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

Authors: Habtamu Tesfaye
Journal: Advances in Materials Science and Engineering
 (2022)

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## 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.


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## Body

## 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.Figure 1
Resistance of a specimen of mercury versus absolute temperature [2].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,εF, and there is a finite density of state at the Fermi level, DεE. 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 as(1)Δ=2ħωDexp−1λ,where ωD is the Debye frequency. The expression for superconducting gap parameter at any temperature between the absolute zero and Tc is given by(2)Δ=3.06kBTc1−TTC1/2.The BCS theory estimates the zero temperature energy gap 2Δ (0) as(3)2Δ0=3.53kBTc.The ratio2Δ0/KBTC is a universal constant. For ErNi2B2C, the superconducting transition temperature is observed at Tc≈11 K [6].So,(4)Δ0=26.58x10−23J.From equations (1) and (4), the critical temperature is expressed as(5)Tc=1.14ħωDKBexp−1λ.The value of the coupling parameter constantλ = 0.4721 [7–9].


## 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,εF, and there is a finite density of state at the Fermi level, DεE. 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 as(1)Δ=2ħωDexp−1λ,where ωD is the Debye frequency. The expression for superconducting gap parameter at any temperature between the absolute zero and Tc is given by(2)Δ=3.06kBTc1−TTC1/2.The BCS theory estimates the zero temperature energy gap 2Δ (0) as(3)2Δ0=3.53kBTc.The ratio2Δ0/KBTC is a universal constant. For ErNi2B2C, the superconducting transition temperature is observed at Tc≈11 K [6].So,(4)Δ0=26.58x10−23J.From equations (1) and (4), the critical temperature is expressed as(5)Tc=1.14ħωDKBexp−1λ.The value of the coupling parameter constantλ = 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,Kp, and electronic thermal conductivity, Ke [10–12].


### 3.1. Heat Current Density (JQ)

The heat current density (JQ) is defined as(6)EK=1Ω∑EK−EFʋKƒK,where 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 atk-mode is given by [15](7)ƒkO=1expEk−μ/KBT+1,where kB is Boltzmann’s constant, T is the temperature, μ is the chemical potential, and the superscript in ƒkO indicates the thermal equilibrium.From the Fermi–Dirac distribution function at the thermal equilibrium, we have relations for partial derivative ofƒkO.(8)∂ƒkO∂Ek=1KBTƒkO1−ƒkO,(9)∂ƒkO∂T=−Ek−μT∂ƒkO∂Ek,(10)∂ƒkO∂μ=−∂ƒkO∂Ek.Assume thatƒkO characterizes local equilibrium in such a way that the special variation of ƒkO arises from the temperature, T, and chemical potential, μ:(11)∂ƒk∂r=∂ƒkO∂T∇T+∂ƒkO∂μ∇μ.Substituting equations (9) and (10) into equation (11), we get(12)∂ƒk∂r=Ek−μT∇T∂ƒkO∂Ek−∂ƒkO∂Ek∇μ.But,(13)∂ƒk∂r=ħʋk∂ƒkO∂Ek,μk=1ħ∂Ek∂KandF=−eE.The description of quasi-particles as wave package allows one to introduce the non-equilibrium distribution functionf (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, ƒkO; 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(14)ƒr,k,t=ƒr−ʋkdt,k+eEħdt,t–dt+∂ƒk∂tcolldt.For small dt, equation (14) can be expanded as [17, 18](15)ƒr,k,t=ƒr,k,t−ʋk∇rƒ+eEħ∇kƒ−∂ƒk∂t+∂ƒk∂tcoll,or(16)∂ƒk∂tcoll=ʋk∂ƒ∂r+Fħ∂ƒk∂k+∂ƒk∂t.At the steady state∂ƒk/∂t=0, equation (16) becomes(17)∂ƒk∂tcoll=ʋk∂ƒ∂r+Fħ∂ƒk∂k.Using equations (12) and (13) in equation (17), we get(18)−Ek−EFTʋk.∇T∂ƒkO∂Ek–eE+1e∇μʋk∂ƒkO∂Ek=∂ƒk∂t.But,(19)∂ƒk∂t=ƒk−ƒkOτ,where τ is the relaxation time. Hence, substituting equation (19) into equation (18) in the absence of magnetic field gives [19, 20](20)ʋk−∇T∂ƒkO∂T+eEʋk∂ƒkO∂Ek=ƒk−ƒkOτ.The non-equilibrium part of distribution function with in the relaxation time approximation becomes(21)ƒk=ƒkO+eτυk∂ƒkOEkE−ʋkτ∂ƒkO∂T∇T.Using equation (21) in equation (6), we get(22)JQ=1ΩΣEK−EFʋKƒkO+eτʋk∂ƒkO∂EkE−τʋk∂ƒkO∂T∇T.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,(23)JQ=1ΩΣEK−EFeμk2τ∂fk0∂EkE–1ΩΣEk−EF2Tμk2τ∂fk0∂Ek∇T.Introducing the new parameter called mean free path,lk=τμ, equation (23) becomes(24)JQ=1ΩΣEK−EFeμklk∂fk0∂EkE–1ΩΣEk−EF2Tμklk∂fk0∂Ek∇T.For thermal conductivity calculation,E and B are taken to be zero. Thus, equation (24) reduces to(25)JQ=1ΩΣEk−EF2Tμklk−∂fk0∂Ek∇T.


### 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 byEk=2tγxy (k), where t is the hopping energy and γ 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:(26)E±=±2txzγxyk−2tzcoskz.where txz denotes the interlayer N-N electron hoping energy and γxy is the structure factor of the inter layer electron hoping to the N-N sites on the honeycomb lattice [8].The structure factor,γxy, is given by(27)γxy=34+12coskxcos32ky.For small values ofkx, kxy, and kxz, cosine functions of equation (27), using Taylor series expansion, can be expanded as(28)coskx≈1−kx22,cos12kx≈1−kx28,cos32ky≈1−3ky28.So, equation (27) becomes(29)γxyk=94+3kx28−3ky28+kx2ky264.For small values ofkx and ky, the product of kx2 and ky2 becomes 0.(30)γxyk=32−kx28−ky28.Similarly,(31)coskz≈1−kz22.Substituting equations (32) and (33) into equation (26), we get(32)Ek=2txy32−kx28−ky28−2tz1−kz22.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](33)txy≈1.6ev,and(34)txyz≈1.25ev.Substituting these values in equation (32), we get(35)Ek=2.3−25kx2−25ky2+54kz2.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:(36)Uel≈NkBT2TF.From the total electronic thermal kinetic energy, we can calculate the electronic heat capacity as [22](37)Cel≈NkBTTF.IfU is the total energy transfer, we can write the ΔU as(38)ΔU=∫0∞dεεDεFfε−∫0εFdεεDεF,where 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](39)N=v3π2k3=v3π22mεFħ23/2,where v 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](40)DεF=dNdεF=3N2εF.Multiply the identity(41)N=∫0∞dεDεFfε=∫0εFdεDεFfε,by εF to obtain(42)∫0εFdεfεDεF+∫εF∞dεεfεDεF=∫0εFdεεFDεF.We use equation (42) to rewrite equation (38) as(43)ΔU=∫εF∞dεε−εFfεDεF+∫0εFdεεF−ε1−fεDεF.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(44)Cel=dUdT=∫0∞dεε−εFdfεdTDεF.ForkBT≪εF, we ignore the temperature dependency of the chemical potential, m, in the Fermi–Dirac function and replace m by the constant εF. Let kBT=τ, and consider the Fermi–Dirac distribution function [25, 26].(45)dfεdτ=ε−εFτ2expε−εF/τexpε−εF/τ+12.Letx=ε−εF/τ, and from equation (44) and (45), we get(46)Cel=kB2TDεF∫−εFτ∞x2exex+12dx.We can replace the lower limit by −∞. Because the factor ex in the integrand is already negligible at −εF/τ for low temperatures such that −εF/τ∼∞, the integral in equation (46) becomes(47)∫−∞∞x2exex+12dx=π23.Thus, the heat capacity of an electron gas is given by(48)Cel=13π2DεFkB2T.Substituting equation (40) into equation (48) gives(49)Cel=π2NkB2T2εF=12π2NkBkBTεF.This is the heat capacity of free electron Fermi gas in the temperature region wherekBT≪εF. For free electron gas approximation at the Fermi level, we have(50)32kBTF≡εF≡12mvF2.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(51)Cel=π2NkB2TmvF2+13x3320π2NkB2TmvF2.Consider(52)Ek=2.3−25kx2−25ky2+54kz2,⇒Ek=2.3−25k2+3320kz2.Let the contribution of thez 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. The electronic heat capacity of ErNi2B2C in its normal state becomes(53)Cel=π2NkB2TmvF2+13x3320π2NkB2TmvF2,Cel≈25π2NkB2TmvF2.The Fermi–Dirac distribution forε−εF≫kBT is approximated as(54)fε∼exp−ε−εFkBT,Dε∼DεFforε−εF>Δ,and(55)Dε∼0forε−εF<Δ.where Δ is the energy gap parameter and εF is the Fermi energy. The electronic heat capacity of ErNi2B2C in the superconducting state is calculated as(56)Cs=DεFkB∫0∞dεε−εFkBT2exp−ε−εFkBT+DεFkB∫εF+Δ∞dεε−εFkBT2exp−ε−εFkBT.Rearranging equation (56) gives(57)Cs=2DεFkB∫εF∞dεε−εFkBT2exp−ε−εFkBT.Letx=ε−εF/kBT.(58)⇒dε=kBTdxSubstituting equation (58) into equation (57) gives(59)Cs=2DεFΔ2Texp−ΔkB.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(60)JQ=−K∇T.From equations (25) and (60), the expression for electronic thermal conductivity of ErNi2B2C in its normal state is given as(61)Kn=1ΩT∑lkvkEk–EF2∂ƒkO∂E.Rearranging equation (9) gives(62)∂ƒkO∂E=−1Ek−EF∂ƒkO∂T.From equations (61) and (62), we get(63)Kn=−1Ω∑lkvkEk–Eƒ∂ƒkO∂T.Changing the summation to integration over the allowedk-space and using the relation for any function F (k) gives(64)∑F2Ω2π3∬FkdEdSKħυK.where dSk is the area element in k-space of constant energy and ħʋk=∂E/∂k, and equation (63) becomes(65)Kn=14π3∫0∞lkʋkdSKħυK∫0∞Ek−EF∂ƒkO∂TdE.The Fermi–Dirac distribution function at equilibrium condition,ƒkO, is described in equation (7). Let x=Ek−EF/KBT. Using these expressions in equation (65), we get(66)Kn=KB2T4π3∫0∞lkʋkdSKħυK∫0∞x2exex+12dx.From the relation of mean free path to the electron velocity, we havelk=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=1/2mħ2k2 as(67)ħk=mυk⇒ħ=mυkk.Substituting these expressions into equation (66) gives(68)Kn=τkB2T4π3m∫0∞kdsk∫0∞x2exex+12dx.Solving equation (68) gives(69)Kn=π2kB2τ3mT.IfN is the total number of conduction electrons involved in a metal, then the electronic thermal conductivity of ErNi2B2C in its normal state becomes(70)Kn=Nπ2kB2τ3mT.If the energy varies withk2, i.e., from the kinetic energy Ek=h2k2/2m, 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(71)Kn=13Nπ2kB2τmT+13x13x3320Nπ2kB2τmT,Kn≈Nπ2kB2τ2mT.SinceNπ2kB2τ/2m 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,Δ, where the relation is given by(72)Ek2˜=Ek2+Δ2.where Ek is the excitation energy measured relative to EF–μ and we use the symbol Ek˜=EK−μ where μ is the chemical potential which is equivalent to the Fermi energy EF.Thus, we have(73)Ek2˜=Ek21+Δ22Ek2.⇒Ek˜=Ek1+Δ22Ek21/2.For small Δ, using the Taylor series, we have(74)Ek˜=Ek+Δ24Ek.From equation (63), we have(75)Ks=−1ΩTΣlkυkEkˇ−EF2∂ƒkO∂E.Hence,(76)Eˇk−EF2=Ek−EF2+2Δ22EkEk−EF+Δ44Ek2+…Ignoring higher-order terms of the gap parameter gives(77)Eˇk−EF2=Ek−EF2+2Δ22EkEk−EF.Substituting equation (77) into equation (75) gives(78)Ks=−1ΩTΣlkυkEk−EF2∂ƒkO∂E,−1ΩTΣlkυk2Δ22EkEk−EF2∂ƒkO∂E.Substituting equation (80) into the ﬁrst term of equation (78) gives(79)Ks=13Nπ2kB2τmT−1ΩTΣlkυk2Δ22EkEk−=EF2∂ƒkO∂EUsing equation (64) in the second term of equation (79) gives(80)Ks=13Nπ2kB2τmT+2Δ22π2T∫lkvkdSkħvk∫∂fko∂EdE,−2Δ22π2T∫EFEk∂fko∂EdE∫lkvkdSkħvk.But the last integral contributes only ifEk=EF; otherwise, it becomes zero. Since Ek≠EF, we neglect the last integral.Furthermore, we have the following relations:(81)∫lkvkdSkħvk=τm∫kdSk=4π3τmanddfkdE=1.Thus, using the above expressions in equation (80) gives(82)Ks=13Nπ2kB2τmT+Δ2τm1T.⇒Ks=αT+Δ2βT.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,δ, is determined for any metallic material as(83)δ=−e2Ω∑lkvk∂ƒkO∂E.Applying similar steps as we did for thermal conductivity, one can arrive at(84)δ=−e24π2∫lkvkdSkħvk∫E−EF2∂ƒkO∂E.Evaluating the second integral yields(85)∫−∞∞E−EF2∂ƒkO∂E=πkBT23.Similarly, interchanging the thermal conductivityKn by electrical conductivity σ gives(86)σ=−e24π2∫υklkhυkdSk,and(87)Kn=−14π3T×πkBT3∫υklkħυkdSk.Taking the ratio of the thermal conductivity to the electrical conductivity gives(88)Kn=σπ2kB23e2T.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,(89)KnσT=13πkBe2≈2.44X10−8V2K2.


## 3.1. Heat Current Density (JQ)

The heat current density (JQ) is defined as(6)EK=1Ω∑EK−EFʋKƒK,where 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 atk-mode is given by [15](7)ƒkO=1expEk−μ/KBT+1,where kB is Boltzmann’s constant, T is the temperature, μ is the chemical potential, and the superscript in ƒkO indicates the thermal equilibrium.From the Fermi–Dirac distribution function at the thermal equilibrium, we have relations for partial derivative ofƒkO.(8)∂ƒkO∂Ek=1KBTƒkO1−ƒkO,(9)∂ƒkO∂T=−Ek−μT∂ƒkO∂Ek,(10)∂ƒkO∂μ=−∂ƒkO∂Ek.Assume thatƒkO characterizes local equilibrium in such a way that the special variation of ƒkO arises from the temperature, T, and chemical potential, μ:(11)∂ƒk∂r=∂ƒkO∂T∇T+∂ƒkO∂μ∇μ.Substituting equations (9) and (10) into equation (11), we get(12)∂ƒk∂r=Ek−μT∇T∂ƒkO∂Ek−∂ƒkO∂Ek∇μ.But,(13)∂ƒk∂r=ħʋk∂ƒkO∂Ek,μk=1ħ∂Ek∂KandF=−eE.The description of quasi-particles as wave package allows one to introduce the non-equilibrium distribution functionf (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, ƒkO; 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(14)ƒr,k,t=ƒr−ʋkdt,k+eEħdt,t–dt+∂ƒk∂tcolldt.For small dt, equation (14) can be expanded as [17, 18](15)ƒr,k,t=ƒr,k,t−ʋk∇rƒ+eEħ∇kƒ−∂ƒk∂t+∂ƒk∂tcoll,or(16)∂ƒk∂tcoll=ʋk∂ƒ∂r+Fħ∂ƒk∂k+∂ƒk∂t.At the steady state∂ƒk/∂t=0, equation (16) becomes(17)∂ƒk∂tcoll=ʋk∂ƒ∂r+Fħ∂ƒk∂k.Using equations (12) and (13) in equation (17), we get(18)−Ek−EFTʋk.∇T∂ƒkO∂Ek–eE+1e∇μʋk∂ƒkO∂Ek=∂ƒk∂t.But,(19)∂ƒk∂t=ƒk−ƒkOτ,where τ is the relaxation time. Hence, substituting equation (19) into equation (18) in the absence of magnetic field gives [19, 20](20)ʋk−∇T∂ƒkO∂T+eEʋk∂ƒkO∂Ek=ƒk−ƒkOτ.The non-equilibrium part of distribution function with in the relaxation time approximation becomes(21)ƒk=ƒkO+eτυk∂ƒkOEkE−ʋkτ∂ƒkO∂T∇T.Using equation (21) in equation (6), we get(22)JQ=1ΩΣEK−EFʋKƒkO+eτʋk∂ƒkO∂EkE−τʋk∂ƒkO∂T∇T.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,(23)JQ=1ΩΣEK−EFeμk2τ∂fk0∂EkE–1ΩΣEk−EF2Tμk2τ∂fk0∂Ek∇T.Introducing the new parameter called mean free path,lk=τμ, equation (23) becomes(24)JQ=1ΩΣEK−EFeμklk∂fk0∂EkE–1ΩΣEk−EF2Tμklk∂fk0∂Ek∇T.For thermal conductivity calculation,E and B are taken to be zero. Thus, equation (24) reduces to(25)JQ=1ΩΣEk−EF2Tμklk−∂fk0∂Ek∇T.


## 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 byEk=2tγxy (k), where t is the hopping energy and γ 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:(26)E±=±2txzγxyk−2tzcoskz.where txz denotes the interlayer N-N electron hoping energy and γxy is the structure factor of the inter layer electron hoping to the N-N sites on the honeycomb lattice [8].The structure factor,γxy, is given by(27)γxy=34+12coskxcos32ky.For small values ofkx, kxy, and kxz, cosine functions of equation (27), using Taylor series expansion, can be expanded as(28)coskx≈1−kx22,cos12kx≈1−kx28,cos32ky≈1−3ky28.So, equation (27) becomes(29)γxyk=94+3kx28−3ky28+kx2ky264.For small values ofkx and ky, the product of kx2 and ky2 becomes 0.(30)γxyk=32−kx28−ky28.Similarly,(31)coskz≈1−kz22.Substituting equations (32) and (33) into equation (26), we get(32)Ek=2txy32−kx28−ky28−2tz1−kz22.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](33)txy≈1.6ev,and(34)txyz≈1.25ev.Substituting these values in equation (32), we get(35)Ek=2.3−25kx2−25ky2+54kz2.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:(36)Uel≈NkBT2TF.From the total electronic thermal kinetic energy, we can calculate the electronic heat capacity as [22](37)Cel≈NkBTTF.IfU is the total energy transfer, we can write the ΔU as(38)ΔU=∫0∞dεεDεFfε−∫0εFdεεDεF,where 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](39)N=v3π2k3=v3π22mεFħ23/2,where v 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](40)DεF=dNdεF=3N2εF.Multiply the identity(41)N=∫0∞dεDεFfε=∫0εFdεDεFfε,by εF to obtain(42)∫0εFdεfεDεF+∫εF∞dεεfεDεF=∫0εFdεεFDεF.We use equation (42) to rewrite equation (38) as(43)ΔU=∫εF∞dεε−εFfεDεF+∫0εFdεεF−ε1−fεDεF.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(44)Cel=dUdT=∫0∞dεε−εFdfεdTDεF.ForkBT≪εF, we ignore the temperature dependency of the chemical potential, m, in the Fermi–Dirac function and replace m by the constant εF. Let kBT=τ, and consider the Fermi–Dirac distribution function [25, 26].(45)dfεdτ=ε−εFτ2expε−εF/τexpε−εF/τ+12.Letx=ε−εF/τ, and from equation (44) and (45), we get(46)Cel=kB2TDεF∫−εFτ∞x2exex+12dx.We can replace the lower limit by −∞. Because the factor ex in the integrand is already negligible at −εF/τ for low temperatures such that −εF/τ∼∞, the integral in equation (46) becomes(47)∫−∞∞x2exex+12dx=π23.Thus, the heat capacity of an electron gas is given by(48)Cel=13π2DεFkB2T.Substituting equation (40) into equation (48) gives(49)Cel=π2NkB2T2εF=12π2NkBkBTεF.This is the heat capacity of free electron Fermi gas in the temperature region wherekBT≪εF. For free electron gas approximation at the Fermi level, we have(50)32kBTF≡εF≡12mvF2.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(51)Cel=π2NkB2TmvF2+13x3320π2NkB2TmvF2.Consider(52)Ek=2.3−25kx2−25ky2+54kz2,⇒Ek=2.3−25k2+3320kz2.Let the contribution of thez 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. The electronic heat capacity of ErNi2B2C in its normal state becomes(53)Cel=π2NkB2TmvF2+13x3320π2NkB2TmvF2,Cel≈25π2NkB2TmvF2.The Fermi–Dirac distribution forε−εF≫kBT is approximated as(54)fε∼exp−ε−εFkBT,Dε∼DεFforε−εF>Δ,and(55)Dε∼0forε−εF<Δ.where Δ is the energy gap parameter and εF is the Fermi energy. The electronic heat capacity of ErNi2B2C in the superconducting state is calculated as(56)Cs=DεFkB∫0∞dεε−εFkBT2exp−ε−εFkBT+DεFkB∫εF+Δ∞dεε−εFkBT2exp−ε−εFkBT.Rearranging equation (56) gives(57)Cs=2DεFkB∫εF∞dεε−εFkBT2exp−ε−εFkBT.Letx=ε−εF/kBT.(58)⇒dε=kBTdxSubstituting equation (58) into equation (57) gives(59)Cs=2DεFΔ2Texp−ΔkB.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(60)JQ=−K∇T.From equations (25) and (60), the expression for electronic thermal conductivity of ErNi2B2C in its normal state is given as(61)Kn=1ΩT∑lkvkEk–EF2∂ƒkO∂E.Rearranging equation (9) gives(62)∂ƒkO∂E=−1Ek−EF∂ƒkO∂T.From equations (61) and (62), we get(63)Kn=−1Ω∑lkvkEk–Eƒ∂ƒkO∂T.Changing the summation to integration over the allowedk-space and using the relation for any function F (k) gives(64)∑F2Ω2π3∬FkdEdSKħυK.where dSk is the area element in k-space of constant energy and ħʋk=∂E/∂k, and equation (63) becomes(65)Kn=14π3∫0∞lkʋkdSKħυK∫0∞Ek−EF∂ƒkO∂TdE.The Fermi–Dirac distribution function at equilibrium condition,ƒkO, is described in equation (7). Let x=Ek−EF/KBT. Using these expressions in equation (65), we get(66)Kn=KB2T4π3∫0∞lkʋkdSKħυK∫0∞x2exex+12dx.From the relation of mean free path to the electron velocity, we havelk=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=1/2mħ2k2 as(67)ħk=mυk⇒ħ=mυkk.Substituting these expressions into equation (66) gives(68)Kn=τkB2T4π3m∫0∞kdsk∫0∞x2exex+12dx.Solving equation (68) gives(69)Kn=π2kB2τ3mT.IfN is the total number of conduction electrons involved in a metal, then the electronic thermal conductivity of ErNi2B2C in its normal state becomes(70)Kn=Nπ2kB2τ3mT.If the energy varies withk2, i.e., from the kinetic energy Ek=h2k2/2m, 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(71)Kn=13Nπ2kB2τmT+13x13x3320Nπ2kB2τmT,Kn≈Nπ2kB2τ2mT.SinceNπ2kB2τ/2m 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,Δ, where the relation is given by(72)Ek2˜=Ek2+Δ2.where Ek is the excitation energy measured relative to EF–μ and we use the symbol Ek˜=EK−μ where μ is the chemical potential which is equivalent to the Fermi energy EF.Thus, we have(73)Ek2˜=Ek21+Δ22Ek2.⇒Ek˜=Ek1+Δ22Ek21/2.For small Δ, using the Taylor series, we have(74)Ek˜=Ek+Δ24Ek.From equation (63), we have(75)Ks=−1ΩTΣlkυkEkˇ−EF2∂ƒkO∂E.Hence,(76)Eˇk−EF2=Ek−EF2+2Δ22EkEk−EF+Δ44Ek2+…Ignoring higher-order terms of the gap parameter gives(77)Eˇk−EF2=Ek−EF2+2Δ22EkEk−EF.Substituting equation (77) into equation (75) gives(78)Ks=−1ΩTΣlkυkEk−EF2∂ƒkO∂E,−1ΩTΣlkυk2Δ22EkEk−EF2∂ƒkO∂E.Substituting equation (80) into the ﬁrst term of equation (78) gives(79)Ks=13Nπ2kB2τmT−1ΩTΣlkυk2Δ22EkEk−=EF2∂ƒkO∂EUsing equation (64) in the second term of equation (79) gives(80)Ks=13Nπ2kB2τmT+2Δ22π2T∫lkvkdSkħvk∫∂fko∂EdE,−2Δ22π2T∫EFEk∂fko∂EdE∫lkvkdSkħvk.But the last integral contributes only ifEk=EF; otherwise, it becomes zero. Since Ek≠EF, we neglect the last integral.Furthermore, we have the following relations:(81)∫lkvkdSkħvk=τm∫kdSk=4π3τmanddfkdE=1.Thus, using the above expressions in equation (80) gives(82)Ks=13Nπ2kB2τmT+Δ2τm1T.⇒Ks=αT+Δ2βT.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,δ, is determined for any metallic material as(83)δ=−e2Ω∑lkvk∂ƒkO∂E.Applying similar steps as we did for thermal conductivity, one can arrive at(84)δ=−e24π2∫lkvkdSkħvk∫E−EF2∂ƒkO∂E.Evaluating the second integral yields(85)∫−∞∞E−EF2∂ƒkO∂E=πkBT23.Similarly, interchanging the thermal conductivityKn by electrical conductivity σ gives(86)σ=−e24π2∫υklkhυkdSk,and(87)Kn=−14π3T×πkBT3∫υklkħυkdSk.Taking the ratio of the thermal conductivity to the electrical conductivity gives(88)Kn=σπ2kB23e2T.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,(89)KnσT=13πkBe2≈2.44X10−8V2K2.


## 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(90)Ek=2.3−25Kx2+3320Kz2.Using this approximation, the calculated electronic heat capacity of ErNi2B2C in its normal state is(91)Cn≈32Nπ2kB2mυF2T.⇒Cn∼T.The expression for electronic heat capacity of ErNi2B2C in its superconducting state is also determined as(92)Cs=2NKBΔT2exp−ΔT.In the superconducting state, the heat capacity falls exponentially as the temperature decreases:(93)Cs∝exp−ΔkBT.Similarly, in its normal state, the electronic thermal conductivity is calculated as(94)Kn=∝T.where(95)∝=12Nπ2kB2m.The electronic thermal conductivity in its normal state varies linearly with temperature (Figure2).Figure 2
The electronic thermal conductivity of ErNi2B2C at the normal state versus temperature.The electronic thermal conductivity of ErNi2B2C in its superconducting state is determined as(96)Ks=∝T+Δ2βT.where(97)∝=13Nπ2kB2τm.and β=τ/m are constants. Figure 2 shows the graph of electronic thermal conductivity of ErNi2B2C in its superconducting state.As can be seen from Figure3, the electronic thermal conductivity of ErNi2B2C has a shoulder (kink) above the superconducting transition temperature.Figure 3
The electronic thermal conductivity of ErNi2B2C in its superconducting state.In a superconductor, the thermal conductivity belowTc 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.



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*Source: 3199173-2022-09-20.xml*