Thermal and Electronic Conductivity in Normal and Superconducting Erbium Nickel Borocarbide (ErNi2B2C)

In this work, the total electronic energy, the electronic thermal conductivity, and the heat capacity of erbium nickel borocarbide, ErNi2B2C, in the normal and superconducting states are calculated using Boltzmann transport equations (BTEs) and energy dispersion relation function. Results from the electronic thermal conductivity versus temperature (T) are presented. From the result, electrical and thermal conductivity at low temperature obey the Wiedemann–Franz law. Moreover, at the normal state, the electronic thermal conductivity of ErNi2B2C is directly proportional to the temperature (T) and reaches its maximum (kink) at the transition temperature, Tc. After the superconducting transition temperature, the electronic thermal conductivity begin to decrease. e drop in electronic thermal conductivity beyond its peak (kink) value is due to the formation of energy gap and the absence of Cooper pairs.


Introduction
e phenomenon of superconductivity has a rich and interesting history, starting in 1911 when Kamerlingh Onnes discovered that upon cooling elemental mercury to very low temperature, the electrical resistance suddenly and completely vanished below a critical temperature, T c , of 4.2 K [1,2] (Figure 1). is resistance-less state enables persistent currents to be established in circuits to generate enormous magnetic elds and to store and transport energy without dissipation.
Superconductors have other unique properties such as the ability to expel and screen magnetic elds (Meissner e ect) and quantum oscillations controlled by the magnetic eld that provide extraordinary measurement sensitivity [2,3].
Electrons of materials at normal state are free to move and provide electrical conduction, but collisions with other electrons, lattice vibrations, impurities, and defects in the material cause resistance and thus energy dissipation in the material. Unlike normal metals, superconducting materials allow the passage of electricity through them without almost any electrical resistance and this was mystery at that time.
Although phenomenological models with predictive power were developed in the 30s and 40s to explain this mystery, the microscopic mechanism underlying superconductivity was not discovered until 1957 by three American physicists John Bardeen, Leon Cooper, and John Schrie er through their BCS theory to explain the mystery in superconductors [3,4].
According to the theory, at very low temperature, two electrons with equal and opposite speeds are glued together in a coordinated fashion, so collisions are not possible, and they move through the material without any resistance [3][4][5].
In this work, concepts of superconductivity and the general properties of ErNi 2 B 2 C in its normal and superconducting states are explained. Using Boltzmann transport equations (BTEs) and energy dispersion relation function, the electronic thermal conductivity and heat capacity of ErNi 2 B 2 C in both the normal and superconducting states are calculated.

Formation of Energy Gap Parameter and Critical
Temperature of ErNi 2 B 2 C. Superconducting energy gap is a measure of how strongly the electrons of a superconductor are bound inside a Cooper pair. e superconducting gap parameter, Δ, plays a role when we consider the electrons at the Fermi surface. In the normal metal, the electron states are filled up to the Fermi energy, εF, and there is a finite density of state at the Fermi level, D(εE). But below T c , the electron density of state acquires a small gap (2Δ) which separates the occupied and unoccupied states [4]. e energy gap, at zero temperature, is evaluated as where ωD is the Debye frequency. e expression for superconducting gap parameter at any temperature between the absolute zero and T c is given by e BCS theory estimates the zero temperature energy gap 2Δ (0) as e ratio 2Δ(0)/K B T C is a universal constant. For ErNi 2 B 2 C, the superconducting transition temperature is observed at T c ≈11 K [6]. So, From equations (1) and (4), the critical temperature is expressed as e value of the coupling parameter constant λ � 0.4721 [7][8][9].

Methods
In this work, Boltzmann transport equations (BTEs) and energy dispersion relation function were used to calculate the electronic thermal conductivity and heat capacity of ErNi 2 B 2 C in both the normal and superconducting states.

Electronic Thermal Conductivity of ErNi 2 B 2 C in Its Normal and Superconducting States
Metals possess both phonon thermal conductivity and electronic thermal conductivity. So, the total thermal conductivity is the sum of both phonon thermal conductivity, Kp, and electronic thermal conductivity, Ke [10][11][12].

Heat Current Density (J Q ).
e heat current density (J Q ) is defined as where Ω is the volume of the metallic material to be crossed by the electrons, f k is the Fermi-Dirac distribution function, ʋ k is the group velocity, E k is the energy at a state k, and E F is the Fermi energy [13,14]. e Fermi-Dirac distribution at k-mode is given by [15] where k B is Boltzmann's constant, T is the temperature, μ is the chemical potential, and the superscript in f kO indicates the thermal equilibrium. From the Fermi-Dirac distribution function at the thermal equilibrium, we have relations for partial derivative of f kO.
Assume that f kO characterizes local equilibrium in such a way that the special variation of f kO arises from the temperature, T, and chemical potential, μ: Substituting equations (9) and (10) into equation (11), we get e description of quasi-particles as wave package allows one to introduce the non-equilibrium distribution function f (k, r) which is the average occupation number for a state k at a point r. In the absence of interactions and external fields, the function is equal to the equilibrium, f kO; otherwise, it becomes time dependent [16].
Let us consider a system, in which only a temperature gradient exists and causes the electron to diffuse with the velocity, ʋ. Since the electron travels a distance dʋdt after dt, the electron distribution, f (r, k, t), at the position (r, k) in the phase space at a time t is expressed as For small dt, equation (14) can be expanded as [17,18] At the steady state zf k /zt � 0, equation (16) becomes Using equations (12) and (13) in equation (17), we get But, where τ is the relaxation time. Hence, substituting equation (19) into equation (18) in the absence of magnetic field gives [19,20] e non-equilibrium part of distribution function with in the relaxation time approximation becomes Using equation (21) in equation (6), we get It is convenient to change summation to integration.
In this case, the first term of equation (22) becomes zero, because the integration of even and odd function is zero, as ʋ k is odd and f k is even function. So, Introducing the new parameter called mean free path, l k � τμ, equation (23) becomes For thermal conductivity calculation, E and B are taken to be zero. us, equation (24) reduces to

Total Electron Energy.
Using tight binding approximation, we can calculate the total electron energy of a given lattice structure. Consider a half-filled Hubbard model on a single layer honeycomb lattice with N-N hoping. e energy dispersion is given by Ek � 2tcxy (k), where t is the hopping energy and c is the energy scale. e energy dispersion has a particle-hole symmetry due to the bi-particle nature of lattice. Since borocarbide is an intermetallic compound, the tight binding fits to the band structure calculations of the following dispersion relation: where txz denotes the interlayer N-N electron hoping energy and cxy is the structure factor of the inter layer electron hoping to the N-N sites on the honeycomb lattice [8]. e structure factor, cxy, is given by For small values of kx, kxy, and kxz, cosine functions of equation (27), using Taylor series expansion, can be expanded as Advances in Materials Science and Engineering 3 So, equation (27) becomes For small values of kx and ky, the product of kx2 and ky2 becomes 0. Similarly, Substituting equations (32) and (33) into equation (26), we get e tight binding approximation site of the states close to the Fermi level, obtained from ab initio band structure calculation, gives the values of parameters as [8,21] t xy ≈ 1.6ev, and Substituting these values in equation (32), we get Hence, equation (35) is an expression for the dispersion relation.

Electronic Heat Capacity.
Near the Fermi level, electrons gain total electronic thermal kinetic energy, U el : From the total electronic thermal kinetic energy, we can calculate the electronic heat capacity as [22] Cel ≈ NkB If U is the total energy transfer, we can write the ΔU as where f (ε) and D (ε) are the Fermi-Dirac function and the number of orbitals per unit energy range, respectively. e total number of electrons, N, inside a sphere of radius, k, is given by [23] N � v where v is the volume of the sphere and m is the mass of the electron. e density of state of electron, D (ε), is calculated as [24] Multiply the identity by εF to obtain We use equation (42) to rewrite equation (38) as e first integral on the right hand side of equation (43) gives the energy needed to take electrons from ε F to the orbitals of the energy ε > ε F and the second integral gives the energy needed to bring the electrons to ε F from orbitals below ε F .
In equation (43), the product of the first integral of f (ε) D (ε F ) dε is the number of electrons evaluated to orbitals in the energy range dε at an energy ε. e factor [1 − f (ε)] in the second integral is the probability that an electron has been removed from an orbital, ε. e heat capacity of electron gas is calculated as For kBT ≪ εF, we ignore the temperature dependency of the chemical potential, m, in the Fermi-Dirac function and replace m by the constant ε F . Let kBT � τ, and consider the Fermi-Dirac distribution function [25,26].
Let x � ε − ε F /τ, and from equation (44) and (45), we get We can replace the lower limit by − ∞. Because the factor ex in the integrand is already negligible at − εF/τ for low 4 Advances in Materials Science and Engineering temperatures such that − εF/τ ∼ ∞, the integral in equation us, the heat capacity of an electron gas is given by Substituting equation (40) into equation (48) gives is is the heat capacity of free electron Fermi gas in the temperature region where kBT ≪ εF. For free electron gas approximation at the Fermi level, we have Substituting (50) into (49) gives the heat capacity of electron gas at the Fermi level in terms of the total number of electrons N, mass of electron gas m, velocity v, and temperature T at the Fermi level as Let the contribution of the z component of the wave vector, kz2, for the heat capacity of the electron gas at the Fermi level be one-third of the contribution of the total wave vector of the electron gas k2. e electronic heat capacity of ErNi 2 B 2 C in its normal state becomes and where Δ is the energy gap parameter and εF is the Fermi energy. e electronic heat capacity of ErNi 2 B 2 C in the superconducting state is calculated as Rearranging equation (56) gives Substituting equation (58) into equation (57) gives Equation (59) is the electronic heat capacity of ErNi 2 B 2 C in its superconducting state.

3.4.
ermal Conductivity. ermal conductivity of ErNi 2 B 2 C is related to the thermal heat current density as JQ � − K∇T. (60) From equations (25) and (60), the expression for electronic thermal conductivity of ErNi 2 B 2 C in its normal state is given as Rearranging equation (9) gives From equations (61) and (62), we get Advances in Materials Science and Engineering Changing the summation to integration over the allowed k-space and using the relation for any function F (k) gives where dS k is the area element in k-space of constant energy and � hʋk � zE/zk, and equation (63) becomes e Fermi-Dirac distribution function at equilibrium condition, f kO, is described in equation (7). Let x � E k − E F /K B T. Using these expressions in equation (65), we get From the relation of mean free path to the electron velocity, we have lk � τvk, where τ is the average time between collision of electrons and lk is the vector displacement of the electron. Moreover, the momentum in k-space can be found from the kinetic term of the electron with Ek � 1/2m� h2k2 as Substituting these expressions into equation (66) gives Solving equation (68) gives If N is the total number of conduction electrons involved in a metal, then the electronic thermal conductivity of ErNi 2 B 2 C in its normal state becomes If the energy varies with k 2 , i.e., from the kinetic energy E k � h 2 k 2 /2m, then the expression of thermal conductivity given by equation (70) holds true also as in a heat capacity.
us, applying similar analogy as equation (53), the thermal conductivity becomes K n � 1 3 T.

(71)
Since Nπ 2 k 2 B τ/2m is constant, the electronic thermal conductivity of ErNi 2 B 2 C in its normal state is directly proportional to temperature.
In a superconducting state, the superconductor is characterized by a weak coupling energy gap parameter, Δ, where the relation is given by where E k is the excitation energy measured relative to EF-μ and we use the symbol E k � EK − μ where μ is the chemical potential which is equivalent to the Fermi energy E F . us, we have For small Δ, using the Taylor series, we have From equation (63), we have Hence, Ignoring higher-order terms of the gap parameter gives Substituting equation (77) into equation (75) gives Substituting equation (80) into the first term of equation (78) gives Using equation (64) in the second term of equation (79) gives But the last integral contributes only if Ek � EF; otherwise, it becomes zero. Since Ek ≠ EF, we neglect the last integral.
Furthermore, we have the following relations: us, using the above expressions in equation (80) gives Equation (82) gives the general expression for electronic thermal conductivity of ErNi 2 B 2 C in its superconducting state.

Electrical Conductivity and Wiedemann-Franz Law.
e electrical conductivity, δ, is determined for any metallic material as Applying similar steps as we did for thermal conductivity, one can arrive at Evaluating the second integral yields Similarly, interchanging the thermal conductivity (K n ) by electrical conductivity (σ) gives and Taking the ratio of the thermal conductivity to the electrical conductivity gives Equation (88) relates thermal conductivity to the electrical conductivity.
At a given temperature, thermal and electrical conductivities of metals are proportional but raising the temperature increases the thermal conductivity while decreasing the electrical conductivity. is relation is known as Wiedemann-Franz law. Hence,

Results
Using Boltzmann Transport Equations (BTE), appropriate energy dispersion relation and tight binding approximation were aplied to calculate the electronic thermal conductivity and heat capacity of ErNi 2 B 2 C in both the normal and superconducting states. e tight binding approximation is given as Using this approximation, the calculated electronic heat capacity of ErNi 2 B 2 C in its normal state is e expression for electronic heat capacity of ErNi 2 B 2 C in its superconducting state is also determined as In the superconducting state, the heat capacity falls exponentially as the temperature decreases: Similarly, in its normal state, the electronic thermal conductivity is calculated as where ∝ � 1 2 e electronic thermal conductivity in its normal state varies linearly with temperature ( Figure 2). e electronic thermal conductivity of ErNi 2 B 2 C in its superconducting state is determined as where ∝ � 1 3 and β � τ/m are constants. Figure 2 shows the graph of electronic thermal conductivity of ErNi 2 B 2 C in its superconducting state.

Advances in Materials Science and Engineering
As can be seen from Figure 3, the electronic thermal conductivity of ErNi 2 B 2 C has a shoulder (kink) above the superconducting transition temperature.
In a superconductor, the thermal conductivity below T c rises sharply and reaches the superconducting transition temperature, T c , near T = 10 k. After it reaches its maximume value, the kink, then begin to decrease. e drop in the thermal conductivity beyond its peak value, kink, is due to the formation of energy gap and the absence of Cooper pairs [4]. Figure 3 shows the electronic thermal conductivity of ErNi 2 B 2 C in its superconducting state.

Conclusion
In this paper, the total electronic energy, the electronic thermal conductivity, and the heat capacity of erbium nickel borocarbide, ErNi 2 B 2 C, in its normal and superconducting states are calculated using the energy dispersion relation and Boltzmann transport equations. From the result, we concluded that the electronic thermal conductivity, at low temperature, of ErNi 2 B 2 C is directly proportional to the temperature in its normal state and obeyed the Wiedemann-Franz law as in most metals but at the transition temperature, T c , near 10 K, a kink is formed below which the state is superconducting.
Above the superconducting transition temperature, increasing the temperature furter decreases the thermal conductivity. e drop in the thermal conductivity beyond its peak value, kink, is due to the formation of energy gap and the absence of Cooper pairs, and erbium nickel borocarbide, ErNi 2 B 2 C, becomes normal conductor.

Data Availability
No data were used to support this study.    Advances in Materials Science and Engineering