Two Band Model of Superconductivity of Magnesium Diboride ...
Transcript of Two Band Model of Superconductivity of Magnesium Diboride ...
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
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DEDICATION
This work is dedicated to the Singularity, the One who knows it all; Supreme God.
We are all but trying.
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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.
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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
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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
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List of Tables
1.1 List of Superconducting parameters of MgB2 . . . . . . . . . . . . . . . . . 17
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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
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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-
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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.
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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)
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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
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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)
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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
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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].
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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
REFERENCES
[1] N . Sacchetti, International Journal of Modern Physics B 14 27 (2000).
[2] J . D . Ketterson, S.N. Song,Superconductivity, Cambridge University Press,
United Kingdom, 1999), p.1.
[3] H . R . Khan, Superconductivity, Encyclopedia of Physical Science and Technol-
ogy,Vol.16, Academic press, 1992, p.221.
[4] B . T . Mathias, J . K . Hulum, Science 208 881 (1980).
[5] J . R . Gavalen, Appl. Phys. Lett. 23 480 (1973).
[6] R . Chevrel, M . Sergent, J . Prigent, J . Soild State Chem 3 515 (1971).
[7] F . Steglich, J . Aarts, C . D . Bredl, W . Lieke, D . Meschede, W . Franz, H .
Schafer, Phys . Rev . Lett 43 1892 (1979).
[8] D . Jerome, A . Mazaud, M . Ribault, K . Bechgaard, J. Phys(Paris) Lett.41 195
(1980).
[9] C . P . Poole Jr, H . A . Farach, R . J . Creswick, Superconductivity, Academic Press
Inc., 1995, p.23.
[10] G . R . Stewart, Z . Fisk, J . O . Willis, T . J . Smith, Phys . Rev . Lett. B 52 679
(1984).
[11] J . G . Bednorz, K . A . Muller, Condensed matter 64 189 (1986).
[12] C . W . Chu, P . H . Hor, R . L . Meng, L . Gao, Z . J . Huang, Y . Q . Wang, Phys
. Rev . Lett . 58 908 (1987).
[13] M . K . Wu, J . R . Ashburn, C . J . Torng, P .H . Hor, R .L . Meng, L . Gao, Z .
J . Huang, Y . Q . Wang, C . W . Chu, Phys . Rev . Lett 58 908 (1987).
55
[14] V . V . Schmidt, P . Muller(Ed), A . V . Ustinov(Ed), The Physics of Superconduc-
tors, Nauka Publishers, Moskau 1982 p.80.
[15] H . Maeda, Y . Tanaka, N . Fukutomi, T . Asamo, Jpn . J .Appl. Phys . Lett . 27
209 (1988).
[16] Y . D . Chuang et al, Electronic Structure of Colossal Magnetoresistive (CMR)
oxides (Unpublished).
[17] D . D . Barkley, E . F . Skelton, N . E . Moulton, M . S . Osofsky, W . T . Lechter,
V . M . Browning, D . H . Liebenberg, Phys . Rev B 47 5524 (1993).
[18] M . Nunez-Regueiro, J . L . Tholence, E . V . Antipov, J . J . Capponi, M . Marezio,
Science 262 99 (1993).
[19] C . P . Poole Jr, H . A . Farach, R . J . Creswick Superconductivity, Academic Press
Inc., 1995, p.88.
[20] J . Nagamatsu, N . Nakagawa, T . Muranaka, Y . Zenitan, J . Akimistu, Nature
410 63 (2001).
[21] Robert Casalbuoni Lecture Notes on Superconductivity, Condensed Matter and QCD
(2003).
[22] W . Meissner, R . Ochsenfeld, Naturewissenchaften 21 787 (1933).
[23] A . A . Abrikosov, Zh Eksp . Teor . Fiz 35 1442 (1957).
[24] J . G . Daunt, K . Mendlesshn, Proc . Roy . Soc. A185 225 (1946).
[25] V . L . Ginzburg, Fortschr . Phys . 1101 (1953).
[26] J . Bardeen, Handbuch der Physik XV (Springer Verlag,Berlin) 1956.
[27] H . Frolich, Phys . Rev 79 845 (1950).
[28] E . Maxwell, Phys . Rev. 78 477 (1950).
[29] C . A . Reynolds, B . Serin, W . H . Wright, L . B . Nesbitt, Phys . Rev . 78 487
(1950).
[30] J . Bardeen, L . N . Cooper, J . R . Schrieffer, Phys . Rev 106 162 (1957).
56
[31] C . Kittle Introduction to Solid State Physics Wiley 5th ed. 1971, p.348.
[32] M . Monteverde, M . N . Regueiro, N . Rogado, K . A . Regan, M . A . Hayward,
T . He, S . M . Loureiro, R . J . Cava, Sicence 292 75 (2001).
[33] C . J . Gorter, H . G . B . Casimir, Z . Tech . Phys 15 539 (1934).
[34] F . London, H . London, Proc . Roy . Soc . A149 71 (1935).
[35] L . Ginzburg, L . D . Landau, Zh . Eksp . Teor . Fiz 20 1064 (1950).
[36] L . P . Gor’kov, Zh . Eksp . Teor . Fiz 36 1918 (1959).
[37] H . B . Schuttler, C . H . Pao, Phys . Rev . Lett . 75 4504 (1995).
[38] P . W . Anderson, Theory of Superconductivity in the High-Tc Cuprates, Princeton
N J (1997).
[39] F . Bouquet, R . A .Fisher, N . E . Philips, Y . Wang, D . G . Hinks, J . D .
Jorgenser, A . Junod,
Europhys Lett.56 856 (2001).
[40] Y . Wang, T. Plackonwski, A . Junod, Physica C 355 179 (2001).
[41] A . Sharoni, I . Felner, O . Millo, Phys . Rev . B 63 220508 (2001).
[42] H . Kotegawa, K . Ishida, Y . Kitaoka, T . Muranaka, J . Akimitsu, Phys . Rev .
Lett . 87 127001 (2001).
[43] E . S. Clementyev, K . Conder, A . Furrer, I . L . Sashin, Eur .Phys . J . B 21 405
(2001).
[44] T . Muranaka, S . Margadonna, I . Maurin, K . Brigatti, D . Colognesi, K . Prassides,
Y . Iwasa, M . Arai, M . Takata, J . Akimitsu , J .Phys . Jpn 70 1480 (2001).
[45] S . L . Bud’ko, G .Lapertot, C . Petrovic, C . E . Cunningham, N . Anderson, P .
C . Canfield, Phys . Rev . Lett . 86 1877 (2001).
[46] D . G . Hinks, H . Claus, J . D . Jorgensen, Nature 411 458 (2001).
[47] G . Karapetrov, M . Iavarone, W . K . Kwok, G . W . Crabtree, D . G . Hinks
Cond-mat/0102312.
57
[48] A . Y . Liu, I . I . Mazin, J . Kortus, Phys . Rev . Lett 87 087005 (2001).
[49] H . J . Choi, M . L . Cohen, S . G . Louie , Physica C 345 66 (2003).
[50] P . Szabo, P . Samuely, J . Kacmarcik, T . Kelin, J . Marcus, D . Fruchart, S .
Miraglin, C . Mercenat, A . G . M . Jansen, Phys . Rev . Lett . 87 177008 (2001).
[51] H . Schmidt, J . F . Zasadzinski, K . E . Gray, D . G . Hinks, Cond-mat/0112144
(2002).
[52] F . Giubileo, D . Roditchev, W . Sacks, R . Lamy, D . X . Thanh, T . Kelin, S .
Miraglia, D . Fruchart, J . Marcus, Ph . Monod, Phys . Rev . Lett.87 177008 (2001).
[53] X . K . Chen, M . J . Korstantinovic, J . C . Irwin, D . D . Lawrie, J .P . Franck,
Cond-mat/0104005
[54] R . E . Lagos, G . G . Cabrera, Braz . J . Phys.33 14 (2003).
[55] E . F . Skelton, D . U . Gubser, J . O . Willis, R . A . Hein, S . C . Yu, I . L . Spain,
R . M . Waterstrat, A . R . Sweedler, Phys . Rev B 20 4538 (1979).
[56] M . Monteverde, M . N . Regueiro, N . Rogado, K . A . Regan, M . A . Hayward,
T . He, S . M . Loureiro, R . J . Cava, Sicence 292 75 (2001).
[57] P . Bordet, M . Mezourar,M . Nunez-Regueiro, M . Monteverde, M . D . Nunez-
Regueiro, N . Rogado, K . A . Regan, M . A . Hayward, T . He, S . M . Loureiro,
R . J . Cava, Phys . Rev B 64 172502 (2001).
[58] C . Buzea, T . Yamashita, Supercond . sci . Tech 14 R115 (2001).
[59] J . Kortus, I . I . Mazin, K . D . Belashchenko, V . P . Antropov, L . L . Boyer,
Phys . Rev . Lett . 86 4656 (2001).
[60] W . N . Kang, C . U . Jang, K . H . P . Kim, M . S . Park, S . Y . Lee, H . J .
Kim, E . M . Choi,K . H . Kim, M . S . Kim, S . I . Lee, Appl . Phys . Lett . 79 982
(2001).
[61] S . Jin, H . Mavoori, R . B . Van Dover, Nature 411 563 (2001).
[62] M . R . Eskildsen, M . Kugler, S . Tanaka, J . Jun, S . M . Kazakov, J . Karpinski,
Ø . Fischer , Phys . Rev. Lett. 89 187003 (2002).
58
[63] T . S . Kayed, Crystal Research and Technology 39, 50 (2004).
[64] B . Gorshunov, C . A . Kuntscher, P . Haas, M . Dressel, F . P . Mena, A . B .
Kuz’menko, D . Van der Marcel,T . Muranaka, J . Akimitsu, Eur . Phys . J . B 21
159 (2001).
[65] J . J . Rodriguez, A . A . Schmidit, Phys . Rev . B 68 224512 (2003).
[66] C . B . Eom, M . K . Lee, J . H . Choi, L . Belenky, X . Song, L . D . Cooley, M . T
. Naus, S . Patnaik, J . Jiang, M . Rikel, A . Polyanskii, A . Gurevich, X . Y . Cai,
S . D . Bu, S . E . Babcook, E . E . Hellstrom, D . C . Larbalestier, N . Rogado, K
. A . Regan, M . A . Hayward, T . He, J . S . Slusky, K . Inumaru, M . K . Haas, R
. J . Cava, Nature 411 588 (2001).
[67] Y . Bugoslavsky, G . K . Perkins, L . F . Cohen, A . D . Caplin, M . Polichetti, T .
J . Tate, R . G . Willam, Nature 411 561 (2001).
[68] D . Kaczorowski, A . J . Zaleski, O . J Zogal, J . Klamut, Cond-mat/0104479 (2001).
[69] D . P . Young, P . W . Adams, J . Y . Chan, F . R . Fronczek, Cond-mat/0104063
(2001).
[70] R . da Silva, J . H . S . Torres, Y . Kopelevich, Cond-mat/0105329 (2001).
[71] T . He, Q . Huang, A . P . Ramirez, Y . Wang, K . A . Regan, N . Rogada, M . A
. Hayward, M . K . Haas, J . S . Slusky, K . Inumaru, H . W . Zandberg, N . P .
Ong, R . J . Cava, Cond-mat/0103296 (2001).
[72] M . I . Eremets, V . V . Struzhkin, H . K . Mao, R . J . Hemley, Science 293 272
(2001).
[73] M . Iavarone, G . Karapetov,A . E . Koshelev, W . K . Kwok, G . W . Crabtree, D
. G . Hinks, W . N . Kang, E . Choi, H . J . Kim, S . I . Lee, Phys . Rev . Lett . 89
187002 (2002).
[74] S . Tsuda, T . Yokoya, T . Kiss, Y . Takano, K . Tagano, H . Kito, H . Ihara, S .
Shin, Phys . Rev . Lett . 87 177006 (2001).
59
[75] F . Giubileo, D . Roditchev, W . Sacks, R . Lamy, D . X . Thanh, J . Kelin, S
. Miraglia, D . Fruchart, J . Marcus, Ph . Monod, Phys . Rev . Lett . 87 177008
(2001).
[76] G . Gorshunov, C . Kuntscher, P . Haas, M . Dressel, F . P . Mera, A . B .
Kuz’Menko, D. van Marcel, T . Muranaka, J . Akimistu, Cond-mat/ 0103164.
[77] W . Pickett, Nature 418 733 (2002).
[78] K . D . Belashchenko, V . P . Antropov, S . N. Rashkeev, Phys . Rev . B 64 132506
(2001).
[79] S . Haas, K . Maki, Cond-mat/0103408.
[80] A . A . Zhukov, L . F . Cohen, A . Purnell, Y . Bugoslavsky, A . Berenov, J . L .
Macmanus Driscoll, H . Y . Zhai, H . M . Christen, M . P . Parathaman, D . H .
Lowndes, M . A . Jo, M . C . Blamire, L . Hao, J . Gallop, Supercond . Sci . Technol
. 14 L13-L16 (2001).
[81] H . Suhl, B . T . Matthias, L . R . Walker, Phys . Rev . Lett . 3 522 (1959).
[82] S . V . Shulga, S . L . Drechsler, H . Eschrig, H . Rosner, W . E . Pickett,
Cond.mat/0103154 (2001).
[83] A . Y . Liu, I . I Mazin, J . Kortus, Phys . Rev . Lett 87 87005 920010.
[84] C . M . I . Okoye, Chinese Journal of Physics 36 1 (1998).
[85] N . Kristoffel, P . Korsin, T . Ord, Rivista del Nnovo Cimento 17 (1994).
[86] N . Kristoffel, T . Ord, Phys . Stat . Solidi B 175 K9 (1995).
[87] V . A . Moskalenko, Fiz . Met . Metalloved 8 503 (1959).
[88] N . Kristoffel, P . Rubin, Physica C 402 257 (2004).
[89] S . Punpocha, R . Hoonsawat, I . M . Tang, Solid State Communications 129 151
(2004).
[90] K . Yamaji, J . Phys . Soc . Jpn 70, 1476 (2001).
[91] T . Ord, N . Kristoffel, Physica C 370, 17 (2002).
60
[92] G . A . Ummarino, R . S . Gonnelli, S . Massidda, A . Bianconi, Cond-
mat/0310284V1.
[93] M . Wang, X . Y . Su, Z . J . Xu, Physica C 386 657 (2003).
[94] C . P . Moca, Phys . Rev . B 65 132509 (2002).
[95] A . A . Golubov, J . Kortus, O . V . Dolgov, O . Jepsen, Y . Kong, O . K . Andersen,
B . J . Gibson, K . Ahn, R . K . Kremer, J . Phys . Condense Matter 14 1353 (2002).
[96] M . E . Zhitomirsky, V . H. Dao, Phys . Rev . Lett . 90 177002 (2003).
[97] T . Yaragisawa, H . Shibata, Physica C 392 276 (2003).
[98] G . M . Elisahberg, Soviet Phys . JETP 11 696 (1960).
[99] M . Calandra, M . Lazzeri, F . Mauri, Physica C 456 38 (2007).
[100] H . Choi, M . L . Cohen, S . G . Louie, Physica C 385 66 (2003).
[101] P . Udomsamuthirun, C . Kumvongsa, A . Burakorn, P . Changkanarth, Pramana
J . Physics 66 589 (2005).
[102] G . Blumberg, A . Mialistin, B . S . Dennis, N . D . Zhigadlo, J . Karpinski, Physica
C 456 75 (2007).
[103] H . J . Choi, D . Roundy, H .Sun, M . L . Cohen, S . G . Louie, Phys . Rev . B66
020513 (2002).
[104] C . Joas, I . Eremin, D . Manske, K . H . Benneman, Phys . Rev . B 65 132518
(2002).
[105] X . Y . Su, J . Shen, L . Y . Zhang, Phys . Lett A 143 9 (1990).
[106] O . C . Abah, G . C . Asomba, C . M . I . Okoye, Solid State Communications 149
1510 (2009) .
[107] J . E . Hirsch, D . J . Scalapino, Phys . Rev . Lett 58 2732 (1986).
[108] J . Labbe, J . Bok, EuroPhys . Lett . 3 1225 (1987).
[109] C . C . Tsuei, D . M . Newns, C . C . Chi, P . C . Pattnaik, Phys . Rev . Lett 65
2724 (1990).
61
[110] P . C . Pattnaik, C . L . Kane, D . M . Newns, C . C . Tsuei, Phys . Rev . B 45
5714 (1992).
[111] S . Sarkar, A . N . Das, Phys . Rev . B 49 13070 (1994).
[112] R . S . Markiewicz, Physica C 177 171 (1991).
[113] D . M . Newns et al, Phys . Rev . Lett . 73 1695 (1994).
[114] M . L . Horbach, H . Kajuter, Int . J . Mod . Phys . B 9 1067 (1995).
[115] D . M . King, Z . -X . Shen, D . S . Dessau, Phys . Rev . Lett . 73 3298 (1994).
[116] K . Gofron, J . C . Campuzano, A . A . Abrikosov, M . Lindroos, A . Bansil, H .
Ding, D . Koelling, B . Dabrowski , Phys . Rev . Lett . 73 3302 (1994).
[117] S . Sarkar, A . N . Das, Phys . Rev . B 54 21 (1996).
[118] S . Sarkar, A . N . Das, Phys . Rev . B 49 18 (1994).
[119] P . Udomsamuthriun, C . Kumvongsa, A . Burakorn, P . Changkanarth, S . Yoksan,
Physica C 425 (2005) 149
[120] P . W .Anderson, Z . Zon, Phys . Rev . Lett . 60 132 (1998).
[121] J . C .Phillips, Phys . Rev . Lett . 59 1856 (1987).
[122] M . C . Anna, C . Noce, A . Romano, Physica C 202 33 (1992).
[123] M . Gurvitch, J . M . Valles Jr, A . M . Cucolo, R . C . Dynes, J . P . Garno, L . F
. Schneemeyer, J . V . Waszczack, Phys . Rev . Lett . 63 1008 (1989).
[124] C . M . I . Okoye, Physica C 313 197 (1999).
[125] N . N . Bogoliubov, Nuovo Cimento 7 794 (1958).
[126] J . G . Valatin, Nuovo Cimento 7 843 (1958).
[127] L . N . Cooper, Phys . Rev . 104 1189 (1956).
[128] J . B . Goodenough, Superconductor High-temperature Encyclopedia of Physical
Science and Technology Vol.16 Academic Press Inc. 1992, p.275.
62
[129] B . Hao, C . D . Gong, Solid State Communications 85 341 (1993).
[130] P . G . DeGennes, Superconductivity of Metals and Alloys Benjamin New York,
1996, p.26.
[131] M . L . McMillan, Phys. Rev . 167 331 (1968).
[132] P . Morel, P . W . Anderson, Phys . Rev . 125 1263 (1962).
[133] I . I . Mazin, V . P . Antropov, Physica C 385 (2003).
[134] I . I . Mazin, O . K . Andersen, O . Jepsen, O . V . Dolgov, J . Kortus, A . A
. Golubov, A . B . Kuz’menko, D . van der Marel, Phys . Rev . Lett . 89 107002
(2002).
[135] A . F . Goncharov, V . V . Struzhkin, E . Gregoryanz, J . Z . Hu, R . J . Hemley, H
. K . Mao, G . Lapertot, S . K . Bud’ko, P . C . Canfield, Phys . Rev . B 64 100509
(2001).
63