Abstract
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. The drop in electronic thermal conductivity beyond its peak (kink) value is due to the formation of energy gap and the absence of Cooper pairs.
1. Introduction
The 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, Tc, of 4.2 K [1, 2] (Figure 1). This resistance-less state enables persistent currents to be established in circuits to generate enormous magnetic fields and to store and transport energy without dissipation.
Superconductors have other unique properties such as the ability to expel and screen magnetic fields (Meissner effect) and quantum oscillations controlled by the magnetic field 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 Schrieffer 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 [3–5].
In this work, concepts of superconductivity and the general properties of ErNi2B2C 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 ErNi2B2C in both the normal and superconducting states are calculated.
1.1. Formation of Energy Gap Parameter and Critical Temperature of ErNi2B2C
Superconducting energy gap is a measure of how strongly the electrons of a superconductor are bound inside a Cooper pair. The 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, , and there is a finite density of state at the Fermi level, D. But below Tc, the electron density of state acquires a small gap (2Δ) which separates the occupied and unoccupied states [4].
The energy gap, at zero temperature, is evaluated aswhere D is the Debye frequency. The expression for superconducting gap parameter at any temperature between the absolute zero and Tc is given by
The BCS theory estimates the zero temperature energy gap 2Δ (0) as
The ratio is a universal constant. For ErNi2B2C, the superconducting transition temperature is observed at Tc≈11 K [6].
So,
From equations (1) and (4), the critical temperature is expressed as
The value of the coupling parameter constant λ = 0.4721 [7–9].
2. 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 ErNi2B2C in both the normal and superconducting states.
3. Electronic Thermal Conductivity of ErNi2B2C 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, and electronic thermal conductivity, [10–12].
3.1. Heat Current Density (JQ)
The heat current density (JQ) is defined aswhere Ω is the volume of the metallic material to be crossed by the electrons, ƒk is the Fermi–Dirac distribution function, ʋk is the group velocity, Ek is the energy at a state k, and EF is the Fermi energy [13, 14].
The Fermi–Dirac distribution at k-mode is given by [15]where kB is Boltzmann’s constant, T is the temperature, is the chemical potential, and the superscript in indicates the thermal equilibrium.
From the Fermi–Dirac distribution function at the thermal equilibrium, we have relations for partial derivative of .
Assume that characterizes local equilibrium in such a way that the special variation of arises from the temperature, T, and chemical potential, :
Substituting equations (9) and (10) into equation (11), we get
But,
The 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, ; 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]or
At the steady state , equation (16) becomes
Using equations (12) and (13) in equation (17), we get
But,where is the relaxation time. Hence, substituting equation (19) into equation (18) in the absence of magnetic field gives [19, 20]
The 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 ʋk is odd and ƒk is even function. So,
Introducing the new parameter called mean free path, , equation (23) becomes
For thermal conductivity calculation, E and B are taken to be zero. Thus, equation (24) reduces to
3.2. 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. The energy dispersion is given by (k), where t is the hopping energy and is the energy scale. The 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 denotes the interlayer N-N electron hoping energy and is the structure factor of the inter layer electron hoping to the N-N sites on the honeycomb lattice [8].
The structure factor, , is given by
For small values of , , and , cosine functions of equation (27), using Taylor series expansion, can be expanded as
So, equation (27) becomes
For small values of and , the product of and becomes 0.
Similarly,
Substituting equations (32) and (33) into equation (26), we get
The 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]and
Substituting these values in equation (32), we get
Hence, equation (35) is an expression for the dispersion relation.
3.3. Electronic Heat Capacity
Near the Fermi level, electrons gain total electronic thermal kinetic energy, Uel:
From the total electronic thermal kinetic energy, we can calculate the electronic heat capacity as [22]
If U is the total energy transfer, we can write the ΔU aswhere f (ε) and D (ε) are the Fermi–Dirac function and the number of orbitals per unit energy range, respectively. The total number of electrons, N, inside a sphere of radius, k, is given by [23]where is the volume of the sphere and m is the mass of the electron. The density of state of electron, D (ε), is calculated as [24]
Multiply the identityby to obtain
We use equation (42) to rewrite equation (38) as
The first integral on the right hand side of equation (43) gives the energy needed to take electrons from εF to the orbitals of the energy ε > εF and the second integral gives the energy needed to bring the electrons to εF from orbitals below εF.
In equation (43), the product of the first integral of f (ε) D (εF) dε is the number of electrons evaluated to orbitals in the energy range dε at an energy ε. The factor [1 − f (ε)] in the second integral is the probability that an electron has been removed from an orbital, ε.
The heat capacity of electron gas is calculated as
For , we ignore the temperature dependency of the chemical potential, m, in the Fermi–Dirac function and replace m by the constant . Let , and consider the Fermi–Dirac distribution function [25, 26].
Let and from equation (44) and (45), we get
We can replace the lower limit by −. Because the factor in the integrand is already negligible at for low temperatures such that , the integral in equation (46) becomes
Thus, the heat capacity of an electron gas is given by
Substituting equation (40) into equation (48) gives
This is the heat capacity of free electron Fermi gas in the temperature region where . 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 and temperature T at the Fermi level as
Consider
Let the contribution of the z component of the wave vector, , 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 . The electronic heat capacity of ErNi2B2C in its normal state becomes
The Fermi–Dirac distribution for T is approximated asandwhere is the energy gap parameter and is the Fermi energy. The electronic heat capacity of ErNi2B2C in the superconducting state is calculated as
Rearranging equation (56) gives
Let
Substituting equation (58) into equation (57) gives
Equation (59) is the electronic heat capacity of ErNi2B2C in its superconducting state.
3.4. Thermal Conductivity
Thermal conductivity of ErNi2B2C is related to the thermal heat current density as
From equations (25) and (60), the expression for electronic thermal conductivity of ErNi2B2C in its normal state is given as
Rearranging equation (9) gives
From equations (61) and (62), we get
Changing the summation to integration over the allowed k-space and using the relation for any function F (k) giveswhere dSk is the area element in k-space of constant energy and , and equation (63) becomes
The Fermi–Dirac distribution function at equilibrium condition, , is described in equation (7). Let . Using these expressions in equation (65), we get
From the relation of mean free path to the electron velocity, we have , where is the average time between collision of electrons and is the vector displacement of the electron. Moreover, the momentum in k-space can be found from the kinetic term of the electron with 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 ErNi2B2C in its normal state becomes
If the energy varies with k2, i.e., from the kinetic energy , then the expression of thermal conductivity given by equation (70) holds true also as in a heat capacity.
Thus, applying similar analogy as equation (53), the thermal conductivity becomes
Since is constant, the electronic thermal conductivity of ErNi2B2C in its normal state is directly proportional to temperature.
In a superconducting state, the superconductor is characterized by a weak coupling energy gap parameter, , where the relation is given bywhere Ek is the excitation energy measured relative to and we use the symbol where is the chemical potential which is equivalent to the Fermi energy EF.
Thus, 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 ; otherwise, it becomes zero. Since , we neglect the last integral.
Furthermore, we have the following relations:
Thus, using the above expressions in equation (80) gives
Equation (82) gives the general expression for electronic thermal conductivity of ErNi2B2C in its superconducting state.
3.5. Electrical Conductivity and Wiedemann–Franz Law
The electrical conductivity, 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 by electrical conductivity givesand
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. This relation is known as Wiedemann–Franz law.
Hence,
4. 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 ErNi2B2C in both the normal and superconducting states.
The tight binding approximation is given as
Using this approximation, the calculated electronic heat capacity of ErNi2B2C in its normal state is
The expression for electronic heat capacity of ErNi2B2C 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 aswhere
The electronic thermal conductivity in its normal state varies linearly with temperature (Figure 2).
The electronic thermal conductivity of ErNi2B2C in its superconducting state is determined aswhereand are constants. Figure 2 shows the graph of electronic thermal conductivity of ErNi2B2C in its superconducting state.
As can be seen from Figure 3, the electronic thermal conductivity of ErNi2B2C has a shoulder (kink) above the superconducting transition temperature.
In a superconductor, the thermal conductivity below Tc rises sharply and reaches the superconducting transition temperature, Tc, near T = 10 k. After it reaches its maximume value, the kink, then begin to decrease.
The 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 ErNi2B2C in its superconducting state.
5. Conclusion
In this paper, the total electronic energy, the electronic thermal conductivity, and the heat capacity of erbium nickel borocarbide, ErNi2B2C, 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 ErNi2B2C 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, Tc, 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. The 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, ErNi2B2C, becomes normal conductor.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.