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Since the discovery of high temperature superconductivity (HTSC), there have been varied results concerning the high pairing energy in the unconventional superconductivity. In an attempt to unearth the mechanism behind the high pairing energy, we study a model in which an excited electron interacts with a Cooper pair. The interaction was studied earlier on using second quantization which depended on a wave function. We now use Bogoliubov-Valatini trans-formation (BVT) that is independent of the wave function to study the thermodynamic properties due to the interaction between an electron and a Cooper pair in a Yttrium based cuprate. The interaction energy increases with temperature and at ; we record the interaction energy of 2.173 meV and the specific heat of 4.669 J·mol · 1 · K · 1. The entropy of this system is found to be 5.11 mJ · g · 1 · K · 1. The Sommerfeld’s coefficient is maximum at where . It has also been observed, for the first time, that the thermodynamic quantities of YBCO123 are constant when temperature varies between 0 K and 18 K. Above 18 K, the thermodynamic quantities rise sharply. This observation has been attributed to the pairing of electrons in the singlet-s and singlet-p states.

The microscopic theory of superconductivity, based on s-wave pairing, has been used to explain conventional superconductivity [

YBa_{2}Cu_{3}O_{7}, with a T_{C} of 93 K, was discovered a year later after the landmark discovery of a lanthanum-based cuprate [_{C} of 203 K, the highest experimental T_{C} that has ever been achieved in the history of superconductivity [_{C}’s as well as other thermodynamic properties of cuprates. This will open avenues for discoveries towards room-temperature superconductivity (RTS). Although many different HTSC cuprates have been discovered, theoretical as well as experimental studies are still conflicting with the observed properties [

The Hamiltonian, H_{0}, of interaction between a Cooper pair in state k and electron in state q for unperturbed system is

H 0 = ∑ q ϵ q a q † a q + ∑ k ϵ k b k † b k (1)

where a q † ( a q ) is the creation (annihilation) operator for an electron in state q, b k † = a k † a − k † ( b k = a − k a k ) is the creation (annihilation) operator for the Cooper pair in state k; a q † a q and b k † b k are the number operators for an electron and a Cooper pair respectively; ϵ q = ℏ k e 2 / 2 m e is the kinetic energy for electron and ϵ k = ℏ k e 2 / 2 m c is the kinetic energy for the Cooper pair.

The Hamiltonian for the perturbation, H_{P}, is given by

H P = ∑ k , q V k , q a q † a q ( b k † − b k ) − ∑ q , k U k a q † a q b k † b k (2)

where V_{k}_{,q}(U_{k}_{,q}) is the interaction potential between electron and the Cooper pair.

Equations (1) and (2), when combined, give the Hamiltonian for a perturbed system as

H = ∑ q ϵ q a q † a q + ∑ k ϵ k b k † b k + ∑ k , q V k , q a q † a q ( b k † − b k ) − ∑ q , k U k a q † a q b k † b k (3)

Equation (3) is then written in terms of Bogoliubov-Valatini operators, γ , using the relations a k = u q γ q + v q γ − q + , a − k = u k γ − k − v k γ k + , a k † = u q γ q + + v q γ − q and a − k † = u k γ − k + − v k γ k . In this case, | u k | 2 is the probability that the pair state {k, −k} within a certain interval around the Fermi level is unoccupied and | v k | 2 is the probability that the pair state {k, −k} within a certain interval around the Fermi level is occupied (k is the wave vector). Thus, Equation (4) gives the total Hamiltonian of the system.

H = ∑ q ϵ q { u q 2 m q + v q 2 ( 1 − m − q ) + u q v q ( γ q + γ − q + + γ − q γ q ) } + ∑ k ϵ k { u k 4 m − k m k − u k 2 v k 2 m k ( 1 − m − k ) + u k 2 v k 2 ( 1 − m k ) m k + u k 2 v k 2 ( 1 − m − k ) m − k − u k 2 v k 2 ( 1 − m − k ) m k + v k 4 ( 1 − m − k ) ( 1 − m k ) + { u k 3 v k ( m − k + m k ) + v k 3 u k ( 2 − m k − m − k ) } ( γ k + γ − k + + γ − k γ k ) } + ∑ k , q V k , q { { u q 2 m q + v q 2 ( 1 − m − q ) } ( γ k + γ − k + − γ − k γ k ) }

− ∑ k , q U k { u k 4 u q 2 m − k m k m q + u k 4 v q 2 m k ( 1 − m − q ) m − k + u q 2 v k 4 ( 1 − m k ) ( 1 − m − k ) m q + v q 2 v k 4 ( 1 − m − k ) ( 1 − m − q ) ( 1 − m k ) + u q 2 u k 2 v k 2 m q [ m k ( 1 − m k ) − 2 m k ( 1 − m − k ) + m − k ( 1 − m − k ) ] + u k 2 v k 2 v q 2 ( 1 − m − q ) [ ( 1 − m k ) m k − 2 ( 1 − m − k ) m k + ( 1 − m − k ) m − k ]

+ [ u k 3 u q 2 v k ( m − k + m k ) m q + u k 3 v q 2 v k ( m − k + m k ) ( 1 − m − q ) + v k 3 u q 2 u k ( 2 − m k − m − k ) m q + v k 3 v q 2 u k ( 1 − m − q ) ( 2 − m k − m − k ) ] ( γ k + γ − k + + γ − k γ k ) + [ u k 2 v k 2 u q v q ( 1 − m − k ) m − k − 2 u k 2 v k 2 u q v q m k ( 1 − m − k ) + u k 2 v k 2 u q v q ( 1 − m k ) m k + u k 4 u q v q m − k m k + v k 4 u q v q ( 1 − m − k ) ( 1 − m k ) ] ( γ q + γ − q + + γ − q γ q ) } + 4 OT (4)

where 4OT are the fourth order terms.

The diagonal part of the effective Hamiltonian represents the energy of the system at equilibrium. Therefore, at equilibrium the energy of the system is given by Equation (5).

E k = ∑ q ϵ q { u q 2 m q + v q 2 ( 1 − m − q ) } + ∑ k ϵ k { u k 4 m − k m k − u k 2 v k 2 m k ( 1 − m − k ) + u k 2 v k 2 ( 1 − m k ) m k + u k 2 v k 2 ( 1 − m − k ) m − k − u k 2 v k 2 ( 1 − m − k ) m k + v k 4 ( 1 − m − k ) ( 1 − m k ) } − ∑ k , q U k , q { u k 4 u q 2 m − k m k m q + u k 4 v q 2 m k ( 1 − m − q ) m − k + u q 2 v k 4 ( 1 − m k ) ( 1 − m − k ) m q + v q 2 v k 4 ( 1 − m − k ) ( 1 − m − q ) ( 1 − m k ) + u q 2 u k 2 v k 2 m q [ m k ( 1 − m k ) − 2 m k ( 1 − m − k ) + m − k ( 1 − m − k ) ] + u k 2 v k 2 v q 2 ( 1 − m − q ) [ ( 1 − m k ) m k − 2 ( 1 − m − k ) m k + ( 1 − m − k ) m − k ] } (5)

At equilibrium, the quasi-particles represented by the operators γ are very few or do not exist and therefore, m k = m − k = 0 and m q = m − q = 0 . Thus, Equation (5) becomes

E k = ∑ q ϵ q v q 2 + ϵ k v k 4 − v k 4 ∑ q U k , q v q 2 (6)

For the electron to interact with the Cooper pair, they must be in the same state i.e. at the time of interaction we only consider the state k of the electron and neglect all the other states in q and therefore Equation (7) becomes,

E k = ϵ q v k 2 + ϵ k v k 4 − v k 4 U k , k v k 2 (7)

For the three-electron interaction to take place, the cooper pair and the electron involved in the interaction must be present. Thus, v k = 1 and u k = 0 ,

E k = ϵ q + ϵ k − U k , k (8)

To introduce temperature dependence, the energy E_{k} is multiplied by the thermal activation factor exp ( − E k K B T ) where E_{k} is energy of the system and K_{B} is the Boltzmann constant. This produces a temperature dependent energy E_{T} given as

E T = E k e ( − E k K B T ) (9)

The sum of kinetic energies ( ϵ q + ϵ k ) and U k , k for Y123 have been determined as 10.8 meV and 26.8 meV respectively [

The specific heat capacity of a system is given by

C V = ∂ E T ∂ T (10)

For the three electron system, we substitute 10 into 9 so that,

C V = [ E k 2 K B T 2 ] e ( − E k K B T ) (11)

The electronic specific heat capacity (γ) is determined from the specific heat capacity using the formula

γ = C v T (12)

Putting (9) into (12) we get

γ = [ E k 2 K B T 3 ] e ( − E k K B T ) (13)

From the first law of thermodynamics temperature is defined as

1 T = d s d Q (14)

From which we have,

d s = d Q T (15)

But recalling that dQ = C_{v}dT and integrating Equation (13), we have

S = ∫ C v T d T (16)

Putting Equation (11) into Equation (14) to substitute for C_{V}, we have

S = ∫ [ E k 2 K B T 3 ] e ( − E k K B T ) d T (17)

So that the entropy becomes

S = ∫ − K B { ( E k K B T − 1 ) e E k K B T } (18)

The total energy of a system results from the interaction between the particles of the system. The energy due to interaction between these particles increases with the temperature of the system. At T_{C}, point the material changes from a superconducting to a normal state.

This half-stretched sigmoid curve has been obtained previously by other researchers when they were relating energy of the system to temperature [

As the temperature rises from 0 K to 18.5 K, the Cooper pair formation is largely due to the singlet s-state because most of the electrons are in the ground state. The energy of a mole of YBCO123 within this temperature range is kept close to 0 K i.e. below 0.0007482 meV. When the temperature rises above 18.5 K, some electrons enter the first excitation state where Cooper pair start forming in the singlet p-state. The result is that the energy of the system starts to increase rapidly from below 0.0007482 meV. As the temperature increases towards the critical temperature, more electrons enter the first excitation state and more interactions between electrons in the singlet p-state increase leading to continued increase in the energy of the system. At T = T C , the energy per unit mass for an

electron interacting with a Cooper pair in Y123 is 314.7 mJ∙g^{−1}. This energy falls within the range for cuprates (240 J∙kg^{−1} - 375 J∙kg^{−1}) at T_{C} as provided by Odhiambo (2016). The energy of a mole of YBCO123 at T_{C} is 2.173 meV. This energy was determined at T_{C} using second quantization method as 2.168 meV [

The measured bulk energy for YBCO123 is 34 meV [

Specific heat capacity of a substance is the amount of energy required to change the temperature of a unit mass of substance by 1 K. The unit of specific heat is J∙kg^{−1}∙K^{−1}. The specific heat of a substance depends on the temperature of the substance and is highest at the critical temperature.

This shape concurs with those observed in previous research work [

T = T C

where C V = 6.757 mJ ⋅ g − 1 ⋅ K − 1 ( 4.669 J ⋅ mol − 1 ⋅ K − 1 ) . Odhiambo (2016) gives the

specific heat of YBCO123 as (4.526 J∙mol^{−1}∙K^{−1}). A theoretical study the pairing symmetry of the singlet and triplet pairing reveals that specific heat capacity of a 3He - 4He mixture at T_{C} is 28.91 J∙mol^{−1}∙K^{−1} [_{C} = 91.68 K is 127.8 J∙mol^{−1}∙K^{−1} and the electronic contribution is about 1% - 2% of the phonon contribution to the total specific heat [_{C}, the specific heat dependence on temperature follows a power law indicating that the energy gap at Fermi surface has nodes in certain directions implying anisotropy in the energy gap [

The Sommerfeld’s coefficient of a material relates the heat capacity of the material to a specific temperature. _{C}.

Using the graph, the Sommerfeld’s coefficient increases with temperature and is optimum at T/T_{C} = 0.6666 where γ = 0.09031 mJ ⋅ g − 1 ⋅ K − 2 , which upon conversion becomes 62.4 mJ∙mol^{−1}∙K^{−2}. This value is again close to 60.4 mJ∙mol^{−1}∙K^{−2} that was obtained by second quantization method [^{−1}∙K^{−2} [

Entropy of a system is the measure of disorder within the system. Increase in temperature of the system increases the agitation of the particles hence increase in entropy. When the entropy is very high, the enhanced disorder hence increased energy within the system destroys Cooper pairs. Consequently, the material changes from being superconducting state to a normal state or vice versa. The lower the entropy of a system, the higher the orderliness of its particles.

The entropy of the system decreases with temperature implying increase in the orderliness of the system with decrease in temperature. At lower temperature, the entropy is zero to imply that the system is highly ordered. Above this entropy, the increased energy of the system agitates the lattice particles and the superconductivity of the system is destroyed. The lower the entropy of a system, the higher the order of its particles. At T = 0, the entropy of the system is zero and the particles within the system, at this point, are highly ordered.

Within the temperature range 0 ≤ T ≤ T C , the entropy YBCO123 is maintained below 0.0002083 meV∙mol^{−1}∙K^{−1}. When the temperature rises above 17.5 K, some electrons are excited into the first excitation state where electrons pair in the singlet p-state. At this point, disorder starts setting in. As the temperature increases up to the critical temperature, more electrons enter the first excitation state and more electrons pair up in the singlet p-state leading to continued decrease in the order of the system. Therefore, a singlet s-state is more ordered than the singlet p-state.

Similar curves were obtained by other authors [_{C}.

^{−1}. The interaction of singlet and triplet state in a heavy fermion superconductor gives an entropy of 3.5 × 10^{−21} J∙K^{−1} [^{−22} J∙K^{−1}) [_{B} (Unit cell)^{−}^{1} when holes were varied

from 0.57 - 0.97 (Unit cell)^{−1} [^{−25} - 3.036 × 10^{−24} J∙(Unit cell)^{−1}∙K^{−1}. When the temperature of a material rises above its T_{C}, the increased energy of the system agitates the lattice vibrations (increased disorder), leading to a higher entropy. In that condition, no Cooper pair exists and thus superconductivity of the material is destroyed. However, below this entropy, the system exhibits orderliness resulting into the formation Cooper pairs.

Within this range of temperature, the entropy YBCO123 is below 0.0002083 meV∙mol^{−1}∙K^{−1}. When the temperature rises above 17.5 K, some electrons gain access to the first excitation state where electrons pair in the singlet p-state. At this point, disorder sets in slowly. As the temperature increases up to the critical temperature, more electrons enter the first excitation state and more electrons pair up in the singlet p-state leading to a less ordered system. Therefore, a singlet s-state is more ordered than the singlet p-state.

We note that the energy of interaction between a Cooper pair and an electron increases with temperature in the range 18 ≤ T ≤ T C . The energy of interaction within Y123 at T = T C , as determined in this study, is 2.174 meV∙mol^{−1}. As the energy of the system increases from zero as the Cooper pairing changes from singlet s-state to singlet p-state. The energy of interaction between the electron and the Cooper pair determined in this study accounts for 6.39% of the total energy.

The specific heat of YBCO123 increases with temperature with the peak at

T = T C

where C V = 4.669 J ⋅ mol − 1 ⋅ K − 1 . In the region T > T C , the specific heat decreases with increase in the temperature of the system. In the range 18 ≤ T ≤ T C , the Sommerfeld’s coefficient increases with temperature and is highest at

T = 2 3 T C

where γ = 0.09036 mJ ⋅ g − 1 ⋅ K − 2 , which upon conversion becomes 62.4 mJ∙mol^{−1}∙K^{−2}. The entropy of YBCO123 at T/T_{C} = 1 was found to be 3.84 meV∙K^{−1}. The electron pairing is more ordered in singlet s-state (below 18 K) than in the singlet p-state (above 18 K).

This study has shown that the energy, specific heat, Sommerfeld’s coefficient and entropy of YBCO123 remain almost constant (and close to zero) as temperature rises from zero to about 18 K due to electron pairing in the singlet-s state. Above T = 18 K the energy of YBCO123 rises sharply owing to the electron pairing in singlet-p state. In this region, high energy of up to 2.174 meV per Mole is involved. The change in electron pairing from a purely singlet-s environment to a combination of both singlet-s and singlet-p states might be an avenue to understanding high temperature superconductivity, owing to the variation of energy with temperature associated with the change in the pairing environment.

The authors declare no conflicts of interest regarding the publication of this paper.

Mukubwa, A.W., Odhiambo, J.O. and Makokha, J.W. (2018) Thermodynamic Properties of Yttrium Based Cuprate Due to Electron-Cooper Pair Interaction Using BVT. Open Access Library Journal, 5: e4880. https://doi.org/10.4236/oalib.1104880