^{1}

^{2}

^{3}

A quantum statistical theory of the superconductivity in MgB
_{2} is developed regarding it as a member of the graphite intercalation compound. The superconducting temperature
T_{c} for MgB
_{2}, C
_{8}K ≡ KC
_{8}, CaC
_{6}, are 39 K, 0.6 K, 11.5 K, respectively. The differences arise from the lattice structures. In the plane perpendicular to the c-axis, B’s form a honeycomb lattice with the nearest neighbour distance
while Mg’s form a base-hexagonal lattice with the nearest neighbour distance
above and below the B-plane distanced by
. The more compact B-plane becomes superconducting due to the electron-phonon attraction. Starting with the generalized Bardeen- Cooper-Schrieffer (BCS) Hamiltonian and solving the generalized Cooper equation, we obtain a linear dispersion relation
for moving Cooper pairs. The superconducting temperature
T_{c} identified as the Bose-Einstein condensation temperature of the Cooper pairs in two dimensions is given by
, where
is the Cooper pair density,
the Boltzmann constant. The lattices of KC
_{8} and CaC
_{6}
are clearly specified.

Nagamatsu et al. [_{2}. MgB_{2} forms a lattice closely related to that of a graphite intercalation compound (GIC). It is similar to NaC_{2} composition- wise, but the lattice structures are distinct as shown below. The superconducting temperatures _{2} and NaC_{2} are 39 K and 5 K, respectively. This difference arises from the lattice structures. Canfield and Crabtree [_{2} is shown in Ref. 2, _{2} [_{2}. We propose a different lattice. The two lattices have the same first neighbour configurations but different second nearest neighbours. Our proposed lattice has a lower Coulomb energy and should be realized in practice.

We shall develop a quantum statistical theory of the superconductivity in MgB_{2}, starting with a generalized Bardeen-Cooper-Schrieffer (BCS) Hamiltonian [^{+} in our lattice is surrounded by six Mg^{+}’s while each B^{+} in Canfield-Crabtree’s lattice is surrounded by three Mg^{+}. Hence our lattice is more stable. In the course of the development, we clearly specify the lattices of C_{8}K º KC_{8} and CaC_{6}.

Following Ashcroft and Mermin (AM) [

where

where

is the electron velocity, where

If the electron is in a continuous energy range (energy band), then it will be accelerated by the electric force

where n is the electron density and

We consider a graphene which forms a 2D honeycomb lattice. The Wigner-Seitz (WS) unit cell, a rhombus, contains two C’s. We showed in our earlier work [

Hence, “electrons” are the majority carriers in graphene. The thermally activated electron densities are given by

where

Graphite is composed of graphene layers stacked in the manner ABAB∙∙∙ along the c-axis. We may choose an orthogonal unit cell shown in

The carbons (circles) in the A (B) planes are shown in dark (light) gray circles.

The unit cell contains 16 C’s. The two rectangles (white solid lines) are stacked vertically with the interlayer separation,

The unit cell has three side-lengths:

Clearly, the system is periodic along the orthogonal directions with the three periods

The negatively charged “electron” (with the charge^{+} when moving vertically up or down in the plane. Then, the easy direction for the “electrons” is vertical. The easy direction for the “holes” is horizontal. There are no hindering hills for “holes” moving horizontally. Hence, the “electron” in graphite has the lower activation energy

We now consider GIC. Let us first take C_{8}K. The K^{+} ions should enter as interstitials and occupy the sites away from the positive ions C^{+}. We see in ^{+} should occupy the midpoint between two graphene layers. The 3D unit cell contains 16 C’s and 2 K’s. Alkali metal GIC, including C_{8}Li, C_{8}Rb, should form similar lattices. Next we condier C_{6}Ca. Carbons (C) in graphite form a honeycomb lattice in the A plane as shown in _{6}Ca. The composition ratio 6:1 is correct. After the C-filling, the C-plane becomes primitive (base)-hexagonal and has a 60˚ rotation symmetry. The primitive unit cell contains six (6) C’s. Two Ca’s are likely to occupy below the centers of the primitive cells located at the two-light gray circles in _{6}Ca is obtained by stacking the C_{6}Ca sheets in the manner ABAB… We note that the structure of C_{6}Ca is significantly more compact than that of C_{8}K. C_{6}Yb should have a similar lattice structure. GIC C_{4}Na (C_{3}K, C_{2}Na) should have the same 12 C-sheets and 3 Na (4 K, 6 Na) intersheets.

Consider now MgB_{2}. It is only natural to start with the B-plane since this plane becomes superconducting at 0 K. Let us look at the top sheet in _{2} is more compact with the smaller lattice constant, the B-plane is likely to become superconducting at the lowest temperatures. Note that all ions position are specified. Ions Mg^{+} and B^{+} are positively charged so that they tend to stay away among and between them.

Our lattice and Canfield-Crabtree’s are different in the second nearest neighbour configuration. Each B^{+} in our lattice is surrounded by six Mg^{+} while each B^{+} in Canfield-Crabtree’s lattice is surrounded by three Mg^{+}. Hence our lattice is more stable. The B-plane contains a honeycomb lattice of B's for both. Our Mg-plane contains a base-hexagonal lattice of the nearest neighbour distance_{2}.

The countability and statistics of the fluxons (magnetic flux quanta) are the fundamental particle properties. We postulate that the fluxon is a half-spin fermion with zero mass and zero charge.

We assume that the magnetic field

with the states

A longitudinal phonon, acoustic or optical, generates a charge density wave, which affects the electron (fluxon) motion through the charge displacement (current). Let us first consider the case of superconductivity. The phonon exchange between two electrons shown in

where

An electric current

where

interaction with the energy denominators

(11). The interaction is attractive (negative) and most effective when the states before and after the exchange have the same energy

BCS [

where

The fluxon number operator

The phonon exchange can create electron-fluxon composites, bosonic or fermionic, depending on the number of fluxons. The CM of any composite moves as a fermion (boson) if it contains an odd (even) numbers of elementary fermions. The electron (hole)-type c-particles carry negative (positive) charge. Electron (hole)-type Cooper-pair-like c-bosons are generated by the phonon-exchange attraction from a pair of electron (hole)-type c-fermions. The pair operators B are defined by

The prime on the summation in Equation (12) means the restriction:

The pairing interaction terms in Equation (12) conserve the charge. The term

pairing strength, generates a transition in electron-type c-particle states. Similarly, the exchange of a phonon

generates a transition between hole-type c-particle states, represented by

can also pair-create (pair-annihilate) electron (hole)-type c-boson pairs, and the effects of these processes are

represented by

The Cooper pair, also called the pairon, is formed from two “electrons” (or “holes”). The pairons move as bosons, which are shown in Appendix. Likewise the c-bosons may be formed by the phonon-exchange attraction from two like-charge c-fermions. If the density of the c-bosons is high enough, then the c-bosons will be Bose- condensed and exhibit a superconductivity.

The pairing interaction terms in Equation (12) are formally identical with those in the generalized BCS Hamiltonian [

The c-bosons, having the linear dispersion relation, can move in all directions in the plane with the constant speed

is given by

where

where

Hence we obtain the Hall resistivity as

For the integer QHE at

The supercurrent generated by equal numbers of ± c-bosons condensed monochromatically is neutral. This is reflected in our calculations in Equation (18). The supercondensate whose motion generates a supercurrent must be neutral. If it has a charge, it would then be accelerated indefinitely by any external electric field because the impurities and phonons cannot stop the supercurrent to grow. That is, the circuit containing a superconducting sample and a battery must be burnt out if the supercondensate is not neutral. In the calculation of

Equation (21), we used the unaveraged drift velocity

averaged drift velocity cancels out

We now extend our theory to include elementary fermions (electron, fluxon) as members of the c-fermion set. We can then treat the 2D superconductivity and the QHE in a unified manner. The c-boson containing one electron and one fluxon can be used to describe the principal QHE. Important pairings and effects are listed below: a) a pair of conduction electrons, superconductivity; b) c-fermions and fluxon, QHE; c) a pair of like- charge conduction electrons with two fluxons, QHE in graphene.

The conduction electrons (“electrons”, “holes”) are excited based on the orthogonal unit cells. As mentioned earlier the “electrons” are the majority carriers in both graphene and graphite. The excitation energy for the “electrons” is smaller than for the “holes”. Phonons are generated based on the same orthogonal unit cells. Phonons are bosons, and hence can be generated with no activation energies. The phonons are distributed, following the Planck distribution function:

which is a sole function of the Kelvin temperature T.

As an example consider acoustic phonons with a linear dispersion relation:

where s is the sound speed. The phonon size may be characterized by the average wave length:

The average size of phonons at the room temperature is greater by a few orders of magnitudes than the electron size.

Cooper solved the Cooper equation [Ref. 12, Equation (1)] with a negative interaction energy constant,

where

The superconductivity occurs only in regular crystals. That is, it occurs only in crystals and not in liquids. C_{8}K and graphene have a 120˚ rotation symmetry. C_{6}Ca has a base-hexagonal (60˚ rotation) symmetry.

C_{8}K has graphene sheets, and each sheet is likely to become superconducting below the critical temperatures_{8}K and C_{6}Ca, the critical temperatures should be different significantly.

BCS [

constructed a ground-state vector and obtained a ground-state energy of an electron-phonon system:

where N is the pairon number per spin and

which is shown below.

The number operator in the k-q representation

has eigenvalues 0 or 1: [

The total number of a system of pairons, N, is represented by

where

represents the number of pairons having net momentum

and obtain, after simple calculations,

Although the occupation number

The present author’s group [

where

is greater several times than the BCS coherence length (pairon size):

where

Thus 2D pairons do not overlap in space. Hence the

Formula (25) is distinct from the BCS formula in the weak coupling limit:

Our Formula (35) obtained after identifying superconducting temperature as the BEC condensation temperature contains familiar quantities, the Fermi speed

For illustration let us take GaAs/AlGaAs. We assume

The neutral supercondensate is generated from the two ranges of energies of “electrons” and “holes”. Hence it is difficult to precisely determine the critical temperature from the theoretical consideration alone. The com- parison between theory and experiment may be carried out as follows. First we find the Fermi speed

which indicates a close connection between the zero temperature gap

We have developed a theory regarding MgB_{2} as a member of GIC. We start with the lattice configuration with all ions locations specified, and find that each B-plane contains B’s forming a honeycomb lattice of the nearest neighbour distance (lattice constant)

We obtain a linear dispersion relation

calculated without introducing the averaging. The superconducting energy gap

Canfield and Crabtree have discussed two energy gaps, which is strange since there is one superconducting state at 0 K. We shall discuss this topic in a separate publication.

S. Fujita,A. Suzuki,Y. Takato, (2016) Quantum Statistical Theory of Superconductivity in MgB_{2}. Journal of Modern Physics,07,1546-1557. doi: 10.4236/jmp.2016.712141

We consider the case of a 2D superconductor. The phonon exchange attraction is in action for any pair of electrons near the Fermi surface. In general the bound pair has a net momentum, and hence, it moves. Such a pair is called a moving pairon. The energy

which is Cooper’s equation in 2D, Equation (1) of his 1956 Physical Review paper [

Equation (41) can be solved simply. We briefly review the calculations and results here. We assume that the energy

Then,

where

is k-independent. Introducing Equation (43) in Equation (44), and dropping the common factor

We now assume a free-electron model in 2D. The Fermi surface is a circle of the radius (momentum)

where

The prime on the k-integral in Equation (45) means the restriction:

We may choose the z-axis along

The k-integral in Equation (45) can then be expressed by

where

After performing the integration and taking the small-q and small-

where

is the pairon ground state energy.

As expected, the zero-momentum pair has the lowest energy

Such a linear dispersion relation is valid for pairs moving in any dimensions (D). However the coefficients slightly depend on the dimension as follows:

where

The velocity

The velocity magnitude

We consider a system of free bosons having a linear dispersion relation:

where n is the 2D boson density. The

where