The discovery of superconductivity by Onnes (1911) raised many questions about the properties of superconducting materials. It was not until 1933, that the interaction between superconductors and magnetic fields was brought to limelight [1] . The magnetic field around a superconductor is related to the current flowing (that results to electric field) through a superconductor by the London equations [2] . The penetration depth into a superconductor and the magnetic fields around the superconductor are related by the Ginzburg-Landau equations [3] . Later on, it was established that superconductivity is hinged on the pairing of electrons at the Fermi surface that lead to the formation of Cooper pairs [4] . The microscopic theory of superconductivity, based on s-wave pairing, laid the general foundation not only for conventional superconductivity, but also for unconventional superconductivity. However, in conventional superconductivity, the Cooper pair is not considered in isolation as is the case with conventional superconductivity [5] [6] . Furthermore, the challenge in high temperature superconductivity (HTS) is that the manner in which the electron-boson interaction occurs is still unknown [6] . Thus, the pairing mechanism in high temperature superconductors below the critical temperature remains unclear.

Secondly, the size of the energy gap is an important factor that contributes towards the superconductivity of a material. In general, the critical temperature of a material increases with the energy gap of the material. In HTSC, many theoretical formulations linking the energy gap to the critical temperature of a HTSC material have been developed, but they do not replicate [7] . The energy gap in HTS is anisotropic along the Fermi surface, unlike that in low temperature superconductors (LTSC) [8] .

HTSC was born with the discovery of a Lanthanum-based cuprate [9] . A yttrium-based cuprate was discovered a year later when lanthanum was replaced with yttrium LBCuO₄ [10] . The highest critical temperature so far reached at ambient pressure 138 K in mercury based cuprate and 203 K in Sulphur hydride, under extremely high pressure of about 200 GPa [11] [12] . Cuprates have been the only known group of high temperature until 2006 when properties of superconductivity were observed in a lanthanum based iron pnictide [13] . Following this discovery, other iron-based superconductors have since been discovered. Iron-based conductors are generally classified as 11 (e.g. FeSe), 111 (e.g. LiFeAs), 122 (e.g. LaFe₂As_1−xP_x), and 1111 (e.g. BaFeAsO_1−xF_x) [14] . Figure 1 below shows the structure of 1111-type (a), 122-type (b), 1111 (c) and 11-type (d).

Though FeAs appears in the 111-, 122- and 1111-type, the layers that separate FeAs vary from one type to another. However, Fe is blocked immediately from both sides in all the four categories. Of interest among the four families is the 1111-type, because the high critical temperatures for these materials have been recorded.

Electron-phonon interactions and spin fluctuations are central to the mechanism of HTS [15] . It has been widely agreed upon that superconductivity in cuprates results from electron-phonon interactions that form Cooper pairs. On the other hand, superconductivity in IBSC has been attributed to spin fluctuations. Spin fluctuations have sufficient strength to mediate the pairing interactions in high-temperature superconductivity except that the estimates of the

strengths of this interaction differ widely [16] . HTS requires a theory, beyond spin fluctuations, to explain the anisotropic nature of its energy gap [17] . Spin fluctuations is a convenient way to describe multiple Coulomb interactions between fermions [18] . IBSCs can be treated as moderately interacting itinerant fermionic systems with multiple FS sheets and effective four-fermion intra-band and inter-band interactions in the band basis [18] . There is general consensus on the unconventional nature of the Cooper pairing state of IBSC in multiple energy gaps and Fermi surfaces [19] . Based on this understanding, the origin of Cooper pairing in IBSC (Ba_1−xK_xFe₂As₂) has been studied using Raman spectroscopy to probe the structure of the pairing interaction at play in the superconductor [20] . The spectra collected in the B_1g symmetry channel reveal the existence of two collective modes which are indicative of the presence of two competing pairing tendencies of d x 2 − y 2 symmetry type. Bohm et al. (2018) has considered this pairing as the basis of the formation of Cooper pairs in the IBSC. The d-wave gap in IBSCs has dependence on the electron Fermi surfaces and may be nodeless in some cases [21] . Other than electrons pairing, interactions between electrons and bosons have also been observed in IBSC.

Experimental results based on time-resolved spectroscopy have also shown that there exists electron-boson coupling in IBSC [22] [23] . While cuprates show a single energy gap with one Fermi surface (FS), a point-contact Andreev-reflection study of Ba(Fe_1−xCo_x)₂As₂ single crystals has revealed that there exists a multigap band and multiple Fermi surfaces with a four-fermion intraband and interband interactions in the band basis [23] . Because the Fermi level of each parent compound is primarily governed by the five 3d orbitals of iron ion, iron plays a primary role in the superconductivity [21] . In most cases, removing the magnetism is an experimental step needed for the emergence of superconductivity. For the 1111 system, T_c appears when the antiferromagnetism (AFM) disappears but the AFM and superconductivity coexist in the 122 system [21] . The energy gaps in HTS are largely anisotropic. It has been argued that the electron correlation effect should be considered to explain the anisotropic transport properties of the general d/f valence electron system [17] [24] . Based on these findings, I argue that interaction between electrons and Cooper pairs may as well be used to study the thermodynamic properties of IBSC just as it has been the case for cuprates. The Cooper pair in this interaction plays the part of a boson [25] . The IBSC materials to be considered under this study are CeFeAsO_0.84F_0.16 with a T_c of 34 K while SmFeAsO_0.8F_0.2 with T_c = 54 K.

High temperature superconductors have a vast applications some of which include maglev trains, Superconducting Quantum Interface Devices (SQUIDs) and Magnetic Resonance Imaging (MRI). Understanding the pairing mechanism in HTSC superconductors will enhance theoretical predictions, with precision, of the T_C’s as well as other thermodynamic properties of high temperature superconductors [26] . This will open avenues to discoveries towards room-temperature superconductivity (RTS). Though the number of High temperature iron-based high temperature superconductors is on increase, theoretical as well as experimental studies are still conflicting with the observed properties [7] [27] [28] .

Superconductivity majorly results from the formation of Cooper pairs, at the Fermi surface, at the critical temperature. However, not all electrons at the Fermi surface take part in the formation of Cooper pairs, giving rise to a phenomenon in which Cooper pairs interact with the free electrons. This model has been used to study cuprates and the resulting entropy and specific heat showed close proximity to the results from previously done experimental and theoretical work [7] [26] . However, the energy due to electron interaction is found to be much less than the measured energy gap in cuprates. The difference arises from the fact that the measured energy gap represents the energy due to both electron-electron interaction and electron-phonon interactions [7] . The BVT has previously yielded results that are in agreement with other methods such as second quantization [26] . In this approach, a Hamiltonian of interaction between the electron and the Cooper pair is first developed using the kinetic energies of both the electron and the Cooper pair, the positive interaction potential and the negative interaction potential. Row reducing the Hamiltonian leads to energy of the system at condensate state. The specific heat of the system has been deduced from the expression of the system’s energy.

To start with, the Cooper pair and the electron are considered to occupy different states. The total Hamiltonian, for the interaction is given by

H = H 0 + H P (1)

where, H 0 is the Hamiltonian of interaction between a Cooper pair in state k and electron in state q for unperturbed system and is given by

H 0 = ∑ q ϵ q a q † a q + ∑ k ϵ k b k † b k (2)

H P is the Hamiltonian for the perturbed system and 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 (3)

From Equations (2) and (3), 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 c 2 / 2 m c is the kinetic energy for the Cooper pair.

Equations (2) and (3), 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 ) − ∑ k , q U k a q † a q b k † b k (4)

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 . Where, | 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 a wave vector). Thus,

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 (5)

where 4OT are the fourth order terms.

1) Energy of the System

The diagonal part of the effective Hamiltonian represents the energy of the system when it is in equilibrium. Therefore, at equilibrium the energy of the system is then given by

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 ] } (6)

At equilibrium, the quasi-particles represented by the operators γ’s are very few or do not exist and therefore, m k = m − k = 0 and m q = m − q = 0 . Thus, Equation (6) becomes

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

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 (6) becomes,

E k = ϵ q v k 2 + ϵ k v k 4 − v k 4 U k , k v k 2 (8)

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 = 1 ,

E k = ϵ q + ϵ k − U k , k (9)

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 ) (10)

2) Electronic Specific Heat Capacity

In finding the electronic specific heat, we first use the specific heat capacity of the system. The specific heat is expressed, as a function of the total energy of the system, using the expression

C v = ∂ E T ∂ T (11)

For the three-electron system, we substitute (11) into (10) so that,

C v = [ E k 2 K B T 2 ] e ( − E k K B T ) (12)

The Sommerfeld’s coefficient is determined from the specific heat as

γ = C v T (13)

Substituting (12) into (13)

γ = [ E k 2 K B T 3 ] e ( − E k K B T ) = [ 4 K B T 2 T 3 ] e ( − 2 T c T ) (14)

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 the temperature T = T_c, the material changes from a superconducting to a normal state. Figure 2 shows the total energy of electron-Cooper pair interaction in of Y123 as a function of temperature.

This half-stretched sigmoid curve has been obtained previously by other researchers when they were relating energy of the system to temperature [29] . From the graph, it is observed that the energy of the system increases with temperature but not linearly. In determining the critical point on the curve, the line x = T c has been used, that is, x = 34 for Ca_0.32Na_0.68Fe₂As₂ and x = 55 for SmFeAsO_0.8F_0.2. The energy of interaction between an electron and a Cooper pair at the critical temperature in SmFeAsO_0.8F_0.2 at is 1.26 meV, while that in Ca_0.32Na_0.68Fe₂As₂ is 0.6 meV. Comparatively, the energy of the three-electron model in thallium based cuprates, Tl2201, Tl2212 and Tl2223 were determined as 2.2 meV, 2.5 meV, and 2.9 meV while that of YBCO123 is found to be 2.2

meV [26] against the experimental 34 meV [5] . These energies are very low compared to the experimental energy values.

Electronic Specific Heat Capacity (γ)

Figure 3 represents graph of the electronic specific heat as a function of temperature for the two IBSCs under study.

Similar results were obtained by other researchers while comparing the variations in the electronic specific heat with temperature in cuprates [7] [26] . From Figure 3, the electronic specific heat for CeFeAsO_0.84F_0.16 and SmFeAsO_0.8F_0.2 are found to be 164.3 mJmol⁻¹K⁻² and 101.6 mJmol⁻¹K⁻² respectively. The electronic specific heat of CeFeAsO_0.84F_0.16 has been determined by measurement as 105 ± 5 mJmol⁻¹K⁻² [30] . The theoretical value is of the same order as the experimental value. The method used in this work has been found to agree with the highest measured value for electronic specific heat of a Yttrium based cuprate, YBCO123 with γ = 60 mJmol⁻¹K⁻² [26] [31] .

The difference in the electronic specific heat between the theoretical value and the measured value may arise due to the quality of the sample used in the experimental approach [30] . For instance, working on the quality of Ca_0.33Na_0.67Fe₂As₂ to Ca_0.32Na_0.68Fe₂As₂ improves its electronic specific heat from 39 mJmol⁻¹K⁻² to 105 mJmol⁻¹K⁻². Therefore, working on the quality of the substance can improve the measured values significantly and is likely to agree with the theoretical values obtained using this approach.

For the first time, the specific heat of iron pnictides has been determined based on a theoretical approach. The interaction between an electron and a Cooper

pair has successfully given values that are in close proximity to the measured values of the specific heat. Despite the differences in the origin of Cooper pairing, the electronic specific heats in both high temperature IBSC and cuprates depend on the electron-Cooper pair interaction. However, the energy obtained using this approach varies significantly from the experimental results confirming that electron-Cooper pair interaction and even spin fluctuations are not sufficient to explain HTSC. These findings reveal that the mechanism behind HTSC in both cuprates and IBSC is likely to be the same though the origin of pairing may vary.

The author declares no conflicts of interest regarding the publication of this paper.

Mukubwa, A. (2018) Electronic Specific Heat of Iron Pnictides Based on Electron-Cooper Pair Interaction. Open Access Library Journal, 5: e5107. https://doi.org/10.4236/oalib.1105107