. 8
( 14)


@E k 20!D°ÞjM j2
0vk ™ 1þ :
@k "
°E k À Þ2 À °0!Þ2

The in¬nities that this expression predicts when E k ¼  Æ 0! are a spurious
consequence of our averaging procedure. The value predicted when k lies in
the Fermi surface, where E k ¼ , is more plausible, and gives us the result

vk ™ v0 °1 À Þ
6.4 Polarons and mass enhancement
where v0 is the unperturbed velocity and

2jM j2 D°Þ
¼ :
This decrease in the electron velocity is equivalent to an increase in the density
of states by the factor °1 À ÞÀ1 . Because v0 is inversely proportional to the
electron mass, m, it is common to discuss the increase in the density of states in
terms of an increase in the e¬ective mass of the electron. One refers to °1 À ÞÀ1
as the mass enhancement factor due to electron“phonon interactions.
It is important to note that the enhanced density of states must only be
used in interpreting experiments in which the Fermi energy is not altered.
This is because the kink in the electron spectrum depicted in Fig. 6.4.1 is in
fact tied to the Fermi energy; if one were to increase the density of electrons
in the metal the kink in the curve would ride up with . We thus predict that
it is the enhanced density of states that determines the electronic speci¬c heat,
since the e¬ect of raising the temperature above zero is just to excite a few
electrons from states below the Fermi energy to states above the Fermi
energy. The semiclassical dynamics of a conduction electron in a magnetic
¬eld considered in Section 4.6 will also involve the corrected electron velocity,
as again the Fermi energy is unchanged. The cyclotron mass m* is thus
enhanced by the factor °1 À ÞÀ1 .
A di¬erent situation is encountered in the theory of the Pauli spin suscept-
ibility p of the electron gas. We recall that a suitably oriented magnetic
¬eld increases the size of the spin-up Fermi surface while diminishing the
spin-down one in the manner shown in Fig. 6.4.2, in which the densities

Figure 6.4.2. A magnetic ¬eld lowers the energy of the spin-up states while increasing
the energy of those with spin down. The magnetization is proportional to the dif-
ference in area between the two halves of this diagram.
222 Electron“phonon interactions

Figure 6.4.3. The electron“phonon interaction changes the shape of the densities of
states of the spin-up and spin-down systems, but does not alter the di¬erence in areas
of the two halves of the diagram. The magnetization due to the electron spins thus
remains unenhanced.

of states, D" and D# , are plotted as functions of energy. The magnetization
is proportional to N" À N# , and hence to the di¬erence in the areas
enclosed on the left and on the right; in a free-electron model p is thus
proportional to D°EÞ. If we now include the electron“phonon interaction
in our calculation the spin-up and spin-down systems remain indepen-
dent of each other, but two kinks are introduced in the densities of
states, as shown in Fig. 6.4.3. The di¬erence in areas enclosed, however,
remains unaltered by this modi¬cation. We conclude that we should
use the unenhanced density of states when discussing such properties as
the Pauli paramagnetism or the change in Fermi surface dimensions on
alloying, since in these experiments the position of the Fermi surface is

6.5 The attractive interaction between electrons
The fact that the energy of the electron“phonon system (as expressed in
Eq. (6.2.3)) contains terms proportional to hnk nk 0 i is an indication that
there is an e¬ective electron“electron interaction mediated by phonons.
The nature of this interaction is illustrated by redrawing Fig. 6.2.1(b) in
the form of Fig. 6.5.1, in which we deliberately fail to notice that the two
intermediate electron states are the same. The passage of the phonon from
one electron to the other contributes an energy E c which from Eq. (6.2.3) may
6.5 The attractive interaction between electrons

Figure 6.5.1. This way of redrawing Fig. 6.2.1(b) looks similar enough to Fig. 2.3.1
that we are led to believe there may be an e¬ective electron“electron interaction
caused by the electron“phonon interaction.

be written as

X hnk nk 0 i
Ec ¼ À jMkk 0 j2
E k À E k 0 À 0!q
k;k 0

1X 20!q
¼ jMkk 0 j2 hÀnk nk 0 i: °6:5:1Þ
°E k À E k 0 Þ2 À °0!q Þ2
2 k;k 0

Comparison with Eq. (2.4.2) suggests that this may be considered as the
exchange energy of the electron gas when interacting through a potential V
whose matrix element Vkk 0 is given by

Vkk 0 ¼ jMkk 0 j2 : °6:5:2Þ
°E k À E k 0 Þ2 À °0!q Þ2

This is signi¬cant in that although E k and E k 0 are much greater than 0!q , and
thus usually give a positive Vkk 0 , there is always the possibility of E k and E k 0
being close enough that the matrix element could become negative. While it is
not necessarily true that an interaction that is negative in k-space is negative
in r-space, this form for Vkk 0 does open the door to the possibility of
an attractive interaction between electrons. Because the wavefunction of a
system always modi¬es itself to maximize the e¬ects of attractive inter-
actions, while it tends to minimize those of repulsive interactions, even a
weak attraction between electrons can lead to important consequences.
224 Electron“phonon interactions
We can verify that such an attractive interaction does exist by performing a
canonical transformation of the Frohlich Hamiltonian (6.1.1). We write

H ¼ H0 þ HeÀp

and then look for a transformation of the form

H 0 ¼ eÀs Hes °6:5:3Þ

that will eliminate HeÀp to ¬rst order. On expansion of the exponentials in
Eq. (6.5.3) we have

H 0 ¼ °1 À s þ 1 s2 À Á Á ÁÞH°1 þ s þ 1 s2 þ Á Á ÁÞ
2 2

¼ H0 þ HeÀp þ ½H0 ; sŠ þ ½HeÀp ; sŠ þ 1 ½½H0 ; sŠ; sŠ þ Á Á Á :

We choose s in such a way that its commutator with H0 cancels the term
HeÀp . With this achieved we have

H 0 ¼ H0 þ 1 ½HeÀp ; sŠ þ Á Á Á ;

the omitted terms being of order s3 or higher. Now since
0!q ay aq
H0 ¼ E k c k ck þ q

X y
Mkk 0 °ay þ aq Þc k ck 0
HeÀp ¼ Àq
k;k 0

we try
X y
°Aay þ Baq ÞMkk 0 ck ck 0 ;
s¼ Àq
k;k 0

with A and B coe¬cients to be determined. We then ¬nd that in order to

HeÀp þ ½H0 ; sŠ ¼ 0

we must have

A ¼ À°E k À E k 0 þ 0!Àq ÞÀ1 ; B ¼ À°E k À E k 0 À 0!q ÞÀ1 :
6.5 The attractive interaction between electrons
Mkk 0 °ay
H ¼ H0 À 1 þ aq Þc k ck 0 ;
ay 0 aq 0 y
þ c 00 c 000 :
Mk 00 k 000
E k 00 À E k 000 þ 0!Àq 0 E k 00 À E k 000 À 0!q 0 k k
k 00 ;k 000

Of the many terms in the commutator, we examine particularly the set that
arise from commuting the phonon operators. Using the fact that Mkk 0 is a
function only of k À k 0 ¼ q we ¬nd

X 0!q y y
H ¼ H0 þ jMq j2 ck 0 þq ckÀq ck ck 0
°E k À E kÀq Þ À °0!q Þ
2 2
k;k 0 ;q

þ °terms involving only two electron operatorsÞ: °6:5:4Þ

If we were now to take the expectation value of H 0 in an eigenstate of H0
we should regain expression (6.2.3). (Because of the transformation we per-
formed, the energy of the system described by H 0 is the same in ¬rst-order
perturbation theory as that of the system described by H in second order.)
The terms that are not displayed explicitly in Eq. (6.5.4) contain one cy and
one c, and have diagonal elements equal to the terms in hnk i and hnk nq i in
Eq. (6.2.3). The diagonal elements of the terms involving four electron opera-
tors have the sum shown in Eq. (6.5.1), and represent the Hartree“Fock
approximation applied to H 0 . (We recall that the direct term is absent as
there is no phonon for which q ¼ 0 in a Bravais lattice.)
The power of the method of canonical transformations which we have just
employed lies in the fact that we are not restricted to using perturbation
theory to ¬nd the eigenstates of H 0 . This is particularly important for the
theory of superconductivity which, as we shall see in the next chapter,
involves a phase transition in the electron gas brought about by attractive
electron interactions and which cannot be accounted for by perturbation
theory. Just such an attractive interaction is present in H 0 whenever
jE k À E kÀq j < 0!q . This may allow pairs of electrons to form a bound
state of lower energy than that of the two free electrons. The existence of
Cooper pairs, in which two electrons of opposite wavenumber and spin
form a bound state, provides the foundation for the BCS theory of super-
226 Electron“phonon interactions

6.6 The Nakajima Hamiltonian
While the Frohlich Hamiltonian (6.1.1) provides a useful model that exhibits
many of the interesting properties of metals, it does have the failing that the
phonon frequencies !q and the electron“phonon matrix elements, Mkk 0 , must
be assumed known. The results of the previous sections cannot then be
thought of as calculations ˜˜from ¬rst principles.™™ A better theory would
start with the Hamiltonian of a lattice of bare ions interacting through a
Coulomb potential. To this would be added the electrons, which would
interact with each other and with the ions.
The total Hamiltonian obtained in this way would di¬er from the Frohlich
Hamiltonian in a number of aspects. Firstly we should have to replace the
phonon Hamiltonian
0!q ay aq
Hp ¼ q

by the phonon Hamiltonian, Hip , for the lattice of bare ions, which, as we saw
in Chapter 1, does not support longitudinally polarized acoustic phonons.
We should have
0q Ay Aq
Hip ¼ q

with Ay the operator creating a phonon whose frequency q in a monovalent
metal approaches p , the ion plasma frequency, °4Ne2 =MÞ1=2 , for small q.
By making use of the de¬nitions (3.6.4) of both types of phonon operator we
can write Hip in terms of the operators ay and a. With neglect of the o¬-
diagonal terms and some constants we have

X °2 À !2 Þ
q q
Hip À Hp ™ 0 °2nq þ 1Þ:

The next correction to the Frohlich Hamiltonian is the replacement of
the screened matrix element, Mkk 0 , by the scattering matrix element, Mkk 0 ,
of an ion. Because of the long-range Coulomb potential due to the ion
we expect Mkk 0 to diverge as q ! 0. Finally we must add the mutual

interaction of the electrons. Our total Hamiltonian, HN , can then
be written as the Frohlich Hamiltonian, HF , plus a set of correction terms.
6.6 The Nakajima Hamiltonian
We have
X y
°Mkk 0 À Mkk 0 Þ°ay þ aq Þc k ck 0
HN ¼HF þ i
k;k 0

1X X °2 À !2 Þ
y y q q
þ Vq c kÀq c k 0 þq ck 0 ck þ 0 °2nq þ 1Þ: °6:6:1Þ
2 k;k 0 ;q 4!q

This model was studied by Nakajima.
In writing the Frohlich Hamiltonian we made the assumption that if !q
and Mkk 0 were correctly chosen, then it was permissible to ignore the e¬ects
of the correction terms in Eq. (6.6.1). If this is really to be the case then the
canonical transformation (6.5.3) must not only eliminate the interaction
terms in Eq. (6.1.1), but also the correction terms in (6.6.1). That is to say,
we must demand that in some approximation

eÀs °HN À HF Þes ¼ 0;

the operator s being the same as before. Let us write Eq. (6.6.1) as

HN ¼ HF þ HeÀp þ HeÀe þ Hp :

Then we impose the condition that all terms linear or bilinear in the phonon
operators must vanish in the expression

HeÀp þ HeÀe þ Hp þ ½HeÀp ; sŠ þ ½HeÀe ; sŠ þ ½Hp ; sŠ þ Á Á Á : °6:6:2Þ

If we neglect ½Hp ; sŠ then the two sets of terms of ¬rst order in the phonon
operators that arise in this expression are the correction to the electron“
phonon interaction, HeÀp , and the commutator, ½HeÀe ; sŠ, of the Coulomb
interaction of the electrons with s. For the latter term we can build on the
calculation of Eq. (2.7.1) in which we saw that
X y y
; °cy cp Þ
Vq 0 ckÀq 0 c k 0 þq 0 ck 0 ck pþq
k;k 0 ;q 0
Vq 0 ½°cy y y y y
¼ pþqÀq 0 cp À c pþq cpþq 0 Þ q 0 þ q 0 °c pþq 0 þq cp À c pþq cpÀq 0 ފ:

We then make the random phase approximation of retaining only those
terms containing number operators, and so we must put q 0 ¼ Æq. We are
228 Electron“phonon interactions
left with just

Vq y °np À npþq Þ:

Since we can write
X ay aq
Mp;pþq cy cp
s¼À þ ;
E p À E pþq þ 0!Àq E p À E pþq À 0!q


HeÀp þ ½HeÀe ; sŠ ¼ 0


X hnpþq À np i
À Mkk ¼ ÀMkk Vq °6:6:3Þ
Mkk 0 0 0
E pþq À E p þ 0!q

where k À k 0 ¼ q as before. In terms of the dielectric constant °q; !Þ de¬ned
in Eq. (2.7.6) and Problem 2.9 this simpli¬es to

Mkk 0
¼ ; °6:6:4Þ
Mkk 0
°q; !q Þ

which is a result we might intuitively have expected. We note that since
°q; !q Þ varies as qÀ2 for small q, the screened matrix element Mkk 0 no longer
diverges as q ! 0.
The phonon frequency !q is ¬xed by the condition that the diagonal ele-
ments of Hp are cancelled by the other terms in the transformed
Hamiltonian of order nq . This leads to the condition

Hp þ 1 ½°HeÀp þ HeÀp Þ; sŠ ¼ 0;

the term in HeÀp occurring since this part of the Frohlich Hamiltonian gives
rise to terms in nq . [These were included in the unspeci¬ed part of Eq. (6.5.4).]
We then ¬nd

X hnpþq À np i
2 À !2 1i
q q
0 ¼ À Mkk 0 Mk 0 k : °6:6:5Þ
E pþq À E p þ 0!q
4!q 2 p
6.6 The Nakajima Hamiltonian
If, following Eq. (6.1.2), we write
jk 0 À kjVkÀk 0 ;
Mkk 0 ¼ i
i i
with VkÀk 0 the Fourier transform of the Coulomb potential of a bare ion, we
can rewrite (6.6.5) as
X hnpþq À np i
N 0
2 À !2 ¼À °k À kÞ VkÀk 0 VkÀk 0
q q
E pþq À E p þ 0!q
M p
1 À °q; !q Þ
N 2 °VkÀk 0 Þ2
¼À :
°q; !q Þ
M Vq

For our monovalent metal we can put

¼ ÀVq ¼ À 2 ;
VkÀk 0
and then
4Ne2 1
!2 ¼ 2 À 1À
q q

2 p
¼ 2 À 2 þ : °6:6:6Þ
°q; !q Þ

Since for small q the phonon frequency q of the Coulomb lattice tends to p
this equation correctly predicts that !q should vanish as q ! 0. An expan-
sion of the right-hand side of Eq. (6.6.6) in even powers of q leads to a
prediction for the velocity of sound. One can write

2 q2
2 À 2 ¼ þ ÁÁÁ
p q
2 2 q2 v2
p p F
°q; !q Þ !p2

with and constants of order unity and !p the plasma frequency. Thus
0!p 2 1=2
!q ™ q vF À °6:6:7Þ
2E F

in rough agreement with the semiclassical arguments of Section 1.7.
230 Electron“phonon interactions
We note that the electron“electron interaction has not been removed from
HN by our canonical transformation, but only modi¬ed. Further transforma-
tions are necessary to exhibit explicitly such consequences of HeÀe as the
plasma oscillations.

6.1 Show that the enhancement of the electron density of states in a metal
by the electron“phonon interaction is independent of the isotopic mass
of the ions.

6.2 Use the method of canonical transformations discussed in Section 6.5 to
diagonalize the magnon“phonon system described in Problem 3.6.

6.3 Investigate to second order the T-matrix for scattering of a free electron
by a single impurity ¬xed in a crystal lattice. Assume the impurity to
participate in the motion of the lattice without perturbing the phonon
spectrum. How is the probability of elastic scattering of the electron
a¬ected by the motion of the impurity?

6.4 What errors would one make if, instead of using Nakajima™s approach,
one used a Frohlich Hamiltonian with matrix elements given by Eq.

6.5 Discuss whether the in¬nite group velocity of phonons predicted in the
theory of the Kohn anomaly violates the principle of special relativity.

6.6 Estimate the e¬ective mass of a single polaron in the model described by
the Frohlich Hamiltonian (6.1.1) with !q equal (for all q) to the ion
plasma frequency :p de¬ned in Eq. (1.5.1) and with Vk 0 k equal to
À4e2 =°k À k 0 Þ2 . Why is this an unrealistic description of a polaron
in an ionic crystal?

6.7 A one-dimensional system consists of electrons and phonons in inter-
action, and is described by a Hamiltonian of the form
X X Xy
y y
0!q ay aq þ M
H¼ E k ck ck þ °aÀq þ aq Þck ck 0 :
k;k 0

Here 0!q ¼ W sin °qa=2Þ and in the electron“phonon interaction term,
q ¼ k À k 0 . The lattice, originally of spacing a, has become dimerized by
the electron“phonon interaction, so that the spacing is now 2a. A gap
has consequently appeared in the electron energy-band structure at
k ¼ =2a, and now E k ¼ ÀA À B cos °2kaÞ in the ¬lled valence band
and E k ¼ A þ B cos °2kaÞ in the empty conduction band. The energies
A and B are both positive, and A À B ) W > 0. The matrix element M
is a constant and M 2 ¼ G=N, with N the number of atoms. Find a
condition on the magnitude of G in terms of W, A, and B in order
for the phonon energy at q ¼ =a to be reduced to zero.

6.8 Why doesn™t benzene dimerize?
(a) Consider a periodic polyacetylene chain of six monomers with lat-
tice spacing a, and evaluate the perturbed frequency of the phonon
of wavelength 2a. Determine the condition for this frequency to
vanish when the Hamiltonian is
X X Xy
°aÀq þ aq Þcy ck 0
H¼ E k c k ck þ 0!q aq aq þ M k
k;k 0

with q ¼ k À k 0 , M ¼ constant, E k ¼ A°1 À cos kaÞ, and 0!q ¼
W sin °qa=2Þ.
(b) In a completely di¬erent approach, calculate the total energy,
E elastic þ E electronic , of a static periodic polyacetylene chain of six
monomers with periodic boundary conditions. The distance
between adjacent monomers alternates between a þ u and a À u.
The Hookian spring constant for a single monomer“monomer
bond is K. The electronic energy E k is given by Eq. (4.3.2)
with g ¼ =a and Vg ¼
u. Find the condition for
in terms of
K; 0; m, and a for dimerization to be energetically favored.

6.9 A hydrogen atom may be absorbed into an interstitial position in pal-
ladium with evolution of energy E s . Make an order-of-magnitude esti-
mate of the amount by which E s is altered by the vibrational motion of
the hydrogen. [Hint: Consider the atom as a three-dimensional harmo-
nic oscillator of frequency !0 moving in a free-electron gas of Fermi
energy E F and carrying its potential Va °rÞ rigidly with it. Calculate to
second order in the electron-vibration interaction in analogy with the
treatment of the electron“phonon interaction of Section 6.1.]

6.10 Find a condition for the minimum value of the electron density in a
metal by requiring that the expression (6.6.7) for the phonon frequency
be real.
Chapter 7

7.1 The superconducting state
The discovery, in Section 6.5, of an attractive interaction between electrons in
a metal has mentally prepared us for the existence of a phase transition in the
electron gas at low temperatures. It would, however, never have prepared us
to expect a phenomenon as startling and varied as superconductivity if we
were not already familiar with the experimental evidence. The ability to pass
an electrical current without any measurable resistance has now been found
in a wide range of types of material, including simple elements like mercury,
metallic alloys, organic salts containing ¬ve- or six-membered rings of carbon
and sulfur atoms, and ceramic oxides containing planes of copper and oxygen
In this chapter we shall concentrate mainly on the simplest type of super-
conductor, typi¬ed by elements such as tin, zinc, or aluminum. The organic
superconductors and the ceramic oxides have properties that are so aniso-
tropic that the theories developed to treat elemental materials are not applic-
able. Accordingly, with the exception of the ¬nal Section 7.11, the discussion
that follows in this chapter applies only to the classic low-temperature super-
Some of the properties of these materials are shown in Fig. 7.1.1, in which
the resistivity, , speci¬c heat, C, and damping coe¬cient for phonons, , are
plotted as functions of temperature for a typical superconductor. At the
transition temperature, Tc , a second-order phase transition occurs, the
most spectacular consequence being the apparent total disappearance of
resistance to weak steady electric currents. The contribution of the electrons
to the speci¬c heat is found no longer to be proportional to the absolute
temperature, as it is in normal (i.e., nonsuperconducting) metals and in
superconductors when T > Tc , but to vary at the lowest temperatures as

7.1 The superconducting state

Figure 7.1.1. The resistivity , the electronic speci¬c heat C, and the coe¬cient of
attenuation for sound waves all change sharply as the temperature is lowered
through the transition temperature Tc .

eÀÁ=kT , with Á an energy of the order of kTc . This leads one to suppose that
there is an energy gap in the excitation spectrum “ an idea that is con¬rmed
by the absorption spectrum for electromagnetic radiation. Only when the
energy 0! of the incident photons is greater than about 2Á does absorption
occur, which suggests that the excitations that give the exponential speci¬c
heat are created in pairs.
A rod-shaped sample of superconductor held parallel to a weak applied
magnetic ¬eld H0 has the property that the ¬eld can penetrate only a short
distance  into the sample. Beyond this distance, which is known as the
penetration depth and is typically of the order of 10À5 cm, the ¬eld decays
rapidly to zero. This is known as the Meissner e¬ect and is sometimes
thought of as ˜˜perfect diamagnetism.™™ This rather misleading term refers
to the fact that if the magnetic moment of the rod were not due to currents
¬‚owing in the surface layers (the true situation) but were instead the conse-
quence of a uniform magnetization, then the magnetic susceptibility would
have to be À1=4, which is the most negative value thermodynamically
permissible. If the strength of the applied ¬eld is increased the superconduc-
tivity is eventually destroyed, and this can happen in two ways. In a type I
superconductor the whole rod becomes normal at an applied ¬eld Hc , and
then the magnetic ¬eld B in the interior of a large sample changes from zero
to a value close to H0 (Fig. 7.1.2(a)). In a type II superconductor, on the other
hand, although the magnetic ¬eld starts to penetrate the sample at an applied
¬eld, Hc1 , it is not until a greater ¬eld, Hc2 , is reached, that B approaches H0
within the rod, and a thin surface layer may remain superconducting up to a
yet higher ¬eld, Hc3 (Fig. 7.1.2(b)). For applied ¬elds between Hc1 and Hc2
234 Superconductivity

Figure 7.1.2(a). A magnetic ¬eld H0 applied parallel to a large rod-shaped sample of
a type I superconductor is completely excluded from the interior of the specimen
when H0 < Hc , the critical ¬eld, and completely penetrates the sample when
H0 > Hc .

Figure 7.1.2(b). In a type II superconductor there is a partial penetration of the
magnetic ¬eld into the sample when H0 lies between the ¬eld values Hc1 and Hc2 .
Small surface supercurrents may still ¬‚ow up to an applied ¬eld Hc3 .
7.2 The BCS Hamiltonian
the sample is in a mixed state consisting of superconductor penetrated by
threads of the material in its normal phase.
These ¬laments form a regular two-dimensional array in the plane perpen-
dicular to H0 . In many cases it is found possible to predict whether a super-
conductor will be of type I or II from measurements of Á and . One de¬nes
a coherence length 0 equal to 0vF =Á with vF the Fermi velocity. (This length
is of the order of magnitude of E F =Á times a lattice spacing.) It is those
superconductors for which  ) 0 that tend to exhibit properties of type II.

7.2 The BCS Hamiltonian
The existence of such an obvious phase transition as that involved in super-
conductivity led to a long search for a mechanism that would lead to an
attractive interaction between electrons. Convincing evidence that the elec-
tron“phonon interaction was indeed the mechanism responsible was provided
with the discovery of the isotope e¬ect, when it was found that for some
metals the transition temperature Tc was dependent on the mass A of the
nucleus. For the elements ¬rst measured it appeared that Tc was proportional
to AÀ1=2 , and hence to the Debye temperature ‚. More recent measurements
have shown a wider variety of power laws, varying from an almost complete
absence of an isotope e¬ect in osmium to a dependence of the approximate
form Tc / A2 in -uranium. It is accordingly reasonable to turn to the
Frohlich electron“phonon Hamiltonian discussed in the preceding chapter
as a simple model of a system that might exhibit superconductivity. The
canonical transformation of Section 6.5 allowed us to write the
Hamiltonian in the form of Eq. (6.5.4), which states that
X y y
H 0 ¼ H0 þ Wkk 0 q ck 0 þq;s 0 c kÀq;s ck;s ck 0 ;s 0 : °7:2:1Þ
k;s;k 0 ;s 0 ;q

Here H0 was the Hamiltonian of the noninteracting system of electrons and
phonons, and Wkk 0 q was a matrix element of the form

jMq j2 0!q
Wkk 0 q ¼ : °7:2:2Þ
°E k À E kÀq Þ2 À °0!q Þ2

The spin s of the electrons has also been explicitly included in this trans-
formed Hamiltonian.
Because this interaction represents an attractive force between electrons,
we do not expect perturbation theory to be useful in ¬nding the eigenstates of
236 Superconductivity
a Hamiltonian like Eq. (7.2.1). In fact, an in¬nitesimal attraction can change
the entire character of the ground state in a way not accessible to perturba-
tion theory. We could now turn to a variational approach, in which we make
a brilliant guess at the form the correct wavefunction jÉi will take, and then
pull and push at it until the expectation value hÉjH 0 jÉi is minimized. This is
the approach that Bardeen, Cooper, and Schrie¬er originally took, and is
described in their classic paper of 1957. We shall take a slightly di¬erent route
to the same result, and turn for inspiration to the only problem that we have
yet attempted without using perturbation theory “ the Bogoliubov theory of
helium discussed in Section 3.4.
The starting point of the Bogoliubov theory was the assumption of the
existence of a condensate of particles having zero momentum. This led to the
approximate Hamiltonian (3.4.1) in which the interactions that took place
involved the scattering of pairs of particles of equal but opposite momentum.
Now we can consider the superconducting electron system as being in some
sort of condensed phase, and it thus becomes reasonable to make the hypoth-
esis that in the superconductivity problem the scattering of pairs of electrons
having equal but opposite momentum will be similarly important. We refer
to these as Cooper pairs, and accordingly retain from the interaction in Eq.
(7.2.1) only those terms for which k ¼ Àk 0 ; in this way we ¬nd a reduced
y y y
HBCS ¼ E k cks cks À °7:2:3Þ
Vkk 0 ck 0 s 0 cÀk 0 s cÀks cks 0
ks kk s

where in the notation of Eq. (7.2.2)

Vkk 0 ¼ À2WÀk;k;k 0 Àk À Ukk 0

with Ukk 0 a screened Coulomb repulsion term which we add to the Frohlich ¨
Hamiltonian. A positive value of Vkk 0 thus corresponds to a net attractive
interaction between electrons.
One other question that must be answered before we attempt to diagona-
lize the Hamiltonian HBCS concerns the spins of the electrons: do we pair
electrons of like spin (so that in Eq. (7.2.3) we put s ¼ s 0 ) or of opposite spin?
The answer is that to minimize the ground-state energy it appears that we
must pair electrons of opposite spin. We shall assume this to be the case, and
leave it as a challenge to the sceptical reader to ¬nd a wavefunction that leads
to a lower expectation value of Eq. (7.2.3) with s ¼ s 0 than we shall ¬nd when
s and s 0 represent spins in opposite directions. With this assumption we can
7.3 The Bogoliubov“Valatin transformation
perform the sum over s in Eq. (7.2.3). Since
X y y y y y y
c k 0 s 0 c Àk 0 s cÀks cks 0 ¼ ck 0 # cÀk 0 " cÀk" ck# þ c k 0 " c Àk 0 # cÀk# ck"
y y y y
¼ ck 0 # cÀk 0 " cÀk" ck# þ c Àk 0 # ck 0 " ck" cÀk#

and Vkk 0 ¼ VÀk;Àk 0 the summation over s is equivalent to a factor of 2. This
allows us to abbreviate further the notation of Eq. (7.2.3) by adopting the
convention that an operator written with an explicit minus sign in the sub-
script refers to a spin-down state while an operator without a minus sign
refers to a spin-up state. Thus

cy 0  cy 0 " ; cy 0  cy 0 Þ# ; etc:
Àk °Àk
k k

We then have
y y y y
HBCS ¼ E k °c k ck þ cÀk cÀk Þ À Vkk 0 c k 0 c Àk 0 cÀk ck : °7:2:4Þ
k k;k

This is the model Hamiltonian of Bardeen, Cooper, and Schrie¬er, of which
the eigenstates and eigenvalues must now be explored.

7.3 The Bogoliubov“Valatin transformation
In the Bogoliubov theory of helium described in Section 3.4 it was found to
be possible to diagonalize a Hamiltonian that contained scattering terms like
ak aÀk by means of a transformation to new operators

k ¼ °cosh k Þak À °sinh k Þay :

The k and their conjugates k were found to have the commutation relations
of boson operators, and allowed an exact solution of the model Hamiltonian
(3.4.1). This suggests that we try a similar transformation for the fermion
problem posed by HBCS , and so we de¬ne two new operators

k ¼ uk ck À vk cy ;
Àk ¼ uk cÀk þ vk ck °7:3:1Þ

with conjugates
y y

y ¼ uk cy þ vk ck :

k ¼ uk c k À vk cÀk ; °7:3:2Þ
Àk Àk
238 Superconductivity
The constants uk and vk are chosen to be real and positive and to obey the

u2 þ v2 ¼ 1
k k

in order that the new operators have the fermion anticommutation relations
k ;
k 0 g ¼ f
k ;
Àk 0 g ¼ f
k ;
Àk 0 g ¼ 0
k ;
k 0 g ¼ f
y ;
Àk 0 g ¼ kk 0 ;

as was veri¬ed in Problem 2.4. Equations (7.3.1) and (7.3.2) comprise the
Bogoliubov“Valatin transformation, which allows us to write the BCS
Hamiltonian in terms of new operators. We do not expect to be able to
diagonalize HBCS completely, as this Hamiltonian contains terms involving
products of four electron operators, and is intrinsically more di¬cult than
Eq. (3.4.1); we do, however, hope that a suitable choice of uk and vk will
allow the elimination of the most troublesome o¬-diagonal terms.
We rewrite the BCS Hamiltonian by ¬rst forming the inverse transforma-
tions to Eqs. (7.3.1) and (7.3.2). These are
ck ¼ uk
k þ vk
y ; cÀk ¼ uk
Àk À vk
k °7:3:3Þ

y y
cy ¼ uk
y À vk
k :
ck ¼ uk
k þ vk
Àk ; °7:3:4Þ
Àk Àk

The ¬rst part of Eq. (7.2.4) represents the kinetic energy HT , and on sub-
stitution from Eqs. (7.3.3) and (7.3.4) is given by
X y y
E k ½u2
k þ v2
y þ uk vk °
y þ
k Þ
HT ¼ Àk Àk
k k
y y
þ v2
k þ u2
Àk À uk vk °
k þ
Àk ފ:
k Àk Àk

The diagonal parts of this expression can be simpli¬ed by making use of the
anticommutation relations of the
™s and de¬ning a new pair of number
mÀk ¼
Àk :
mk ¼
k ; Àk

X y
E k ½2v2 þ °u2 À v2 Þ°mk þ mÀk Þ þ 2uk vk °
y þ
k ފ:
HT ¼ °7:3:5Þ
k k k
7.3 The Bogoliubov“Valatin transformation
We note here the presence of three types of term “ a constant, a term contain-
ing the number operators mk and mÀk , and o¬-diagonal terms containing the
Àk or
k . The potential energy HV is given by the second part
of HBCS and leads to a more complicated expression. We ¬nd
X y y
HV ¼ À Vkk 0 °uk 0
k 0 þ vk 0
Àk 0 Þ°uk 0
Àk 0 À vk 0
k 0 Þ
k;k 0
‚ °uk
Àk À vk
k Þ°uk
k þ vk
y Þ Àk
¼À Vkk 0 ½uk 0 vk 0 uk vk °1 À mk 0 À mÀk 0 Þ°1 À mk À mÀk Þ
k;k 0
þ uk 0 vk 0 °1 À mk 0 À mÀk 0 Þ°u2 À v2 Þ°
k þ
y ފ
k k

þ °fourth-order off-diagonal termsÞ: °7:3:6Þ

We now argue that if we can eliminate the o¬-diagonal terms in HBCS by
ensuring that those in Eq. (7.3.5) are cancelled by those in Eq. (7.3.6), then we
shall be left with the Hamiltonian of a system of independent fermions. We
¬rst assume that the state of this system of lowest energy has all the occupa-
tion numbers mk and mÀk equal to zero; this assumption may be veri¬ed at a
later stage of the calculation. To ¬nd the form of the Bogoliubov“Valatin
transformation that is appropriate to a superconductor in its ground state we
then let all the mk and mÀk vanish in Eqs. (7.3.5) and (7.3.6) and stipulate
that the sum of o¬-diagonal terms also vanish. We ¬nd
y y
2E k uk vk °
y þ
k Þ À Vkk 0 uk 0 vk 0 °u2 À v2 Þ°
y þ
k Þ
Àk Àk
k k
k;k 0

þ °fourth-order termsÞ ¼ 0: °7:3:7Þ

If we make the approximation that the fourth-order terms can be neglected
(Problem 7.6) then this reduces to
2E k uk vk À °u2 À v2 Þ Vkk 0 uk 0 vk 0 ¼ 0: °7:3:8Þ
k k

Because uk and vk , being related by the condition that u2 þ v2 ¼ 1, are not
k k
independent it becomes convenient to express them in terms of a single vari-
able xk , de¬ned by

uk ¼ ° 1 À xk Þ1=2 ; vk ¼ ° 1 þ xk Þ1=2 : °7:3:9Þ
2 2
240 Superconductivity
Then Eq. (7.3.8) becomes
2E k ° 1 À x2 Þ1=2 þ 2xk Vkk 0 ° 1 À x2 0 Þ1=2 ¼ 0: °7:3:10Þ
k k
4 4

If we de¬ne a new quantity Ák by writing
Ák ¼ Vkk 0 ° 1 À x2 0 Þ1=2 °7:3:11Þ

then Eq. (7.3.10) leads to the result
xk ¼ Æ : °7:3:12Þ
2°E 2 þ Á2 Þ1=2
k k

Substitution of this expression in Eq. (7.3.11) gives an integral equation for
Ák of the form
1X Ák 0
Ák ¼ : °7:3:13Þ
Vkk 0 2
°E k 0 þ Á2 0 Þ1=2
2 k0 k

If Vkk 0 is known then this equation can in principle be solved and re-
substituted in Eq. (7.3.12) to give xk . In doing so we once more note that
the zero of energy of the electrons must be chosen to be the chemical poten-
tial  if the total number of electrons is to be kept constant. To see this we
note that
°c k ck þ cy cÀk Þ
N¼ Àk
X y
½2v2 þ °u2 À v2 Þ°mk þ mÀk Þ þ 2uk vk °
y þ
k ފ;
¼ Àk
k k k

and so the expectation value of N in the ground state of the system is just
hNi ¼ 2vk
¼ °1 þ 2xk Þ: °7:3:14Þ

In the absence of interactions
hNi ¼ 2;

and so we deduce that xk ¼ 1 if E k  and xk ¼ À 1 if E k > , as illustrated
2 2
7.3 The Bogoliubov“Valatin transformation
in Fig. 7.3.1(a). This shows that xk is an odd function of E k À  in the
noninteracting case, and that if we make sure xk remains an odd function
of E k À  in the presence of interactions, then Eq. (7.3.14) tells us that hNi
will be unchanged (the energy dependence of the density of states being
neglected). We also want the form of xk given by Eq. (7.3.12) to reduce to
the free-electron case when Vkk 0 vanishes, and so we choose the negative
square root and obtain a form for vk and xk like that shown in Fig.
7.3.1(b). To remind ourselves that E k is measured relative to  we use the
symbol E k ¼ E k À  to rewrite Eq. (7.3.12) as
xk ¼ À :
2°E 2 þ Á2 Þ1=2

Figure 7.3.1. In this diagram we compare the form that the functions vk and xk take
in a normal metal at zero temperature (a) and in a BCS superconductor (b).
242 Superconductivity

Figure 7.3.2. This simple model for the matrix element of the attractive interaction
between electrons was used in the original calculations of Bardeen, Cooper, and

To make these ideas more explicit we next consider a simple model that
allows us to solve the integral equation for Ák exactly. The matrix element
Vkk 0 has its origin in the electron“phonon interaction, and, as Eq. (7.2.2)
^ ^
indicates, is only attractive when jE k À E k 0 j is less than the energy 0!q of
the phonon involved. In the simple model ¬rst chosen by BCS the matrix
element was assumed to be of the form shown in Fig. 7.3.2, in which
¼ V if jE k j < 0!D
Vkk 0
¼ 0 otherwise;

with V a constant and 0!D the Debye energy. It then follows that Ák is also a
constant, since Eq. (7.3.13) reduces to
11 Ák 0
Ák ¼ D°E k 0 Þ dE k 0 2 Vkk 0
°E k 0 þ Á2 0 Þ1=2
2 À1 k
° 0!D
1 ^
¼ VD°Þ d E;
^2 þ Á2 Þ1=2
À0!D °E

the energy density of states D°EÞ here referring to states of one spin only and
again being taken as constant. This has the solution
Á¼ : °7:3:16Þ

The magnitude of the product VD°Þ can be estimated by noting that from
Eq. (7.2.2)

jMq j2
7.4 The ground-state wavefunction and the energy gap
and from Eq. (6.1.2)

jMq j $
jVk j2
$ NjVk j2
M 0!D

with Vk the Fourier transform of a screened ion potential. Since D°Þ $ N=
we ¬nd that
m NVk 2
VD°Þ $ :
M 0!D

While 0!D might typically be 0.03 eV, the factor NVk is something like the
average of the screened ion potential over the unit cell containing the ion, and
might have a value of a few electron volts, making °NVk =0!D Þ2 of the order
of 104 . The ratio of electron mass to ion mass, m=M, however, is only of the
order of 10À5 , and so it is in most cases reasonable to make the approxima-
tion of weak coupling, and replace Eq. (7.3.16) by

Á ¼ 20!D eÀ1=VD°Þ ; °7:3:17Þ

the di¬erence between 2 sinh (10) and e10 being negligible. In strong-coupling
superconductors such as mercury or lead, however, the electron“phonon
interaction is too strong for such a simpli¬cation to be valid; for these metals
the rapid damping of the quasiparticle states must also be taken into account.
The important fact that this rough calculation tells us is that Á is a very
small quantity indeed, being generally about one percent of the Debye
energy, and hence corresponding to thermal energies at temperatures of the
order of 1 K. The parameter xk thus only di¬ers from Æ 1 within this short
distance of the Fermi energy and our new operators
k and
Àk reduce to
simple electron annihilation or creation operators everywhere except within
this thin shell of states containing the Fermi surface.

7.4 The ground-state wavefunction and the energy gap
Our ¬rst application of the Bogoliubov“Valatin formalism must be an
evaluation of the ground-state energy E S of the superconducting system.
We hope to ¬nd a result that is lower than E N , the energy of the normal
system, by some amount which we shall call the condensation energy E c . The
244 Superconductivity
ground-state energy of the BCS state is given by the sum of the expectation
values of Eqs. (7.3.5) and (7.3.6) under the conditions that mk ¼ mÀk ¼ 0. As
we have already eliminated the o¬-diagonal terms we are only left with the
constant terms, and ¬nd
^k v2 À
ES ¼ 2E k Vkk 0 uk 0 vk 0 uk vk
k;k 0
¼ E k °1 þ 2xk Þ À Vkk 0 ½° 1 À x2 0 Þ° 1 À x2 ފ1=2 : °7:4:1Þ
4 4
k;k 0

It is interesting to pause at this point and note that we could have considered
the BCS Hamiltonian from a variational point of view. Instead of eliminating
the o¬-diagonal elements of HBCS we could have decided to choose the xk in
such a way as to minimize E S . It is reassuring to see that this approach leads
to the same solution as before; if we di¬erentiate Eq. (7.4.1) with respect to xk
and equate the result to zero we obtain an equation that is identical with Eq.
(7.3.10). We write expression (7.4.1) as
ES ¼ ½E k °1 þ 2xk Þ À ° 1 À x2 Þ1=2 ÁŠ

with xk and Á de¬ned as before in Eqs. (7.3.12) and (7.3.11). In the normal
system x2 ¼ 1 for all k and so the condensation energy, de¬ned as E S À E N , is
k 4
^k °2xk À 1Þ þ ^k °2xk þ 1Þ À
Ec ¼ E E °1 À x2 Þ1=2 Á
k<kF k>kF

° 0!D  
2E k þ Á2
^ ^
¼ 2D°Þ Ek À d Ek
2°E k þ Á2 Þ1=2

¼ D°Þf°0!D Þ2 À 0!D ½°0!D Þ2 þ Á2 Š1=2 g
¼ °0!D Þ2 D°Þ 1 À coth :

In the weak-coupling case this becomes

E c ™ À2°0!D Þ2 D°ÞeÀ2=VD°Þ
¼ À 1 D°Þ Á2 : °7:4:2Þ

This condensation energy is surprisingly small, being of the order of only
10À7 eV per electron, which is the equivalent of a thermal energy of about a
millidegree Kelvin. This is a consequence of the fact that only the electrons
7.4 The ground-state wavefunction and the energy gap
with energies in the range  À Á to  þ Á are a¬ected by the attractive
interaction, and these are only a small fraction of the order of Á= of the
whole. We note that we cannot expand E c in a power series in the interaction
strength V since the function exp½À2=VD°ފ has an essential singularity at
V ¼ 0, which means that while the function and all its derivatives vanish as
V ! þ0, they all become in¬nite as V ! À0. This shows the qualitative
di¬erence between the e¬ects of an attractive and a repulsive interaction,
and tells us that we could never have been successful in calculating E c by
using perturbation theory.
The wavefunction É0 of the superconducting system in its ground state
may be found by recalling that it must be the eigenfunction of the diagona-
lized BCS Hamiltonian that has mk ¼ mÀk ¼ 0 for all k, so that

k jÉ0 i ¼
Àk jÉ0 i ¼ 0: °7:4:3Þ

Now since
k ¼
Àk ¼ 0 we can form the wavefunction that satis¬es Eq.
(7.4.3) simply by operating on the vacuum state with all the
k and all the

Àk . From Eq. (7.3.1) we have
Y  Y 

Àk j0i ¼ °uk ck À vk c Àk Þ°uk cÀk þ vk c k Þ j0i
k k
°uk vk þ v2 cy cy Þ j0i:
¼ k k Àk

To normalize this we divide by the product of all the vk to obtain
jÉ0 i ¼ °uk þ vk c k c Àk Þ j0i: °7:4:4Þ

This wavefunction is a linear combination of simpler wavefunctions contain-
ing di¬erent numbers of particles, which means that it is not an eigenstate of
the total number operator N. Our familiarity with the concept of the chemi-
cal potential  teaches us not to be too concerned about this fact, however, as
long as we make sure that the average value is kept constant.
The quasiparticle excitations of the system are created by the operators
y acting on É0 . By adding Eqs. (7.3.5) and (7.3.6) one can write the
Hamiltonian in the form
HBCS ¼ E S þ °mk þ mÀk Þ °uk À vk ÞE k þ 2uk vk
Vkk 0 uk 0 vk 0

þ higher-order terms;
246 Superconductivity
which on substitution of our solution for uk and vk becomes
HBCS ¼ E S þ °E k þ Á2 Þ1=2 °mk þ mÀk Þ þ Á Á Á :

The energies Ek of these elementary excitations are thus given by
Ek ¼ °E 2 þ Á2 Þ1=2 : °7:4:5Þ

These excitations cannot be created singly, for that would mean operating on
É0 with a single
y , which is a sum containing just one c and one cy . Now any
physical perturbation that we apply to É0 will contain at least two electron
operators, since such perturbations as electric and magnetic ¬elds act to
scatter rather than to create or destroy electrons. For instance
y y y
c k ck 0 jÉ0 i ¼ °uk
k þ vk
Àk Þ°uk 0
k 0 þ vk 0
Àk 0 ÞjÉ0 i
¼ uk vk 0
Àk 0 jÉ0 i;

the other terms vanishing. We thus conclude that only pairs of quasiparticles
can be excited, and that from Eq. (7.4.5) the minimum energy necessary to
create such a pair of excitations is 2Á. This explains the exponential form of
the electronic speci¬c heat at low temperatures and also the absorption edge
for electromagnetic radiation at 0! ¼ 2Á.
It is interesting to compare these quasiparticle excitations with the particle“
hole excitations of a normal Fermi system. In the noninteracting electron gas
at zero temperature the operator ck ck 0 creates a hole at k 0 and an electron at
^ ^ ^ ^
k provided E k 0 < 0 and E k > 0. The energy of this excitation is E k À E k 0 ,
^ ^
which can be written as jE k j þ jE k 0 j, and is thus equal to the sum of the
lengths of the arrows in Fig. 7.4.1(a). In the superconducting system the
y yy
operator ck ck 0 has a component
Àk 0 , which creates an excitation of total
energy Ek þ Ek 0 equal to the sum of lengths of the arrows in Fig. 7.4.1(b).
The density of states is inversely proportional to the slope of E °kÞ in the
normal metal, and this leads us to think of an e¬ective density of states in
the superconductor inversely proportional to dE=djkj. As Ek and E k are
related by Eq. (7.4.5) we ¬nd this e¬ective density of states to be equal to
D°EÞ ¼ 2D°EÞ
< 2D°Þ if jEj > Á
°E 2 À Á2 Þ1=2
™ °7:4:6Þ
if jEj Á:
7.5 The transition temperature

Figure 7.4.1. In a normal metal (a) an electron“hole pair has an excitation energy
equal to the sum of the lengths of the two arrows in the left-hand diagram; the
density of states is inversely proportional to the slope of E °kÞ and is thus roughly
constant. In a superconductor (b) the energy of the excitations is the sum of the
lengths of the arrows when Ek is plotted; an e¬ective density of states can again be
drawn which is inversely proportional to the slope of Ek , as shown on the right.

The factor of 2 enters when D°Þ is the normal density of states for one spin
direction because the states
k jÉ0 i and
y jÉ0 i are degenerate.

7.5 The transition temperature
Our method of diagonalizing HBCS in Section 7.3 was to ¬x uk and vk so that
the sum of o¬-diagonal terms from Eqs. (7.3.5) and (7.3.6) vanished. The
resulting equation was of the form
2E k uk vk À Vkk 0 uk 0 vk 0 °1 À mk 0 À mÀk 0 Þ°u2 À v2 Þ
k k
‚ °
y þ
k Þ ¼ 0; °7:5:1Þ
248 Superconductivity
and to solve this we ¬rst put mk 0 ¼ mÀk 0 ¼ 0. While this approach remains
valid in the presence of a few quasiparticle excitations it clearly needs mod-
i¬cation whenever the proportion of excited states becomes comparable to
unity, for then the terms in mk 0 and mÀk 0 will contribute signi¬cantly to the
summation over k 0 . This will be the case if the temperature is such that kT is
not much less than Á.
We resolve this di¬culty by ¬rst of all assuming that it is possible to
eliminate these o¬-diagonal terms. We are then left with a Hamiltonian
that is the sum of the diagonal parts of Eqs. (7.3.5) and (7.3.6), so that
^k ^
HBCS ¼ 2E k v2 þ °u2 À v2 ÞE k °mk þ mÀk Þ
k k
k k
À Vkk 0 uk 0 vk 0 uk vk °1 À mk 0 À mÀk 0 Þ°1 À mk À mÀk Þ: °7:5:2Þ

The energy Ek necessary to create a quasiparticle excitation will be

@hHBCS i
Ek ¼
@hmk i
¼ E k °u2 À v2 Þ þ 2uk vk Vkk 0 uk 0 vk 0 °1 À hmk 0 i À hmÀk 0 iÞ: °7:5:3Þ

Now if we had a system of independent fermions at temperature T then we
should know from Fermi“Dirac statistics just what the average occupancy of
each state would be; from Eq. (3.3.3) we could immediately write

" "
mk ¼ mÀk ¼ : °7:5:4Þ
exp °Ek =kTÞ þ 1


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