. 9
( 14)


(No chemical potential appears in this function because the total number of
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
2E k uk vk À °u2 À v2 Þ Vkk 0 uk 0 vk 0 ½1 À 2f °Ek 0 ފ ¼ 0:
k k
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
Ák °TÞ ¼ Vkk 0 °1 À x2 0 Þ1=2 ½1 À 2f °Ek 0 ފ °7:5:5Þ

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 ފ: °7:5:6Þ
Vkk 0 2
2 k0 ½E k 0 þ Ák 0 °Tފ 1=2

This equation still contains the excitation energy Ek which we evaluate by
taking the thermal average of Eq. (7.5.3). We ¬nd
Ek ¼ E k °u2 À v2 Þ þ 2uk vk Vkk 0 uk 0 vk 0 ½1 À 2f °Ek 0 ފ

¼ ½E k þ Á2 °Tފ1=2 ; °7:5:7Þ

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
1X Ák 0 °E k 0 þ Á2 0 Þ1=2
Ák ¼ : °7:5:8Þ
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 =2kTŠ ^
d E ¼ 1: °7:5:9Þ
°E 2 þ Á2 Þ1=2

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

° 0!D =2kTc
0! =2kTc
½ln x tanh xŠ0 D À sech2 x ln x dx ¼ :

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
0!D 1
À sech2 x ln x dx ¼ :
2kTc VD°Þ

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

¼ 3:50;
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
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
^2 ^
2 2
E°TÞ ¼ ½2E k v k þ °u k À vk Þ2fk E k À °1 À 2fk Þuk vk ÁŠ

¼ E k À Ek À °1 À 2fk Þ

from which

Á2 df 1 2 d°1=EÞ
¼ 2 EÀ þ °1 À 2f ÞÁ
2E dT 2 dT
1 À 2f
df 2d
¼ þÁ
dT dT

X  dÁ2 2E 2  @f
¼ À : °7:5:11Þ
dT T

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

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
Hq ¼ Mq °ay þ aq Þ °c kþq ck þ cy cÀ°kþqÞ Þ
Àq Àk
X y
¼ Mq °ay þ aq Þ ½°ukþq
kþq þ vkþq
À°kþqÞ Þ°uk
k þ vk
y Þ
Àq Àk
þ °uk
y À vk
k Þ°ukþq
À°kþqÞ À vkþq
kþq ފ
X y
½°ukþq uk À vkþq vk Þ°
k þ
À°kþqÞ Þ
¼ Mq °aÀq þ aq Þ Àk
þ °ukþq vk þ uk vkþq Þ°
y þ
k ފ: °7:6:1Þ
7.6 Ultrasonic attenuation
The terms in
y and
k lead to the simultaneous creation or
destruction of two quasiparticles and, as we have seen, cannot represent
energy-conserving processes when 0!q is negligible. We are thus left
with the terms
k and
À°kþqÞ , which can lead to energy-conserving
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
Pa / °ukþq uk À vkþq vk Þ2 ½ fk °1 À fkþq Þ °Ekþq À Ek À 0!q Þ

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

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

þ ° fÀ°kþqÞ À fÀk Þ °EÀk À EÀ°kþqÞ À 0!q ފ; °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

fk À fkþq ¼ Àq Á
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
°  ^ 2  
0!q E
E @f ^
¼ D°Þ j cos j  cos  À d°cos Þ d E
@E q0vk E
!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
n f °0Þ
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
j¼À °7:7:1Þ
7.7 The Meissner effect
with c the speed of light, and then take the curl of this relation we obtain the
London equation,

r‚j¼À B:

The use of the Maxwell equations
r‚H¼ rÁB ¼ 0

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

À1 r Á v ¼ À ;

of which the Fourier transform is

@q 1
iÀ1 q Á vq ¼ À ¼ ½H; q Š:
@t i0

We thus ¬nd

q Á jq ¼ À ½°H0 þ H1 Þ; q Š:

The commutator of the zero-¬eld Hamiltonian H0 with q was evaluated in
Section 2.7, and ½H1 ; q Š can be similarly calculated. The result is that

X  e0 
e2 Aq y
jq ¼ °2k À qÞckÀq;s ck;s À cc : °7:7:3Þ
mc k;s 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

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

h jq i ¼ hÉ0 j jq jÉ0 i þ hÉ0 j jq H jÉ i
E 0 À H0 1 0
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)
7.7 The Meissner effect
and drop terms of order A2 to ¬nd

X e0B Aq
Ne2 Aq
Á °2k 0 À qÞ
h jq i ¼ À À hÉ0 j
mc k;s;k 0 ;s 0

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

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-
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
h jq i ¼ À Aq À 2 Aq D°Þ8k2 dE:
4m c 3
258 Superconductivity
Since D°Þ may be written as 3Nm=202 k2 this becomes

Ne2 @f
h jq i ¼ À A 1þ dE
mc q
Ne2 E df
¼À A 1þ2 °7:7:5Þ
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
h jq i ¼ À A
42 q

if  is chosen such that


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
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 =LÞzŠ. The energy
di¬erence 2T will then be given by
2T ¼ kz þ À k2 ; °7:8:1Þ
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


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;


E ¼ 1 °E a þ E s Þ:
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

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

X y
Hr ¼ E k d ks dks

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
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
jTj2 ½ fk °1 À fk 0 Þ À fk 0 °1 À fk ފ °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 eVފ 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

D°E k Þ ¼
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 eVފ 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 ÁŠg dE k

™ 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.
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.
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
þ °uk 0
k 0 þ vk 0
Àk 0 Þck þ °uk 0
y 0 À vk 0
k 0 ÞcÀk Š: °7:8:4Þ

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
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
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
^ ^ ^
I/ jTj2 °u2 0 f f °E k Þ½1 À f °Ek 0 ފ À f °Ek 0 Þ½1 À f °E k ފg °Ek 0 À E k À eVÞ
k;k 0

^ ^ ^
þ v2 0 f f °E k Þ f °Ek 0 Þ À ½1 À f °E k ފ½1 À f °Ek 0 ފg °E k þ Ek 0 þ eVÞÞ
^ ^
¼ jTj2 fu2 0 ½ f °E k Þ À f °Ek 0 ފ °E k À Ek 0 þ eVÞ
k;k 0

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

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 ƽ°E k þ eVÞ2 À Á2 Š1=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
perform the integration. Similar arguments applied to the second -function
264 Superconductivity
and the coe¬cients v2 0 give us
^ ^ ^ ^ ^
I / T 2 Dl °E k ÞDr °E ¼ E k þ eVÞ½ f °E k Þ À f °E k þ eVފ 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
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.
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 :
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
7.9 Flux quantization and the Josephson effect
with energies
02 e*A
E¼ kx À þ ky þ kz :
2 2

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
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
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

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
2N02 eÈ
ÁE total ¼ À ;

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
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

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
!¼ ;
which is of the order of magnitude of 1 GHz per microvolt.
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
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
EM ¼  Á dH

H2 0
¼ :
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

Hc 1
 ¼ 2 D°ÞÁ2 °0Þ:

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
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

Ekþq À Ek ™ q Á

becomes invalid. This occurs when

@E k
qÁ $Á

which is equivalent to the condition

q0 $ 1

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

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°rފ2 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°rފ2 , 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
(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
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
FS À FN ¼ ajÁ°rÞj þ bjÁ°rÞj þ
2 4
2 8

þ c r À Á°rÞ dr °7:10:3Þ


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
1 À j j2 À rÀA ¼0 °7:10:4Þ

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

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,
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
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

7.11 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
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
7.11 High-temperature superconductivity

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

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


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