_{1}

The discovery of iron pnictides in 2006 added on the number of materials that have the potential to transmit electricity with close zero d.c resistance. High-temperature iron-based superconductors have been obtained throu
gh modification, mostly by doping, of the initially low-temperature iron-based superconductors. Unlike in LTSC, the energy gap in HTSC requires a theory, beyond spin fluctuations, to explain its anisotropy. This study seeks to establish a common ground between iron pnictides and cuprates towards explaining high temperature superconductivity. There is a general consensus on the existence of Cooper pairs in these systems. In addition to this, experimental results have revealed the existence of electron-boson coupling in iron pnictides. These results make it viable to study the interaction between an electron and a Cooper pair in iron based superconductors (IBSC). In this study, Bogoliubov-Valatini transformation has been used in determining the electronic specific heat based on the interaction between an electron and a Cooper pair in high-temperature IBSC, namely, Ca
_{0.33}
Na
_{0.6}
Fe
_{2}
As
_{2}
and SmFeAsO
_{0.8}
F
_{0.2}
. We record the theoretical electronic specific heat of CeFeAsO
_{0.84}
F
_{0.16}
and SmFeAsO
_{0.8}
F
_{0.2}
as 164.3 mJ mol
^{-1}
K
^{-2}
and 101.6 mJ mol
^{-1}
K
^{-2}
respectively.

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 [

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 [

HTSC was born with the discovery of a Lanthanum-based cuprate [_{4} [_{2}As_{1−x}P_{x}), and 1111 (e.g. BaFeAsO_{1−x}F_{x}) [

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 [

strengths of this interaction differ widely [_{1−x}K_{x}Fe_{2}As_{2}) has been studied using Raman spectroscopy to probe the structure of the pairing interaction at play in the superconductor [_{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 [

Experimental results based on time-resolved spectroscopy have also shown that there exists electron-boson coupling in IBSC [_{1}_{−}_{x}Co_{x})_{2}As_{2} 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 [_{c} appears when the antiferromagnetism (AFM) disappears but the AFM and superconductivity coexist in the 122 system [_{0.84}F_{0.16} with a T_{c} of 34 K while SmFeAsO_{0.8}F_{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 [

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 [

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.

This half-stretched sigmoid curve has been obtained previously by other researchers when they were relating energy of the system to temperature [_{0.32}Na_{0.68}Fe_{2}As_{2} and x = 55 for SmFeAsO_{0.8}F_{0.2}. The energy of interaction between an electron and a Cooper pair at the critical temperature in SmFeAsO_{0.8}F_{0.2} at is 1.26 meV, while that in Ca_{0.32}Na_{0.68}Fe_{2}As_{2} 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 [

Electronic Specific Heat Capacity (γ)

Similar results were obtained by other researchers while comparing the variations in the electronic specific heat with temperature in cuprates [_{0.84}F_{0.16} and SmFeAsO_{0.8}F_{0.2} are found to be 164.3 mJmol^{−1}K^{−2} and 101.6 mJmol^{−1}K^{−2} respectively. The electronic specific heat of CeFeAsO_{0.84}F_{0.16} has been determined by measurement as 105 ± 5 mJmol^{−1}K^{−2} [^{−1}K^{−2} [

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 [_{0.33}Na_{0.67}Fe_{2}As_{2} to Ca_{0.32}Na_{0.68}Fe_{2}As_{2} improves its electronic specific heat from 39 mJmol^{−1}K^{−2} to 105 mJmol^{−1}K^{−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