Elements of Modern Physics
240
The theory of superconductivity was given by Barden, Cooper and
Schrieffer, and is known as the BCS theory. In this theory the electrons experience
a special kind of mutual attraction which at large distances dominates over the
Coulomb repulsion between them. It is the lattice of the material which provides
the necessary medium for producing the attractive forces. An electron moving
in the metal disturbs the lattice, producing phonons (quanta of vibrational
motion). These phonons may be absorbed by another electron which may, at
the microscopic level, be far away (about 10
–6
m) from the first electron. In
effect, the two electrons interact and the electron-electron interaction energy
due to a phonon exchange is negative. If at low temperatures this attraction
exceeds the Coulombic repulsion, the two electrons form a weakly bound state
called a Cooper pair (a Cooper pair has an energy of about 10
–3
eV in
superconductors). If such pairs are created, the conductor becomes a
superconductor.
Cooper pairs are spin-zero bosons and hence can be in the same state. They
are in the ground state and are described by a wave function which extends over
the entire body of the metal. In a sense, this is quantum mechanics on a
macroscopic scale.
The Energy Gap
The specific heat of a superconductor at very low temperatures, is observed to
be of the form
C v =
3
– /
b kT
T
A
a e
+
θ
(7.107)
where the first term is the lattice specific heat [Eq. (7.67)]. The second term is
the electronic contribution to specific heat. The exponential form of this term
suggests the presence of an energy gap [see Eq. (7.82) which indicates that if
there is an energy gap ∆, the leading behaviour of the total energy of bosons at
low temperatures is given by E ~ e
–∆/kT
]. The energy gap comes from the binding
energy of Cooper pairs. Since energy is required to break the bond in Cooper
pairs, a sharp jump in the specific heat is observed at the transition temperature.
Furthermore, when T~ T c , there is a substantial number of electrons in the normal
state so that the energy gap is
∆ ~ kT c
(7.108)
The detailed BCS theory gives the binding energy of the Cooper pair as
E b (T) ≈ 3.5 kT c for T → 0
→ 0 for T → T c
(7.109)
and this is the energy gap between the ground state and the dissociated state.
For T c = 4K, and energy gap of the order of 3 × 10
–4
eV is obtained. The smallness
240
The theory of superconductivity was given by Barden, Cooper and
Schrieffer, and is known as the BCS theory. In this theory the electrons experience
a special kind of mutual attraction which at large distances dominates over the
Coulomb repulsion between them. It is the lattice of the material which provides
the necessary medium for producing the attractive forces. An electron moving
in the metal disturbs the lattice, producing phonons (quanta of vibrational
motion). These phonons may be absorbed by another electron which may, at
the microscopic level, be far away (about 10
–6
m) from the first electron. In
effect, the two electrons interact and the electron-electron interaction energy
due to a phonon exchange is negative. If at low temperatures this attraction
exceeds the Coulombic repulsion, the two electrons form a weakly bound state
called a Cooper pair (a Cooper pair has an energy of about 10
–3
eV in
superconductors). If such pairs are created, the conductor becomes a
superconductor.
Cooper pairs are spin-zero bosons and hence can be in the same state. They
are in the ground state and are described by a wave function which extends over
the entire body of the metal. In a sense, this is quantum mechanics on a
macroscopic scale.
The Energy Gap
The specific heat of a superconductor at very low temperatures, is observed to
be of the form
C v =
3
– /
b kT
T
A
a e
+
θ
(7.107)
where the first term is the lattice specific heat [Eq. (7.67)]. The second term is
the electronic contribution to specific heat. The exponential form of this term
suggests the presence of an energy gap [see Eq. (7.82) which indicates that if
there is an energy gap ∆, the leading behaviour of the total energy of bosons at
low temperatures is given by E ~ e
–∆/kT
]. The energy gap comes from the binding
energy of Cooper pairs. Since energy is required to break the bond in Cooper
pairs, a sharp jump in the specific heat is observed at the transition temperature.
Furthermore, when T~ T c , there is a substantial number of electrons in the normal
state so that the energy gap is
∆ ~ kT c
(7.108)
The detailed BCS theory gives the binding energy of the Cooper pair as
E b (T) ≈ 3.5 kT c for T → 0
→ 0 for T → T c
(7.109)
and this is the energy gap between the ground state and the dissociated state.
For T c = 4K, and energy gap of the order of 3 × 10
–4
eV is obtained. The smallness