Two Band Model of Superconductivity of Magnesium

Diboride(MgB2) Using Three-square-well Potential

and Linear-energy-dependent Electronic Density of

States: Application to Isotope Effect.

Ogbuu Okechukwu Anthony

PG/MSc/06/41698

BEING

PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE AWARD OF THE MASTER

OF SCIENCE DEGREE OF THE UNIVERSITY OF NIGERIA

MAY, 2011

CERTIFICATION

This is to certify that Ogbuu Okechukwu Anthony, a postgraduate student with registra-

tion number PG/M.Sc/06/41698 has satisfactorily completed the requirements for course

and research work for the degree of Master of Science(M.Sc) in Theoretical Solid State

Physics, Depatment of Physics and Astronomy.

The work contained in this project report is original and has not been submitted in full

or part for any other diploma or degree of this or any other University.

Prof. C.M.I OKOYE —————————— ————————

(Supervisor) Signature Date

Prof.S PAL —————————— ————————

(Supervisor) Signature Date

Prof C.M.I OKOYE —————————— ————————

(HOD, Physics and Astronomy, UNN) Signature Date

————————- —————————— ————————-

(External Examiner) Signature Date

i

DEDICATION

This work is dedicated to the Singularity, the One who knows it all; Supreme God.

We are all but trying.

ii

ACKNOWLEGEMENTS

My most unalloyed gratitude goes to one of my supervisors, Prof. C.M.I. Okoye, for

his sacrifices, suggestions and continued interest towards a successful completion of this

work. I sincerely thank him particularly for providing most of the literature used and for

being generous with his time. I thank also my other supervisor, Prof. S. Pal for very

useful encouragement and advices throughout this work.

It is also a pleasure to acknowledge the moral support and encouragement of Dr. A.E.

Chukwude and Prof. R U Osuji. I am indebted to all the academic staff of Department

of Physics and Astronomy, for the knowledge I gained from them as well as for their

encouragment. I thank them all.

I have to express my gratitude to Mr. Abah Obinna(Oga Obi) whose kind gesture,

criticism and scrutiny contributed to the completion of this work. I also wish to appreciate

my friends; Mr Obodo Joshua, Mr Onah Emeka (Oga Emy), Madam Nnaji Oluchi for

their criticism, encouragement.

My parents and my siblings; Mr.Ogbuu Joseph Udoka and Mrs.Ogbuu Felicia, Ms.Ogbuu

Chinyere, Ms.Ogbuu Ifeoma, Ms.Ogbuu Amara for their massive support throughout my

educational career. They are my rock. I must particularly thank my friend, Ms. Obiala-

sor Uche for practically dragging me to finishing the final type-setting of this work.

iii

Abstract

We have derived the expressions for the transition temperature and the iso-

tope effect exponent within the framework of Bogoliubov-Valatin two-band

formalism using a linear-energy-dependent electronic density of states assum-

ing a three-square-well potentials model. Our results show that the approach

could be used to account for a wide range of values of the transition temper-

ature and isotope effect exponent. The relevance of the present calculations

to MgB2 is analyzed.

Contents

1 General Introduction 1

1.1 Introduction And Discovery of Superconductivity . . . . . . . . . . . . . . 1

1.2 Properties of Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Electromagnetic Properties . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.2 Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.3 Isotope Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.4 Tunelling and Joesphson Effect . . . . . . . . . . . . . . . . . . . . 7

1.3 Models of Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Phenomenological Model . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 Microscopic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Magnesium diboride(MgB2) and Its properties . . . . . . . . . . . . . . . . 14

1.5 Outline of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Review Of Literature On Occurrence Of Two-Band Energy Gaps in

Magnesium Diboride, Isotope Effect and Influence of linear-energy-

dependence on the Density of States In Magnesium Diboride(MgB2)

Superconductor As Well As BCS Theory Of Isotope Effect 19

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Existence Of Two-Band Energy Gaps In MgB2 Superconductor . . . . . . 20

2.3 Occurrence of Isotope Effect in High-Tc MgB2 Superconductor . . . . . . . 22

2.4 Influence of Linear-Energy-Dependent Electronic Density of State on Two-

Band Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5 Bardeen-Cooper-Schrieffer(BCS)Theory and the Two-Square-Well Theory

of Isotope Effect Using Linear-Term Energy Dependent Electronic Density

of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

i

3 Two-Band Model of Superconductivity of MgB2 Using Three-Square

-Well Potential with Linear-Energy-Dependent Electronic Density of

states 38

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Derivation Of Transition Temperature And Isotope Effect in One-Band

Model Using Three Square Well Potential With Linear Term Energy De-

pendent of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Derivation of Transition Temperature And Isotope Effect in Two-Band

Model Using Three-Square-Well Potential with Linear-Energy-Dependent

Electronic Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Discussions and Conclusion 51

4.1 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

ii

List of Tables

1.1 List of Superconducting parameters of MgB2 . . . . . . . . . . . . . . . . . 17

iii

List of Figures

1.1 Crystal Structure of MgB2. . . . . . . . . . . . . . . . . . . . . . . . 16

4.1 The plot of variation of transition temperature with the effective linear-

energy-dependent acoustic-electron phonon coupling, λ11 +λ1

12 using Eq.(3.54). 52

4.2 The plot of variation of isotope effect exponent with the effective linear-

energy-dependent acoustic-electron phonon coupling, λ11 +λ1

12 using Eq.(3.58). 53

4.3 The plot of variation of isotope effect exponent with the transition temper-

ature for λ11 + λ1

12 = - 0.0002 to 0.0008 using Eq.(3.59) . . . . . . . . . . . 54

iv

Chapter 1

General Introduction

1.1 Introduction And Discovery of Superconductiv-

ity

In July 1908, Kammerling Onnes succeeded in liquidfying helium gas and this en-

abled him to measure the resistivity of metals down to 4.2 K. This succees prompted

Onnes into a research program which was centered on the investigation of the electrical

resistivites of metals at lower temperature.

In 1911, he discovered superconductivity in a sample of mercury at about few degrees

kelvin. He observed, while studying the electrical resistivity of mercury that the resis-

tivity decreased more or less smoothly with temperature until at 4.2 K an unexpected

dramatic plunge in resisitivty estimated at least six orders magnitude lower than the room

temperture value occured [1]. In his word, “Mercury has passed into a new state which

in account of its extraordinary electrical properties may be called the superconducting

state”.

Superconductivity is a property of some materials(Metal, Metallic alloys and ceramic

oxides, etc) characterised by an abrupt and complete disappearance of resistance to direct

current when the materials are cooled below a certain temperature known as the critical

or transition temperature, Tc, of the material. Thus superconductivity is a zero electrical

resistance [2] and materials exhibiting this phenomeno are called superconductors.

Since its discovery, superconductivity has found many application in technology.

These applications are found in nuclear magnetic resonance, tomogramphy, magnetic

sensors(magnetometer), digital signal and data processing(Geological survey), supercon-

1

ducting magnetically levitated train [3], superconducting magnet used to detonate mines,

superconducting cables, superconducting fault limiting device, superconducting magnetic

energy storage(SMES), magnetohydrodynamic propulsion of boats. Exploring that fact

the electrical resistance in a superconductor is almost zero, large homogeneous field can

be generated by simply winding the coils of the wires made from the high transition

temperature and critical magnetic field superconducting materials.

However, these applications of superconductors have suffered because of the need to

maintain their temperature with a refrigerant (liquid helium). Although liquid hydrogen

and nitrogen can be relatively cheaper, more abundant and easier to handle than helium,

no superconductor existed with the right parameter to operate in them. Consequestly,

physicist and material scientist have been working relentlessly to improve the low tem-

perature nature of this phenonmeno and obtain superconductivity at high temperatures.

Matthias and Hulum [4] pioneered the search for the high Tc superconductors in transi-

tion metal alloys and compounds in the 1950’s. This led to the independent discovery of

superconductivity in thin films of the A12 compound Nb3Ge at 23 K [5].

Superconductivity has been discovered in several other classes of materials such as

the cheveral phases [6], heavy fermion systems [7], organic superconductors and more

recently diborides. The cheveral phases AxMO6X8 [8] are mostly tenary transition metal

chalcogenides where X is sulphur (S), selenium (Se) or tellurium (Te) and A can be

almost any element. The first organic superconductor, quasi 1-D(TMTSF)2PF6 (Tc =

0.9 K at P = 12 Kbar), was discovered in 1979 and required high pressure to supress a

metal-insultor antiferromagnetic ordering transition at approximately 16 K [9]. The study

of heavy fermion system led to the discovery of CeCu2Si2 and UPtc [10] systems.Heavy

fermion system often exhibit two ordering transitions: a superconducting transition at Tc

and an antiferromagnetic ordering transition at Neel temperature.

In 1986, Bednorz and Muller [11], reported observation of superconductivity with Tc ≈

30 K in the tenary(La1−xBax)2CuO4 otherwise known as 214 compound. Before the end

of 1986, superconductivity at up 57 K in La-Ba-Cu-O under pressure and improved stio-

chiometry was reported [12]. It was found that Tc of La-Ba-Cu-O increases with pressure

at an unprecedented rate. This led to the substitution of barium by smaller stronium

in this class of oxides in order to simulate the pressure effect and this gave Tc = 42.5 K

without the application of pressure [12]. Exploring the concept that smaller atoms tend

to favour high Tc under high pressure, lanthanum(La) which has a larger atomic radius

2

was completely replaced with smaller but chemically similar yttrium(Y). This resulted to

a Tc ≈ 93 K in a sample of multiphase quarternary YBa2Cu3O7−x(YBCUO) compound

at ambient pressure [13]. Subseqently, superconductivity was discovered in bismuth ox-

ide(BiSrCaCuO) [14, 15] with Tc ≈ 110 K and in thallium oxide(TiBaCuO)[9] wth Tc

up to 125 K. Many other oxide superconductors have been discovered over these years.

Some of them were made by substitution of element in already known superconducting

compounds.

Chuang et al [16] used the angle mode of the high energy resolution spectrome-

ter(HERS) and cleaved single crystallne samples of the layered Manganite La1.2Sr1.8Mn2O7

which had a Tc ≈ 126 K. In 1993, Berkley et al [17] reported Tc = 131.8 K for Ti2Ba2Ca2Cu3O10−x

at a pressure of 7 GPa. The recently discovered homologous series HgBa2Can−1O2n+2+δ

possesses remarkable properties. A superconducting temperature Tc as high as 133 K has

been found to be attributable to the Hg-Ba-Ca-Cu-O compound. Temperature dependent

electrical resistivity measurements under pressure on a(>95 percent) pure Hg-1223 phase

are reported. The report shows that Tc increases steadily with pressure at a rate of about

1 K per Giga-Paschal up to 150 GPa,then more slowly and reaches a Tc = 150 K, with

the onset of the transition at 157 K for 23.5 GPa HgBa2Ca2Cu3O8+d [18].

Again, the compound C60 called buckminfullerenes or Fullerenes, consisting of 60

carbon atoms and alkalis metals were found to be superconducting in 1979 [10]. The

transition temperature of several doped fullerenes range from 19 K to 45 K [19].

The search for new high temperature superconductors has proceeded by following sim-

ple trends in the periodic table which provide insight into the correct theoritical model for

the superconductors. In the light of this tremendous progress that been made in raising

the transition temperature of the copper oxide superconductors, it is natural to know how

high the Tc can be increased in other classes of materials. The discovery of supercon-

ductivity with T ≈ 39 K in Magnesium diboride(MgB2) in 2001 [20] caused excitement

in the solid state physics community because it introduced a new simple(three atoms per

unit cell) binary intermetallic superconductor with a record high superconducting Tc for

non-oxide and non-C60 based compound.

3

1.2 Properties of Superconductors

Superconductors have interesting varied and perculiar properties. The superconducting

state is distinguished from the normal state by its electromagnetic, thermodynamic, iso-

tope effect and tunnelling properties. We shall briefly discuss some of these properties.

1.2.1 Electromagnetic Properties

The electromagnetic properties of a superconductors were first observed experimentally.

The basic observation was the disappearance of electrical resustance of various metals

(mercury, lead, tin) and alloys in a very small range of temperature around a critical

temperature Tc, charateristic of the material. Critical temperature for typical supercon-

ductors range from 4.15 K for mercury, to 3.9 K for tin, 7.2 K and 9.2 K for lead and

niobium respectively. This is particularly clear in experiments with persistent current in

superconducting rings as a result of zero resistance leading to infinite conductivity. These

currents have been observed to flow without measurable decay up to 105 years. Good

conductors have resistivity at a temperature of several degrees kelvin of the order of

106 Ωcm [21].

In 1933 Messiner and Ochenfeld [22] discovered the perfect diamagnetism, that is the

magnetic field penerates only a depth λ w 500 Angstroms and is excluded from the body

of the material. One could think that due to vanishing of the electrical resistance, the

electrical field is zero within the material, therefore due to the Maxwell equation

∇XE =−1

C

∂B

∂t(1.1)

the magnetic field is frozen, but it is expelled. This implies that supeconductivity will be

destroyed by a critical magnetic field Hc such that

fs(T ) +H2c

8π(T ) = fn(T ) (1.2)

where fs,n(T) are the densities of free energy in the superconducting phase at zero magnetic

field and in the normal phase. The behavoiur of the critcal magnetic field with the

temperature was found emprically to be a parabolic and is by Tuyn’s law:

Hc(T ) ≈ Hc(0)

[1−

(T

Tc

)2]

(1.3)

4

The critical field at zero temperature is of order of few hundred guass for type I (Soft)

superconductor. Example Al, Sn, In, Ph etc. For hard or type II superconductors such as

Nb3Sn, superconductivity stays up to the value of 105 gauss. What happens is that up to

a lower critical value Hc1, we have the complete Messiner effect. Above Hc1,the magnetic

flux penetrates into the bulk of the material in the form of vortices(Abrikosov vortices)

[23] and the penetration is complete at H = Hc2 >Hc1,Hc2 is called “upper critical field ”.

1.2.2 Thermal Properties

Similar to the electromagnetic propertiess such as Gibbs free energy, entropy and elec-

tronic specific heat of a metal also change sharply as the transition temperature for

superconductivity.

The Gibb’s free energy of any system in a magnetic field is given by

G(P, T,H) = U − TS −MH (1.4)

where U is the enthalpy,S is the entropy, M is the magnetization, P is the Pressure and

T is the absolute temperature. If the enthalpy is fixed then

dG = SdT −MdH (1.5)

In normal metal, Gn is independent of H, then

dGn = −SndT (1.6)

In a superconductor

dGs = −SsdT −(−H4π

)dH (1.7)

or

Gs =H2

8π−∫SsdT

Therefore

Gs −Gn =H2

8π≈ 107eV

At the phase boundary separating the normal and the superconducting state

Ss − Sn =1

8π

dH2c

dT(1.8)

It shows that entropy in the superconducting state is always less than the entropy in the

normal state

5

W.H Keesom observed in 1932 that the transition to the superconducting state is

accompanied by a jump in the specfic heat. The transition from the superconducting

state to normal state is second order in zero magnetic field at Tc = T. This means

that there is no discontinuity at Tc in either entropy or thermal hystersis(volume) but

there is a sharp discontinuity, ∆C, in the heat capacity. The specific heat in the normal

state varies linearly with temperature T, while specific heat in the superconducting state

initially shoots above the normal state,Cn, and drops below it before finally vanishing

expontentially as T→ 0. Theoretically,it is found that the specific heat below Tc, Cs is

given by

Cs ≈ exp

(−∆

KβT

)(1.9)

where ∆ is the energy gap. This dependence indicates the existence of an energy gap

in the energy spectrum separating the exicted state from the ground state(or energy gap

of the elementary exictations or quasi-particle). The presence of an energy gap in the

spectrum of the quasi-particles has been observed directly in various other ways. For

instance the threshold for the absorption of electromagnetic radiation or through the

measure of the electron tunelling current between two films of superconducting material

separated by thin (≈ 20 Angstroms) oxide layer. The presence of an energy gap of order

Tc was suggested by Daunt and Mendelssohn [24] to explain the absence of thermoelectric

effect, but it was also postulated theoretically by Ginzburg [25] and Bardeen [26]

1.2.3 Isotope Effect

An interesting property of superconductors leading eventually to appreciating the roles

of phonon in superconductivity [27] is the isotope effect. It was found [28, 29] that the

critical field at zero temperature and transition temperature Tc vary with the isotopic

mass of the material as

Tc ∝M−β (1.10)

where M is the ionic mass of the material, β is the isotope effect exponent. The isotope

effect exponent is given by

β = − ∂lnT

∂lnM(1.11)

6

This is for single component system. For multicomponent system, the total isotope effect

exponent is the sum of the individual atoms with mass M∑i

βi = −∑i

∂lnT

∂lnM

It has been found that β ≈ 0.45 - 0.5 for many superconductors although there are several

expections such as Ru, Mo, Zr etc. The discovery of isotope effect indicates the importance

of electron-phonon interaction which provides the basis for the microscopic theory [30] .

1.2.4 Tunelling and Joesphson Effect

Consider two metals separated by an insulator. The insulator normally acts as a barrier to

the flow of conduction electrons from one metal to the other. If the barrier is sufficiently

thin(less than 10 or 20 Angstroms) there is a significant probability that an electron which

impinges on the barrier will flow from one metal to the other. This is called Tunelling

[31]

If both metals are superconductors, two types of particles may tunnel; single quasi-

particle and paired superconducting pair. Tunnelling of single quasi-particle has been

used to measure the energy gap in superconducting state. Tunnelling of superconducting

paritcles, called Josephson Tunnelling , exhibit unusual quantum effect that has been

exploited in a variety of quantum devices. The effects of superconductive pair tunnelling

include:

DC Josephson Effect

A direct current flow across the junction in the absence of an electric or magnetic field.

The current, J, of the superconducting pair depends on the phase difference ϕ as

J = J0 sinϕ = J sin(θ2 − θ1) (1.12)

where J0 is the maximum zero voltage current that can be passed by the junction. With

no applied voltage, a dc current will flow across the junction with the value between J0

and -J0 according to the value of the phase difference (θ2 − θ1)

AC Josephson Effect

If the current supplied by an external source exceeds the critical value Ic of a super-

conductor it causes a voltage V to appear across the junction. Thus, the current of

normal electrons In starts flowing through the Josephson junction. This leads to the

7

so called resistively shunted model of the Josephson junction(RSJ) [14] which is con-

sidered as a circuit made up of the Josephson junction itself and a normal resistance

connected in parallel. The total current is then a sum of the normal current(I=VR

) and

the supercurrent(Is=Ic sinϕ):

I = Ic sinϕ+V

R(1.13)

where R is the normal-state resistance of the junction. The presence of the voltage V

across the weak link suggests that Cooper pair energies in superconductors on either side

of the junction, E1 and E2 are related by

E1 − E2 = 2eV

Thus, the second fundamental relation of Josephson is

2eV = ~∂ϕ

∂t(1.14)

putting (1.13) into (1.14) we have

I = Ic sinϕ+~

2eR

∂ϕ

∂t(1.15)

Integrating this differential equation and substituting the solution to equation (1.14), we

obtain the voltage across the junction as

V (t) = RI2− − I2

c

I + Ic cosωt(1.16)

and frequency

ω =2e

~R√I2− − I2

c

Thus we found a fasctinating property of the Josephson junction. If an external direct

current, I, through the junction exceeds the critical value Ic, it causes a voltage to appear

across the junction which oscillates periodically with time. This phenomenon is often

referred to as Josephson Radiation [14]. The frequency of the AC voltage depends on

the amount by which the current through the junction exceeds the critical value. The

first experimental observation of the Josephson radiation was reported in 1964 Yanson,

Svistunov, Dmitrenko [32].

8

1.3 Models of Superconductivity

1.3.1 Phenomenological Model

It is clear that in a superconductor a finite fraction of electrons form a sort of condensate or

“macroelectrons”(superfluid) capable of motion. At zero temperature the condensation is

complete over all the volume, but when increasing the temperature part of the condensate

evaporates and goes to form a weakly interacting normal fluid liquid. At the critical

temperature all the condensate disappears. We shall commence to review the theoritical

approaches to understanding the convectional superconductor.

Gorter-Casimir Model

This model was first formulated in 1934 [33] and it consists a simple anasatz for the free

energy of the superconductor. Let x represent the fraction of electrons in the normal fluid

and 1-x the ones in the superfluid. Gorter and Casimir assumed the following expression

for the energy of the electrons

F (x, T ) =√xfn(T ) + 1− xfs(T ) (1.17)

with

fn(T ) = −γ2

2T 2, fs(T ) = −β

The free energy for the electrons in a normal metal is fn whereas fs gives the condensation

energy associated with the superfluid. Minimising the free energy with respect to x, one

finds the fraction of normal electrons at temperature T as

x =1

16

γ2

β2T 4 (1.18)

At x=1, the critical temperature Tc is

T 2s =

4β

γ2(1.19)

Therefore the fraction of electron at temperature T is

x =

(T

Tc

)4

(1.20)

The corresponding value of the free energy is

fs(T ) = −β

(1 +

(T

Tc

)4)

(1.21)

9

and

fn(T ) = −2β

(T

Tc

)We easily find that

H2c (T )

8π= fn(T )− fs(T ) = β

(1−

(T

Tc

)2)2

The specfic heat in the normal phase is

Cn = −T ∂2fn(T )

∂T 2= γT (1.22)

whereas in the superconducting phase, it is

Cs = 3γTc

(T

Tc

)3

(1.23)

This shows that there is a jump in the specific heat and that the ratio of the two specific

heats at the transition is 3.

The London Theory

The brothers H . and F. London [34] gave a phenomenological description of the basic

facts of superconductivity by proposing a scheme base on two fluid model concept with

superfluid and normal fluid densities ns and nn associated with velocities Vs and Vn. The

densities satisfy

nc + ns = n (1.24)

where n is the average number per volume. The two current densities satisfy

∂Js∂t

=nse

2E

m, (Js = −ensVs) (1.25)

Jn = σnE, (Jn = ennVn) (1.26)

Equation (1.25) is the Newton equation for particles of charge (-e) and density ns.

From EM theory,

∇XE = −∂B∂t

(1.27)

combining equations (1.25) and (1.27), we deduce that

∇X(m

ne2.∂Js∂t

)= −∂B

∂t(1.28)

This gives the other London equation

∇XJs = −nse2

mcB (1.29)

10

Considering the Maxwell’s equation

∇XB =4π

cJs (1.30)

where we have neglected the displacement current and the normal fluid current.

Taking the curl of Eq.(1.30), we get after evaluation that

∇XB =4πnse

2

mc2B =

B

λ2L

(1.31)

with the penetration depth defined by

λL(T ) =

(mc2

4πnse2

) 12

(1.32)

Applying equation(1.31) to a plane boundary located at x = 0, we obtain

B(x) = B(0) exp

(−xλL

)(1.33)

showing that the magnetic field vanishes in the bulk of the material. Notice that as

T →Tc, one expects ns → to 0 and therefore λ(T) → ∞ in the limit.

On the other hand for T→ 0, ns →n, we obtain

λL(0) =

(mc2

4πnse2

) 12

(1.34)

In the two fluid model of Gorter and Casimir, one has

nsn

= 1−(T

Tc

)4

, (1.35)

and

λ(T ) = λ(0)

1

1−(TTc

)4

12

(1.36)

This agrees well with experiments.

Another characteristic length in a superconductor is the coherence length. It is a

measure of the correlated distance of the superconducting electrons and is denoted by ξ0.

The coherence length given in terms of the Fermi velocity VF , Boltzmann constant Kβ

and the superconducting transition temperature Tc is

ξ0 =~VFKβTc

(1.37)

Ginzburgh-Landau Theory

In 1950, Ginzburgh and Landau [35] formulated their theory of superconductivity by

11

introducing a complex wave function as an order parameter. This is in contex of Landau

theory of second order phase transitions and as such this treatment is strictly valid only

around the second order critical point. The wave function is related to the superfluid

density by

ns = |Ψ(r)|2 (1.38)

Furthermore, it postulated a difference of free energy between the normal and supercon-

ducting phase of the form

fs(T )− fn(T ) =

∫d3(r)

(1

2m∗Ψ∗(r)|∇+ iAe∗|2∇(r) + α(T )|Ψ(r)|2 +

1

2β(T )|Ψ|4

)(1.39)

where m∗ and e∗ are the effective mass and charge that turned out to be 2e and 2m

respectively in microscopic theory. Minimizing the free energy, we find in the absence of

field and gradient that the wave function is

|Ψ|2 = −α(T )

β(T )(1.40)

and the free energy density becomes

fs(T )− fn(T ) =−1

2

α2(T )

β(T )=−H2

c (T )

8π(1.41)

Recalling that in London theory,

ns = |Ψ|2 ≈ 1

λ2L(T )

(1.42)

We find thatλ2L(0)

λ2L(T )

= − 1

n

α(T )

β(T )(1.43)

combining equations(1.41) and (1.43), we get

nα(T ) =−H2

c

4π

λ2L(T )

λ2L(0)

(1.44)

and

n2β(T ) =−H2

c

4π

λ4L(T )

λ4L(0)

(1.45)

Solving the equation of motion at zero EM field, we obtain the lowest order in free energy

that1

4m∗|Ψ|(T )∇2f − f = 0 (1.46)

12

This shows an exponential decrease which we will write as

f = exp

(√2r

ε(T )

)(1.47)

where ε(T ), the Ginzburgh-Landau(GL) coherence length is

ε(T ) =1√

2m∗|α(T )|(1.48)

using the expression (1.40) for α(T ), we have

ε(T ) =λL(0)

λL(T )

√2πn

m∗H2c (T )

(1.49)

We see that as ε(T )→∞ for T→Tc

ε(T ) =1

Hc(T )λL(T )(1.50)

The ratio of the two characteristic lengths define the GL parameter

k =λ(T )

ε(T )

The Ginzburgh-Landau theory was able to explain the intermediate state of superconduc-

tors in which the superconducting and normal domains coexist in the presence of critical

magnetic field H ≈ Hc. Also Abrikosov in 1957 [23] classified superconductor into type

I and type II using Ginzburgh-Landau(GL) parameter, where k < 1√2

indicates type I

and k > 1√2

indicates type II. However, despite the success of Ginzburgh-Landau theory

it failed to account for the basic interaction mechanism for superconductivity. Gor’kov

[36] in 1959 was able to show that Ginzburgh-Landau theory was a limiting form of the

microscopic theory of BCS model near transition temperature.

1.3.2 Microscopic Model

The microscopic theory of superconductivity, formulated in 1957 by Bardeen, Cooper

and Schrieffer [30] now known as the BCS theory, gave a successful of most of the basic

features of the superconducting state. The theory was initiated on the idea that the

carriers of electric current in a superconductor are bound pairs of electrons. These bound

pairs are formed when the electron-electron phonon mediated interaction is attractive and

dominates the screened coulomb interaction of the electron.

13

The expression for the superconducting transition temperature, Tc is

KβT = 1.14~ωD exp

(− 1

N(0)V

)(1.51)

where N(0) is the electron density of states, V is the net attractive potential between the

electrons and ωD is the Debye frequency.

1.4 Magnesium diboride(MgB2) and Its properties

Magnesium diboride(MgB2) is sp bounded material which was first synthesized in 1953

but its superconducting properties were discovered until half a century later. The dis-

covery of superconductivity in MgB2 with Tc at 39 K sparked great interest with respect

fundamental physics and practical application of this material.

This recently discovered high temperature superconductor has many similarities with

the convectional superconductor which are understood on the basis of the theory proposed

in 1957 by Bardeen, Cooper and Schrieffer known as the BCS theory of superconductiv-

ity [30]. Magnesium diboride is an inexpensive and simple superconductor. Its critical

temperature of 39 K is the highest among convectional superconductors and also higher

than those of some cuprate high-Tc, where pairing driving forces other than phonons have

been speculated [37, 38]. It has a hexagonal crystal structure with space group p6/mmm

where boron atoms form graphite-like sheets separated by hexagonal layers of Mg atoms.

The boron atoms form honeycombed layers and the magnesium atom are located above

the centre of hexagons in-between the boron planes. Specific heat [39, 40] and Tunnelling

spectroscopy measurements [41] as well as nuclear magnetic resonance(NMR)studies [42]

show that MgB2 is an s-wave superconductor. The phonon density of states of MgB2

has been obtained by inelastic neutron scattering [43, 44, 45]. These results indicate that

phonons play a vital role in the superconductivity of MgB2. Most experiments in MgB2

such as the presence of isotope effect [45, 46], Tc pressure dependence [47] indicate that

the superconductivity of MgB2 points towards phonon-mediated BCS electron pairing.

The Fermi surface of MgB2 consists of four sheets: two 3D sheets from the π-bonding

and antibonding(B-2Pz)and two nearly cylindrical sheets from 2D σ-bonding(B-2Px,y)

[48, 49].

Experiments such as point-contact spectroscopy [50, 51], specific heat measurement

[39, 40], scanning tunnelling microscopy [52] and Raman spectroscopy [53] clearly explains

14

the existence of two distinct superconducting gap with small gaps ∆s(0) = 2.8 ± 0.05

meV and large gap ∆L(0) = 7.1 ± 0.1 meV [54]. The ratio ∆s/∆L is estimated to be

around 0.3 - 0.4 [47]. Both gaps close near the bulk transition temperature Tc = 39 K.

This case has been predicted theoretically by Liu et al [48]. With Tc = 39 K and two

distinct superconducting gaps, MgB2 serves as an important test case for Density Func-

tional Theory(DFT) for superconductors. For simple BCS metal the critical temperture

decreases under pressure due to the reduced electron-phonon coupling [55]. For MgB2 the

transition temperature also decreases with pressure up to the highest pressure studied[56].

Though the Tc decreases with pressure, MgB2 is still superconducting up to the highest

pressure studied and there is no structural transition in MgB2 up to 40 GPa [57]. Ther-

mal expansion demonstrates the out-of-plane Mg-B bonds are much weaker than in-plane

Mg-Mg bonds [58].

Band structure calculation clearly reveals that, while strong B-B covalent bonding is

retained, Mg is ionised and its two electrons are fully donated to B-derived conduction

band [59]. Then it may be assumed that the superconductivity in MgB2 is essentially due

to the metallic nature of the 2D sheets of Boron and the high vibrational frequencies of

the light boron atoms lead to the high Tc of this compound. There are only three reports

about the Hall effect in MgB2 until now [60, 61]. All reports agree with the fact that

the normal state Hall coefficient RH is positive, therefore the charge carriers in MgB2 are

holes with a density at 300 K of between 1.7 - 2.8x1023 holes/cm3, about two orders of

magnitude higher than the charge carrier density for Nb3Sn and YBCO.

The coherence length at zero temperature, ξ(0), of the diboride superconductor in the

high Tc material is small are comparable with the interatomic distance(d) with an average

value of about 0.49 Angstrom [62]. It was concluded that MgB2 is an extreme type II

superconductor with Ginzburg-Landau parameter K ≈ 23 [63]. The observed isotope

effect is reduced substantially from BCS value of 0.5. In MgB2 Tc is sensitive to boron

isotopic substitution while Mg isotope substitution does not make a significant change in

Tc. Tc is higher by ∼ 1 K for the Mg10B2 compared to Mg11B2 [46]. The boron isotope

coefficient(αB) is only significant and the Mg isotope coefficient(αMg) is very small but

still non-zero. Bud’ko et al measured an αB of 0.26 [45]. Measurement by Hinks et al

shows an αB of 0.30 and αMg of 0.02 [46], altogether a total isotope coefficient α of 0.32

for MgB2 with a high Debye temperature of θD = 750 K. Optical measurement [64] and

the specific heat measurement [40] for MgB2 roughly estimated 2∆0/KBTC ' 2.6 and

15

2∆0/KBTC ' 4.2 for MgB2 which deviates from the BCS value of 3.53

This material is interesting because it is a solid metallic superconductor and made of

very light and cheap materials. It is a good metal where there is no high contact resistance

between the grain boundaries thereby eliminating the weak link problem that has avoided

the widespread commercialization of high temperature cuprate superconductors [65]. Un-

like the cuprates, MgB2 has lower anisotropy, larger coherence length, transparency of

the grain boundaries to current flow makes it a good candidate for applications. MgB2

promises a higher operating temperature and higher device speed than the present elec-

tronics based on Nb. Moreover, higher critical current densities(Jc) can be achieved in

a magnetic field by oxygen alloying [66] and irradiation shows an increase on Jc values

[67]. The discovery of MgB2 superconductivity has spurred the search for other related

MgB2 superconductors. It served as a catalyst for MgB2 related superconductors: TaB2

with Tc = 9.5 K [68], BeB2.75 with Tc = 0.7 K [69], graphite sulfur composites, similar to

MgB2 electronically and crystallographically, with Tc = 35 K [70] and another not related

but “inspired“ by it, MgCNi3 with Tc = 8 K [71]. Probably, the most impressive is the

recent report related to superconductivity under pressure of B, with a very high critical

temperature for a simple element of Tc = 11.2 K [72].

Figure 1.1: Crystal Structure of MgB2.

16

Table 1.1: List of Superconducting parameters of MgB2

Parameter Values

Critical Temperature TC = 39K-40K

Hexagonal Lattice Parameters a =0.3086nm,c =0.3524nm

Theoretical Density ρ = 2.5g/cm3

Pressure Coefficient dTC /dP=-1.1 -2K/GPa

Carrier Density ns = 1.7-2.8x1023 holes/cm3

Isotope effect αT = αB + αMg = 0.3+0.02

Resistivity near TC ρ(40K)= 0.4 - 0.6µΩcm

Resistivty ratio RRρ/ρ(300K)= 1-27

Upper critical field Hc2 //ab(0)=14 - 39 T

Hc2 //c(0)=2 - 24T

Lower critical field Hc1 (0)= 27 - 48mT

Irreversibility field Hirr (0)=6 - 35T

Coherence lengths ξab (0)=3.7 - 12 nm

ξc (0)=1.6 - 3.6 nm

Penetration depths λ(0)=85 - 180nm

Energy gap ∆(0)=1.8 - 7.5 meV

Debye temperature θD =750 - 880K

Critical current densities Jc (4.2K,0T)>107 A/cm2

Jc (4.2K,4T)=106 A/cm2

Jc (4.2K,10T)>105 A/cm2

Jc (25K,0T)>5x106 A/cm2

Jc (25K,2T)>105 A/cm2

17

1.5 Outline of Problem

In this study, we shall employ both the experimental results that support the existence of

two energy gap and theoretical approaches that suggest the existence of two overlapping

band at the Fermi level. The outline of the project is as follows:

In chapter 1 , we shall review the discovery of superconductivity, the properties of a

superconductor and the models of superconductivity. This permits us to present a review

of the current theoretical approaches to understanding conventional superconductors. In

chapter 2, We review the BCS theory which incorporates two-square-well potential theory

of isotope effect since this equips us with relevant tools that will be used in this study.

Also, in that chapter, we shall present a review of theoretical studies on the isotope

effect of the high-Tc MgB2 superconductor especially the effect of a shift on the isotope

effect exponent of the two-band high-Tc superconductor by introducting a linear term

to its electronic density of states. Chapter 3 deals with the development of transition

temperature and isotope effect for one-band as well as two-band case using three-square-

well potential with linear-energy-dependent electronic densty of states. The three-square-

well potential corresponds to the electron-acoustic phonon, electron-optical phonon and

electron-electron interactions. Finally,in chapter 4, we shall deal with possible discussion

of results and conclusion.

18

Chapter 2

Review Of Literature On Occurrence

Of Two-Band Energy Gaps in

Magnesium Diboride, Isotope Effect

and Influence of

linear-energy-dependence on the

Density of States In Magnesium

Diboride(MgB2) Superconductor As

Well As BCS Theory Of Isotope

Effect

2.1 Introduction

The discovery of superconductivity at 40 K in MgB2 Nagamatsu [20] has a notable ex-

itement in the people of solid state community. For numerous reasons MgB2 is a very

unusual superconductor. Though it is a non-Copper oxide and non C60- based compound

yet having a Tc of 40 K is a remarkable feature. Even after years of of discovery of

19

sperconductivity in MgB2 , the question of high-Tc is still unresolved.

2.2 Existence Of Two-Band Energy Gaps In MgB2

Superconductor

A large number of experimental data and theoretical arguments favour a two gap model for

superconductivity in MgB2. The study of the anisotropic superconducting MgB2 using

a combination of scanning tunnelling microscopy and spectroscopy reveal two distinct

energy gaps ∆1 = 2.3 meV and ∆2 = 7.1 meV [73]. More recent experiments such as

High Resolution Photoemission Spectroscopy(HRPS) [74], STM tunneling Spectroscopy

[75], Far-Infra Red Transmission Studies (FIRT) [76],specific heat measurement [39, 40]

point towards the existence of two distinct gaps. Directional point-contact spectroscopy

in magnetic field provided direct evidence of the energy gaps, ∆σ =7.1± 0.1 meV or ∆π

= 2.80± 0.05 meV [50, 51, 77](the subcripts refer to the gaps being for σ− electrons and

π−electrons),that are respectively larger or smaller than the expected weak coupling value.

Magneto-Raman spectroscopy [48] experimental data points towards the existence of two

distinct gaps associated with two separate segments of the Fermi surface [78] in MgB2. The

gap sizes of 1.7 meV and 5.6 meV were obtained at 5.4 K, which provides spectroscopic

evidence for the multi-gap of MgB2 superconductor. Specific heat measurements [39, 40]

suggest that it is necessary to involve either two gap or a single anisotropic gap [79] to

explain the data. Microwave measurement results can be explained by the existence of

anisotropic superconducting gap or the presence of a secondary phase with a lower gap

width in some MgB2 samples [80].

There has been several theoretical studies that used the two-band model to investi-

gate the superconductivity in High-Tc superconductors. More than fifty years ago Suhl,

Matthias and Walker [81] predicted the existence of multi-gap superconductivity, in which

a disparity of the pairing interactions in different bands such as s and d bands in tran-

sition metal, leads to different order parameters and to an enhancement of the critical

temperature. The existence of multi-band gap superconductivity in MgB2 was first pro-

posed theoretically by Shulga et al [82] to explain the behaviour of the upper critical

magnetic field. This scenario has also been predicted theoretically by Liu et al [83] in

order to explain the magnitude of Tc and to establish the importance of Fermi surface

20

sheet dependent superconductivity in MgB2. Superconductivty in two band model has

been known for a long time. Okoye [84] employed the two band model to study the isotope

effect of high-Tc superconductors. Kristoffel et al [85] also employed it in superconduct-

ing oxide and fullerenes [86] to study their isotope effect. Moskalenko [87] and Suhl et

al [81] used the concept of multi-band superconductors in the case of large disparity of

the electron-phonon interaction for different Fermi surface sheets. In various cases, such

approach have been applied to study the cuprate high Tc superconductivity [88]. First

principle calculations show that the Fermi surface of MgB2 consists of 2D cylindrical

sheets arising from σ-antibonding states of Boron Pxy orbitals and 3D tubular networks

arising from the π-bonding and antibonding states of Boron Pz orbitals. In this theoretical

framework [86] two different energy gaps exist; the smaller one being an induced gap with

3D bands and the large one associated with the superconducting 2D bands. Punpocha et

al [89] calculates the Tc and 2∆i(0)/KBTc(i = σ, π) within the framework of a two band

Elishaberg formalism. Based on the observed values, the ratio 2∆i(0)/KBTc lie between

4.2 and 5.

Buzea and Yamashita [58] have presented a review of the superconducting properties

of MgB2 known up to the middle of 2001. In their review, they briefly mentioned that

more precise measurements done on MgB2 indicated the existence of two energy gaps

in this superconductor. Pickett [77] interpreted the double gap nature of MgB2 as the

existence of two Tc’s in the superconductor one at 43 K and the other at 13 K, which act

in concert to yield the observed Tc at 39 K. He also pointed out that MgB2 might be the

long sought after(Theorists) two band superconductor. Yamaji [90] has used tight-binding

model to explain a two-band type superconducting instability in MgB2. He incorporated

the Hubbard on-site Coulomb interaction on two inequivalent Boron orbitals to the tight-

binding model. He finds that the amplitude of the interband pair scattering between two

π bands diverges if the interband polarization function in it becomes large enough. These

results lead to a divergent interband pair scattering which implies that two-band type

superconducting instability leads to enhanced Tc.

Ord et al [91] have also developed a two-band model for the description of MgB2 two

gap superconductivity. They suggested that two-band model including interband scatter-

ing of intraband pairs describes the MgB2 superconducting two gap behaviour. Ummarino

et al [92] proposed that MgB2 is a weak coupling two-band phononic system where the

Coulomb pseudopotential and the interchannel pairing mechanism are the key terms to in-

21

terpret superconductivity two gap behaviour. Wang et al [93] proposed a two-band model

with electron-phonon and non electron-phonon interaction for the superconductivity of

MgB2 system within the framework of BCS theory using self-consistent method.

Several properties of MgB2 superconductor has been studied using the two-band

model. Moca [94] has calculated the penetration depth in MgB2 using Elisaberg the-

ory of superconductivity for two-bands. Calculation of specific heat of MgB2 from first

principle shows that a two-band model agrees with experimental data more than the one-

band model [95]. Zhitomirsky et al [96] derived the Ginzburg-Landua function for two gap

superconductors from microscopic BCS theory and then investigated the magnetic prop-

erties. Yanagiwsawa et al [97] examined the transmittance, optical conductivity and the

jump in the specific heat of MgB2 using two-band model. All these assert that two-band

model approach may be adequate to describe superconductivity in MgB2 superconductor.

2.3 Occurrence of Isotope Effect in High-Tc MgB2

Superconductor

The first direct experimental indication in favour of a phonon-related mechanism for

superconductivity goes back to the discovery of the isotope effect by Maxwell [28] and

Reynold et al [29]. They showed that the critical temperature Tc depends strongly on

the average isotopic mass M of the constituents and more precisely that Tc ' M−1/2.

The discovery by Bardeen, Cooper and Schrieffer [30] (BCS) that Tc ' ~ ωD ' M−1/2

further clarified this issue. Since the discovery of 40 K [20] superconductivity in MgB2,

researchers has shown that isotope effect play a vital role in indicating that phonon are

relevant element for superconductivity in this superconductor.

It has been revealed by Bud’ko et al [45] that the isotope effect of MgB2 is β '

0.26, with respect of boron using thermodynamic measurement within the framework of

BCS model. Hinks et al [46] measured the isotope effect of both boron and magnesium.

They found a boron isotope effect exponent,βB = 0.30,consistent with the measurement

of Bud’ko et al [45] and a small isotope effect exponent for magnesium,βMg = 0.020. The

result suggests a significant isotope effect exponent value of 0.32. This observation of

a weak but non-zero isotope effect in this compound suggests a phonon-mediated BCS

superconductivity mechanism.

22

It is well known that a small isotope effect in the convectional low Tc superconductors

can occur for a strong electron-phonon interaction and can be understood with Elisahberg

theory [98] by explicitly including the on-site replusive electron-electon interaction of

strength Uc in addition to the attractive electron-phonon coupling parameter in the pairing

mechanism. The effect of these pairing mechanism in MgB2 using one-band or two-bands

Migdal-Elisahberg approach yielded an isotope effect exponent of about 0.4 - 0.45 [99].

Ord et al [91] has theoretically calculated the isotope effect exponent of MgB2 =

0.34 by inclusion of interband scattering pairs in a two-band for the description of MgB2

superconductor. Choi et al [100] calculated the isotope effect exponent of boron, βB =

0.32 and that of magnesium, βMg = 0.03, using the anisotropic Elisahberg theory of MgB2

with anharmonic phonon frequencies. Also, isotope effect exponent, β = 0.3 at Tc = 40 K

[101] has been found in the weak coupling regime of the two-band BCS model considering

an attractive electron-phonon interaction.

Theorists have adopted various approaches in explaining the derivation of the isotope

effect exponent based on the BCS theory. It suggested that low isotope effect exponent

is primarily due to impurity effect [102], phonon anharmonicity [103] and presence of

multi-band gap [104].

2.4 Influence of Linear-Energy-Dependent Electronic

Density of State on Two-Band Superconductors

Researchers have studied the effects of density of states on the isotope effect, transition

temperature and other properties of superconductors. Xi-yu Su et al [105] implemented

the free-carrier-negative-U-center interaction model to investigate the isotope effect of the

oxide superconductors using constant density of states. Okoye [84] studied a two-band

model of isotope effect of the high-Tc superconductors using a constant density of states

approach. Abah et al [106] investigated the interband interactions and three-square-well

potentials on the superconductivity of MgB2 using a constant density of states approach

to derive the isotope effect exponent and the transition temperature.

Numerous studies have been carried out to understand the influence of density of states

on isotope effect, transition temperature and other properties of superconductor within

van Hove singularity (VHS). The importance of a van Hove singularity in the electronic

23

density of states for enhancing the transition temperature over the BCS value is well

known [107, 108]. Several papers [109, 110, 111, 112, 113, 114] suggested that the high

transition temperature, anomalous isotope effect, linear resistivity, and thermoelectric

power behaviour of high-Tc systems might be understood assuming a presence of a VHS

in te density of states(DOS) and its proximity to the Fermi level near optimum doping.

Angle-resolved photoemission experiments [115, 116] have provided direct evidence for the

presence of an extended VHS in the DOS of the high-Tc systems. Sujit et al [117] inves-

tigated the jump in the specific heat at Tc, the specific heat in both the superconducting

and normal states, and the Knight shift in the superconducting state within van Hove sin-

gularity scenario considering density of states for a two-dimensional tight-binding system

and with an extended saddle-point singularity. He also derived the exact expression [118]

for the isotope-shift exponent and the pressure coefficient of the transition temperature

from the BCS gap equation using van Hove singularity (VHS) in density of states (DOS).

Tsuei et al [109] investigated the role of the VHS in DOS within the BSC phonon-mediated

mechanism and proposed an interesting explanation for the anomalous isotope effect in

the high-Tc cuprate oxide systems. They showed that a maximum transition temperature

with minimum isotope-shift exponent (β) occurs when the Fermi level lies at the energy

of the VHS and Tc decreases while β increases as the Fermi level is displaced from the

VHS. This behavior is in good agreement with the experimental results of high-Tc oxide

systems. Udomasamuthriun [119] derived exact formula of Tc’s equation and the isotope

effect exponent of two-band s-wave superconductors in weak-coupling limit by considering

the influence of two kinds of density of state: constant and van Hove singularity. He finds

that the interband interaction of electronphonon show more effect on isotope exponent

than the intraband interaction and the isotope effect exponent with constant density of

state can fit to experimental data, MgB2 and high-Tc superconductor, better than van

Hove singularity density of state.

Tunnelling experiments performed on high-Tc superconductors show an unusual be-

haviour of the background conductance, which up to very high biases (∼ 200 meV) follows

linear dependence on the absolute value of the applied voltage. Several authors have given

theoretical explanations of density of states effects. Anderson and Zou [120] have derived

a normal state linear conductance from simple assumptions on the spectrum of holon

and spinon excitations in the bi-dimensional resonating valence bond(RVB) state. Philip

[121] within a quantum percolation theory, separates the density of states into an ex-

24

tended part, responsible for the superconducting properties, and a localized part linear

in energy,responsible for the normal properties.The temperature dependence of the zero-

bias conductance, the electronic specific heat and the ultrasonic attenuation in high-Tc

superconductors have been analyzed within the framework of a phenomenological model

based on a density of states expressed as a superposition of a linear [122] to BCS stan-

dard one. Tunnelling experiments in the high-Tc oxide superconductors reveal a linear-

energy-dependent density of states of the quasiparticles [109]. At low temperatures, the

tunnelling conductance of a normal metal/insulator/superconductor tunnel junction is

proportional to the DOS of the superconductor. Conductance backgrounds measured in

high-temperature ceramic oxide material [123] often increases linearly with voltage over

hundreds of millivolts in both normal and superconducting states. This is interpreted

to imply a linear-energy-dependent DOS of the quasiparticle of form N(E) = N0+ N1|E|

where E is the energy measured from the chemical potential, N0 and N1 are constants.

Okoye [124] derived the expressions for the critical temperaure (Tc) and isotope effect

exponent(β) within a two-band model approach using three-square-well potential and

linear-energy-dependent density of states.

2.5 Bardeen-Cooper-Schrieffer(BCS)Theory and the

Two-Square-Well Theory of Isotope Effect Using

Linear-Term Energy Dependent Electronic Den-

sity of States

The microscopy theory of superconductivity which was formulated by Bardeen, Cooper

and Schriffer [30], currently known as BCS theory, gave a remarkably successful account

of most of the basic features of the superconducting state. The theory was based on the

idea that the carrier of electric current in superconductors are not individual electrons but

bound pairs of electrons. These bound pairs,known as Cooper pairs, are formed when the

electron-phonon interaction is attractive and dominates the screened Coulomb interaction

of the electrons. The net interaction is interpreted to arise due to the constant emission

and reabsorption of virtual phonons by electrons.

The BCS theory can be seen to arise from the earlier indication by Cooper that the

25

ground state of a normal metal(non-superconducting) was unstable at zero temperature

with respect to an arbitrary weak interaction between the electrons near the Fermi surface.

This shows that the normal metal at sufficiently low temperature prefers to be in another

state; superconducting state.

In their formulation, BCS considered that the ground state(Ψ0) of a superconductor

is made up of states of electrons excited above the normal ground state by a wave number

of the order of energy gap 4 '10−4KF , where KF is the Fermi wave number. Based on

Pauli exclusion principle, electrons can only be excited into unoccupied states, therefore

it is obvious that only the electronic states within a wave number range 10−4KF of the

Fermi surface are involved in the superconducting phase transition.

These lines of reasoning led to a reduction of the problem of determining the ground

state of many Cooper pairs to the model(BCS) Hamiltonian;

HBCS =∑kσ

ξkb+σ (k)bσ(k) +

∑kk′

Vkkb+↑ (k)b+

↓ (−k)b↓(−k′)b↑(k′) (2.1)

We can write the above Hamiltonian as

HBCS = H0 +Hres

where

H0 =∑kσ

ξkb+σ (k)bσ(k) +

∑kk′

[b+↑ (k)b+

↓ (−k)Γk′ + b↓(−k′)b↑(k′)Γ∗k − Γkk′ ]

and

Hres =∑kk′

Vkk[b+↑ (k)b+

↓ (−k)− Γ∗k][b↓(−k′)b↑(k′)− Γk′ ]

with Γk =< b↓(−k)b↑(k) > the expectation value of the fermion operator b↓(−k)b↑(k) in

the BCS ground state. Neglecting Hres as a toy model, we then define

4k = −∑k′

Vkk′Γk′ (2.2)

The value of the reduced Hamiltonian is

H0 =∑kσ

ξkb+σ (k)bσ(k)−

∑k

[4kb+↑ (k)b+

↓ (−k) +4∗kb↓(−k′)b↑(k′)−4kΓ∗k] (2.3)

The operators b+↑ , b

+↓ , b↑, b↓ obey the commutation relation of imperfect Bose gas

The ground state of a superconductor and the quasi-particle spectrum can be described

by introducing a linear combination of the creation and annihilation operators of the

26

normal fermions using the transformation introduced independently by Bogolibov [125]

and Valatin [126]. The transformation are:

A↑(k) = Ukb↑(k)− Vkb+↓ (k) (2.4)

A↓(−k) = Vkb+↑ + Ukb↓(−k) (2.5)

where A↑(k) and A↓(−k) are called Bogolibov-Valatin operators, Uk and Vk are real.

The inverse transformation of equations(2.4) and (2.5) are found to be

b↑(k) = U∗kA↑(k) + VkA+↓ (−k) (2.6)

b+↓ (−k) = −V ∗k A∗↑(k) + UkA

+↓ (−k) (2.7)

with

|Uk|2 + |Vk|2 = 1 (2.8)

in order to get cannonical anticommutation relation among the Ai(k) oscillators.

Expressing H0 through the Bogolibov-Valatin operators gives

H0 =∑kσ

ξk[(|Uk|2 − |Vk|2)A+σ (k)Aσ(k)] + 2

∑k

ξk[|Vk|2 + UkVkA+↑ (k)A+

↓ (−k)

−U∗kV ∗k A↑(k)A↓(−k)] +∑k

[(4UkV ∗k +4∗kU∗kVk)(A+↑ (k)A↑(k) + A+

↓ (k)A↓(k)− 1)+

(4∗kU2k −4kV

∗2k )A↑(k)A↓(k)− (4kU

2k −4∗kV 2

k )A+↑ (k)A+

↓ (k) +4kΓ∗k] (2.9)

Reducing H0 to a cannonical form, we must cancel terms of terms of type A+↑ (k)A+

↓ (-k)

and A↑(k)A↓(k) by choosing

2ξkUkVk − (4kU2k −4∗kV 2

k ) = 0 (2.10)

Multipying equation(2.10) by 4∗k/U2k , we obtain(

4∗kVkUk

+ ξk

)= ξ2

k + |4k|2 (2.11)

Introducing Ek=√ξ2k + |4k|2 which is the energy of quasiparticles, we get

4∗kVkUk

= Ek − ξk

or ∣∣∣∣VkUk∣∣∣∣ =

Ek − ξk|4k|

(2.12)

27

Combining equation(2.12) and |Uk|2 + |Vk|2=1, we get

|Vk|2 =1

2

(1− ξk

Ek

), |Uk|2 =

1

2

(1 +

ξkEk

)(2.13)

Using these relations, we easily evaluate the coefficients of the other terms in H0. As far

as the bilinear term in the creation and annihilation operators, we get

ξ(|Uk|2 − |Vk|2) +4kUkV∗k +4∗kU∗k = ξ(|Uk|2 − |Vk|2) + 2|Uk|2(Ek − ξk) = Ek (2.14)

showing that indeed Ek is associated to the new creation and annihilation operators.

Therefore, H0 reduces to

H0 =∑kσ

EkA+σ (k)Aσ(k)+ < H0 > (2.15)

with

< H0 >=∑k

[2ξk|Vk|2 −4∗kU∗kVk −4kU∗kVk +4kΓ

∗k] (2.16)

Evaluating Γk we get,

Γk =< b↓(−k)b↑(k) >= U∗kVk <(1− A+

↑ (k)A↑(k)− A+↓ (−k)A↓(−k)

)>= U∗kVk (2.17)

From the complex conjugate of equation(2.12) we can write

4kUkV

∗k

|Uk|2= Ek − ξk

Using equation(2.13) we obtain

UkV∗k =

1

2

4∗kEk

(2.18)

From the thermal average at T 6=0

< O >T=Tr[e−

HT O]

Tr[e−HT ]

(2.19)

The thermal average of a Fermi Hamlitonian H = Eb+b is obtained easily since

Tr[e−Eb+b/T ] = 1 + e−E/T

and

Tr[b+be−Eb+b/T ] = e−E/T

Therefore the operation

< b+b >T=1

eE/T + 1(2.20)

28

It follows from equation(2.17) that

Γk(T ) =< b↓(−k)b↑(k) >T= U∗kVk(1− 2f(ε)) (2.21)

Therefore the gap equation is given by

4k = −∑k′

Vkk′U∗kVk(1− 2f(ξ)) (2.22)

This equation, first obtained by BCS is a non-linear integral equation for gap parameter

4k which is clearly temperature dependent. The integral equation has a trivial solution

4k = 0, which corresponds to the normal state. A non-trivial solution exists if the normal

state becomes unstable and in this case the system will be found in the superconducting

state.

Equation(2.22) yield the superconducting state as long as the gap parameter 4 is

non-zero. Equation(2.22) can be written as

4k = −∑k′

Vkk′4k′

2Ek′tanh(

Ek′

2T) (2.23)

Considering the gap function in the Cooper model [127]

Vkk′ =

−V |ξ − ξk′ | ≤ ~ωD,

0 |ξ − ξk′ | > ~ωD(2.24)

Equation(2.23) may be written as

1 = N(0)V

∫ ~ωD

−~ωDdξ

1− 2f(ξ2 +42(T ))1/2

2(ξ2 +42(T ))1/2(2.25)

The transition temperature Tc corresponds to 4(Tc) = 0 which from Equation(2.25)

1 = N(0)V

∫ ~ωD

−~ωDdξ

1− 2f(ξ, Tc)

2ξ

= N(0)V

∫ ~ωD

0

dξ tanh

(ξ

2kBTc

)(2.26)

For large ξ, tanh(ξ/2kBTc) → 1 and the integral has the asymptotic form ln(~ωD/kBTc)

+C, where C=1.14. Therefore equation(2.26) reduces to

1 = N(0)V ln1.14~ωDkBTc

(2.27)

which can be rewritten as

kBTc = 1.14~ωDe−1/N(0)V (2.28)

29

This yields the expression for the superconducting transition temperature(Tc)

for (kB = ~ = 1) is

Tc = 1.14ωDe−1/N(0)V (2.29)

We note that superconductors are characterised according to the magnitude of electron-

phonon coupling constant, λ = N(0)V. λ 1 is weak coupling regime, λ ∼ 1 is interme-

diate coupling regime and λ 1 is strong coupling regime.

The BCS theory discussed in this section deals with weak coupling superconductors.

The phenomenon of high-Tc superconductivity cannot be accounted for by BCS theory

if the the theory is restricted to binding of superconductive electron pairs by a dynamic

coupling to phonon. This breakdown of the existing theory has split theorists into two

camps: one camp extends the BCS theory by introducing into the pair-binding potential

energy an electronic-phonon mechanism(weak coupling), the other camp would construct

a ”strong-coupling“ theory in which electron pairs form as a disordered array of bipo-

larons at temperatures T > Tc [128]. In the weak coupling limit(λ 1), the Cooper

model potential can be modified to include the effects of Coulomb repulsion. Using the

Bogoliubov model potential shown in the figure which may be expressed in the form

V (ξ − ξ′) =

−Vp + Vc, |ξ − ξ′| < ~ωDVc, |ξ − ξ′| < ~ωc0, |ξ − ξ′| > ~ωc

(2.30)

where Vc is a constant repulsive potential and ωc is a Coulomb cut-off frequency. The

transition temperature under weak coupling condition can be obtained from the BCS

energy gap as

4 = −∫V (ξ − ξ′)4(ξ′)

2ξ′N(ξ)(1− 2f(ξ′))dξ′ (2.31)

Substituting for V(ξ − ξ′)

4 = −∫

(−Vp + Vc)4(ξ′)

2ξ′N(ξ)(1− 2f(ξ′))dξ′ (2.32)

where N(ξ) is the density of states on the Fermi level and ξ′ is the excited energy measured

from the Fermi level. Introducing a linear term energy-dependent electronic density of

states [129]:

N(ξ) = N0 +N |E|

Evaluating equation(2.32) yields

4 = −(N0 +N |E|)∫

(−Vp + Vc)4(ξ′)

2ξ′(1− 2f(ξ′))dξ′ (2.33)

30

Equation(2.33) can be solved following the standard procedure [105, 130]. The net at-

tractive electronic part is separated and its contribution is called A. From equation (2.33)

B ' −(N0 +N |E|)∫ ωD

−ωDVpB

(1− 2f(ξ′))

2ξ′dξ′ + A (2.34)

Simplifying the above equation we obtain

B ' N0B

∫ ωD

−ωDVp

(1− 2f(ξ′))

2ξ′dξ′ +N |E|B

∫ ωD

−ωDVp

(1− 2f(ξ′))

2ξ′dξ′ + A

' BN0Vp ln

[1.14~ωDkBTc

]+BN |E|VpωD + A (2.35)

Defining

ZD = ln

[1.14~ωDkBTc

], λ1 = N0Vp, λ2 = N |E|Vp

we have

B ' Bλ1ZD +Bλ2ωD + A (2.36)

The contribution from the electronic part is

A ' −(N0 +N |E|)(∫ ωD

−ωcA+

∫ ωD

−ωDB +

∫ ωc

ωD

A

)V c

1− 2f(ξ′)

2ξ′dξ′

' −BN0Vc ln

[1.14~ωDkBTc

]− AN0Vc ln

[ωcωD

]−BN |E|VcωD − AN |E|Vcωc + AN |E|VcωD

(2.37)

Defining

Zc = lnωcωD

, µ1 = N0Vc, µ2 = N |E|Vc

we obtain

A ' −Bµ1ZD − Aµ1Zc −Bµ2ωD − Aµ2ωc + Aµ2ωD (2.38)

Equations(2.36) and (2.38) can be written as

A+B(λ1ZD + λ2ωD − 1) = 0 (2.39)

− A(µ1Zc + µ2(ωc − ωD) + 1)−B(µ1ZD + µ2ωD) = 0 (2.40)

Equation(2.39) and (2.40) are two homogeneous equation expressed in matrix form as 1 (λ1ZD + λ2ωD − 1)

−(µ1Zc + µ2(ωc − ωD) + 1) −(µ1ZD + µ2ωD)

A

B

= 0 (2.41)

Non-trivial solution of the matrix is obtained when the secular equation given by∣∣∣∣∣∣ 1 (λ1ZD + λ2ωD − 1)

−(µ1Zc + µ2(ωc − ωD) + 1) −(µ1ZD + µ2ωD)

∣∣∣∣∣∣ = 0 (2.42)

31

Solving the determinant of equation(2.42), we obtain

ZD [−µ1 + λ1(µZc + µ2(ωc − ωD) + 1)] = −(λ2ωD − 1)1 [µ1Zc + µ2(ωc − ωD) + 1] + µ2ωD

The above expression can be written as

ZD =−(λ2ωD − 1) [µ1Zc + µ2(ωc − ωD) + 1] + µ2ωD

[−µ1 + λ1(µ1Zc + µ2(ωc − ωD) + 1)](2.43)

Dividing the numerator and the denominator of equation(2.43) by

[µ1Zc + µ2(ωc − ωD) + 1] we get

ZD =1− λ2ωD + µ2ωD

[µ1Zc+µ2(ωc−ωD)+1]

λ1 − µ1[µ1Zc+µ2(ωc−ωD)+1]

(2.44)

If we define the Coulomb repulsive pseudopotential

µ∗ =µ1

[µ1Zc + µ2(ωc − ωD) + 1]

and

K∗ =µ2

[µ1Zc + µ2(ωc − ωD) + 1]

we obtain

ZD =1− ωD(λ2 −K∗)

λ1 − µ∗(2.45)

This yields after simplification

kBTc = 1.14~ωD exp

[−(

1− ωD(λ2 −K∗)λ1 − µ∗

)](2.46)

If kB = ~ = 1, equation(2.46) becomes

Tc = 1.14ωD exp

[−(

1− ωD(λ2 −K∗)λ1 − µ∗

)](2.47)

Equation(2.47) reduces to the McMillian[131, 132] expression for the transition tempera-

ture Tc as λ2 = K∗ and to Xi-Yu et al [105] as

µ∗ = −(

µ1

[µ1Zc + µ2(ωc − ωD) + 1]

)Equation(2.47) accounts to the fact that the Coulomb repulsion described by ωc is not

very efficient in counteracting superconductivity and for deviation of the isotope effect

coefficient from the BCS predicted value of 0.5

Suhl et al [81] and Moskalenko [87] extended the BCS approach to account for the

superconductivity in materials with overlapping bands such as transition metals. In their

32

approach, the BCS Hamiltonian is written for the two overlappping bands and an in-

teraction term that couples the Cooper pairs in each band. The Hamiltonian is of the

form

H = H1BCS +H2

BCS +Hint (2.48)

where H iBCS(i = 1, 2) are the BCS effective Hamiltonian for the respective bands given

by

H iBCS =

∑ikσ

ξikσC+ikσCikσ −

∑ikk′σ

VikkC+ik′↑C

+ik′↓Cik↓Cik↑

where ξ1k and ξ2k are the BCS kinetic energies of the bands measured relative to the

Fermi level, k is the Bloch wave vector,V1kk′ and V2kk′ are the interband interaction

matrix elements and the suffix σ is a spin (↑ and ↓) index.

Hint =∑kk′

V12kk′ [C+1k↑C

+1k↓C2k′↓C2k′↑ + C+

2k↑C+2k↓C1k′↓C1k′↑]

is the contribution from the interband tnteraction with a phonon mediated matrix element

V12k′ . In this term, Cooper pairs from different bands interact.

Using the standard Bogoliubov-Valatin transformation approach [125, 126] to BCS

theory, we obtain the following gap equations:

∆1k = −∑k′

V1kk′∆1k′

2ξ1k′(1− 2f(ξ1k′))−

∑k′

V12kk′∆2k′

2ξ2k′(1− 2f(ξ2k′)) (2.49)

∆2k = −∑k′

V2kk′∆2k′

2ξ2k′(1− 2f(ξ2k′))−

∑k′

V12kk′∆2k′

2ξ2k′(1− 2f(ξ1k′)) (2.50)

Where ∆1k and ∆2k are the effective gap parameters for the bands 1 and 2, f(ξ1k′)and

f(ξ2k′) represent the number of the quasiparticles deriving from bands 1 and 2 that are

excited to energies ξ1k and ξ2k above the Fermi level respectively. Using the Bogoliubov

model potential of the form:

Vikk′ =

−Vip + Vic, −ωD < ω < ωD

Vic, ωc < ω < ωc

0, ωc > ω > ωc

(2.51)

in equations(2.49) and (2.50) and replacing the summation over k′ with the integration

over energy, thus introducing the linear term energy dependent density of states N i(E),

under weak coupling approximation (kβ = ~ = 1) we have:

41k = −∫N1(E)(−V1p + V1c)

41k′

2ξ1k′(1− 2f(ξ1k′))dξ1k′

33

−∫N2(E)(−V12p + V12c)

42k′

2ξ2k′(1− 2f(ξ2k′))dξ2k′ (2.52)

42k = −∫N2(E)(−V2p + V2c)

42k′

2ξ2k′(1− 2f(ξ2k′))dξ2k′

−∫N1(E)(−V12p + V12c)

41k′

2ξ1k′(1− 2f(ξ1k′))dξ1k′ (2.53)

where N1(E) and N2(E) are the densities of states in band 1 and 2 respectively. To

illustrate the effects of the linear-energy-dependent density of state[129]:

N i(E) = N i0 +N i

1|E| (2.54)

i = 1 , 2 represents bands 1 and 2.

Defining the gap parameters in each band in the form [90]:

4ik = 4i0ηi(k) +4iα(1− ηi(k)), i = 1, 2 (2.55)

where ηi(k) is equal to one when |εik| < ωD and zero otherwise, 4i0 and 4iα represent the

phonon and the electronic parts of the effective order parameter. We solve the following

standard procedure [105, 130] and separate the attractive electronic parts. The phonon

part for band 1 from equation(2.52) is

B ' (N10 +N1

1 )

∫ ωD

−ωDV1pB

(1− 2f(ξ1k′)

2ξ1k′dξ1k′+A+(N2

0 +N21 )

∫ ωD

−ωDV12pD

(1− 2f(ξ2k′)

2ξ2k′dξ2k′

(2.56)

simplifying the expression above we obtain

B ' BN10V1p ln

[2ωD410

]+BN1

1V1pωD + A+DN20V12p ln

[2ωD420

]+DN2

1V12pωD (2.57)

Defining

ZD = ln

[2ωD410

]= ln

[2ωD420

], λ1 = N1

0V1p, λ′1 = N1

1V1p, λ212 = N2

1V12p, λ′12 = N2

1V12p

we obtain

B(1− ZDλ1 − λ′1ωD)− A−D(ZDλ212 + λ′12ωD) = 0 (2.58)

The Coulomb electronic part

A ' −(N10 +N1

1 )

(∫ −ωD−ωc

A+

∫ ωD

−ωDB +

∫ ωc

ωD

A

)V1c

(1− 2f(ξ1k′)

2ξ1k′dξ1k′ − (N2

0 +N21 )

(∫ −ωD−ωc

C +

∫ ωD

−ωDD +

∫ ωc

ωD

C

)V12c

(1− 2f(ξ2k′)

2ξ2k′dξ2k′ (2.59)

34

Simplifying the above expression we obtain

A ' −BN10V1c ln

[2ωD410

]−AN1

0V1c ln

[ωcωD

]−BN1

1V1cωD−AN11V1cωc+AN1

1V1cωD−DN20

V12c ln

[2ωD420

]− CN2

0V12c ln

[ωcωD

]−DN2

1V12cωD − CN21V12cωc + CN2

1V12cωD (2.60)

Defining

Zc = ln

[ωcωD

], µ1 = N1

0V1c, µ212 = N2

0V12c, µ′1 = N1

1V1c, k2 = N21V12c

we obtain

A(1+Zcµ1+µ′1(ωc−ωD))+B(ZDµ1+ωDµ′1)+C(Zcµ

212+k2(ωc−ωD))+D(ZDµ

212+k2ωD) = 0

(2.61)

Similarly,the phonon part for band 2 from equation(2.53) is

D ' (N20 +N2

1 )

∫ ωD

−ωDV2pD

(1− 2f(ξ2k′)

2ξ2k′dξ2k′+C+(N1

0 +N11 )

∫ ωD

−ωDV12pB

(1− 2f(ξ1k′)

2ξ1k′dξ1k′

(2.62)

This can be written as

D ' DN10V2p ln

[2ωD420

]+DN2

1ωDV2p + C +BN10V12p ln

[2ωD410

]+BN1

1V12pωD (2.63)

Defining

λ2 = N20V2p, λ

112 = N1

0V12p, λ′2 = N2

1V2p, λ′12 = N1

1V12p

we obtain

−B(ZDλ112 + ωDλ

′12)− C +D(1− ZDλ2 − ωDλ′2) = 0 (2.64)

The Coulomb electronic part contribution is

C ' −(N20 +N2

1 )

(∫ −ωD−ωc

C +

∫ ωD

−ωDD +

∫ ωc

ωD

C

)V2c

(1− 2f(ξ2k′)

2ξ2k′dξ2k′ − (N1

0 +N11 )

(∫ −ωD−ωc

A+

∫ ωD

−ωDB +

∫ ωc

ωD

A

)V12c

(1− 2f(ξ1k′)

2ξ1k′dξ1k′ (2.65)

Simplifying we get

C ' −DN20V2c ln

[2ωD420

]−CN2

0V2c ln

[ωcωD

]−DN2

1V2cωD−CN21V2cωc+CN2

1V2cωD−BN10

V12c ln

[2ωD410

]− AN1

0V12c ln

[ωcωD

]−BN1

1V12cωD − AN11V12cωc + AN1

1V12cωD (2.66)

Defining

µ2 = N20V2c, µ

′2 = N2

1V2c, k1 = N11V12c, µ

112 = N1

0V12c

35

we obtain

A(Zcµ112+k1(ωc−ωD))+B(ZDµ

112+k1ωD)+C(1+Zcµ1+µ′2(ωc−ωD))+D(ZDµ2+µ′12ωD) = 0

(2.67)

These homogeneous equations(2.58),(2.61),(2.64) and(2.67) can be rewritten in matrix

form as−1 (1−ZDλ1−ωDλ′1) 0 −(ZDλ

212+ωDλ

′12)

(1+Zcµ1+µ′1(ωc−ωD)) (ZDµ1+ωDµ′1) (Zcµ212+k2(ωc−ωD)) (ZDµ

212+k2ωD)

0 −(ZDλ112+ωDλ

′12) −1 (1−ZDλ2−ωDλ′2)

(Zcµ112+k1(ωc−ωD)) (ZDµ112+ωDk1) (1+Zcµ1+µ′2(ωc−ωD)) (ZDµ2+ωDµ

′12)

A

B

C

D

= 0

(2.68)

For non-trivial solution, the determinant of the 4x4 matrix in equation(2.68) must

vanish. Let us for one moment treat the simpliest case in which the two bands have

identical charateristics[90] ,that is:

λ1 = λ2 = λ; µ1 = µ2 = µ; µ′1 = µ′2 = µ′; µ112 = µ2

12 = µ12; λ112 = λ2

12 = λ12; λ′12 = λ′12 = λ′12

k1 = k2 = k; λ′1 = λ′2 = λ′; A = C = 40; B = D = 4α

Under this condition,equation (2.68) reduces to: −1 (1− ZD(λ+ λ12)− ωD(λ′ + λ′12))

(1 + Zc(µ+ µ12) + (ωc − ωD)(µ′ + k)) (ZD(µ+ µ12) + ωD(µ′ + k))

40

4α

= 0

(2.69)

Solving the secular equation we have

−(ZD(µ+µ12)+ωD(µ′+k)) = (1−ZD(λ+λ12)−ωD(λ′+λ′12))(1+Zc(µ+µ12)+(ωc−ωD)(µ′+k))

(2.70)

Simplifying we obtain

ZD =(1− ωD(λ′ + λ′12))(1 + Zc(µ+ µ12) + (ωc − ωD)(µ′ + k)) + ωD(µ′ + k)

(λ+ λ12)(1 + Zc(µ+ µ12) + (ωc − ωD)(µ′ + k))− (µ+ µ12)(2.71)

Dividing the Numerator and Denominator by (1 + Zc(µ + µ12) + (ωc − ωD)(µ′ + k)) we

obtain

ZD =1− ωD(λ′ + λ′12) + ωD

(µ′+k)(1+Zc(µ+µ12)+(ωc−ωD)(µ′+k))

(λ+ λ12)− (µ+µ12)(1+Zc(µ+µ12)+(ωc−ωD)(µ′+k))

(2.72)

This can be written in reduced form as

ZD =1− ωD(λ′ + λ′12 +K∗)

λ+ λ12 − µ∗(2.73)

36

where

K∗ =(µ′ + k)

(1 + Zc(µ+ µ12) + (ωc − ωD)(µ′ + k))

µ∗ =(µ+ µ12)

(1 + Zc(µ+ µ12) + (ωc − ωD)(µ′ + k))

Replacing

ZD = ln

[2ωD410

]= ln

[2ωD420

]with ln

[1.14ωD

Tc

].

Equation(2.73) becomes

ln

[1.14ωDTc

]=

1− ωD(λ′ + λ′12 +K∗)

λ+ λ12 − µ∗(2.74)

This yields simplification that

Tc = 1.14ωD exp−(

1− ωD(λ′ + λ′12 +K∗)

λ+ λ12 − µ∗

)(2.75)

Equation(2.75) gives the superconducting transition temperature when a linear energy

dependent density of states is used. This result reduces to usual expression obtained

using a constant density of states in a two band model. In such a limit, N1 = 0 as such

λ′ = λ′12 = k∗ = 0. It is observed that the presence of λ′, λ′12, k∗, µ∗ enhance Tc.

37

Chapter 3

Two-Band Model of

Superconductivity of MgB2 Using

Three-Square -Well Potential with

Linear-Energy-Dependent Electronic

Density of states

3.1 Introduction

In this section, we derived the the expressions for the transition temperature and the

isotope effect exponent within the frame work of Bogoliubov-Valatin two-band formalism

using a linear-energy-dependent electronic density of states assuming a three-square-well

potential model made up of contributions which arise from three interactions such as

acoustic-electron phonon (Va), optical-electron phonon (Vp) and the repulsive electron-

electron (Coulomb) interactions (Vc).

38

3.2 Derivation Of Transition Temperature And Iso-

tope Effect in One-Band Model Using Three Square

Well Potential With Linear Term Energy Depen-

dent of States

Similarily, we assume that the interaction matrix element(Vkk′) are made up of contribu-

tion which arise from three interaction namely: electron-acoustic phonon(Va), electron-

optical phonon (Vp) and the Coulomb repulsive interaction(Vc). Therefore

V (kk′) =

−Va − Vp + Vc, ωa < ω < ωa

−Vp + Vc, ωp < ω < ωp

Vc, ωc < ω < ωc

(3.1)

where ωa is the cut-off frequency for the attrctive electron-acoustic phonon part, ωp is

the cut-off frequency for the attractive electron-optical phonon part and ωc is the cut-

off frequency for the on-site repulsive electron-electron part(ωc > ωp > ωa). Empolying

equation(3.1) in equation (2.31) and assuming approximate solution, thus introducing

the linear-energy-dependent density of states under weak-coupling condition(kB = ~ = 1)

yields

4K = −(N0 +N |E|)∫

(−Va − Vp + Vc)4(ξ′)

2ξ′(1− 2f(ξ))dξ′ (3.2)

We solve the resulting eqautions following the standard procedure[105,130] and sepa-

rate the interacting parts as follows:The electron-acoustic parts yields

B ' (N0 +N |E|)∫ ωa

−ωaVaB4(ξ′)

2ξ′(1− 2f(ξ))dξ′ + A (3.3)

Simplifying yields

B ' BN0Va ln

[1.14ωaTc

]+BN |E|Vaωa + A (3.4)

Defining

Za = ln

[1.14ωaTc

], λ1 = N0Va, λ2 = N |E|Va

we obtain

B(1− λ1Za − λ2ωa)− A = 0 (3.5)

The electron-optical phonon part:

A ' (N0 +N |E|)

(∫ −ωa−ωp

A+

∫ ωa

−ωaB +

∫ ωp

ωa

A

)Vp4(ξ′)

2ξ′(1− 2f(ξ))dξ′ + C (3.6)

39

Simplifying yields

A ' BN0Vp ln

[1.14ωaTc

]+AN0Vp ln

[ωpωa

]+BN |E|Vpωa +AN |E|Vpωp −AN |E|Vpωa +C

(3.7)

Defining

Zp = ln

[ωpωa

], k1 = N0Vp, k2 = N |E|Vp

we obtain

A(1− k1Zp − k2(ωp − ωa))−B(k1Za + k2ωa)− C = 0 (3.8)

The Coulomb electronic part:

C ' −(N0+N |E|)Vc

(∫ −ωp−ωc

C +

∫ −ωa−ωp

A+

∫ ωa

−ωaB +

∫ ωp

ωa

A+

∫ ωc

ωp

C

)4(ξ′)

2ξ′(1−2f(ξ))dξ′

(3.9)

This can be written as

C ' −BN0Vc ln

[1.14ωaTc

]−AN0Vc ln

[ωpωa

]−CN0Vc ln

[ωcωp

]−BN |E|VcωD−CN |E|Vcωc

+ CN |E|Vp − AN |E|Vcωp + AN |E|Vcωa (3.10)

Defining

Zc = ln

[ωcωp

], µ1 = N0Vc, µ2 = N |E|Vc

we obtain

A(µ1Zp + µ2(ωp − ωa)) +B(µ1Za + µ2ωa) + C(1 + µ1Zc + µ2(ωc − ωp)) = 0 (3.11)

Equation (3.5),(3.8) and (3.11) are three simultaneous homogeneous equations.They can

be written in the matrix form as:−1 (1− λ1Za − λ2ωa) 0

(1− k1Zp − k2(ωp − ωa)) (k1Za + k2ωa) −1

(µ1Zp + µ2(ωp − ωa)) (µ1Za + µ2ωa) (1 + µ1Zc + µ2(ωc − ωp))

A

B

C

= 0

(3.12)

For non-trival solution, the determinant of the matrix formed by these equations must

vanish.Solving the secular equation arising from the coefficients A,B and C we have

− (((k1Za + k2ωa)(1 + µ1Zc + µ2(ωc − ωp)) + (µ1Za + µ2ωa))− (1− λ1Za − λ2ωa)

((1− k1Zp − k2(ωp − ωa))(1 + µ1Zc + µ2(ωc − ωp)) + (µ1Zp + µ2(ωp − ωa))) = 0 (3.13)

40

This can be expressed after simplification in the form of

Za =(1− λ2ωa)Ψ + k2ωa(1 + µ1Zc + µ2(ωc − ωp)) + µ2ωa

λ1Ψ− µ1 − k1(1 + µ1Zc + µ2(ωc − ωp))(3.14)

where

Ψ = (1− k1Zp − k2(ωp − ωa))(1 + µ1Zc + µ2(ωc − ωp)) + (µ1Zp + µ2(ωp − ωa))

Dividing the numerator and denominator by Ψ we deduce

Za =(1− λ2ωa) + ωa(k2+µ2)

Ψ

λ1 − µ1−k1(1+µ1Zc+µ2(ωc−ωp))

Ψ

(3.15)

This can written in reduced form as

Za =1− ωa(λ2 +K∗)

λ1 − µ∗(3.16)

where

K∗ =(k2 + µ2)

Ψ

µ∗ =k1(1 + µ1Zc + µ2(ωc − ωp)) + µ1

Ψ

But Za = ln[

1.14ωaTc

]. we obtain after simplification

Tc = 1.14ωa exp

(−1− ωa(λ2 +K∗)

λ1 − µ∗

)(3.17)

Equation(3.17) is the expression for the superconducting transition temperature, Tc, in

one-band using three-square-well potential and linear-energy-dependent density of states.

The Isotope effect exponent, β, is given by

β = −Md lnTcdM

We assume that Tc ∝M−β and recall that ωa ∝M−1/2 to obtain:

d lnTcdM

=d lnωadM

− d

dM

(1− ωa(λ2 +K∗)

λ1 − µ∗

)(3.18)

The Isotope effect exponent(β) can be derived from the expression for Tc in equation(3.17)

β = −M[d lnωadM

− d

dM

(1− ωa(λ2 +K∗)

λ1 − µ∗

)](3.19)

41

We deduce after simplification that

β =1

2+M

(−(λ1 − µ∗)[ωa dK

∗

dM+K∗ dωa

dM] + (1− ωa(λ2 +K∗))dµ

∗

dM

(λ1 − µ∗)2

)(3.20)

dK∗

dM=

(k2 + µ2)[(k1 + k2ωa)(1 + µ1Zc + µ2(ωc − ωp)) + (µ1 + µ2)]

2MΨ2

dµ∗

dM=k1(1− µ1Zc + µ2(ωc − ωp) + µ1)[(k1 + k2ωa)(1 + µ1Zc + µ2(ωc − ωp)) + (µ1 + µ2)]

2MΨ2

We can easily re-arrange the above expressions in the form:

dK∗

dM=

(k2 + µ2)∇2MΨ2

=K∗∇2MΨ

dµ∗

dM=k1(1− µ1Zc + µ2(ωc − ωp) + µ1)∇

2MΨ2=

µ∗∇2MΨ

where

∇ = [(k1 + k2ωa)(1 + µ1Zc + µ2(ωc − ωp)) + (µ1 + µ2)]

Replacing the expressions for dK∗

dMand dµ∗

dMin equation (3.20) we obtain

β =1

2+M

(1

2M

[K∗ωa

(λ1 − µ∗)(1− ∇

Ψ) +

(1− ωa(λ2 +K∗))µ∗∇Ψ(λ1 − µ∗)

])Simplifying yields

β =1

2

(1 +

ωa(λ1 − µ∗)

[K∗(1− ∇

Ψ)− (λ2 +K∗)µ∗∇

Ψ(λ1 − µ∗)

]+

µ∗∇Ψ(λ1 − µ∗)2

)(3.21)

Equation(3.21) is the expression for isotope effect, β, in one-band using three-square-well

potential and linear-energy-dependent density of states.

3.3 Derivation of Transition Temperature And Iso-

tope Effect in Two-Band Model Using Three-

Square-Well Potential with Linear-Energy-Dependent

Electronic Density of States

We consider the model Hamiltonian of the form

H = H1BCS +H2

BCS +Hint (3.22)

where

H iBCS =

∑ikσ

ξikσC+ikσCikσ −

∑ikk′σ

VikkC+ik′↑C

+ik′↓Cik↓Cik↑

42

Hint =∑kk′

V12kk′ [C+1k↑C

+1k↓C2k′↓C2k′↑ + C+

2k↑C+2k↓C1k′↓C1k′↑]

Using the standard and straight foward Bogoliubov-Valatin transformation approach

∆1k = −∑k′

V1kk′∆1k′

2ξ1k′(1− 2f(ξ1k′))−

∑k′

V12kk′∆2k′

2ξ2k′(1− 2f(ξ2k′)) (3.23)

∆2k = −∑k′

V2kk′∆2k′

2ξ2k′(1− 2f(xi2k′))−

∑k′

V12kk′∆1k′

2ξ1k′(1− 2f(ξ1k′)) (3.24)

Where ∆1k and ∆2k are the effective gap parameters for the bands 1 and 2, f(ξ1k′) and

f(ξ2k′) represent the number of the quasiparticles deriving from bands 1 and 2 that are

excited to energies ξ1k and ξ2k above the Fermi level respectively.

The interaction matrix elements, Vikk′ , is approximated by using the three-square well

potential made up of contributions which arise from three interaction such as electron-

acoustic phonon(Va), electron-optical phonon(Vp) and the Coulomb repulsive interaction

(Vc). Therefore

Vikk′ =

−Via − Vip + Vic ,ωa < ω < ωa

−Vip + Vic ,ωp < ω < ωp

Vic , ωc < ω < ωc

(3.25)

where ωa is the cut-off frequency for the attracitve electron-acoustic part, ωp is the cut-off

frequency for the attractive electron-optical phonon part and ωc is the cut-off frequency

for the on-site repulsive electron-electron part all in the range of ωc > ωp < ωa; i = 1 , 2

and 12.

Empolying equations(3.25) in (3.23) and(3.24) and replacing the summation over k′

with integration over energy thus introducing the energy density of states, N(E), under

the weak coupling limit(kβ = ~ = 1) we deduce

∆1k = −∫N1(E1)V1kk′

∆1k′

2ξ1k′(1− 2f(ξ1k′))dξ1k′ −

∫N2(E2)V12kk′

∆2k′

2ξ2k′(1− 2f(ξ2k′))dξ2k′

(3.26)

∆2k = −∫N2(E2)V2kk′

∆2k′

2ξ2k′(1− 2f(ξ2k′))dξ2k′ −

∫N1(E1)V12kk′

∆1k′

2ξ1k′(1− 2f(ξ1k′))dξ1k′

(3.27)

replacing Vikk with −Via − Vip + Vic we obtain

∆1k =

∫N1(E1)(V1a + V1p − V1c)

∆1k′

2ξ1k′(1− 2f(ξ1k′))dξ1k′

+

∫N2(E2)(V12a + V12p − V12c)

∆2k′

2ξ2k′(1− 2f(ξ2k′))dξ2k′ (3.28)

43

∆2k =

∫N2(E2)(V2a + V2p − V2c)

∆2k′

2ξ2k′(1− 2f(ξ2k′))dξ2k′

+

∫N1(E1)(V12a + V12p − V12c)

∆2k′

2ξ1k′(1− 2f(ξ1k′))dξ1k′ (3.29)

where N1(E1) and N2(E2) are the densities of states in band 1 and 2 respectively. To

illustrate the effects of the linear-energy-dependent of density of states [129]

N i(Ei) = N i0 +N i

1|E| (3.30)

i=1,2 represent bands 1 and 2 repectively. We substitute equation(3.30) into equa-

tions(3.28) and (3.29). We solve the resulting equations following the standard pro-

cedure and separate the electronic parts as well as the electron-phonon parts. From

equation(3.28),we have the electron-acoustic phonon part for band 1 as

B = (N10 +N1

1 )

∫ ωa

−ωaBV1a

(1− 2f(ξ1k′))

2ξ1k′dξ1k′ + A

+ (N20 +N2

1 )

∫ ωa

−ωaEV12a

∆2k′

2ξ2k′(1− 2f(ξ2k′))dξ2k′ (3.31)

This yields

B = BN10V1a ln

[2ωa∆10

]+BN1

1V1aωa + A+ EN20V12a ln

[2ωa∆20

]+ EN2

1V12aωa

Assuming

Za = ln

[2ωa∆10

]= ln

[2ωa∆20

]= ln

[1.14~ωakβTc

], λ1 = N1

0V1a, λ11 = N1

1V1a, λ12 = N20V12a, λ

212 = N2

1V12a

B = BλZa +Bλ11ωa + A+ Eλ12Za + Eλ2

12ωa

Rearranging the above equation we obtain

− A+ (1− λ1Za − λ11ωa)B − (λ12Za + λ2

12ωa)E = 0 (3.32)

The electron-optical phonon part is given by

A = (N10 +N1

1 )V1p

(∫ −ωa−ωp

A+

∫ ωa

−ωaB +

∫ ωp

ωa

A

)1− 2f(ξ1k′)

2ξ1k′dξ1k′ + C

(N20 +N2

1 )V12p

(∫ −ωa−ωp

D +

∫ ωa

−ωaE +

∫ ωp

ωa

D

)1− 2f(ξ2k′)

2ξ2k′dξ2k′ (3.33)

Evaluating equation (3.33) we deduce

A = BN10V1p ln

[2ωa∆10

]+ AN1

0V1p ln

[ωpωa

]+BN1

1V1pωa + AN11V1pωp − AN1

1V1pωa + C

44

+EN20V12p ln

[2ωa∆20

]+DN2

0V12p ln

[ωpωa

]+EN2

1V12pωa+DN21V12pωp−DN2

1V12pωa

Assuming

Zp = ln

[ωpωa

], k1 = N1

0V1p, k11 = N1

1V1p, k12 = N20V12p, k

212 = N2

1V12p

We obtain after rearranging

[1− k1Zp − k11(ωp − ωa)]A− (k1Za + k1

1ωa)B − C −[k12Zp + k2

12(ωp − ωa)]D

−[k12Za + k2

12ωa]E = 0 (3.34)

The electronic part yields

C = −(N10 +N1

1 )V1c

[∫ −ωp−ωc

C +

∫ −ωa−ωp

A+

∫ ωa

−ωaB +

∫ ωp

ωa

A+

∫ ωc

ωp

C

](1− 2f(ξ1k′))

2E1k′dξ1k′

− (N20 +N2

1 )V12c

[∫ −ωp−ωc

F +

∫ −ωa−ωp

D +

∫ ωa

−ωaE +

∫ ωp

ωa

D +

∫ ωc

ωp

F

](1− 2f(ξ2k′))

2ξ2k′dξ2k′

(3.35)

integrating equation(3.35) gives

C = −CN10V1c ln

[ωcωp

]−BN1

0V1c ln

[2ωa∆10

]− AN1

0V1c ln

[ωpωa

]−BN1

1V1cωa − CN11V1cωc

+CN11V1cωp − AN1

1V1cωp + AN11V1cωa − FN2

0V12c ln

[ωcωp

]− EN2

0V12c ln

[2ωa∆20

]−DN2

0

V12c ln

[ωpωa

]− EN2

1V12cωa −DN21V12cωp +DN2

1V12cωa − FN21V12cωc + FN2

1V12cωp

Assuming

Zc = ln

[ωcωp

], µ1 = N1

0V1c, µ11 = N1

1V1c, µ12 = N20V12c, µ

212 = N2

1V12c

We deduce after rearranging

[µ1Zp + µ1

1(ωp − ωa)]A+

[µ1Za + µ1

1ωa]B +

[1 + µ1Zc + µ1

1(ωc − ωp)]C

+[µ12Zp + µ2

12(ωp − ωa)]D+

[µ12Za + µ2

12ωa]E +

[µ12Zc + µ2

12(ωc − ωp)]F = 0 (3.36)

For Band 2: The electron acoustic phonon part

E = (N20 +N2

1 )V2a

∫ ωa

−ωaE

(1− 2f(ξ2k′))

2ξ2k′dξ2k′+D+(N1

0 +N11 )V12a

∫ ωa

−ωaB

(1− 2f(ξ1k′))

2ξ1k′dξ1k′

(3.37)

45

Integration equation (3.37) we have

E = EN20V2a ln

[2ωa∆20

]+ EN2

1V2aωa +D +BN10V12a ln

[2ωa∆10

]+BN1

1V12aωa

Assuming

λ2 = N20V2a, λ

22 = N2

1V2a, λ′12 = N1

0V12a, λ112 = N1

1V12a

we obtain after rearranging that

− (λ′12Za + λ112ωa)B −D + (1− λ2Za − λ2

2ωa)E = 0 (3.38)

The electron-optical phonon contribution is given by

D = (N20 +N2

1 )V2p

(∫ −ωa−ωp

D +

∫ ωa

ωa

E +

∫ wp

wa

D

)(1− 2f(ξ2k′))

2ξ2k′dξ2k′ + (N1

0 +N11 )

V12p

(∫ −ωa−ωp

A+

∫ ωa

−ωaB +

∫ ωp

ωa

A

)(1− 2f(ξ1k′))

2ξ1k′dξ1k′ + F (3.39)

Evaluating equation(3.39) and assuming that

k2 = N20V2p, k

22 = N2

1V2p, k′12 = N1

0V12p, k112 = N1

1V12p

we obtain after rearrangment that

−[k′12Zp + k1

12(ωp − ωa)]A−

[k1

12ωa + k′12Za]B +

[1−K2Zp −K2

2(ωp − ωa)]D

−[k2Za + k2

2ωa]E − F = 0 (3.40)

The electronic Coulomb part gives

F = −(N20 +N2

1 )V2c

(∫ −ωp−ωc

F +

∫ −ωa−ωp

D +

∫ ωa

−ωaE +

∫ ωp

ωa

D +

∫ ωc

ωp

F

)(1− 2f(ξ2k′))

2ξ2k′dξ2k′

− V12c(N10 +N1

1 )

(∫ −ωp−ωc

C +

∫ −ωa−ωp

A+

∫ ωa

−ωaB +

∫ ωp

ωa

A+

∫ ωc

ωp

C

)(1− 2f(ξ1k′))

2ξ1k′dξ1k′

(3.41)

Assuming that

Zc = ln

[ωcωp

], µ2 = N2

0V2c, µ22 = N2

1V2c, µ′12 = N1

0V12c, µ112 = N1

1V12c

Evaluating and rearranging equation (3.41) yields

[µ′12Zp + µ1

12(ωp − ωa)]A+

[µ′12Za + µ1

12wa]B +

[µ′12Zc + µ1

12(ωc − ωp)]C

46

+[µ2Zp + µ2

2(ωp − ωa)]D +

[µ2Za + µ2

2ωa]E +

[1 + µ2

2(ωc − ωp) + µ2Zc]F = 0 (3.42)

Writing out equations (3.32),(3.34),(3.36),(3.38),(3.40),(3.42)

− A+ (1− λ1Za − λ11ωa)B − (λ12Za + λ2

12ωa)E = 0 (3.43)

[1− k1Zp − k11(ωp − ωa)]A− (k1Za + k1

1ωa)B − C −[k12Zp + k2

12(ωp − ωa)]D

−[k12Za + k2

12ωa]E = 0 (3.44)[

µ1Zp + µ11(ωp − ωa)

]A+

[µ1Za + µ1

1wa]B +

[1 + µ1Zc + µ1

1(ωc − ωp)]C

+[µ12Zp + µ2

12(ωp − ωa)]D+

[µ12Za + µ2

12ωa]E +

[µ12Zc + µ2

12(ωc − ωp)]F = 0 (3.45)

− (λ′12Za + λ112ωa)B −D + (1− λ2Za − λ2

2ωa)E = 0 (3.46)

−[k12Zp + k1

12(ωp − ωa)]A−

[k1

12ωa + k′12Za]B +

[1− k2Zp − k2

2(ωp − ωa)]D

−[k2Za + k2

2ωa]E − F = 0 (3.47)[

µ′12Zp + µ112(ωp − ωa)

]A+

[µ′12Za + µ1

12ωa]B +

[µ′12Zc + µ1

12(ωc − ωp)]C

+[µ2Zp + µ2

2(ωp − ωa)]D +

[µ2Za + µ2

2ωa]E +

[1 + µ2

2(ωc − ωp) + µ2Zc]F = 0 (3.48)

These six simultaneous homogeneous equations can be written in matrix form:

−1 (1−λ1Za−λ11ωa) 0 0

(1−k1Zp−k11 (ωp−ωa)) −(k1Za+k11 ωa) −1 −(k12Zp+k212(ωp−ωa))

(µ1Zp+µ11 (ωp−ωa)) (µ1Za+µ11ωa) (1+µ1Zc+µ11 (ωc−ωp)) (µ12Zp+µ212 (ωp−ωa))

0 −(λ′12Za+λ112ωa) 0 −1

−(k12Zp+k112 (ωp−ωa)) −(k′12 Za+k112 ωa) 0 (1−k2Zp−k22 (ωp−ωa))

(µ′12Zp+µ112(ωp−ωa)) (µ′12Za+µ112ωa) (µ′12Zc+µ112(ωc−ωp)) (µ2Zp+µ22 (ωp−ωa))

−(λ12Za+λ212ωa) 0

−(k12Za+k212ωa) 0

(µ12Za+µ212ωa) (µ12Zc+µ212 (ωc−ωp))

(1−λ2Za−λ22 ωa) 0

−(k2Za+k22ωa) −1

(µ2Za+µ22ωa) (1+µ22 (ωc−ωp)+µ2Zc)

A

B

C

D

E

F ‘

= 0. (3.49)

The non-trival solution exist when the determinant of the matrix (6x6) in equation

(3.48) vanishes. We shall treat the simpliest case in which the two bands have identical

47

characteristics [48, 73, 77, 84]. This is similar to isotropization of the Fermi surfaces due

to strong coupling. The two gaps merge into one [132, 133]

A = D ≡ ∆0, B = E ≡ ∆∝, C = F ≡ ∆θ;

λ1 = λ2 = λ; λ11 = λ2

2 = λ11; λ12 = λ′12 = λ12; λ1

12 = λ212 = λ1

12;

k1 = k2 = k; k11 = k2

2 = k11; k12 = k′12 = k′12; k1

12 = k212 = k1

12;

µ1 = µ2 = µ; µ11 = µ2

2 = µ11; µ′12 = µ12 = µ′12; µ2

12 = µ112 = µ1

12

Under this condition, equation(3.49) reduces to:−1 (1−(λ+λ12)Za−(λ11+λ112)ωa) 0

(1−(k1+k12)Zp−(k11+k112)(ωp−ωa)) (−(k1+k12)Za−(k11+k112)ωa) −1

((µ1+µ′12)Zp+(µ11+µ112)(ωp−ωa)) ((µ1+µ′12)Za+(µ11+µ112)ωa) (1+(µ1+µ′12)Zc+(µ11+µ112)(ωc−ωp))

∆0

∆∝

∆θ

= 0

(3.50)

By solving the secular equation arising from the coefficient of ∆0,∆∝,∆θ in equation

(3.50) we deduce

−[−(k1 + k12)Za

1 + (µ1 + µ′12)Zc + (µ11 + µ1

12)(ωc − ωp)− (k1

1 + k112)ωa

1 + (µ1 + µ′12)Zc + (µ1 + µ112)(ωc − ωp)

+ (µ1 + µ′12)Za + (µ1

1 + µ112)ωa]

−[1− (λ+ λ12)Za − (λ11 + λ1

12)ωa)][(1− (k1 + k12)Zp − (k11 + k1

12)(ωp − ωa))

(1 + (µ1 +µ′12)Zc + (µ11 +µ1

12)(ωc−ωp)) + (µ1 +µ′12)Zp + (µ11 +µ1

12)(ωp−ωa))] = 0 (3.51)

This can be expressed after simplification in the form of

Za =−(k1

1 + k112)ωa

[1 + (µ1 + µ′12)Zc + (µ1

1 + µ112)(ωc − ωp)

]+ (µ1

1 + µ112)ωa + [1− (λ1

1 + λ112)ωa]Γ

(k1 + k12)[1 + (µ1 + µ′12)Zc + (µ11 + µ1

12)(ωc − ωp)]− (µ1 + µ′12) + (λ+ λ12)Γ(3.52)

where

Γ = [1−(k1 +k12)Zp−(k11 +k1

12)(ωp−ωa)][1+(µ1 +µ′12)Zc+(µ11 +µ1

12)(ωc−ωp)]+(µ1 +µ′12)

Zp + (µ11 + µ1

12)(ωp − ωa)

Dividing the numerator and the denominator with Γ gives

Za =1− ωa(λ1

1 + λ112 +K∗)

λ1 + λ′12 + µ∗(3.53)

K∗ =(k1

1 + k112)[1 + (µ1 + µ′12)Zc + (µ1

1 + µ112)(ωc − ωp)]− (µ1

1 + µ112)

Γ

48

µ∗ =(k1 + k12)[1 + (µ1 + µ′12)Zc + (µ1

1 + µ112)(ωc − ωp)]− (µ1 + µ′12)

Γ

Replacing Za = ln[

2ωa∆10

]= ln

[2ωa∆20

]with ln

[1.14~ωakβTc

]in weak coupling limit(kβ = ~ = 1)

ln

[1.14ωaTc

]=

1− ωa(λ11 + λ1

12 +K∗)

λ+ λ12 + µ∗

Tc = 1.14ωa exp

−[

1− ωa(λ11 + λ1

12 +K∗)

λ+ λ12 + µ∗

](3.54)

Equation(3.54) gives the expression for the superconducting transition temperature when

linear-energy-dependent density of states and three square well potentials; attractive

electron-optical phonon, attractive electron-acoustic phonon and repulsive electron-electron

(Coulomb)interaction are used. The result for Tc similar to Okoye[124] is recovered when

ωa = ωp in two-band two-square-well potential. And the well known McMillan [131] ex-

pression is recovered as we neglected the contributions of the energy-dependent part of

DOS (N|E| = 0).

The Isotope effect exponent, β, is given by

β = −Md lnTcdM

Assuming that Tc ∝ M−β, and recall that ωa ∝ M−1/2 we obtain from equation(3.54)

thatd lnTcdM

=d lnωadM

− d

dM

[1− ωa(λ1

1 + λ112 +K∗)

λ+ λ12 + µ∗

](3.55)

The Isotope effect exponent(β) can be derived from the expression for Tc given in equa-

tion(3.54)

β = −M[d lnωadM

− d

dM

[1− ωa(λ1

1 + λ112 +K∗)

(λ+ λ12 + µ∗)

]](3.56)

Assuming that

η = (λ+ λ12 + µ∗) , τ = (λ11 + λ1

12 +K∗)

We deduce after simplification that

β =1

2+M

η ddM

[1− ωa(τ)]− [1− wa(τ)] dηdM

η2

(3.57)

dK∗

dM=

XΥ

2MΓ 2

49

dµ∗

dM=

YΥ

2MΓ 2

where

X = (k11 + k1

12)[1 + (µ1 + µ′12)Zc + (µ11 + µ1

12)(ωc − ωp)]− (µ11 + µ1

12)

Y = (k1 + k12)[1 + (µ1 + µ′12)Zc + (µ11 + µ1

12)(ωc − ωp)]− (µ1 + µ′12)

Υ = (1 + (µ1 + µ′12)Zc + (µ11 + µ1

12)(ωc − ωp))((k1 + k12) + (k11 + k1

12)ωa)−

((µ11 + µ1

12)ωa + (µ1 + µ′12))

We can easily re-arrange the above equations in the form:

dK∗

dM=

1

2MΓ 2[XY +X2ωa] =

1

2M[K∗µ∗ + (K∗)2ωa]

dµ∗

dM=

1

2MΓ 2[Y 2 +XY ωa] =

1

2M[(µ∗)2 + µ∗K∗ωa]

Replacing the expression for dk∗

dMand dµ∗

dMin equation(3.56),

we obtain after simplifcation that

β =1

2

(1 +

1

η

[ωaτ −K∗µ∗ωa − (K∗ωa)

2 − [1− ωaτ ]((µ∗)2 + µ∗K∗ωa)

η

])(3.58)

Equation(3.58) can be rewritten in terms of Tc as

β =1

2

(1− 1

η

[ωa(K∗µ∗ + (K∗)2ωa

)]+

((µ∗)2 + µ∗K∗ωa)

η2

)

− 1

2

([(1 + (µ∗)2 + µ∗K∗ωa)

η2

] [1 + η ln

(Tc

1.14ωa

)]). (3.59)

Equation(3.54) and Equation(3.58) gives the expression for the superconducting tran-

sition temperature and isotope exponent in two-band, three-square-well potential model

using linear-energy-dependent electronic density of states (DOS). Equation (3.58) shows

that the energy-dependent electronic DOS influences the isotope effect exponent and may

account for the anomalous isotope exponent observed in high-Tc superconductors. If we

neglect the contributions of the linear-energy-dependent DOS, Ni1(E) = 0, in both bands

and interband, we recover the results of Abah O.C et al [106] is recovered.

50

Chapter 4

Discussions and Conclusion

4.1 Discussions

In this section, we will proceed to compute numerically the effect of linear-energy-dependent

DOS on transition temperature and isotope effect exponent of two-band high-Tc su-

perconductor. Considering the well known two-band superconductor, MgB2, we shall

use measured acoustic-electron phonon cut-off frequency, ωa = 750 K [45] and optical-

electron phonon cut-off frequency, ωp = 812 K [135]. Assuming an electron-electron

(Coulomb) cut-off frequency, ωc = 5000 K , λ1 = 0.34, λ12 = 0.01, k11 + k1

12 = 0.0001,

k1 = 0.01, k12 = 0.001, µ1 = 0.14, µ12 = 0.01 and µ11 + µ1

12 = 0.004, we use Eq. (3.54) to

compute numerically and show graph of the variation of transition temperature with the

effective acoustic-electron phonon coupling associated to linear-energy-dependent DOS,

(λ11 + λ1

12), in Fig. 1. The Fig. 1 shows that the transition temperature increases with

the energy-dependent coupling constant. In Fig. 2, the isotope effect exponent, β shows

a linear dependent on the effect of energy-dependent DOS. It can be observed from the

figure that the isotope exponent that is larger or smaller than BCS value of 0.5 is possi-

ble depending on the sign of the coupling parameter. Also, Fig. 3 shows the numerical

result of the variation of β with Tc using the same set of data. For MgB2, the transition

temperature, Tc ∼ 40 K [20] corresponds to β ∼ 0.36 and λ11 + λ1

12 ∼ 0. And neglect-

ing all the energy-dependent DOS contribution yields Tc ∼ 14 K and β ∼ 0.41 with the

same parameters. Our analysis suggest that the linear-energy-dependent electron-acoustic

phonon coupling has a good effect on the superconducting properties of MgB2.

Furthermore, lets consider the limiting cases of isotope effect exponent, Eq. (3.58):

51

Figure 4.1: The plot of variation of transition temperature with the effective linear-energy-

dependent acoustic-electron phonon coupling, λ11 + λ1

12 using Eq.(3.54).

1. In a pure electron-phonon mechanism: λ1, λ12, λ11, λ

112 6= 0, and µ∗ = K∗ = 0.

β =1

2

1 +

ωa (λ11 + λ1

12)

(λ+ λ12)

(4.1)

In this limit, the isotope effect exponent can either be larger or smaller than the

original BCS value depending on the signs and relative sizes of the coupling param-

eters.

2. In a pure repulsive electron-electron (Coulomb) mechanism: µ∗ = K∗ 6= 0, and

λ1 = λ12 = λ11 = λ1

12 = 0;

β =1

2

1 +

1

µ∗

[K∗ ωa −K∗ µ∗ ωa − (K∗ ωa)

2 − [1− ωaK∗]((µ∗)2 + µ∗K∗ ωa)

µ∗

](4.2)

The expected result in this limit is that β = 0. However the non-zero β value given

by Equation(4.2) arises because of the energy dependence of the DOS [124]. This

contribution may have important implications with regard to the values of β which

deviate from 0.5.

52

Figure 4.2: The plot of variation of isotope effect exponent with the effective linear-energy-

dependent acoustic-electron phonon coupling, λ11 + λ1

12 using Eq.(3.58).

4.2 Conclusion

In conclusion, we have derived the expressions for the transition temperature (Eq.(3.54))

and isotope effect exponent (Eq.(3.58)) of two-band MgB2 superconductor using three-

square-well potential and linear-energy-dependent DOS. The plots of the derived expres-

sions show that the transition temperature (Tc) increases with the effective linear-energy-

dependent coupling constant while the isotope effect exponent(β) is linearly dependent on

energy-dependent DOS. Our analysis show that linear-energy-dependent DOS influences

the transition temperature and isotope effect exponent of the two-band superconductor.

Finally, various limits were considered for isotope effect exponent(β) within our model

whose results are :

1. For pure electron-phonon mechanism :

β =1

2

1 +

ωa (λ11 + λ1

12)

(λ+ λ12)

2. For pure repulsive electron-electron mechanism :

β =1

2

1 +

1

µ∗

[K∗ ωa −K∗ µ∗ ωa − (K∗ ωa)

2 − [1− ωaK∗]((µ∗)2 + µ∗K∗ ωa)

µ∗

]

53

Figure 4.3: The plot of variation of isotope effect exponent with the transition temperature

for λ11 + λ1

12 = - 0.0002 to 0.0008 using Eq.(3.59)

Our results suggest that, perhaps, more experimental investigations are needed to iden-

tify and explore in more details, the role of energy dependent density of states on the

superconductivity of MgB2.

54

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