 ;s

e0 X y

hkj À A Á °k þ k 0 Þjk 0 ick;s ck 0 ;s :

¼

2mc k;k 0 ;s

When the vector potential is of the form Aq eiq Á r we have

X y

H1 ¼ ÀB Aq Á °2k À qÞck;s ckÀq;s °7:7:2Þ

k;s

256 Superconductivity

with B ¼ e0=2mc. The current density ev= is found from the continuity

equation

@

À1 r Á v ¼ À ;

@t

of which the Fourier transform is

@q 1

iÀ1 q Á vq ¼ À ¼ ½H; q :

@t i0

We thus ¬nd

e

q Á jq ¼ À ½°H0 þ H1 Þ; q :

0

The commutator of the zero-¬eld Hamiltonian H0 with q was evaluated in

Section 2.7, and ½H1 ; q can be similarly calculated. The result is that

X e0

e2 Aq y

y

jq ¼ °2k À qÞckÀq;s ck;s À cc : °7:7:3Þ

mc k;s k;s

2m

k;s

The e¬ect of the magnetic ¬eld is to perturb the wavefunction of the super-

conductor from its initial state jÉ0 i to the new state jÉi that is given to ¬rst

order in H1 by the usual prescription

1

jÉi ¼ jÉ0 i þ H jÉ i:

E 0 À H0 1 0

The current in the presence of the applied magnetic ¬eld will then be

1

h jq i ¼ hÉ0 j jq jÉ0 i þ hÉ0 j jq H jÉ i

E 0 À H0 1 0

1

þ hÉ0 jH1 j jÉ i:

E 0 À H0 q 0

We substitute in this expression for jq and H1 from Eqs. (7.7.2) and (7.7.3)

257

7.7 The Meissner effect

and drop terms of order A2 to ¬nd

q

X e0B Aq

Ne2 Aq

Á °2k 0 À qÞ

h jq i ¼ À À hÉ0 j

2m

mc k;s;k 0 ;s 0

1

y y y

‚ °2k À qÞ c kÀq;s ck;s ck 0 ;s 0 ck 0 Àq;s 0 þ c k 0 ;s 0 ck 0 Àq;s 0

E 0 À H0

1 y

‚ jÉ0 i: °7:7:4Þ

c c

E 0 À H0 kÀq;s k;s

In this expression the operator H0 is the BCS Hamiltonian in the absence of

the magnetic ¬eld. To evaluate h jq i it is then necessary to express the electron

operators in terms of the quasiparticle operators that change jÉ0 i into the

other eigenstates of H0 . We accordingly use Eqs. (7.3.3) and (7.3.4) to write

y y y

c kÀq" ck" ¼ °ukÀq

kÀq þ vkÀq

À°kÀqÞ Þ°uk

k þ vk

Àk Þ

and other similar expressions. There then follows a straightforward but

lengthy set of manipulations in which the ¬rst step is the argument that

y

whatever quasiparticles are created or destroyed when ck 0 " ck 0 Àq" acts on É0

y

must be replaced when c kÀq" ck" acts if the matrix element is not to vanish.

Similar reasoning is applied to all such combinations of operators so that the

summation over k and k 0 is reduced to a single sum. When q is very small

ukÀq and vkÀq are replaced by uk and vk and a large amount of cancellation

occurs. Eventually one reduces Eq. (7.7.4) to

2NeB e0B

h jq i ¼ À Aq À

0 2m

X mk À mkþq

‚ Aq Á 2k°uk þ vk Þ4khÉ0 j

2 2

jÉ0 i:

Ek À Ekþq

k

The summation over k is then replaced by an integral over energy multiplied

by D°Þ=3, the factor of 1 arising from the angular integration. The expecta-

3

tion of the quasiparticle number operators mk is replaced by their averages fk ,

while for small q the ratio of the di¬erences mk À mkþq and Ek À Ekþq

becomes the ratio of their derivatives so that one has

°

Ne2 e 2 02 @fk

1

h jq i ¼ À Aq À 2 Aq D°Þ8k2 dE:

F

@Ek

4m c 3

mc

258 Superconductivity

Since D°Þ may be written as 3Nm=202 k2 this becomes

F

°

Ne2 @f

h jq i ¼ À A 1þ dE

@E

mc q

°1

Ne2 E df

¼À A 1þ2 °7:7:5Þ

dE

mc q Á °E 2 À Á2 Þ

1=2 dE

with f °EÞ the Fermi“Dirac function.

That this expression is of the expected form can be veri¬ed by examining

the cases where Á ¼ 0 and where T ¼ 0. At temperatures greater than Tc the

gap parameter Á vanishes and the integration may be performed immedi-

ately: the two terms in parentheses cancel exactly and no Meissner e¬ect is

predicted. At T ¼ 0, on the other hand, the integral over E itself vanishes and

Eq. (7.7.5) may be written as

c

h jq i ¼ À A

42 q

if is chosen such that

r¬¬¬¬¬¬¬¬¬¬¬¬¬

mc2

¼

4Ne2

c

¼

!p

with !p the plasma frequency. A Meissner e¬ect is thus predicted with a

penetration depth which is of the correct order of magnitude and which

depends only on the electron density. At intermediate temperatures

Eq. (7.7.5) predicts a penetration depth that increases monotonically with

T and becomes in¬nite at Tc , in accord with experiment.

7.8 Tunneling experiments

Shortly after the development of the BCS theory it was discovered that a

great deal of information about superconductors could be obtained by

studying the current“voltage characteristics of devices composed of two

pieces of metal separated by a thin layer of insulator. A typical device of

this kind might consist of a layer of magnesium that had been exposed to the

atmosphere to allow an insulating layer of MgO to form on its surface. After

˚

this layer had reached a thickness of about 20 A a layer of lead might be

259

7.8 Tunneling experiments

deposited on top of the oxide, and the di¬erential conductance, dI=dV, mea-

sured at some temperature low enough for the lead to be superconducting.

The result would be of the form shown in Fig. 7.8.1; a sharp peak in dI=dV is

observed when the potential di¬erence V between the magnesium and the

lead is such that the gap parameter Á of the lead is equal to eV.

To discuss e¬ects such as these we choose a simpli¬ed model in which the

two halves of a box are separated by a thin potential barrier so that

the variation of potential in the z-direction is as shown in Fig. 7.8.2. The

Figure 7.8.1. A device consisting of a normal metal and a superconductor separated

by a thin layer of insulator has a di¬erential conductance that exhibits a sharp peak

at a voltage V such that eV is equal to the gap parameter Á.

Figure 7.8.2. In the simplest model of a tunneling junction the two halves of a box

are separated by a narrow potential barrier.

260 Superconductivity

wavefunction of a single electron in this system will then be of the form

¼ eikx xþiky y °zÞ;

where, from the symmetry of the potential, °zÞ will be either an odd function

a of z or an even function s . For each state s , there will exist a correspond-

ing a that has just one more node (which will be located at the center of the

barrier). The di¬erence in energy of these states will be governed by the fact

that the phase of the wavefunction at the edge of the barrier will be di¬erent

in the two cases by some amount ; if in the left-hand side of the box s is of

the form sin kz z then a will be of the form sin ½°kz þ =LÞz. The energy

di¬erence 2T will then be given by

2

02

2T ¼ kz þ À k2 ; °7:8:1Þ

z

2m L

and will be proportional to vz , the component of the electron velocity per-

pendicular to the barrier, when is small.

In the discussion of tunneling experiments it is more useful to work in

terms of the wavefunctions s Æ a rather than s and a themselves, as

the sum or di¬erence is largely localized on one side or the other of the

barrier. We thus do not write the Hamiltonian of the system as

H ¼ E s cycs þ E a cy ca ; °7:8:2Þ

s a

with

cyj0i ¼ js i; cy j0i ¼ ja i;

s a

instead we form new fermion operators

cy À cy cy þ cy

y y

c ¼ ap¬¬¬ s ; d ¼ ap¬¬¬ s :

2 2

In terms of these the Hamiltonian (7.8.2) becomes

H ¼ E°cy c þ d y dÞ þ T°cy d þ d y cÞ;

where

E ¼ 1 °E a þ E s Þ:

2

261

7.8 Tunneling experiments

We now have a picture of two independent systems connected by a perturba-

tion term. The Hamiltonian

X y

Hl ¼ E k cks cks

k;s

describes electrons in the left-hand side of the box, while

X y

Hr ¼ E k d ks dks

k;s

describes those on the right. The perturbation

X y y

Ht ¼ Tkk 0 °cks dk 0 s þ d k 0 s cks Þ °7:8:3Þ

0

k;k ;s

acts to transfer electrons through the insulating barrier from one side to the

other, the transition probability in lowest order depending on the square of

the modulus of the matrix element T.

When a voltage V is applied across this device the energies of those states

on the left of the barrier will be raised by an amount eV. Such an electron of

kinetic energy E k can then only be elastically scattered by Ht to a state k 0 on

the right of the barrier of kinetic energy E k þ eV. The net ¬‚ow of electrons

from left to right will then be proportional to

X

jTj2 ½ fk °1 À fk 0 Þ À fk 0 °1 À fk Þ °E k 0 À E k À eVÞ;

k;k 0 ;s

from which the current I can be considered as governed by the relation

°

I/ T 2 ½ f °E k Þ À f °E k þ eVÞ D°E k ÞD°E k þ eVÞ dE k

with T 2 the square of some average matrix element. The fact that T is, as we

have seen in Eq. (7.8.1), proportional to @E=@kz , while

°dN=dkÞ

D°E k Þ ¼

°dE=dkÞ

262 Superconductivity

leads to a cancellation in the energy dependence of the integrand when

eV ( . One is left with a relation of the form

°

I / ½ f °E k Þ À f °E k þ eVÞ dE k

°

¼ f f °E k Þ À ½ f °E k Þ þ eVf 0 °E k Þ þ 1 °eVÞ2 f 00 °E k Þ þ Á Á Ág dE k

2

™ eV

and the device is predicted to obey Ohm™s law. We might picture the calcula-

tion of this kind of tunneling as in Fig. 7.8.3, where the densities of states of

the two halves of the device are plotted horizontally and energy is measured

vertically. The density of occupied states per unit energy is the product

f °E k ÞD°E k Þ, and is represented by the shaded areas. The tunneling current

is then proportional to the di¬erence in the shaded areas on the two sides.

We now consider how this calculation should be modi¬ed if the metal on

the right of the barrier becomes superconducting, so that our system becomes

a model of the Mg“MgO“Pb device whose conductance was shown in Fig.

y

7.8.1. Our ¬rst step must be to replace the electron operators d ks and dks by

the quasiparticle operators that act on the BCS state. We accordingly rewrite

Figure 7.8.3. In this diagram the densities of states in the two halves of a tunneling

junction are plotted to left and right. The densities of occupied states are found by

multiplying these by the Fermi“Dirac distribution function to give the shaded areas.

A voltage di¬erence raises the energy of one side relative to the other and leads to a

current ¬‚ow proportional to the di¬erence in areas.

263

7.8 Tunneling experiments

the tunneling perturbation (7.8.3) as

X y y

T½c k °uk 0

k 0 þ vk 0

y 0 Þ þ cy °uk 0

Àk 0 À vk 0

k 0 Þ

Ht ¼ Àk Àk

k;k 0

y

þ °uk 0

k 0 þ vk 0

Àk 0 Þck þ °uk 0

y 0 À vk 0

k 0 ÞcÀk : °7:8:4Þ

Àk

This perturbation no longer simply takes an electron from one side of the

barrier and replaces it on the other; the ¬rst term, for instance, creates an

electron on the left of the barrier and either creates or destroys a quasiparticle

excitation on the right. If the normal metal is at a voltage V relative to the

y

superconductor then energy conservation demands that for the term ck

k 0 to

^

cause a real scattering process we must have E k ¼ Ek 0 À eV (all energies now

yy

being measured relative to the chemical potential ). For the term ck

Àk 0 to

^

cause scattering, on the other hand, we must have E k ¼ ÀEk 0 À eV. Applying

these arguments to all the terms in expression (7.8.4) we ¬nd that

X

^ ^ ^

I/ jTj2 °u2 0 f f °E k Þ½1 À f °Ek 0 Þ À f °Ek 0 Þ½1 À f °E k Þg °Ek 0 À E k À eVÞ

k

k;k 0

^ ^ ^

þ v2 0 f f °E k Þ f °Ek 0 Þ À ½1 À f °E k Þ½1 À f °Ek 0 Þg °E k þ Ek 0 þ eVÞÞ

k

X

^ ^

¼ jTj2 fu2 0 ½ f °E k Þ À f °Ek 0 Þ °E k À Ek 0 þ eVÞ

k

k;k 0

^ ^

þ v2 0 ½ f °E k Þ À 1 þ f °Ek 0 Þ °E k þ Ek 0 þ eVÞg:

k

^

We change the sums over k and k 0 to integrals over E k and Ek 0 by multiplying

^

by Dl °E k ÞDr °Ek 0 Þ, with Dr °Ek 0 Þ the e¬ective density of states de¬ned in

^

Eq. (7.4.6). The ¬rst -function vanishes unless Ek 0 ¼ E k þ eV, which from

Eq. (7.5.7) means that k 0 must satisfy the condition

^ ^

E k 0 ¼ Æ½°E k þ eVÞ2 À Á2 1=2 :

But since from Eqs. (7.3.9) and (7.3.12)

^ ^

u2 °E k 0 Þ þ u2 °ÀE k 0 Þ ¼ 1;

the coe¬cients u2 0 will vanish from the expression for the current when we

k

perform the integration. Similar arguments applied to the second -function

264 Superconductivity

and the coe¬cients v2 0 give us

k

°

^ ^ ^ ^ ^

I / T 2 Dl °E k ÞDr °E ¼ E k þ eVÞ½ f °E k Þ À f °E k þ eVÞ d E k

where use has been made of the fact that

f °ÀEÞ ¼ 1 À f °EÞ:

Because of the form of Dr °EÞ, which was shown in Fig. 7.4.1(b), the integral

is no longer proportional to V, but leads to a current of the form shown in

^

Figs. 7.8.4 and 7.8.1. For small V little current will ¬‚ow, since f °E k þ eVÞ will

^ ^

di¬er appreciably from f °E k Þ only when jE k j < kT, and in this region the

density of quasiparticle states, Dr °EÞ, vanishes. This is illustrated in Fig.

7.8.5, where the occupied states are again represented by shaded areas. It is

only when eV > Á that a large current ¬‚ows, giving rise to the observed peak

in dI=dV at this voltage.

One e¬ect that does not emerge from this elementary calculation concerns

y

the need to distinguish between the bare electron created by the operator ck

and the electron in interaction with the phonon system. The canonical

transformation of Section 6.5 is applied only to the electrons in the super-

conducting half of the device, and this must be allowed for in any more

careful calculation of the tunneling current. One ¬nds that the observed

characteristics of the junction are greatly a¬ected by the shape of the phonon

Figure 7.8.4. This current“voltage characteristic is the same as that shown in Fig. 7.8.1,

and is typical of a superconductor“insulator“normal metal junction.

265

7.9 Flux quantization and the Josephson effect

Figure 7.8.5. In this generalization of Fig. 7.8.3 the density of states on the right has

been replaced by the e¬ective density of states Dr °EÞ of the superconductor.

density of states in strong-coupling superconductors, and may even give

useful information about phonon modes in alloys that is not easily obtained

by other means.

7.9 Flux quantization and the Josephson effect

The third possible type of tunnel junction is that in which the metal on both

sides of the insulating barrier is superconducting. The calculation of the

characteristics of this device is more di¬cult than the previous examples in

that the total number of electrons is now not well de¬ned on either side of the

barrier; special operators must be de¬ned that add pairs of electrons to one

side or the other of the device. One result of such a calculation is that a

current is predicted to ¬‚ow that varies with applied voltage in the way shown in

Fig. 7.9.1. and which is in accord with the simple interpretation of Fig. 7.9.2 for

a device composed of two dissimilar superconductors. There is, however, also

another type of current that may ¬‚ow in such a device “ a current associated

with the tunneling through the barrier of bound pairs of electrons.

We can gain some insight into the nature of these currents by returning to a

consideration of the e¬ect of magnetic ¬elds on the current carried by an

electron. We have seen (in Eq. (3.10.8), for example) that the Hamiltonian of

a free particle of mass m* and charge e* in a magnetic ¬eld H ¼ r ‚ A is of

the form

°p À e*A=cÞ2

H¼ :

2m*

266 Superconductivity

Figure 7.9.1. The current“voltage characteristic of a tunnel junction of two super-

conductors shows a discontinuity at a voltage V such that eV equals the sum of the

gap parameters of the two materials.

Figure 7.9.2. The results shown in Fig. 7.9.1 can be interpreted with the aid of this

diagram of the e¬ective densities of states.

If the vector potential were of the form

A ¼ °A; 0; 0Þ °7:9:1Þ

with A a constant, then r ‚ A would vanish and there would be no magnetic

¬eld. The eigenstates of H would be

¼ exp i°kx x þ ky y þ kz zÞ °7:9:2Þ

267

7.9 Flux quantization and the Josephson effect

with energies

2

02 e*A

E¼ kx À þ ky þ kz :

2 2

0c

2m*

In the absence of applied ¬elds we imposed periodic boundary conditions

should be equal at the points °x; y; zÞ, °x þ L; y; zÞ,

by stipulating that

°x; y þ L; zÞ, and °x; y; z þ LÞ. As long as no external electric or magnetic

¬elds were acting, this was a permissible step whose only e¬ect was to

make the counting of states a little easier. If applied ¬elds are present, how-

ever, this joining of the wavefunction on opposite faces of a cubical box must

be treated more carefully. We cannot, for instance, apply a uniform electric

¬eld to the system and impose periodic boundary conditions, as such a

procedure would result in the particle being continuously accelerated in

one direction! In the particular case where only a vector potential °A; 0; 0Þ

acts on the particle we can commit the topological sin of imposing periodic

boundary conditions if we remain awake to the physical implications.

Because A is equal at the points °x; y; zÞ and °x þ L; y; zÞ the Hamiltonian

itself is periodic in the x-direction. Periodicity of in this direction then

demands that kx ¼ 2n=L with n an integer. The contribution of the motion

in the x-direction to the energy is thus

02 2n e*A 2

Ex ¼ À : °7:9:3Þ

0c

2m* L

The motion in the y- and z-directions is una¬ected by A and so we can also

retain periodicity in these directions. The physical picture of this situation is

that we have a closed loop of superconducting material, as shown in Fig. 7.9.3.

Since A acts in the x-direction we physically join these two opposite faces of the

material. Because the electrons occupy pair states we interpret the charge e* of

the current carriers as 2e.

While we assume that the Meissner e¬ect obliges the magnetic ¬eld to

vanish within the superconductor (we assume dimensions large compared

with the penetration depth) the magnetic ¬‚ux È threading the ring does

not necessarily vanish, since

°

È ¼ r ‚ A Á dS

þ

¼ A Á dl

¼ AL; °7:9:4Þ

268 Superconductivity

Figure 7.9.3. The free energy of a superconducting ring like this is a minimum when

the magnetic ¬‚ux threading it is quantized.

the line integral being taken around a closed path within the ring. From

Eq. (7.9.3) the additional energy of the electron pair due to the vector potential

will be

2eA 2

02 4n 2eA

ÁE x ¼ À þ : °7:9:5Þ

L 0c 0c

2m*

If for every state of positive n the corresponding state of negative n is also

occupied then there will be no contribution to the total energy of the electron

gas from the term linear in A, and the total energy change will be

2N02 eÈ 2

ÁE total ¼ :

m*L2 0c

The presence of a ¬nite magnetic ¬‚ux threading the ring thus increases the

energy of the system.

If the ¬‚ux is greater than 0c=2e it becomes energetically favorable for an

electron pair with kx ¼ À2n=L to make a transition to a state for which

kx ¼ 2°n þ 1Þ=L. The total energy change due to A is then

2

2N02 eÈ

ÁE total ¼ À ;

0c

m*L2

which takes on its minimum value of zero when È ¼ 0c=e. These arguments

can be extended to show that the minimum possible energy of the system is in

269

7.9 Flux quantization and the Josephson effect

fact a periodic function of È of the form shown in Fig. 7.9.4. There is thus a

tendency for the magnetic ¬‚ux È threading the ring to be quantized in units of

the ¬‚ux quantum 0 , equal to 0c=e. This is veri¬ed experimentally in a

number of delicate measurements in which minute hollow cylinders of super-

conductor have been cooled down through Tc in the presence of applied

magnetic ¬elds. Subsequent measurements of the ¬‚ux trapped in this way

show values that unmistakably cluster around integral multiples of 0 .

The current carried by a single pair of electrons will be proportional to

0kx À 2eA=c, which from Eqs. (7.9.4) and (7.9.5) is in turn proportional to

@°ÁE x Þ=@È. The total current I ¬‚owing in the ring is thus proportional to

dE total =dÈ; one di¬erentiates the curve of Fig. 7.9.4 to ¬nd the sawtooth

graph of Fig. 7.9.5. This graph leads to two results of great importance in the

theory of superconductivity when we realize that this theory will apply not only

to a closed ring of superconductor but also to a device such as that shown in

Fig. 7.9.6, in which the ring is broken and a thin insulating layer inserted.

Provided the gap is narrow enough that an appreciable number of electron

pairs can tunnel through and that no ¬‚ux quanta are contained in the gap itself,

then the current will still be a periodic function of È with period 0 .

The ¬rst experiment we consider is a measurement of the current I when

the ¬‚ux È is held constant. Since the electromotive force in the ring is equal

Figure 7.9.4. This construction shows the energy of a ring of superconductor at zero

temperature to be a periodic function of the magnetic ¬‚ux È, and to have minima at

integral multiples of the ¬‚ux quantum 0 .

Figure 7.9.5. This sawtooth curve is the derivative of that shown in Fig. 7.9.4, and

represents the current I.

270 Superconductivity

Figure 7.9.6. In this simple version of a Josephson junction a superconducting ring

has been cut through at one place and has had a thin insulating layer inserted at the

break.

to ÀcÀ1 dÈ=dt we know that any current ¬‚owing must do so in the absence of

an electric ¬eld. If È is maintained at a value di¬erent from n0 it is thus

energetically favorable for a small supercurrent to ¬‚ow, and the I“V char-

acteristic of Fig. 7.9.1 should exhibit a delta-function peak in the current at

V ¼ 0. This phenomenon is known as the dc Josephson e¬ect (dc standing for

˜˜direct current™™).

The second experiment consists of causing the ¬‚ux to increase uniformly

with time. This constant value of ÀcÀ1 dÈ=dt represents a constant electro-

motive force V acting in the circuit. From Fig. 7.9.5 we then see that I will in

fact alternate in sign as È is increased, the frequency of this alternating

current being equal to the number of ¬‚ux quanta introduced per unit time.

The angular frequency ! of the current in this ac Josephson e¬ect is thus

2eV

!¼ ;

0

which is of the order of magnitude of 1 GHz per microvolt.

2

It is interesting to note that this is just the frequency di¬erence that we

would associate with the wavefunction of a single pair of electrons placed in

this device; the time-dependent Schrodinger equation tells us that a particle

¨

of energy E has a wavefunction of the form

É°r; tÞ ¼ °rÞeÀiEt=0

while the wavefunction for a particle of energy E þ 2eV varies with time as

eÀi°Eþ2eVÞt=0 . For a single particle such as this we can never expect to measure

271

7.10 The Ginzburg“Landau equations

Figure 7.9.7. A wavefunction consisting of a linear combination of di¬erent har-

monic-oscillator states describes a system in which both the number of particles and

the phase of the wavefunction can be partially speci¬ed.

the phase of the wavefunction, as physically measurable quantities like the

particle density always involve j j2 . This may be thought of as another form

of the Uncertainty Principle, and states that if the number of particles is

known then the phase is unknown. We can, however, form a wavefunction

of known phase if we form a wave packet of di¬erent numbers of particles.

Adding equal amounts of the n ¼ 0 ground state and the n ¼ 1 state of a

harmonic oscillator, for example, gives a wavefunction that oscillates back

and forth with the classical oscillator frequency (Fig. 7.9.7). We thus look at

the Josephson junction as a device in which the uncertainty in the number of

pairs on each side of the barrier allows us to measure the relative phase of the

wavefunction of the two parts of the system. This concept can be extended to

the theory of super¬‚uid liquid helium, where e¬ects analogous to the ac

Josephson e¬ect have been detected when a pressure di¬erence is maintained

between two parts of a container separated by a small hole.

7.10 The Ginzburg“Landau equations

The destruction of superconductivity by a strong enough magnetic ¬eld may

be understood by considering the energy associated with the Meissner e¬ect.

The expulsion of all magnetic ¬‚ux from the interior of a long sample held

parallel to an applied ¬eld H0 gives it an e¬ective magnetization per unit

volume of ÀH0 =4. Since the magnetic moment operator is given by À@H=@H

one ¬nds the energy of the sample due to its magnetization to be

° H0

H

EM ¼ Á dH

4

0

H2 0

¼ :

8

272 Superconductivity

Figure 7.10.1. The magnetic energy of a superconductor varies as the square of the

applied magnetic ¬eld H0 . For ¬elds greater than Hc it is energetically favorable for

the sample to make a transition to the normal state.

When this energy becomes greater than the condensation energy E c then, as

illustrated in Fig. 7.10.1, it is no longer energetically favorable for the sample

to remain in the superconducting state. In a weak-coupling superconductor

at zero temperature, for instance, the critical ¬eld Hc above which the metal

could not remain uniformly superconducting would be given by

2

Hc 1

¼ 2 D°ÞÁ2 °0Þ:

8

At ¬nite temperatures, Hc °TÞ can be found by identifying the condensation

energy with the di¬erence in Helmholtz energies of the normal and super-

conducting phases.

If the sample geometry is changed to that shown in Fig. 7.10.2 the magnetic

energy is greatly increased, for there is now a large region outside the sample

from which the applied ¬eld is partially excluded. Since the condensation

energy remains constant the specimen starts to become normal at an applied

¬eld well below Hc . The sample does not become entirely nonsuper-

conducting, but enters what is known as the intermediate state, in which a

large number of normal and superconducting regions exist side by side

(Fig. 7.10.3). In this way the magnetic energy is greatly reduced while a

273

7.10 The Ginzburg“Landau equations

Figure 7.10.2. The magnetic ¬eld at the rim of this disc-shaped sample of super-

conductor is greater than the applied ¬eld; the superconductivity thus starts to be

destroyed at weaker applied ¬elds than is the case for a rod-shaped sample.

Figure 7.10.3. The intermediate state.

274 Superconductivity

large part of the condensation energy is retained. This behavior is similar

to that described at the beginning of this chapter, where a type II super-

conductor was de¬ned. The di¬erence is that while the type I superconductor

only forms a mixture of normal and superconducting regions when the mag-

netic energy is magni¬ed by geometric factors, the type II superconductor

will enter the mixed state even when in the form of a long sample held parallel

to a strong enough applied ¬eld. In the mixed state the distance between

normal regions is typically 0.3 mm, which may be compared with the coarser

structure of the intermediate state, which is characterized by distances typi-

cally of the order of 100 mm.

The analysis of these phenomena in terms of the BCS theory is very com-

plicated. Because the magnetic ¬eld will be a rapidly varying function of

position within the sample we must return to the arguments of Section 7.7

and ask for the response to the vector potential Aq eiq Á r when q is no longer

vanishingly small. There we saw in Eq. (7.7.5) that the current h jq i due to the

vector potential could be expressed as the sum of two parts “ a negative (or

diamagnetic) part and a positive (or paramagnetic) part. In a superconductor

at zero temperature the paramagnetic part vanished for q ! 0, while in a

normal metal it exactly cancelled the diamagnetic part and left no Meissner

e¬ect. A more careful investigation shows that as q is increased from zero the

paramagnetic response of a superconductor also increases, until at large

enough q it approximates the response of a normal metal. If one expresses

this result in the form

h jq i ¼ ÀLq Aq °7:10:1Þ

then one ¬nds that Lq ¬rst becomes appreciably lower than its zero-q value

when the approximation

@Ek

Ekþq À Ek ™ q Á

@k

becomes invalid. This occurs when

@E k

qÁ $Á

@k

which is equivalent to the condition

q0 $ 1

with 0 ¼ 0vF =Á, the coherence length discussed in Section 7.1. The Fourier

275

7.10 The Ginzburg“Landau equations

transform of Eq. (7.10.1) is an equation of the form

°

L°r 0 ÞA°r À r 0 Þ dr 0

h j°rÞi ¼ À °7:10:2Þ

0

with L°r 0 Þ a function that can be reasonably well approximated by L0 eÀr =0 .

An equation of this kind had been suggested by Pippard on macroscopic

grounds before the development of the BCS theory.

When the coherence length 0 is very much smaller than the penetration

depth then A°r À r 0 Þ will not vary appreciably within the range of L°r 0 Þ.

Equation (7.10.2) then tells us that the current density at a point will be

approximately proportional to the vector potential in the gauge that we

have chosen, and the London equation (7.7.1) will be valid. Under these

circumstances it will be energetically favorable for type II superconductivity

to occur, as the magnetic ¬eld can penetrate the superconductor and reduce

the magnetic energy without reducing the condensation energy. If, on the

other hand, ( 0 then Eq. (7.10.2) predicts a nonlocal relation between the

magnetic ¬eld and the current density. The wavefunction of the supercon-

ductor may then be modi¬ed up to a distance 0 from the surface of the

specimen, with a consequent reduction in the condensation energy. Because

the magnetic ¬eld only penetrates a short distance , little magnetic energy is

gained, and the sample will be a type I superconductor. We can illustrate this

situation if we make the generalization that in a spatially varying magnetic

¬eld the gap parameter Á should be considered as a function of position. The

variation of B°rÞ and Á°rÞ at the boundary separating normal and super-

conducting regions of a metal can then be depicted as in Figs. 7.10.4(a)

and 7.10.4(b) for type I and II superconductors respectively, where Á1 is

the value of Á deep inside the superconductor.

A prediction of the geometry of the mixed state in a type II superconductor

can be obtained by considering the Helmholtz energy F S in a superconductor

in which the gap parameter Á varies with position. We saw in Eq. (7.4.2) that

the condensation energy is proportional to ÀÁ2 in a homogeneous super-

conductor at zero temperature, and so it is natural to expect the dominant

term in the Helmholtz energy to vary as ÀÁ2 °rÞ in the more general case. The

fact that Eq. (7.10.2) shows the current (and hence the wavefunction) at a

point r to depend on the conditions at points distant 0 from r suggests that a

term proportional to ½0 rÁ°rÞ2 should be included in F S . In the presence of a

magnetic ¬eld this contribution must be modi¬ed to preserve gauge in-

variance to ½0 °r À ie*A°rÞ=0cÞÁ°rÞ2 , with e* again chosen equal to 2e; a

magnetic energy density of H2 =8 must also be added.

276 Superconductivity

Figure 7.10.4. The area under the curve of B2 °rÞ represents the magnetic energy

gained in forming the normal“superconducting interface, while the area under the

curve of Á2 À Á2 °rÞ is related to the condensation energy lost. In a type I material

1

(a) the net energy is positive but in a type II superconductor (b) there is a net negative

surface energy.

While the terms we have discussed so far present a fair approximation to

F S , this form of the Helmholtz energy does not allow one to discuss which

particular two-dimensional lattice of normal regions gives the mixed state of

lowest Helmholtz energy; this is due to the linearity of the equation one

obtains by trying to minimize F S with respect to Á. One must accordingly

include the term of next highest power in Á, which in this case will be

277

7.10 The Ginzburg“Landau equations

proportional to Á4 . The expression for the di¬erence in Helmholtz energies

between the superconducting and normal states will then be

°

H2

1

FS À FN ¼ ajÁ°rÞj þ bjÁ°rÞj þ

2 4

2 8

2

2ieA

þ c r À Á°rÞ dr °7:10:3Þ

0c

with a; b, and c temperature-dependent constants and where the possibility of

a complex Á has been allowed for in the spirit of the discussion of the phase

of the superconducting wavefunction given at the end of Section 7.9.

Minimization of F S with respect to both A°rÞ and Á°rÞ yields the

Ginzburg“Landau equations, which in principle enable one to calculate A

and Á as functions of position and of the constants a; b, and c. These con-

stants can be evaluated in terms of the parameters of the homogeneous

material when F S is derived from the BCS Hamiltonian using some rather

di¬cult procedures ¬rst applied by Gorkov. It is then possible to remove all

but one of these parameters from appearing explicitly in the Helmholtz

energy by working in terms of the dimensionless quantities in which H is

measured in units proportional to Hc , distances are measured in units of the

London penetration depth , and in which °rÞ is the ratio of Á°rÞ to its value

in the homogeneous material. The Ginzburg“Landau equations then take on

the form

2

1

1 À j j2 À rÀA ¼0 °7:10:4Þ

i

1

j j2 A þ r ‚ °r ‚ AÞ ¼ ° *r À r *Þ °7:10:5Þ

2i

where is =0 . The fact that is the only parameter of the material to enter

these equations con¬rms the idea that it is this quantity alone that determines

whether a superconductor will be of type I or type II.

The detailed solution of Eqs. (7.10.4) and (7.10.5) leads to a variety of

qualitatively correct predictions of the behavior of type II superconductors.

The lower critical applied ¬eld Hc1 at which it ¬rst becomes energetically

favorable for a thread of normal material to exist in the superconductor

can be shown top¬¬¬ less than Hc , the bulk critical ¬eld calculated from Á,

be

provided > 1= 2. Similarly the upper critical ¬eld Hc2 below which a

regular two-dimensional triangular lattice of threads of normal material

278 Superconductivity

Figure 7.10.5. This diagram shows which of the three ˜˜phases™™ of a superconductor

has the lowest free energy for a given Ginzburg“Landau parameter and applied

¬eld H0 .

forms the state of lowest energy can be calculated to be approximately

p¬¬¬

2Hc . This is illustrated in the ˜˜phase diagram™™ of Fig. 7.10.5, in which

the state of the superconductor is shown as a function of the Ginzburg“

Landau parameter and the applied magnetic ¬eld H0 . The existence of a

superconducting surface layer in type II materials up to an applied ¬eld Hc3

equal to about 1.7 Hc2 may also be shown to follow from Eqs. (7.10.4) and

(7.10.5).

7.11 High-temperature superconductivity

The technological promise of superconductivity is so rich that there has been

a continual search for materials with higher critical temperatures. In high-

voltage transmission lines, ohmic resistance losses consume about one per-

cent of the power carried for every 100 km traveled, and even the best electric

motors made of nonsuperconducting materials waste as heat several percent

of the energy they use. If the need for refrigeration could be eliminated or

reduced, the economic bene¬ts ¬‚owing from the adoption of superconducting

materials would be substantial.

In the decades before 1986, the progress made in this search was slow, for

reasons that we can appreciate from the BCS expression (7.5.10) for the

critical temperature. We might think that we could increase Tc by making

the lattice more rigid, and thereby increasing !D . Unfortunately the interac-

tion term V would simultaneously be reduced, as we see from Eq. (7.2.2), and

little advantage would be gained. The remaining component of expression

(7.5.10) is the density of states D°Þ, and this can be increased by using

transition metals and choosing the most favorable crystal structure. In this

way a transition temperature of 23 K was achieved in Nb3 Ge. However, if we

279

7.11 High-temperature superconductivity

travel too far along the path of increasing the density of states we ¬nd a new

obstacle. The electron“phonon interaction reduces the phonon frequencies

through the process shown in Fig. 6.2.2, and a large electronic density of

states enhances this e¬ect. As we saw in the case of the Peierls transition

described in Section 6.3, softening of the phonon modes eventually leads to a

lattice instability.

Among other possible routes to high-temperature superconductivity we

might look for pairing that involves a stronger force than arises from the

electron“phonon interaction. One possibility could be a pairing between an

electron and a hole in a material in which electrons and holes are present in

equal numbers and with similar masses. However, systems of bound electron“

hole pairs have a tendency to form spatially inhomogeneous structures in

which the electric charge density or spin density varies periodically in space.

Superconductivity is not favored in these spatially modulated structures.

It was thus a delightful surprise when, in 1986, Bednorz and Muller dis-

¨

covered that superconductivity occurred at 35 K in a ceramic compound of

lanthanum, barium, copper, and oxygen. This delight was magni¬ed the

following year with the revelation by Chu and Wu that replacement of the

lanthanum by yttrium raised Tc to 92 K. This allowed the use of liquid

nitrogen, which boils at 77 K, for cooling. Many other ceramic compounds

were subsequently found to be high-temperature superconductors. A feature

common to most, but not all, of these was that they contained parallel layers

of CuO2 , each layer separated from its neighbors by ionizable metallic atoms.

Another characteristic was the increase in Tc that occurred when each single

CuO2 layer was replaced by two contiguous layers, and then by three,

although a further increase to four layers produced a decrease in Tc .

The original motivation of Bednorz and Muller for looking at conducting

¨

oxides was the thought that the electron“phonon interaction could be

strengthened if the copper ions were in nearly unstable positions in the lattice.

The Jahn“Teller theorem states that, except in some special circumstances, a

degeneracy in electronic states can be lifted by a distortion in which an atom

moves to a less symmetric position. Since lifting the degeneracy moves the

energies of the states apart, the lowest-lying state will be reduced in energy,

and the system is unstable. The existence of nearly degenerate states on the

copper ions could thus enhance the electron“lattice interaction.

The structure of the CuO2 planes is as shown in Fig. 7.11.1, with each

copper atom having four oxygen neighbors. The oxygen atoms are hungry

for electrons, and succeed in removing not only the single electron in the

outer 4s state of copper, but also one of the electrons from the ¬lled 3d-shell.

The copper is thus doubly ionized to Cu2þ . One might then expect to have a

280 Superconductivity

Figure 7.11.1. In many high-temperature superconductors there are planes of copper

atoms (large spheres) and oxygen atoms (small spheres) arranged in this way.

metal, since the uppermost band should be only half ¬lled. However, the

mutual Coulomb repulsion of the 3d electrons now intervenes and splits

this band. It does so by arranging the spins of the holes on the Cu sites in

an antiferromagnetic ordering, in which the spin on every Cu site is aligned

antiparallel to that of each of its four Cu neighbors. The size of the unit cell

of the CuO2 lattice is e¬ectively doubled, and so the size of the ¬rst Brillouin

zone is correspondingly halved. An energy gap now appears between the new

¬rst Brillouin zone and the new second Brillouin zone. The number of holes is

now exactly equal to the capacity of the ¬rst Brillouin zone, making the

compound an insulator. A ¬lled valence band, consisting of states predomi-

nantly located on the oxygens, is separated by an appreciable gap from the

empty conduction band in which electrons, had they been present, would be

mostly concentrated on the copper sites. This is the situation for the undoped

material, which is commonly referred to as a Mott insulator.

The antiferromagnetism, like all types of ordering, can be destroyed if the

temperature is raised su¬ciently. It can also be destroyed by increasing the

number of holes in the CuO2 layer, as this eliminates the one-to-one corre-

spondence between holes and lattice sites. The doping process that adds extra

holes can be achieved by either replacing some of the atoms with those of

lower valency or modifying the amount of oxygen in the compound. A dop-

ing level (measured as the number of extra holes per CuO2 unit) of a few

percent is generally su¬cient to destroy antiferromagnetism. As the number

of holes is increased beyond this point one enters what is known as the

pseudogap region. Here the electronic speci¬c heat does not ¬t a picture of

electrons as a gas or liquid of quasiparticles, but shows some traits charac-

teristic of superconductors. A further increase in doping takes us into the

superconducting state, which is often optimized at a doping level of about

281

7.11 High-temperature superconductivity

Figure 7.11.2. The generic phase diagram of a typical high-temperature super-

conductor.

20 percent. The generic phase diagram for high-temperature superconductors

thus appears as in Fig. 7.11.2.

The existence of planes of copper and oxygen atoms suggests that the lower

e¬ective dimensionality of the system may be the factor that overcomes the

di¬culties that had been predicted in reaching high transition temperatures. In

fact, there had been suggestions as early as 1964 that one-dimensional organic

molecules might be a possible route to room-temperature superconductivity.

One aspect of low e¬ective dimensionality is that the screening of the Coulomb

interaction is reduced. This has the two opposite e¬ects of increasing the

strength of the electron“lattice interaction, which should favor superconduc-

tivity, and increasing the mutual repulsion of the electrons, which should

inhibit the formation of Cooper pairs and thus disfavor superconductivity.

The response of the system to this combination of a stronger indirect force

of attraction (which may be due to e¬ects other than the electron“phonon

interaction) and a stronger direct repulsion could be a contributory factor in

encouraging the electrons (or holes) to form Cooper pairs having d-wave

symmetry rather than the isotropic s-wave symmetry of the elemental super-

conductors. The spins would again be antiparallel, but now the wavefunction

would vanish as the particle separation tends to zero, reducing the energy

cost of the strong short-range repulsion. Convincing experimental support

for d-wave pairing has been found in a number of ingenious experiments.

One of the most impressive pieces of evidence is seen in the beautiful picture

on the cover of this book, which shows a scanning-tunneling-microscope

image of the surface of a high-temperature superconductor. One of the

copper atoms in a CuO2 plane just below the surface has been replaced by a

zinc atom. Scattering of the d-wave quasiparticles from the zinc atom results

in a distribution that re¬‚ects the structure of the superconducting state. In

d-wave superconductors the gap parameter Á is no longer a constant, but