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Michigan State
University

__.- _

       

This is to certify that the

thesis entitled

Theory of Transition Element

Superconductors

presented by

Paul B. Williams

has been accepted towards fulfillment
of the requirements for ?

Ph.D. Physics

degree in

M/%

Mdor professor

DateWO

0-169

ABSTRACT

THEORY OF TRANSITION ELEMENT
SUPERCONDUCTORS

by Paul B. Williams

A theoretical expression for electron-phonon coupling
is derived and then used in an attempt to explain differ-
ences in superconducting transition temperatures for groups
VB and VIB transition metals.

The bare ionic potential, based upon the Herman-
Skillman atomic structure calculations, is_used to determine
unscreened coupling. Screening is then calculated using the
Bardeen self-consistent field method modified for exchange.
The net result is an expression for the screened electron-
phonon coupling, gK K" exhibiting a positive dependence
upon the electronic density of states at the Fermi level.
Such a dependence is due entirely to exchange and is thought
to stem from the preference of conduction electrons to group
between nearest neighbors in some resonant structure remnant
of atomic "d"—electrons.

A suitable computer program is developed to determine
the product of coupling times phonon density of states as a
function of phonon frequency, a2(cn)F(¢n). The method first

uses interatomic force constants to determine the phonon

Paul B. Williams

density of states, F(m0, via the Born-von Karman method.
The expression for screened electron-phonon coupling is
then incorporated to include umklapp as well as normal
scattering processes. Finally, within the currently
accepted theory of superconductivity, a2(m)F(cn) is used to
determine an expression for the superconducting transition

temperature of tantalum and tungsten,

THEORY OF TRANSITION ELEMENT
SUPERC ONDUCTORS

/U9
Paul B? Williams

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Department of Physics
1970

ACKNOWLEDGMENTS

I wish to thank Dr. Michael Harrison for his
excellent guidance and encouragement throughout the
course of this research. Special thanks to Dr.
‘William.Hartmann for providing a copy of the Gilat
and Raubenheimer program, G(NU), and for some very
helpful advice in the area of accoustical properties
of solids.

Also, I wish to thank Jean Strachan for a
truly outstanding Job of typing.

11

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . 11
LIST OF TABLES . . . . . . . . . . . . . . . . . . IV
LIST OF FIGURES . . . . . . . . . . . . . . . . . . V
Chapter

I INTRODUCTION........_......l
II THE'I‘HEORY.’...............14
III THEGAPEQUATIONS............ 29
Iv THE ELECTRON-PHONON COUPLING CONSTANT . . 46
v EVALUATION OF MATRIX ELEMENTS . . . . . . 66
VI THEORETICAL INVESTIGATION OF
SUPERCONDUCTING TAN'I‘ALUM AND TUNGSTEN . . 7%
.VII CONCLUSION...............91
Appendices
A EVALUATIONOFInlm...........96
B EVALUATIONOFINTEGRALS.........100
c chPtJTERPRoanN.............105

LIST OF WRENCESO O O O O O O O 0 O O O O O O O O 139

111

Table
1.

2.

LIST OF TABLES

Page

IsotOpe effect (experimental for some
simple and transition metals . . . . . . . 5

Summary of physical parameters
influencing superconductivity . . . . . . lO

Character table of the cubic group O . . . 20

List of potential coefficients and
force constants for tantalum.. . . . . . . 84

List of potential coefficients and
force constants for tungsten . . . . . . . 85

A vs Kf for tantalum.and tungsten 92

Figure

1.

2.

5.
6.
7.
8.

10.
11.

LIST OF FIGURES

Qualitative behavior of superconducting
transition temperatures through the ‘
periodic system . . . . . . . . . . . .

Areas of maximum electron density
within the B~C.C. primitive cell

for electrons having

group symmetry. . . . . . . . . . . . .

Important diagrams contributing to self
enem O O O O O O O O O O O O O O O O O

Contours in com lex plane showing some

.polesatoddffl ..........

Theoretical density of states for
tungsten from band structure data . . .

Density of phonon states for tantalum .
Density of phonon states for tungsten .

Coupling times density of states for
tarltalumoe_eoeee'oeeoeoeee

Coupling times density of states for"
tlmsa ten 0 C O O O O C C C O O O O I O O

A vs Kf for tantalum. . . . . . . . . .

A vs Kf for tungsten. . . . . . . . . .

Page

12

23
35
38

45
87
88

89

2‘88

CHAPTER I
INTRODUCTION

The first successful microscopic theory of supercon-
ductivity was presented by Bardeen, Cooper, and Schrieffer
(3.0.8.) in 19571 and then later refined into a more ele-

2 and Bogoliubov’. In

gant and accepted form by valatin
simplest terms, the theory assumes attractive Cooperll elec-
trons forming pairs of opposite momentum and spin which re-
side in a condensed ground state '15; :> . A

reduced hamiltonian (fired ) which considers only the elec-

tron pairs is used in conjunction with a parameterized
trial ground state wavefunction subject to the constraint
that the expectation value of the total number of particles
r
is e b . Let d
som fixed num er No Ck? an Ckl

be creation and annihilation operators for electrons in

state k(spin 1/2) and state -k(spin -l/2) respectively.

Then Hr“? El: 6 le-(T jI-4C—k J'CH
”1+2

            

was C—kl Cm

    

 

 

     

Ir, IE
kZSCkTC -|<l C -k1C|<T l2fl>= N
with
’fl’l + N. CT |<
|W>:fl k V1 + XL?—
The parameter X“ in the ground state wavefunction

is then determined by normal variational methods for a
minimum ground state free energy (see Schrieffer5). The
key results are that the pairing phenomenonis effective
within a small range of energies of approximately 2"“ mo
( COD =- Debye frequency) about the Fermi energy and that
quasiparticle excitations from the ground state always have
an energy greater than or equal to an energy gap A. The

B.C.S. result for A is, in the weak coupling limit (small

. -1 0)
V) AEZ’fiwDe /N( \/

2 Assskfig

where N(o) is the density of electronic states at the Fermi
energy and'v is an effective pairing potential acting only
within the range of energies about the Fermi level where
pairing is approximately assumed to exist. Although the
B.C.S. theory was highly successful in predicting qualita-
tive features of superconductors, it soon became apparent
that a more powerful theory was due. One had to consider
the retarded nature of the electron-phonon interaction and
the screened coulomb repulsion between electrons before
serious attempts could be made to calculate quantitative
properties of superconductors from first principles.

A particularly useful feature of the B.C.S. theory is
the isotope effect. If one considers the isotopes of any
particular element, each arranged in its own lattice, the
normal frequencies of vibration will be (.0200 k/|Vl
in the harmonic approximation where M is the mass of the
isotope in the lattice in question and k is an effective
spring constant common to all the isotopes of that partic-

ular element. Since the transition temperature To satis-

fies Eq. 4, then the ratio of the transition temperatures

between any two isotopes of a given element is

%=\/m— .

 

This important result is known as the isotope effect. Al-
though it does not include the effects of mutual coulomb
repulsion between pairs, it is supported quite well by the
simple metals including such strong coupled superconductors
(V large) as lead and mercury. Hewever, transition.metal
superconductors are a different story altogether. They
seem to exhibit little or no isotope effect and agreement
exists in too few cases to produce any firm conclusions.
The results of measurements of the superconducting tran-
sition temperatures of various isotopes is shown in table 1
where TOO M-F 6. It would seem that for the
transitidfi metals, the B.C.S. theory fails because either
the electron pairing potential is not phonon mediated but
possibly due to some overscreening by nonsuperconducting
electrons in different bands as suggested by several auth-
ors7 or that some electronic mechanism, intermediate to
the superconducting electrons and the ions, serves to
alter the bare electron-phonon coupling constant in ways
more complex than previously described.

Not long after the advent of the B.C.S. theory, theo-
reticians attempted to apply Green function methods to the
superconducting many body problem. Their goal was to dee
velop a theory that would predict the superconducting pro-
perties of metallic superconductors from the most basic

properties of the metals themselves. Ordinary Green

Table 1. Isotope effect (experimental) for some
simple and transition metals.*

 

 

Simple Transition
Metal 8 Metal 8
Pb 0.48 Os 0.21
H8 0.50 Ru 0
Sn 0.51 Mo 0.3}
T1 0.49
Zn 0.50
Cd O.#O

* The quantity 8 is defined with TC~M‘3.

 

functions soon met with failure due to infinities resulting
from summing over certain Feynman. diagrams of a given
order. The culprits are those diagrams where two elec-
trons of opposite momenta and spin repeatedly scatter
against each other. These difficulties were soon sur-
mounted, however, with the introduction of "anomalous prop-
agators" (see Gorkov8) which destroy and create Cooper4
pairs in the ground state. The entire Green function prob-
lem was reformulated into a matrix form (Nambug) and was

successfully applied by Eliashberg10

to the problem of
strong coupled superconductors.

Strong coupling was particularly troublesome because
a large electron-phonon coupling constant would tend to
produce such energetic quasiparticle excitations that the
resulting lifetimes would be considerably shortened (be;
cause of increased available phase space for their decay)
to the point where the width (due to uncertainty considere
ations) of the quasiparticle levels is comparable to the
energy of the excitations themselves. The problem was

acute until an important result by Migdal11

10

was applied
by Eliashberg yielding a set of coupled integral equa-

tions which form the basis of the theory as it stands
today. Important coulomb effects were soon incorporated12
giving a set of equations (known as the Eliashberg gap

equations) which should enable one to calculate the super-

conducting properties of metals and semiconductors to a
relatively high degree of accuracy. These equations form
a vital part of this research and, therefore, will be de-
veloped in detail later.

Use of the Eliashberg equations to find the supercon-

ducting transition temperature Tc was successfully applied

to aluminum, sodium, and potassium by Carbotte and Dynes13.
Essentially, they began by using the Heine-Abarenkov form

1“ from which they

of the electron-ion pseudopotential
could obtain the electron-phonon coupling constant. The
necessary phonon characteristics were represented through
the use of a Born-yon Karman force constant fit to phonon
dispersion data in the high crystalline symmetry directions
where the number of nearest neighbors considered depended
upon the complexity of the phonon dispersion curves forthe
superconductor in question. The Eliashberg equations were
then solved self consistently on a computer yielding re-
sults that were quite successful. The transition temper-
ature for both nonsuperconducting sodium and potassium was
determined to be less thanAESOK. For aluminum, the pre-
dicted transition temperature differed by less than 10%
from the experimental value. In addition, the first strong
coupled superconductor to be investigated using the

Eliashberg equations was lead which has the pecularities

as described earlier. Experimental results for lead strong-
ly support the Eliashberg form as opposed to previous mod-
els. Finally, little or no progress has been made in deter?
'mining the superconducting properties of transition metals
from first principles. Their electronic band structure is
of such complexity that it is difficult, if not impossible,
to make use of pseudopotential methods to determine values
of electron-phonon coupling.

If the Eliashberg equations are considered a truly
accurate set of solutions to the problem or superconduc-
tivity and if superconducting properties can be accurately
measured, then it would be logical to ask if these equa-
tions could be applied in reverse to determine some of the
more elusive properties of solids from their superconducting
properties. Such a problem was undertaken by McMillan15
who applied rit to a variety of materials including transi-
tion metal superconductors and their alloys. The primary
objectives werean empirical electron-phonon coupling con;
stant >\ and a coulomb coupling constant ,5”: as

16

defined by Mbrel and Anderson . From these empirical con-

2
stants the quantity <G. > (the average squared elec-

17 was found for several

tronic transition matrix element
transition metal superconductors. All of this was accom-
plished as follows. The average electron-phonon coupling

2
0C (w) as a function of frequency and the phonon

density of states F(w) are determined as the product
OC 2(a)) Fan) directly through the insertion

of experimentally determined superconducting properties into

the Eliashberg equations. Then the quantity (:(2; is:

<6»: N(o) <w >

where )( ::MZU/‘-£L£¥2,ififouz)F:2CL»

 

«02>: wa oc2(w)l——(oe) co
fiwflz(w)F-(w)

The results are summarized in table 2. In this scheme
McMillan has demonstrated that, for group'V and VI B.C.C.

transition metals (except magnetic chromium), the factor

2
N(O) <G > remains essentially constant
2
and that To depends upon M <60 > (the stiff-

ness of the lattice). These results are uncommon and, as
he pointed out, no satisfactory theoretical explanation has
been given for such phenomena.

The present theory of superconductivity seems to be

satisfactory with the possible exception of the transition

10

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11

metals. These metals can be among the best superconductors
and, when properly alloyed, the transition temperature and

critical field can be greatly increased (i.e. Nb3 8n and
V381) which seems to indicate that physical features which

give rise to superconductivity are very sensitive to the
electronic structure in the case of Nb and V. If one plots:

the transition temperature Tc versus the number of valence

electrons per atom for transition metals and compares it
with a similar plot for simple metals, as was done by,
flatthiasle, (Figure 1), there appears to be little agree-

ment as to how'l‘c varies with the number of valence elec-

trons per atom. This certainly suggests that, as in many
other phenomena, transition metal superconductors must re-
quire a modified or perhaps totally different treatment'
_than was used for the case of simple metals. In fact, if
one compares the transition temperatures of the group V
transition metals with those of group VI transition metals,
it becomes apparent that, with the addition of one more non-
core electron per atom, the excellent superconducting prep-
erties of group v are lost in group VI.’ Assuming that the
theory of superconductivity is correct, it is then logical
to try to find a means to determine the degree of electron-
phonon coupling and the best value for the density of states
(including only those electrons that can take part in pair-

s-dl

IMPLE
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ANSITION
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VALENCE ELE CTRONS PER

13
ing) for these two groups. It certainly would not be unrea-
sonable to suppose that not all of the remnants of the s and
d atomic states are itinerant. Perhaps a sizable fraction
are not available for pairing but do act as an intermediate
device to enhance coupling between the paired electrons and
the ions. This would most likely reduce the density of.
available states but, at the same time, could substantially
increase the effective electron-phonon interaction. Such
a concept is worthy of consideration and, therefore, will

be explored in as much detail as present knowledge of the
group'V and VI transition.metals will allow.

CHMPTER II
THE THEORY

This problem is primarily concerned with the elec-
tronic structure of the group V and group VI transition
metals. As completely free atoms, transition elements
are characterized by two partially filled atomic subshells
(Bass, #dSs, or 5d6s) as opposed to the usual ggg_(or com-
pletely filled shells as in the case of the inert gases).
The number of electrons residing in both the s and d sub-
shells determines the chemical and metallurgical charact-
eristics of the element in question. For groups V and VI,

the atomic Epnfigurations are:
ELI
V-3cl “rsz Cr—3d S
L+ES' ' 5'
Nt-LH s [Via-LN 55
I 3 2 W L)
G's-Ed 6S -5d 652
group V' group V14
The objective is to determine how these electrons in par-
tially filled subshells (valence electrons) interact with
valence electrons of adjacent atoms in a metallic solid.
It is often stated that the unfilled s and d subshclls

transform into overlapping s and d bands in the solid state.

moreover, a frequent assumption is that the relative propor-

14

15
tion of electrons in each of these bands is the same as it
was for the partially filled atomic subshells. A more rea-
listic viewpoint, however, is one proposed by several au-
thoritieslg’20 in this area which is that the valence
electrons collect into two fairly distinct groups. One
such group is quite localized and may possibly be repre-
sented by some linear combination of the former d orbital
wavefunctions. The second group would be an itinerant con-
ductionelike band which could probably be represented by
plane waves properly orthogonalized to both core states and
the localized states of the first group. It seems plausible
that these two bands overlap with the localized band being
the narrower of the two. Moreover, the number of electrons
per atom residing in each of these bands would, in all like;
lihood, not be in the same proportion as the relative num-
bers of electrons in the s and d atomic subshells respec-
tively. Such are the foundations to be later utilized in
setting up an electronic model of the group V and VI tran-
sition'metals.

If one is to base a model on the concept of two over-
lapping bands where one is localized and the other itinerant
and conduction-like, then it is fruitful at first to inves-
tigate how such a configuration might affect the pairing
interaction in superconductors. The conditions for pair-

ing are most favorable with a strong electron-phonon

l6
coupling constant and a high available electronic density
of states at the Fermi surface. The question then arises
as to whether all_of the valence electrons are available
for pairing. Perhaps either the more localized electrons
or the conduction electrons do not take part in the pairing
process at all and hence one must use a density of states
that accurately reflects this reduction of available elec-
tronic states. Certainly, then, if a given proportion of
the valence electrons do not contribute to the pairing pro?
cess, transition metal superconductivity may be enhanced
through a mechanism.other than a high electronic density of
states. Another possibility would be to examine how these
non superconducting valence electrons might tend to modify
ion (phonon) coupling to the superconducting electrons." If
this modified coupling were quite sensitive to the number
of electrons in localized bands, then the difference of a
valence electron per atom between transition groups V and
VI could account for the gross contrast between their
ability to superconduct provided the extra electron in group
VI were in the localized d band.

The model to be used will be one in which nd valence

electrons per atom reside in localized d-like band states

and n0 valence electrons per atom form an itinerant conduc—

tion band. The question naturally arises as to which band

17

contains the superconducting electron pairs. In the past,
it has been quite customary to consider the itinerant elec-
trons as those available for pairing 21’22. The localized
states are then usually treated as either intermediate
states for electron pair scattering by the lattice ions or
as states acting directly between the pairs by providing an
additional attractive coupling mechanism. McMillan15 seems
to avoid this question altogether when he uses a density of
states based upon the electronic specific heat 8 . Al-
though the argument for d-like electrons over-screening
the conduction electrons would enhance superconductivity,‘
it makes little sense to consider electrons with a suppos-
edly high effective mass as capable of screening the effec-
tively lighter conduction electrons. However, it would not
suffice to say that electrons with high effective mass can-
not be screening agents in the case of the ions due to the
fact that they are also strongly coupled (source of high
effective mass). Whether or not screening of one band by
a second could enhance the electron-ion interaction of the
first should be analyzed. These arguments give rise to
two possibilities:

(l) The conduction band electrons form superconduct-
ing pairs. The d-band electrons screen the motion of the
ions in such a way so as to enhance electron-phonon coupling

in group V and diminish it in group VI. The difference in

r1-

l8
superconducting properties between groups V and‘VI is
attributed to the difference of one electron per atom in
the d-band.

(2) The d-band electrons form superconducting pairs.
The conduction band electrons serve as a screening mechanism
to alter electron-phonon coupling and reduce the effective
coulomb repulsion between d band electrons. The high den-
sity of states of the group V d band at the Fermi level
greatly assists superconductivity whereas a small density
of states in group VI all but destroys it. Again, there is
a difference of one electron per atom in the d bands of
group V and group VI.

One of the most crucial portions of the model will be
the structure of the localized electrons. One must know
their density of states distribution, energies, and, if
possible, their wavefunctions. unlike core electrons, the
localized electrons cannot be associated solely with a par-
ticular icn within the crystal. The characteristic that
distinguishes them from conduction electrons is that their
distribution varies grossly within the structure of the‘
unit cell. Localized electrons will tend to bunch up mainly
between nearest neighbors and next nearest neighbors_respece
tively. Such electrons should'besensitive to thetnotion of
the ions even though they cannot be associated solely with
any one particular lattice site. Since both group V and

19
group VI transition metals have BCC crystal structure, it
is best to examine splitting of d-orbital wavefunctions in
a cubic field before attempting to use them in constructing

localized band states. Basis functions from which the

localized states can be formed are as follows:

xy=\/% {fies-Kerb?

YZ :V‘WQEXVGDCPH )2/1(e,d>)}

z xm 3X‘1<e¢)-W
XZ—Y-Ztvg/i—g {yiechyfew
3 22+ Rd» :V1% f We. <19}

where >/ (e are the orthonormal spherical har-
2.

 

monics forming a basis for the angular part of atomic d
wavefunctions. These new basis functions are all mutually
orthogonal and therefore must span the same space as X .
The purpose of the particular basis becomes quite clear
when one examines the full cubic group OhfiTible 3) .

Z 2 Z
The functions X —->/ and 3 Z —R2 form a basis
for the two diminsional representation E 8 ant?< y) y Z
J

andZ X form a basis for the three diminsional represen-

20

MN

macaw sonao>GH on» uH H cams: Hadinno asoam oHoso HHsm moss

 

H- H H- o n me said:

H H- H- o n Ha Nada

H- H- H H H we

I I q I
0 0 m H N m mm «NM N» «x
H
H H H H H 4 mm
soHuwpsomoamom mHmom

zoo om mom mom m mmoHo

 

 

*.o moonw oano on» no oHnmu nopomaono .m oHooB

21

tation 1—2 8 . This means that under spatial opera-
tions which preserve the symmetry of the full cubic group,
these basis functions do not mix but transform only into
linear combinations of the members belonging to their own
particular basis. For small cubic crystal fields, the
five-fold degenerate atomic d-levels would be split into a

doubly degenerate E- level and triply degenerate T 'level

8 28
respectively. Stronger cubic fields (as in a crystal) will
further split these levels in every atom of the crystal thus
forming two distinct bands. These bands are in accord with
Fig. 2 showing electronic structures within the BCC primia
tive cellen.

These structures as shown in.Fig. 2 can be inferred
from examination of the angular dependence of X2, Y2, and

zx for the T28 case or 12-Y2 and 323-32 for the E8 case pro-

vided that the wavefunctions are formed from proper linear
combinations of the basis functions sc.as to preserve cubic
symmetry. Such wavefunctions will be referred to as hy-

bridized T or hybridized E respectivelyzn. Since, in

8

both groups V and VI,there are insufficient valence elec-

28

trons to fill both T28 and BS states, it is natural to de-

termine which of these bands would lie at the lowest energy
and hence begin filling first. Because it is generally true

that nearest neighbor correlations are stronger than next

22

.hhumfifihm mzoum.~mmv 0N9 mnw>oa moonuoon How HHoo

T>HuHEHHm .U.U

 

.m on» :HnuH3 huHmoop conuome Enfiflxmfi mo mmoH<

am am».

 

.m Tasman

Itux

 

(”0'

’fi

5‘?

pl.
1}-

A

25

nearest neighbors and so forth for more distant ions, then
the states representing correlations between nearest neigh-
bers might be expected to receive electrons at a lower en-

ergy than the next nearest neighbor correlated Eg states.

Although band structure data is seriously deficient for
groups V and VI transition metals, one can investigate the
band structure of one particular metal (tungsten) in hope
of getting a handle on the essential features of band
structures of chemically similar materials. Mattheiss25
preposes that the band structures of the BCC transition
metals (group V'and VI) should be quite similar to one
another with the Fermi energy being determined from the
total number of valence electrons. Upon examination of the
theoretical density of states curve as plotted by Mattheiss
for tungsten (Fig. 5), there appears to be four fairly dis-
tinct peaks with the lower three appearing quite similar.
Furthermore, it seems that for up to six valence electrons
per atom, the lower portion would provide a sufficient num-
ber of states to accomodate all of the valence electrons.
Although this problem will be reconsidered later inthe
discussion of the filling of the bands, it will suffice to

say now that the T states will be used to represent the

23
localized electrons and the E8 states can be disregarded

for the use of six or less valence electrons per atom.

....

2”

Proper tight binding wavefunctions representing T28 states

are 8

.. J —‘ 15 £520
A1553— ‘LW 26 tar-mm) 1

where the sum.over 5 is over the entire lattice and

49-400 ><xt=s->yz and AHA-Zx 2

The convergence factor }1(h) satisfies the normalization

‘9 2 .
condition that j f0) Ir‘J- :: l and is
best guessed at from the atomic d-electron radial wave-
26

functions or the band structure itself. If .F(V9 is
normalized as it is here, then these wavefunctions represent
a continuum of states in k-space and, hence, must be used
with an electronic density of states. Later it may become
necessary to modify the spin density of //<:(I)
to reflect the occurance of antiferromagnetiesgiructure in
this band. Such a modification can be included in the
spinor factor :><§I) ‘which may be set to vary spa-
tially from spin up to spin down with a period one half of
that of the lattice.

Representation of the conduction states should be much

simpler. For one thing, the density of these electrons
should not vary grossly within the unit cell since they are

25

free electron like and hence are most easily represented by
some variety of plane wave. Ordinary plane waves would
suffice but should require that a large number of thembe used
in order to cope with strongly varying potentials throughout
the crystal. On the other hand, augmented plane waves (APW)
are particularly advantageous in the sense that they are
‘most frequently used in band structure calculations making
it easier to apply the results of such calculations to this
particular investigation. Orthogonal plane waves (OPW) of-
fer a third alternative which should be most compatible with
using tight binding functions to represent the localized

states.

E's-i:
7434:) — \__i}____ _
V—Nzg</\k' (fi'e k"/\_k (_ )} XCr) 3

For g— _. 1, 2’ or 3 g A represents a localized d
state whereas other values of [‘ imply core states. Again,
it may become necessary to include a spatially dependent
spinor factor -;X<§I9 if antiferromagnetic effects‘
appear to become important.

Up until now, the important parameters nc and nd (the

numbers of conduction electrons and localized electrons per

atom respectively) have remained unspecified. Since the

0‘)

g

.51

‘42..

I:

26

localized T28 electrons are nearest neighbor correlated,

they should contribute substantially to lattice bonding.
The reason is that there exists a critical atomic separa-

h24

tion no as noted by Goodenoug such that when nearest

neighbor separations are less than Rc’ correlated electrons

T should exhibit a net bonding. Although it is usually

28
difficult to predict just how many the localized d-band
would accept for optimum bonding, some relevant informa-
tion can be gained through examination of the magnetic prop-
erties of the system. Slater27 has preposed the possibility
of spin correlations between nearest neighbors. Such cor-
relations would be antiferromagnetic if the band were half
or less filled and ferromagnetic if it were more than half
filled. nereover, any spin correlations would tend to split

the T band into a bonding and an antibonding part with

28
each capable of accepting three electrons. The bonding
states would naturally be lower and hence transition metals

with nearly half-filled T bands would be among the strong-

28
est structurally, i.e. groups V and VI. It is interesting

to note that the only ordered magnetic material in groups
'V and VI is chromium having a net antiferromagnetic moment
of O.# Bohr magnetron per atom. As pointed out by Jensen

and Suh128, even weak magnetic ordering can seriously dis-

27

rupt superconductivity and, although the coexistance of
ferromagnetism.and the superconducting state has been pre-
dicted29, it has yet to be observed. All of this means that
it may not be strictly valid to say that chromium is a poor
superconductor for the same reason as molybdenum and tung-
sten. more than likely the processes which tend to dis-
courage superconductivity in molybdenum and tungsten are
present in chromium so that it will still be instructive to
consider it along with the others. Slater27 seems to just-
ify such unique behavior of chromium by supposing that the
Fermi level of the itinerant electrons does not lie within

the gap of the split T band. Hence, no net reduction in

28
energy could be realized in the first place by splitting of

the T states since an equal number of electrons would

28
both raise and lower their energy upon appearance of the gap
producing no net energy change required for stabilization.
Additional light is shed on this problem by the density of
states calculation for tungsten (Fig. 5-). In this case it
is quite apparent that the T28 states are probably no more

than half-filled providing that there are approximately the
same number of itinerant electrons as localized ones. In
fact, it is important to observe at this point, if group V

had two T electrons per atom and group VI had three, then

2a
changing from a partially filled bonding half band to a

 

It».

ti:

28

full one may be just the mechanism that effects electron-
phonon coupling and the electronic density of states in
such a way as to diminish superconductivity in group VI.

As for the itinerant electrons, strong arguments
for three such electrons per atom are presented by
Goodenoughas. These range from observation of the for-
bidden soft x-ray transitions (indicating a strong admix-
ture of p-states in this band) to experiments in solid
sclubility. In any event, when it isassumed that there
are three itenerent electrons per atom in groups V and VI,

the localized T states cannot be more-than half filled.

28

Moreover, three electrons per atom would still result in
a more than adequate density of available itinerant con-
duction states to form a strongly superconducting band in

the case of group V.

CHAPTER III
THE GAP’EQDATIONS

The key to successful quantitative investigation of
superconducting phenomena is to realize that the energy
difference associated with the superconducting phase change

is miniscule (about 10"8

eV’ per conduction electron). With
the methods presently available it would be impossible to
predict superconductivity from a first principles calcula-
tion. Thus it is necessary to devise a model which treats
only the subtle interactions leading to this phenomenon.

To accomplish this one must already have at hand the cor-
rect qualitative picture which assumes the existence of
electron pairs4 whose momenta are correlated as the result
of a not attractive electron-phonon interaction36. Hence,
when writing down an interaction Hamiltonian, one must in-
clude only‘the mechanisms specifically leading to supercon-
ductivity whilst all other interaction terms can only be
included in the unperturbed Hamiltonian. An additional
consideration must be included. This is that the electron
pairs are "condensed" into the ground state (excited pairs
are ruled out in metals as having energies very much higher

than observed for excitations in superconductors). Thus,

in the unpaired remainder of the system being treated by

29

3O

many-body techniques, provision must be made for the effect
of pairs entering and leaving the condensed state.» Hares
over, in the unpaired system the total number of particles
will not be a quantum number but the average number for such
a system at equilibrium will, of course, be the expectation
value of the number Operator. This allows terms in the
Hhmdltonian which do not conserve the total number of par-
ticles, i.e. those operators that create or destroy one
electron pair.

Before applying the Green function technique, the
Nambu30 formalism will be specified. (This technique in-
volves essentially a matrix Green functionwhose off di-
agonal elements represent Gorkovllanomalous propagators
which transfer electron pairs in and out of the condensate
and has become the more or less standard form of notation
in the theory as it stands today. In order to find the
exact solution from Dyson's equation one uses the full
Hamiltonian which is: V
He 2 (Bloch energies of electrons) + z (energies of phonons)
+ z (interactions between electrons) + z (interactions be-

tween electrons and phonons) + (other terms, i.e. phonon-

phonon interactions, etc.)

31

Or, in the notation of second quanti tization

H: :L—Ek CbsC ks +Z’Kw‘l‘ Clara?"

i"

gs’fik‘ (CL awe—F 01*“) CE" Ck»:

+1161 '<|§+Ll\/Il‘>|k> Ck—L-SCIS -"C 3' WCk +'

where ‘ 6k are the Bloch energies relative to the Fermi

     

level and V is the screened coulomb interaction as a func-
tion of the distance between two electrons. using Nambu's
notation, define new field operators in terms of column and

row matricies .

w_ E“ SDI: Ct C

.k: C‘kil' 3 l; lgl jkl

where the; arrows T and ‘L denote spin up and down re-
spectively. Next using . 5-2 :(3 _i , express the
Hamiltonian in terms of these field operators noting that
the spin is handled automatically.

2

32‘

:;5:Cn:n=+C:C.kfi1)—§:ek fl“; gj+ZeK ,

,Zik‘( 1”“? kka5

kkscc
:gfl,( “W” ”)(CETCW‘CELCM)

:kZig ...(a-°y+% ) (fig %

kk’.

Z<l<4l<k+lMls>|k>CH3Cmgl@1st

2!; lg’s.s'l
-1Z<‘<-U<<+l|V|k>l 1&sz -L.f Clan CanI +
CH: («1+qu ’JCwCELCIJCk 11%;}qu
=;z<_k—L|<E+LIV\E>\k>(SI£lOE@fififi),

k,k'.L

33

or, neglecting the term Zék (unneeded in Dyson self

energy),

H =25 %+Ggqé+;€s@w%~) fig; 1%
2% 6% Z«+%gE§E-U<E+1 Iv mm; {Z M ,

The following development of the gap equations is

 

largely taken from the finite temperature approach given
by Schrieffer, Scalapino, and Wilkins12(ssw). The finite

temperature phonon propagator is

D: (02 1 T): "‘ <T{ C1314 T) 43;: (0)}> :1;

where the brackets <--- > denote the grand canonical aver-

age, T is the flick time jjfering operator)6
: T ‘i ' +
CP%‘(T):C H cp‘fum T: C ‘H’T(a.1¢+a_$e) ,
and 74,: 211'th . In the Na‘mbu scheme the
finite temperature electron propagator becomes
;) h

Gash): —; <Tcn‘7’cm‘°’> {ENC-$05
<TC3kf”C§f°>> <T CZ”) C. gt»)

C‘erfl) Fag) a. _Wfl
‘ ,; (WET) (MM) =26 C(kM)

wh = ( 2 h +1) 11/? whack“): e eHTCk(o)e- 11—17.

where

It can be seen that G(kfl’) accounts for the conden-

11

sate with anomalous propagators 1"." and F which create

and destroy electron pairs respectively. Dyson's equa-

tion is

k.ew.)=[G§1(k.iwn)-Z(kww~)]_l

G;‘(|s.iwv~)=‘wn 1—51; 0’

The problem, then, is to determine the self energy

E (Kiwn) . Illustrated with Feynman diagrams, the
self energy is shown in Fig. 3. Using the diagram rules

given in Mattuck35, the expression (containing terms up
to and including Hartree-Fock) for the self energy is

X(kwiwh)=:}9- 20—2— GQS’: ‘ wn') 6‘ x

          

E n'
.(IS-L L’stwh_ Lwh’)+\/(k.k]
where V(k:E) is the screened coulomb potential

assmned to be independent of bl)“ . The spectral repre-

sentation of the phonon propagator is

D.('12\‘7’r‘)=%[é MTDJETNT
___+2 e “‘ZTW (T) 42;. {08>} “r

3S

.>Hm>wuommmmu.mwmmwooum coconm can nfioasoo wumoflcwfl mmcaa
a>m3 can pounce .wmuoao «How on mcwusnfluusoo mfimummwo ucmuuomfiH

 

O
I
l
O
O
O
O
O
I
Q
Q
‘
‘\
.0
00-.....-

.m ousmflm

. as _
:éfCMT fl-éHTCPIJomiHqugawchT) 7>o
:. £1,3qu eiH‘T 475* (0)6"HT4>:I°°‘(08>4 T, Té 0
gamma "”e' H" o >‘e wane»
Z < I e :H I > . ’r‘>°
dz; Lnem‘rZ—QK—FHecHTfiAO) €-'iHT¢;,(o)\I>
" ' Z-.<¢I (MI) “‘9
ifevnTZ<k HE “323140 ($qu H¢L~(°)‘z>CIT
:2, Z&; I 6 PH I > . 7no

2! ’ev‘ZfiIEF r18H<I>Té°>l><le ¢1<0I>AT
* ‘ Z<WI€PHI> *9

f‘c} 14786 JE: +( E EJIZ‘ d)?‘ 0)“) <5‘ 493413“) cl TH”,
Z (ilee ’H|‘> ‘

2g. >2ngKHEVEJKII-(flw-(E‘EEI)
z Z;€"E‘ x

4|¢gé°4><| flaw» i> (PE

.3
z

_-L
“'J—l’z-

Nc
ov

P8

37

Define the spectral density

A (k _Iév)=(1;(_§:)ZJKJI 43:”; >LE86V‘E +3) ‘9

O where (1 =k_‘k/ .

k-IZ,£VA)=[°A¢(JS'EIV) 7—73.7— —‘ 3; $77-14? 10

Now substitute Eq. 10 into Eq. 7 and transform the sum
over h’ into an integral representation using the J

I residues of poles within “an. infinite contour C to pro-
Ject out terms of the infinite sum, i.e. integrating over
the contour will give infinite series of residues of the
‘poles contained within thecontour C (see Fig. ll).

Functions tanh —E-t2—— and (1+Ct‘921 , one has

the residues for values of F2‘ - o dd ¢ 1T

pz _Lt 2—ch _+c;_‘_“_'.__ 1
ResI1+ 6* ”mm: 73— 7;-

62/: 42/2(
RestahIILzz_ —Lm (Z- L 11:;
2"? (2-75“ 626 —6_‘ e

Now, making use of these residues, let LWhI—, Z to get

(tam-2%: gtfdtzIQIZIépzfaRt-m
Av+\fl<_k’)ttnkgjgz n

 

1
{(1 w..-z -‘VX1+€”}+( I Mr 2+ lWI‘HCin 1}}

38

m_x<
44mm

.m\: 660 um modem mfiom mcfi3onm wasam Xmamfiou ca musoucou

.v mhsmflm

 

 

 

 

 

 

39

,/l I
:When the contour is deformed to C , two additional

poles must be included whose residues are

RQSI Gaze) _ GIEIIwhtv)

W " mam-£17

:‘GIk: wni'V)_Cr(k.iwni’1/)
1+6 “M " 1- e“
m, usinsGIk’,w'+iO—G(k’w'-i8)=z «'lmGQéw')

and the fact that the contribution on the circle at in-

’fAAk E. I

 

finity vanishes, the self energy becomes
2 (kLLM-‘1472;loilmc(kw

, l 1 , _
{(fwh‘ w’ “ W)+(éwn-w’+ 10(1 +3 WBVJ'VIV-zgk ”4%?de

Q 1'; ‘V I". _ z z _’) . 12
OEGQ, m )1+F(ek- UM v’Bo-Igggtl Nk IsVM‘V

         

 

 

Letting i a)“ —-> a) +1 5 , the self energy can
be. expressed in terms of the real analytic variable 00
(see Schrieffers, P. 197). Its most general form is

:05» “0):{1’ZIkuw8w1+¢<KIw)fi9((Kw)q +437qu . 13

Since the Hamiltonian is invariant with respect to rota-
tions about 0':- in E space ( 0;? and
O; are not contained in H), then it is possible
to choose phases such that 4575, w): 0 . Util-
izing Eq; 6, Eq. 7, and Eq. 8'one finds the most general

#0

expression for the Green function is

G(kwI- Z( wII+e 90m 0— + 15’2wa 1.

‘ 0” Z (kwI-[euXIM +<I> (kw) '
This expression must then be substituted into the free
energy yielding a set of four coupled integral equations.

Several steps can be taken to simplify these equa-
tions. First, set the quantityé’flgflU) =€k+X(k)w)
and let AQSJU); CMQQ’VZ (kW) . Then
the following criteria can be used to reduce the algebra
considerably:

(l) X(k,w), being due to nomal coulomb pro-
cesses, is slowly varying for changes in (J) of the Debye
frequency and,hence, independent of CL) .

(2) ZLkJ») and (Pagan) are important

‘mainly at the Fermi level. Since they vary quite slowly
with k , one can set |k‘:‘<}_ .

(3) Since transition metal superconductors are us-
ually impure, it will suffice to determine the spherie’
cally averaged self energy because impurity scattering
reduces the effects of isotropy.

(4) The chief contribution to the sum overE is
when 6.2/3 )6 (DD . Following essentially the
technique of (SSW) the sum over E can be replaced by
continuous integration with the following definitions

It-
0

41

2:... 121%,ggg—TTI1‘3 [ale-:34 “k 15

where m dek= If dk’ . Since the chief contri-

’ I
bution to the sum over k is near 6k =5 ’R 0),; ,
the limits on the integral over Ek’ can be approx-

imated at + 00 respectively and, hence, by residues

fdéfl' C(19)) oériTrL—EUIZM’) 7)1_¢(w’:-] 16
k z szI‘Z‘Iw'F CM

with the one pole being above the real axis. By avera-

 

ging Eq. 12 over the spherical part of k (and remem-
bering that ((1).. —9 (Di-i5) , Eq. 15 and Eq. 16 can
be applied to yield for the phonon part of the self

energy at k 2k}

III r—f Ive—em
4521; [Ad (lg-Ky) 'ngch Zia-wLV+zJX1+e‘PV'T+¢U-I”'*W‘XP e "55]

I I zIr—Ifi—I II I" I—t—EEI

 

 

 

 

42

where (Us: w+(—1):'V+ i 5 . Equation 17 can
be simplified with the transformation of the negative
part of the integral over to the positive portion. Then

:35“ w) :EEWkIWWFKWWQ
{Irv—If III III-III w

 

where the upper signs belong to factors multiplying the
0'2 component only, lower signs belong to factors

multiplying the I compOnent only, and

W9=M7flz W712i: )NlS-Is, twiwcs gm; 19

where the Fermi-Dirac distribution functionf-(w)

 

The coulomb self energy is

)=jg?\<)a‘_[g$ImEvg G(ls'w'kgjwkk'flanhgg'. ' 20

An approximation is developed for the 0'; component
of E c. in (SSW) and, hence, only the necessary re-
sults will be stated here.

#3
These are

the death-N(oflfifie £333 Ly Ia.»

where the "potential" Lax; is approximately
U = \é__ I
c’ 1. + N(O)\/C (.h (6%0 00,)

and is the average of the coulomb interaction over

 

 

the Fermi surface. This finally completes the develop-
ment of the finite temperature gap equations. It is
apparentmthat, even in reduced form, they are intractable
and, thus, some further approximation is needed to bring
forth a reasonably accurate set of solutions.

In practice, the gap equations can be selved by com-
puter for the zero temperature case. Finite temperature
properties can often then be accurately found from the
zero temperature solutions by rules obtained with exper-
imental evidence. From Eq. 18 and Eq. 20, the zero tem-

perature gap 1equations become.

AIwFawo Jaw’ RQ{V_TA_(_Z‘”_')_ }K( on ('w)

 

 

_N__(_o) AI (10') I
2“» U R {Maw

21

22

23

MI

and

[1" Zhui] (mi-[211‘ Rafi/“7%) \(_(w‘,w) 24
where Am”: (hwy/Z (10) and N(o) is

the electronic density of states at the Fermi level.

 

 

45

. .mump wusuosuum
Icahn Eouw soummgsu How moumum mo huwmcoc Hmowuouomna .m enamam

.mecmmo>m. >ommzu
- h. . n._ n._ 3. 3 ho no

 

 

 

 

 

 

 

 

 

 

 

 

 

O
3
_l
a
l. .v
8
mm
3 m
d
, o

E N.
W. m w.
0
H .

0. ON

NOLV 83d
SUBBOAH 83:! 83111.8 NldS

CHAPTER IV
THE ELECTRON-PHONGN COUPLING CONSTANT

Derivation of the finite temperature Eliashberg equa-
tions begins with the irreducible self energy

{III-é- ErfiGIk’MIo;{ZnggIzQII-I’I III +VI—IEI

30 Green

where G and D are the electron and phonon Nwmbu
functions respectively,‘v is the screened coulomb repul-
sion, and 3kg)!" is the screened electrOn-phonon coupling
constant. The most important and also most elusive quan-
tity to determine theoretically is 61%,; . This cone-"1
stant will be derived as rigorously as possible utilizing
Bardeen's self consistent calculation including exchange32
and other necessary modifications needed to tackle the
previously discussed model of groups V and VI transition
elements.

Consider the displacement operator of an ion at
point due to a lattice wave of wave vector db and po-

. .,, i" m 40.9.
Sfimzzefi'fi—(dewa “Ian e B!"

where gang» is the polarization unit vector for wave-

47

vector % and polarization index 06:1,2 D 3

The normal coordinates are given as

QT: :VZMwT‘ a“? (Iaj Tc

where “1‘ and 0.1:: are the creation and

 

 

annihilation operators for phonons of wave vector x and
polarization index DC . Thus the lattice displacement

operator at lattice point becomes

€13,"
SR m =2- 4‘er écrce (av: +£_T¢)+hc

Now assuming that \/Cr) is the external potential at

distance I: from the center of a single ion, the poten-
tial at point B due to displacement of the ion by the
amount 83 will be approximated as V“ “’83) .
Then the change in lattice potential at point I for a

wave or wave VQG‘COI‘ i and polarization index ‘3: 18
Z [Sfim' {-WII III]
M

where M is summed over the entire lattice. Substitut-

ing Eq. 3 into Eq. 4 gives

48

 

 

, ITEM
\/;=‘§1’ IANME 60?: e (QTId-Tt)§3awr'fim)
+hQ

This is the so-called. rigid ion model of Nordheim”. Let
- (Lin—BM)
\/(r-B_mI- ZM Q .
.L
Then,
eel! e‘IQj) Em

Iéf szI “I” “'“I‘ IEVI‘L-IIE E) + Inf
Since 2; Gui-L). Bm_ =ZN 6b$+§h

where GUN is the Ihth reciprocal lattice vector. Then

gym-1h
‘4' " 2mm I‘I“+“I‘Z‘4+II INT“ (

The total electron-phonon interaction Hamiltonian for

 

 

scattering of an electron from state k)$ to state
k S is Just the matrix element of VT, be-
tween states 195 and E,S multiplied by
CEIS Ck; which destroys an electron in state
.3 and recreates it in IS 3 . So one can then

write

\0

 

(A

49

He-Ph —gflE¢ CESCKS (a? ‘+a'"f‘ “I

letting

I<'-IS/=°_1_)+.Gh

glqgf‘i‘I/‘zwr—WIIs-EIQQILE (ISI Q. - )r

Subscripts i may be replaced by kk since q

 

         

is unique for a given kt; and the sum over reciprocal
lattice vectors is inferred in the sum over k _|_< be-
cause Gh is also unique for a given 13 E in

                

.umklapp processes“ . The best way to determine 5kg“
is to find
[VII k EL: <“v O h '9‘
-I j
which gives

MOSIS’Iec: §I§J§°‘V “T“ 3 .

To begin the Bardeen calculation it is necessary to

return to

10

H

1

12

13

 

5O

‘\€;é19VAEEEEQ&Eé;éste+UCfifIzéggfgliag;1f5C7IEEISIIIC 55

01‘

IgffI/z—N—IM—Zyze I”; TVI/(r-BwIJIIIC

‘ whose phonon number eigenvalue has been determined in

phonon occupation number space. Let ice“

upon a conduction electron state ”5)

V19- :{v‘m’ILeTQw +\[I"dlk-oo I»

where

\/‘“’— <k+<jp+gnIV Talk)

and

Vii? <k—4—g l\/_¢Ik>.

Doing the same to a

VJ Is>= Z{\/_

d- electro on state I_I<_>

(hd) (Md)

e IISIIIQ I’Vvl Ii

operate

49>}

u

H

16a

16b

7

, H

51

where

\/‘I-—a;:d):<.|<+ j+gnI\/¢1;)¢II£> 18a
\[_.f““)=<|_<-I-Gn Wang) ,

This represents the action upon a core electron by a

8b

H

 

lattice wave of wave vector i and polarization at .
To be self-consistent, one must also determine how the
electron states are perturbed by such a lattice wave.

For the screening effect, expand the perturbed

electronic states Ik» and |K>> in terms of
I I? no @1454anWQWIroIk-fig}

H

9a

H

IK>>IK>+XIA (K Is + I 334 Ing— Q} 9"
h “12" n ‘32 (.3 I IS “I

to first order where a, b, A and B are small pertur-
bation coefficients. These latter two equations, when
summed over all of the states of the band in question,
are the net response of that particular band to first
order due to the motion of the ions plus the screening
effect of the other band. Next, call the Hartree-Fock a

potential energy of the d plus conduction-electrons \4‘.
0
Operating with \é upon a perturbed state using

52

WII>>=EXIEII<l§IIIIE>IIS>>-<KI“/s|k>>IE>I
+€28K’){<L<’|*/RII:>|I>>-<KII/RlIs>>lK>J

V; |p>=e2XIKII<KI14II KIIKI‘IKII/III K>>I IQ
sflkImsms>-<IIVRIK>>II>I

20b

with the specification that the eigenvectors span spin
space as weall seHilb rst space and that the sums include
spin. Call the Harte -Fock potential energy in the per-

tu Mb dl th ”\4; .. n
I; Il_<>>= e ZEII<<IEIIRIK>>|D¥<<KPIIII§>>IE>>I
KZZIKKKIIfiIKIIIIsI-IKIVRIQIKII

III©=€ZXKI<<K|I IK>I§>IKI§IK>|K>§
MEEIII'IIII’IIKI-IEPIIK>IE>>}

53

is the effect upon a particular bandstate perturbed by
the bare ion and screened by the other non-core electrons.

Now substitute Eq. 19 into Eq. 21 and subtract Eq. 20 to

\; |I>>=(\I*-\;°I\k>>
i.ezwgag‘ghmwQIR'IQmTIMIIR‘IIWQ
4:;th (IE-1'94 R"Ik’>+I>oD-II’IIII€IR'IE‘I'EII‘II
{a}. New R“ III» +ayIIII<If III New
ITIIILTQIR‘IIIEIIYIIQIRIIIE-TQII
I‘IIKIIIIIKIIKIIGIR‘IK>+Ar‘KI<IIIIII-PEI
+B}IKhI<K’3gnI R'IK)+BTIKIII<KIR1IK‘TG'II‘D> a
-{A"ZD.(KII<K'+ iIGIR‘IDIKIIATIKIIIK'IR'IIQKIEI

+B}(KI<K-IGIR“I|5>|K>>+ 814mm bl rap—GI] “2

 

n5

5n

and an identical expression with K and k, a and A, and b
and B interchanged.

Equation 22 represents the change in potential energy
of an electron within a given band interacting with a bare
lattice wave of wavevector a and polarization as
when properly screened by all of the other electrons in
both bands.

It is now necessary to solve the Schrodinger equation
for the perturbation expansion coefficients. For the un-
perturbed lattice, Schrodinger's equation is

I—IOIDIVD IIgI =EOIISIIK> 23s

for the conduction electrons and

HoIK>+IAIKI=EOIKIIIs>

for the d-band electrons where \VL includes all of the
potential fields acting upon an electron within a particu-'
lar band in an undistorted crystal. Now, for a lattice

perturbed by a wave of wavevector Cb ‘ and polarization

0C , the Schrodinger equation ficomes

H. |k>+(\;+vg +\/,IIIg>> =EIIsIIIs» 2“

for the conduction electrons and

HOIK>>+(\;+\;.+\;IK>>=E<KIK>> as

55

for d-band electrons. Substitution of the perturbation
expansions Eq. 22 into equations Eq. 24 and then sub-
tracting Eq. 23 and including the assumption that

E(k) - Eo(k) and E(K) = EO(K).gives ‘

Ham—Lime»tetanus—T9}

+\/,, fia1.(kn)lk+fb+Q>>+bT(k"‘lbib-Cw}

+ :{Wm’lkjg+g,>+\[‘l"lk-i-Gn>}

+_ af‘lm \[mc [kRTZGQJerWfl

n: M ‘1“ +bgqu'fwdlk)*\4:d'k23-ZGS}]

+ ea;(k)lé}ll£n)<lf+i+§ran'1lk>+bj,(EMF-1&4 R"|k>
af(gn)<k’lfi*|1g+5+gp+br(lzni<léIR“ll§—T.G>]|D
-e‘ 231; )ka «E + lfilfi‘l|1§>+b*f«“)<k‘3§»'RAM‘D
+1¢r<m<k| Rik)lK+3+G>+LTAEA<HR1|D|FT9]
WWW .(KnKKwGJ R‘1K>+B§.(En)<K~i‘G~|R“\K>
who KIR‘HLGrG; + 81m (KI R‘1 “.6 are] k>
.eag ”WWW R-1\k>+B’;.(K;)<K-ea\r m
WKQQIR“|1<>\K+1+Gny(m<kIRWWWQ]

-=E(hh:{af(h>\h wows ug— W} .5

56

for conduction electrons and an identical expression with
K and-k, A and a, and B and b interchanged for the s --

d-i—ban‘d ‘ electrons. Now operate with bravectors

<k+c1§+GHI and (k—fi-Gn‘ upon Eq. 25a and
with bravectors <K+$+Qn I and <K‘%‘Gn‘

upon Eq. 25b to get four equations containing the expan-

sion coefficients where the remaining terms containing

H o and V0 in Eq. 25 are removed by sub-
tracting the Schrodinger equation containing them. These

equations are

0 AG? ’+ {E(k+3+Gh}-E(k)} qué M

+92 glinlaaefls'm} <k+1+GJ<E+1+§mI Rflb‘b
+ 8‘1..meth Kk’qgmlR“! E>lts>
+ meg’kaW‘cMQhKE‘R1|k+ib+§rv>|k>

+ LllkakfipGnlflélRam-Gab]

.92 ;( k’)[a’fr(lg’m)<k¢ fp+Q»l<E+1+Gm| WIDE
+ LBJEMKLGEQl<E—i-Q.|Rilk>l_|<’>
+ aidk'ka'TQrKk’l R‘Wkflk’tycé

57

5%. (km)<k+1+G_ul<k’|R‘1lk>|E- $199]
+6 ;(K')[A}(Kn)<k+3§nl<K+TQ~l R‘l K> I19

+81} (Km\<k+oD+QnJ<Kf‘1-leFWD
+A1n(Km)<kq+Qul<Klfi“lK*i‘QQHQ
+314 Km)<k+jg+Q4<1§|R“lE‘i‘G~>lk>]
+8 gm [A34 Km1<k+ TQJ<K+1+QMIWIJS>IK>
+B}i|@m\<k+i+9~l<§w+gmIR‘MQ
iAmevnKbyQ-JQIR“|K>\K+$+Q»>

+ch(_K;n)<k+1QnKkl RAM) | K‘CD‘QQJ 27

 

N!

58

and, again, a similar expression for conduction electrons.
'The expressions for '31.: and Bid are identical
with sq. 26 and Eq. 27 respectively except that + + G.“
becomes - Ci- G“ and “Tr becomes ’09.“ .
Moreover, they contain no additional information. Equa-
tions 26 and 27 may be rearranged to give an explicit
expression for the coefficient multiplying the energy
difference in these equations. The perturbation co-

efficients are

aTmrr/‘(ManL/{EmMam
WWM (k. k- ,ch E(k\-E(k- «3.;- an}

AM M<K.K-i'G0~/iE(K)—Et+i+d>}

WM K K— swam- 19.)}

Now substitute these expressions for the coefficients into
Eq. 26 and Eq. 27 and solve. The result is

8

'0

MLK.l<_+<j1+9n)a=V411 +6 BK)

M

FKKKFEQKGUKWQ KK+°I+QMIRISIOIK>
—(K+§+Q,.KK+TQM "31'9““

MKK K ”i GmIa 1
WORK?" G JKKW‘QRK KleIfi KKK)

-<IS+TGnI<K'- l‘CwIR IK>IK>I

M +
Lug qug +(3°Q(<IS*1I*G~I<KIR ‘IK at.» up

-<I_<+R*G~I<KIR IK>IK+i+G»I)

mK Kira.» J, . ’_ ..
El“ ) HK m QI<I§i+QI<KRIKi Glue
—<Ig+rQI<KI R IK>K- $39)]

+€Z’)

WK r9 |<k+‘I*G»IR ‘Ik’>llg>

—(K+$G»Kk+i+G~JR IIQID)

6O

+WKETQKK’MM lfiflp‘m
‘<K*1*Gn'<|<’ i‘GmlR“IK>II£>)
KEW‘KKI’K GI<KIRIK+M>IK>
~0é+l+§nI<IéIR“IK)IK'+-i+g»))
MW WKR IK-K-sIK)
~<K+K+GI<KIR IR‘IKRK Key] 32

for d electrons coupled with an identical expression for
conduction electrons with k and K, a and A and b and B
interchanged. Before writing down an expression for the

electron-phonon coupling constant, the equations

( c iifim .- :
Vi: l“ *1/sz M KKZQ é KKKKQIVVKKKKK K. K
M

 

 

\[4T3m-gv €16“??qu m C'JVWI‘ R m“k\+hc 16b

“Win ‘3‘5‘ K - +I.
VIT-é {WOQZQ éfk (EGAIVWI 3.0K) 0 8

61

 

\[iid’ %VZNMW Xe 322:“ (LC-1 SJVV‘E’BQIKHC 18b

can be simplified. First, Bloch’s theorem states that

Wk(£+3m)=€ k8“ /%(r)‘
(K lVWK- -K...)I K) =f’tlfcrfl7v7r-Bmmmctr

#fiikcme.ém"v WDEQMEEJQihflgt-R‘)
_____é\‘E‘k“5me:—m)VWBJ%(bBMW»)

zetw-flfiv'fwfivvtwgu) A (I)

- k EDBWAVWK) )lK’> ’3

62
Next, multiply by Q‘a‘BM and sum over all
lattice sites

a . (G‘U‘B
:3 ‘b Ewe <1lvv<r)lk’>=<k|V\/(K)IK+<1+G>N 3*

Equation 34 can now be used in Eq. 16 and Eq. l8 andthese
equations in turn are substituted into Eq. 32 and Eq. 33.
using Eq. 13, the expression for the electronéphonon coup-
ling constant is

EKK 3a.,«=‘\/2§_—'EKTY€,¢'<H.V\/K ) l Kw Q}
e2 ;(K)x
[{&W(Q+1@KI<E+1GKIR‘HD\B
'<k+‘K+QKKK’+1+QKlR"IK>|K’>B
*{gfiwmwgfllRWK+1+G>1K>
“<k*‘1)+G..Kk’|R“IK>ll§+°K-+GK>)} \

 

63

‘{( gkfiékai QMQ+TQMKJS 32 GMIFflIOID
—<K+<)+G.J<K’-i-G~|R 1|K>\K’>)}

*{fiWWanKKIR“|1K’-fl)-CK>IK>-

—(k+1+QKKKIR”1|k)|K'- K993]
+ 62 SK)"

mm KG KKKWKKK)
~(K+}*GKI<K'+GD+QKIR“|K>|K>)}

{(ng'i-KK-mKKK *GKKK IR"| K KKKK
~<K+1)*GKI<KIR‘IK>IK'+‘])+GKE

64
K;

K. W) (ISquQKKK-wAR'flDID

K

—(K+K+QK|<K-°K-GKIR“IK>\K )3

K‘KKKKK—a.) <k+l+9w\<K\R"\K-j-G)\K>
'<‘-<*°fi*G)|Q§|R"|K>\K’—<y§m>) ,5

for conduction electrons coupled with a similar expression

     

for the-Ku‘d-Kbandm electrons. Only the first term in Eq.
35 represents the unscreened electron-phonon_coupling con-
stant whereas the remaining terms under the sum are en-
tirely due to screening.

The problem now is to determine the bare ion poten-
tial \/fikfl . This potential may be broken down into two
parts, one of which is the nuclear coulomb potential and
the other being due to the screening effect (including ex-
change) of the core electrons. Since the core electrons
all lie within closed shells, their potential is spheri-
cally symmetric about the nucleus and hence the vector
notation (r) may be dropped. Since five or six atomic

electrons enter band states in the transition metals in

65

question, then the remaining core will be quite small and
should resemble their atomic counterparts quite closely.
In fact the use of atomic potentials to represent core po-
tentials may very well be a better approximation for tran-
sition metals than for normal metals. The calculation of
these potentials is largely unnecessary due to the efforts
of Herman and Skillmanas. Essentially, they have deter-
mined a set of self-consistent Hartree-Fock potentials for
neutral atomic elements. It is from these numerical poten-
tials and radial densities that the necessary core poten-
tials must be determined.

Up until now the possibility of phonon induced inter-

band transitions has not been considered. ‘The reason so is

that such transition probabilities would be proportional to
matrix elements of the type <L<lVVYHIk+$+Qn> and
<k|VV(r)|K1-$+Qn> . Since the potential V6.)

and hence its gradient is spherically symmetric, these
‘matrix elements can be non zero only if the bra and ket
contain at least one identical spherical harmonic. However,
in specifying the wavefunctions, such a possibility was
eliminated. Thus, for this particular model, interband

transitions are ruled out.

CHAPTER V
EVALUATIW OF MATRIX EIIEMENTS

The purpose of this section is to develop specific

expressions for matrix elements of the type

< 302’) l<B<r>| 143:!) |.g(r))|fi(:9>
and (EMIVVYHI 31m)

where WV) . is the gradient of the bare ion. potential.
The form of the wavefunctions will be linear combinations
of atomic orbitals and plane waves from which any ortho-

gonalized plane wave can be formed.
Part A - Fourier Coefficients of Wavefunctions

Consider a linear combination of atomic orbitals

MM 1 Z ik‘U ~
A v- ="'" e l (II-1’ ) 1
kL) W 3 4)" m I
where N is the total number of lattice points, r: is the

Jth lattice point, and 43 is a given atomic wavefunction
having quantum numbers n, l, and m respectively. The

Fourier transform is

m . tf’I Mm
CLG)-‘=éi}nsf€ A (NA: 2

66

67
Performing the integration, this coefficient then becomes

Mm '(Jr+f)r M ‘
Q, (E) firm—XE ’Iuiia) 3-

Mm Mm -
Ck (P):"_‘(q (ZHP JHIUD) 9r) 4

where
M if- I:
13:36,) =f E (PMMQ‘H’: « 5

Orthogonalized plane waves may now be treated using

COK‘E’izifmfeiP-t 4%“! ‘ 6,

aw “7. ‘fr Mk
0C8): 7211?); e- e

_techrt-r n mgr'nfik “Mr-133“

The integrations are then performed to give

.1 .
W V‘, %n a! (m
G(Ekg—Tjs he; [1 X1“?- “090 CEU”. 8

hca-i(-ohl

68

Likewise, proper linear combinations of d-‘states having

T6 symmetry possess the coefficients
#122 “2- -Z i
Cx y:"(E) Ck _ C4—— ) 9a
'5 V2"

 

 

 

 

n21 h2-1
CZEF-gfi r—CL- 3 9"
and C1121 “2-1
+ C }_ 9°
C7649 V‘é‘ .

To evaluate IhUM , let

(phlyir :RMU') (
and

€{E-r: 5‘77 Z(Z(“)+1)V:, 3‘.(PV‘)X(Q’) 11

('a o

10

Transform from the primed to unprimed coordinate system
letting 6? be the angle between 2' and Z. Then (see
EdmondsBB),

( , ~ '
Y.0(6’)=XX.M(9'¢) (fitm'fi! PM (COSGP) 12'
m':-(

 

 

69

Thus (

( (U-
12€5A=V91T Z} (2‘ .)1/+1 Lars”)! I'D.“ ((056?) 13

'34

«7.1,,

:]3/[R“ (HjtJPfly (e,¢)>’(a,¢)r Séheclegié 4'.

Performing the integrations over angles, the final result

 

 

‘ _ ml
1.?fim,)'-"47((:h1)foh)/P\ C c as o J R (r) 14M '_2 4 b- 11;

If the radial wavefunctions are assumed to have hydrogen

atom-like form, then

KW 2mm ‘ 15

(‘0

h ffe i f h e t on i
were/(istheectvemassoteelcrz, EM s
the state energy, and the coefficients, [3“ are set
to give the best fit to the currently accepted radial wave-

70

functions for the atomic state in question. Equation 15
can then be substituted into Eq. 14 to yield the required

1 MM (see appendix A for the first few co?
efficients). I

Part B - Evaluation of Screening Matrix Elements

A typical screening matrix element is of the form

M. E: {R (h’)'<l?.(r)|1/\r-t’l I ELK)» firm)

' “WV -
flEJEJBm Chimes my 1% 16

Letting Bat - r ' yields

M E3445 c1124 _g A}; as) C33.) 030;) cm ,7
i .. . “3+ - - 9 .

Performing the integrations over h and _13 gives

MES=L|.T]’(21T)73 foPZ DC *1”) f) 18

71

where ‘ ':):: LE;"J:;I .

Transforming the integrals to sums, the final result yields

Wing? CIYEJCXECaflPgE)Cam-E2 19
V 1:3: Pa

‘Part C - Evaluation of the Ionic Matrix Elements

The ionic matrix element is of the type

M 51 = ([203] VVMI BM)
Taking the Fourier transform gives

MEI-=1 Jfidaamcmyeflwm 6'3: 2,

Let My) be spherically symmetric and of the following

form

V(r):§z{Z €£'(1+h:anrh)+(Z—Zc)} 21

72

where Z0 is the number of electrons in the ion core and,
thus Z - Zc is the positive charge of the ions in the
metal. The coefficients ah and E are set to best repre-

sent the experimental ionic potential. Substituting the
gradient of Eq. 21 into Eq. 20 and transforming integrals

to sums yields

4 2
“£96 261(5)qu 22
3.12

fedfi Eh: ZC e "+‘°E‘(1 *‘W Z‘Z¢}4t

nwi

Letting (E;- B)'-t=|f1'f‘-l" case

and performing the integration over r gives

MEL-t _Z_____i|\\I/e;)7(2U XC1(P)C2(P)X

[SEA-E? 4- Eifgmkf 453-5.!) +

 

73

if

 

_L".

a— BI 3-13%?) [“3 "'5 5'; (945439] 3

In practice, an infinite number of coefficients
need not be used in HM and V—(H

Hewever, a sufficient number should be determined to in-

sure the desired accuracy.

CHAPTER ,VI
THEORETICAL INVESTIGATION or
SUPERCONDUCTING TANTALUM AND TUNGS‘I'EN

In an earlier discussion of group 'V and group VI
B.C.C. transition metals it was noted that these two similar
periodic groups had very contrasting properties as super-
conductors. Thus, it would be appropriate to investigate
both groups with the theoretical apparatus developed earlier
in hOpe of finding some reasonable explanation. Tantalum. I
and tungsten are chosen since experimental data is more
readily available for these two metals. The calculations
should emphasize differences between tantalum and tungsten
consistent with being kept as simple as possible.

The first step is to reduce the expression for the
electron-phonon coupling constant to a more soluble form.
For a single band, the real part of the electron-phonon

coupling constant is

albah-tQm-cz- $§§c éT (k IWFMK +14"ny

+LHrCZZ(k')—J—&k+ a“

7R

75

 

Efifllfi'cjfi"

{<k+c,+g,t<k’d+.a,IR*IE>uk>
’<k"}2*§~|0é+‘b+gml R"l k>IE>
+<k+j+G.I<KlR“IE+OD+Q»>Lb
~<k+y+§u<lzlwm>ag+cy$3

 

+{E 1
(aft—1,9,3}
{<k+}+el<k-j,-Q.IR"1K>\D

‘. +1? QmKK- VG”! R’llkflk’)
+<k+fi+§nl<fl R'il K'E‘GQIK)

_ergnlalfi“lk>|k"l'Q~>>}]

76
Where 0 is the volume of the unit cell and the sums are

over all occupied states. Substituting in the correct ex-

pressions for the matrix elements from chapter'v and using

C,<>r>--(N1’TW5M 10’,

d

 

(SEMVGM alga” wit él‘ <k'VW>H<_+:b*§.>
+L‘1TQZ;( — k’) Eliza/mtg“ X
Ban E10; wit"

{2: We )1 m we mks

 

 

 

s5?

-I(k+1,+G,\I(E)I(k}I(E+T§»D 5e.,£rmlk_k+é -GB

 

 

law-firm}

77

www- ,- am)1<k’)1(k)5%fi§mz

 

“ELYQJKJK’ GbI'QMka Rwémfiig‘kfg - 3‘;

 

Equation 3 represents a Fredholm integral equation of the
second kind. For the purpose of simplicity, the screened
coupling, é , will be assumed to depend only upon the
initial and final momentum at the Fermi energy. Since

the screening term is either less than or aboutthe_same
magnitude as the unscreened term, the relatively small f
error that has been introduced should not have a devastat-
ing effect upon the accuracy of the calculation. The form
of the actual wavefunctions, represented by the quantities
If~fl, will be approximated as in the free electron case.
In this simplification the quantitievaGZ) are set equal to
ol/e due to the fact that they are normalized to the unit
cell volume 0. Thus taking the average over polarization,

 

78 l
q:

the electron-phonon coupling constant is reduced to

~ 2 m2 - x
61;.kq’t6, [COS 6'5; + V game.

 

 

, 1 _LL 1 __ 1 1
2(k)E(k’+<yGh)-E(k)62 III—IF Mia-Eli

%=\IZNM W1) 516$ Erm+ +48%? Vi)

 

 

 

+§__8( ‘4' 4+1 ”1 . 3+
(514:2)31 [(E ‘01)) (E+“b) fl . 5

5‘0

fmhe factor 68 represents the number of core electrons in
tauntalum and tungsten and seven coefficients are sufficient

to» represent the ionic potential. Solving for the electron-
Dhcanon coupling constant explicitly and converting from

79

1M
1
"3......

summation to integration over the Fermi sphere, gives

 

 

 

5 C05 83 55
Isl-twig. 1+?- ea Ak I |+ 1 1
Erie) E'QIyGJ' ||.<- k’l' \Iuygn’léb

Where a factor of two is inserted before the integral to
include electrons of both spins. Using E(k)9!’h2l(2-/zm ,
the integrals are performed (Appendix B) for the. three terms
in the integrand. Briefly, for the first term one finds

 

ITGJZJI4E+5+QJ E(l_<’) '—

 

 

,a.§+"c§j?-{21*{W WE

which is the classical Bardeen31 result. The other two
terms are due to exchange and each yield identical results

which are equal to

80

43'? _i___
«Enamel-HE) (lk- a la)
__ (amafl J M '2'. > S

— 4 131+ Q, '{r\/ 1- COS 2(Sén‘1Ke72k

 

 

 

Where [0(E}) is the density of states at the Fermi
energy and Kc represents the correlation momentum.
Such momentum represents a.minimum scattering momentum
associated with a correlation hole about each electron.
Electrons at relatively large distances from each other
are effectively screened so small momentum transfers do
not take place between them. Substituting these integrals
into Eq. 6 yields

3m

‘3'

[1+ ;§B&in d {'21:- +

' '1
(8k:$§%|2)(“%fil%%)} IT:+Q,II<,(FV(E 9‘“ (Nil

for the electron-phonon coupling.

=00: 61)::ng

 

\O

81

An expression for the superconducting transition tem-
perature is derived using a technique similar to that used
by Carbotte and Dynesl3. The desired quantity is the coup-
ling times density of states function which is

ccz(w)F(w)= $52622 Biflg It, '%*§n|| 2%) "m

[141“sz

where the sum over modes of vibration is implied. A com-
puter program developed by Gilat and Raubenheimer39 pro-

vides

RWz‘i'j—EL Zen? SW14” w) n

in histogram form for the phonon densityof states F(m)Am .
Comparing Eq. 11 with Eq. 10, one finds that, for each value
of the phonon wavevector q, the proper_value of c2(mo must
be computed for all possible umklapp processes and then
multiplied by the number of states placed in each histogram
channel for that particular q. Thus, the histogram will
then reflect a2(co) F(m) Am for channel width Am. The coup-
ling factor for phonon wavevector q and the nth umklapp

82

process is then

 

EPH’I‘IIQIEC—LE} l we“ 12

A computer subroutine, EPH, which calculates Eq. 12 is
added to the Gilat and Raubenheimer program (see Appendix
C). Another subroutine, SWEEP, which constructs the phonon
density of states histogram, is modified to find the coup-
ling factor for all umklapp processes involving phonon wave-
vector q. In the same subroutine, the number of states
placed into each histogram channel, F(m)AA(n, is then multi-
plied by EPH for all umklapp processes to construct a new
histogram containing c2(m) _F(co) Mm) in each channel.

The main program.then calculates

A: 2[é‘ma(wlf(w) AI» 13

 

for comparison with McMillan15 together with the transition

83

temperature using

€‘[,\3'_}%L(tig.ié+2>.\))\) }

T: 365

for strong coupled superconductors. The value of )3 repre-
sents effects of electron-electron repulsion whose values
are taken from ncMillan. Plots of both F(m0 Am and
c.2(co) F(ai) 4(a) are constructed and shown in Tables 6
through 9.

Two crucial quantities are Kf and Ké. Upon examina-

tion of calculations for Kc for sodiumuo, one can reason

that,for the greatly increased density of electrons in a
transition.metal, Kc = Kf should be a fairly good approx-

imation. On the other hand Ff is freely varied about its

free electron value to see what results can occur. Since,
for the case of a real metal, Ff. changes from point to
point on the Fermi surface. Other parameters utilized by

the program.are the Debye temperature on, the inter-

atomic force constants, and the coefficients for the func-

tions representing the ionic potential. Such data is

84

Table 4. List of potential coefficients and force constants

for tantalum.

.—

 

 

Ionic Potential Coefficients Force Constants
and Density of States (dynes/cm)
‘mr 1J725xlo9 ,1 16983 AA
a1 -9.774x10‘2 31 11201
10
a2 3.894x10 ,2 1182
a3 -4.77ox102° 32 1u83
a, 3 .oo8xlo30 "3 3546
a5 -1.069x10”'O a -5#27
he 3
a6 2.227x10 Y} 1943
a7 —2.698x1058 ,4 3577
an -718
n(er) 1.52 x1034 ,4 983
”'5 '1’93
35 812
”6 -3705
86 134
"7 558
B7 -237
Y7 106
-683

 

85

Table 5. List of potential coefficients and force constants

for tungsten.

 

 

Ionic Potential Coefficients Force Constants

 

and Density of States (dynes/cm)

E 1.780x109 ,1 23000
a1 1.061x10”1 31 19200
a2 3.926x1010 ,2 #7300
a3 -4.795x102° 32 800
a, 3.023x1030 "3 3200
a5 -l.076x10“O B3 1100
a6 2.248x10u9 y} 4900
a7 -2.734x1058

o(E) 2.56 x1033

 

86

summarized in tables # and 5.

The final results include Figs. 8 and 9 of which Fig.
9 agrees quite well with an experimental investigation by_
L. Y. L. Shenhl. The dimensionless coupling parameter, A,
and the superconducting transition temperature stemming from

it show reasonable values which upon freely varying Kf, can

be made to agree with experimental values.

87

.ESHMNMM—m “Mam—ownmwvsosonm mo muflmsoo
o.m on 06 on o.~ o._
- u q d , q

 

 

 

 

.m ousmwm

 

 

0..

O.N

QM

O.¢

I.-zu 3,0: mm

88

OK

 

.soummcsu Hon mmumum sosonm mo husmsma

2.. so}...

.5 magmas

0d 0.0 03V QM O.N O.—
. sq I1 q . hflJ-I

 

 

-o._ H.
W
H
O
h.
.6
n1
-o.~ 2.

 

.Edasussu Hem moumum no wuwmcop mesa» madamsoo .m munmflm

.2223...

89

0.0 0.? 0.» 0.N 0..

-

 

 

Id I d

 

 

 

 

(093 Die Zl-O' “(and (m) as

90

OK

0.0

.soummssu How madman mo muflmsmv moawu mswamsou

.2 N.95...

0d

0.¢

0.m

0.N

 

 

 

 

 

  

  

 

u

-

 

0..

.m ousoqm

 

 

..0

«.0

mac .

v.0
m.o
ed

5.0

(cos bio 2|_OIX)("’).-! (")z"

CHAPTER‘VII
CONCLUSION

The objective of this research was to examine, theo-
retically, transition elements in order to determine
whether or not their superconductivity is due to a phonon‘
mediated interaction between electrons. The task was to
find an expression for electron-phonon coupling from an
appropriate model and, then, with the theoretical apparatus
used by Carbotte and Dynes13 for simple metals, find the
superconducting properties of the transition metal in
question. It may be recalled that all of the conduction
electrons were assumed to lie in a single band similar to
the free electron case. Although such simplifications
cannot be expected to yield exact expressions, the results
can, nevertheless, tell a great deal about the system at
hand.

First to be considered are the coupling times phonon
density of states curves for tantalum and tungsten (Figures
8 and 9). Comparing these two figures to the corresponding
phonon densities of states shows about twice the effective
coupling at lower (transverse) frequencies compared to that

at the highest (longitudinal) frequencies for both metals.

in the case of tantalum, the curves compare favorably with

91

92

Table 6. 1 vs K for tantalum and tungsten.

 

 

 

r
Tantalum Tungsten
Kf (x108) 1 Kf (x108) 1
1.601 32.8 0.75 0.405
2.0 124.34 0.8 0.342
2.1 5.414 0.9 0.266
3.3 2.358 1.0 0.213
5.5 2.207 1.4 0.183
4.0 1.933 1.779 0.164
5.0 l .041 2.0 0.154
6.0 0.671
Tc=4.48°K 0.69 Tc=0.012°K 0.29

 

 

93

experimental results obtained by L. Y. L. Shennl. Thus,

it appears that the qualitative features of the theoretical
screening are basically correct for tantalum. Moreover,
treating all conduction electrons as members of a single
band with no distinction for s-like or d-like states has
not produced any serious difficulties.

The curves (Figures 10 and 11) giving the dimension-
less coupling parameter, 1, versus the Fermi momentum Kf’
require deeper interpretation. For tantalum it is apparent
from the results that in the theory one must use a value of
K1. about twice as large as would be expected for an elec-
tron gas of similar density (1.6 x 108 qm-cm/sec) to
achieve the correct values of 1 (0.69) and, hence, Tc
(4.48OK). The situation with tungsten is somewhat reversed.
Here the theory requires a value of K? of about one half
that expected in an electron gas of similar density
(1.8 x 108 qm-cm/sec) to get the correct 1 (0.29) and To
(0.012OK). Thus, if the theory is essentially correct for
group V transition metals like tantalum, then what pr0per-
ties of group VI transition metals such as tungsten prove
to be incongruent with these calculations? It must be
remembered that all electrons were assumed to be capable
of pairing and no distinction was made between s-like and

d-like carriers. One such interpretation is that the

conduction electrons in tungsten do separate into both an

100.0

10.0

1.0

0.1

0.01

1.0

0.3

0.1

0.01

94

F

-l _ -'-

-----

 

 

 

 

1.0 2.0 3.0 ”.0 5.0 .0
Kr
Figure 10. A vs Kf for tantalum.
\
I
0.0 0.5 1.0 1.5 2.0
Kr _

Figure 11. A vs Kf for tungsten.

95

 

sélike and a d-like group. If only the more weakly coupled
srlike electrons could form Cooper pairs then both the
abnormally low value of Kf required in the theory and the
low superconducting transition temperature can be Justi-
fied. The assumption that the theory is correct for tanta-
lum is supported, in addition to agreement with the Shen
results, by reasoning that all of the conduction electrons
can form Cooper pairs and that strong coupling results from
the tendency of these electrons to form into a group of
d-like electrons strongly correlated to the positions of
the ions together with an itinerent s-like group.

In conclusion, a theoretical investigation of the
superconducting properties of transition metals using the
same phononrmediated theory successful for simple metals

hasfyielded credible results.

 

APPENDICES

APPENDIX A

EVALUATION or I "1‘“

(1) 1a Ocase

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v- -
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96

 

97

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(2) 1= lease

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i

 

 

APPENDIX B

EVALUATION OF INTEGRALS

Part- I Exchange Screening

[00:11) : 1+ km with EL” a_ function of

 

V E— scalar k'only
d \5 .
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A316 _1__ . ’
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100

 

101

 

 

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99

3 2
243372 C(Ekh (1- C0560 \

 

102
Finally,

L L’ _L __

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( WE

$2;ka P) (h (1- cos ZPLSCHWc/ZKY) )

Where the correlation hole, Kb is given by

3° 9"- .S (n-1(Kc\Z Kr)

 

103

Part- II NOrmal Screening

 

 

 

 

 

 

 

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C

LIST 01" REFERENCES

2.

3.
4.

5o

8.

9.

10.

11.

12.

13.

11".

15.

_Rev. 106,162 (1957);

LIST 01" REFERENCES

J. Bardeen, L. 11.0 and J. R. Schrieffer, Phys.
00p106.1175 (1957)..

J. G. Valatin, Nuevo Cimento I, 843 (1958).
N. N. Bogoliubov, Nuevo Gimento Z, 79lI (.1958).
L. N. Cooper, Phys. Rev. ‘___104, 1189 (1955). 1

J. R. Schrieffer, Theo of Sn erconductivit

(w. A. Benjamin, Inc., Eew YorE, I961), 12.15.
G.Rickayzen, Theo of Sn erconductiv1t (Interscience
Publishers, Inc. , Kev: Yorfi, 1955) £157.

J. J. Engelhardt, B. T. Matthias, G. W. webb, Science
_22. 191 (1967)

L. P. Gorkov, Zh. Eksperim. i Teor. Fiz. fig, 735%1358).
[English transl: Soviet. Phys.- JETP 1, 5 (19 ) .

Y. Nambu, Phys. Rev. 111, 648 (1960).

G. M. Eliashberg, Zh. Eksperim 1 Teor. Fiz. 38, 966
'513‘60 [English transl: Soviet Phys. - JE'I‘F 11, 696
1 O] .

A. B. Higdal, Zh. Ekaperim 1 Tear. Fize 1H38551958).
[English transl.: Soviet Phys.- JETP 7H 1 (19

D. J. Scalapino, J. R. Schrieffer, and J. w. Wilkins,
Phys. Rev. 148, 263 (1966). .

J. P. Carbotte and R. C. Dynes Phys. Rev. 1 2 476
(1968). ’ ' 'JL"

J. 0. Phillips and L. Kleinman, Phys. Rev. ll__6_, 287

,(1959).

w. L. McMillan, Phys. Rev. 161. 331 (1968).

139

l6 .
17.
18.
19.
20.
21.
22.
23.

.24.
25.
26.

27.
28.

29.

30.

31.
32.

140

{.916403'el and P. W. Anderson, Phys. Rev. 125, 1263
1 2 .

J. M. Ziman, Electrons end Phonons, (Oxford University
Press, London, IEU), P. I82.

B. Matthias, Pro ress in Low Tem erature Ph sics,
(North-Holland PEBIIshIng 60., er am, , Vol. 2.

w. M. Lomer, Proc. Phys. Soc. 8_0, 489 (1960). '

N. l". Mott and H. Jones, The Theo and Pr erties 211%
Metals and A110 s Dover Pfi'fil ca ons, Inc. , ew or ,
T933). ‘11- 3.9-

%.ggondo, Pros. of Theoretical Physics (Japan) 22, 1
1 3 . ' -

B. T. Geilikman, Usp F12.,Nauk. 88, 327 (1966) [English
trsnsl: Soviet Phys. -‘ UBPEKRI E: 142 (1966) .

M. Tinkham Grou Theo and uantum Mechanics
(McGraw-Hiil EOE o., ew YorE, 1954), P. 329:

J. B. Goodenough, Phys. Rev. 120, 67 (1960).

L. P. Mattheiss, Phys. Rev. 132A, 1893 (1965).

P. Hennsn and S. Skillman, Atomic Structure Calcula-
tions,(Prentice-Hsll, IncI, EfigIewooa CIITfs, New
Jersey. 1963)- .

J. c. Slater, Phys. Rev. 82, 538 (1951).

M. A. Jensen and H. Suhl, netism, ('G. T. Redo and
H. Suhl, Academic Press, Inc., ew ork, 1966),
Vol. II B, P. 183. .

L. P. Gorkov and A. I. Rusinov, Zh. Eksperim i Teor.
P12. 46, 1363 1964;. [English transl.: Soviet Phys.
- JEEP-lg, 922 1964 J.

Y. Nambu, Phys. Rev. 117, 648 (1960).

J. Bardeen, Phys. Rev. 22, 688 (1937).

M. Bailyn, Phys. Rev. _l_l_'z_, 974 (1960).

55.

34- '
' Benjamin, New York, 1964), P. 234.

35-

37-

39-

40.
41.
42.
43.

141

L. Nordheim, Ann. Phys. 9, 607 (1931).
D. Pines, Elementary Excitations in Solids (w. A.

R. D. Mattuck, A Guide _t_9_ mDiagrams in the
j[Maggy-Bod 'Problem (Mcfiraw- k 50., London,
)‘151. . 5r -

Electrons distort a small region of the lattice about
their position. It is this virtual phonon cloud which
is felt by another electron that forms the attractive
mechanism. Normal coulomb repulsion is, of course,
still present.

The reason that g. is indicated instead of g' is that
the screened electron-phonon coupling constant g will
be sought self-consistently instead of the bare ion

coupling g.

A. R. Edmonds, ular Momentum _i_n_ Quantum Mechanics
(grsiincetonsgnivers y Press, Princeton, New Jersey,
’ Po 0

((1 ggélatand L. J. Raubenheimer, Phys. Rev. 144, 590

D. Pines, Phys. Rev. 9_2_, 626, (1953).
L. Y. L..Shen, Phys. Rev. Letters 24, 1104 (1970).
A. n. 3. Woods, Phys. Rev. 12, A781 (1964). ”

s. H. Chen and a. N. Brockhouse, Solid State Communi-
cations g, 73 (1964).

136

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. aqcc.c.czcc.azcc.z:u xa405cu

u

LIST OF REFERENCES

2.

3.
4.

50

9.

10.

11.

12.

13.

14.

15.

LIST OF REFERENCES

J. Bardeen, L. N.C and J. R. Schrieffer, Phys.
,Rcv. 106, 162 (1957). 00p108, 1175 (1957).

J. G. Valatin, Nuevo Cimento Z, 843 (1958).
N. N. Bogoliubov, Nuevo Cimento '_7_, 794 (1958).

J. R. Schrieffer, flea Su erconductivit
(W. A. Benjamin, Inc., flew: or ,

G.Rickayzen, Theor of Sn erconductivit (Interscience
Publishers, Inc., ew YorE, 1965), P. 157. '

J. J. Engelhardt, B. T. Matthias, G. W. Webb, Science
._25. 191 (1967)

L. P. Gorkov, Zh. Eksperim. i Teor. Fiz. g, 731.31% 358).
[English transl: Soviet Phys.- JETP 1, 5 (1)

Y. Nambu, Phys. Rev.111, 648 (1960).

G. M. Eliashberg, Zh.: Eksperim 1 Teor. Fiz. 38,966
3'3ng [English transl: Soviet Phys. -JE'1"P l_;l._, 696
1 O] .

A. B. Migdal, Zh. Eksperim i Teor. 1712.211 1438531958).
[English transl.: Soviet Phys. - JETP 7% 1 (19

D. J. Scalapino, J. R. Schrieffer, and J. W. Wilkins,
Phys. Rev. 14__8_, 263 (.1966) ,

J. P. Carbotte and R. C. Dynes Phys. Rev. 1 2 476
(1968). ' ' ”'7'”

J. C. Phillips and L. Kleinman, Phys. Rev. l_l_6, 287

((1959)-

W. L. McMillan, Phys. Rev. £61, 331 (1968).

139

l6 .
17.
18.
19.
20.
21.
22.
23.

.24.
25.
26.

27.
28.

29.

30.

31.
32.

140

{.9goziel and P. W. Anderson, Phys. Rev. 125, 1263
1 2 .

J. M. Ziman, Electrons and Phonons, (Oxford University
Press, London, 1965), P. I82.

B. Matthias, W Low Tem erature sics,
(North-Hollan 3 ng 60., fiferaam, , V01. 2.
w. n. Lomer, Proc. Phys. Soc. _8_9_, 489 (1960). '

N. F. Mott and H. Jones, The Theo and Pro erties of
Metals and A110 3 (Dover 'PEEl ca ons, Inc. , ew 051?,
195-3). P. 39-

.‘(I.9g0r)1do, Prog. of Theoretical Physics- (Japan) 22, 1
1 3 . ‘ -

B. T. Geilikman Usp Fiz..Nauk. 88, 327 (1966 [English
transl: Soviet’Phys. -'- USPEKHI E: 142 (1966)).

M. Tinkham Grou Theo and uantum Mechanics
(McGraw-Hiil 50E 0., ew YorF, 1964), P. 329:

J. B. Goodenough, Phys. Rev. 120, 67 (1960).

L. P. Mattheiss, Phys. Rev. 132A, 1893 (1965).

P. Herman and S. Skillman, Atomic Structure Calcula-
tions,(Prentice-Ha11, Inc. , EngIest, New
Jersey, 1963). .

J. C. Slater, Phys. Rev. 82, 538 (1951).

M. A. Jensen and H. Suhl, netism, (a. 'r. Rado and
H. Suhl, Academic Press, Inc., ew orlc, 1966),
V01. II B, P. 183. -

L. P. Gorkov and A. I. Rusinov, Zh. Eksperim i Teor.
Fiz. 46, 1363 1964;. [English transl.: Soviet Phys.
- JETP-Ig, 922 1964 1.

Y. Nambu, Phys. Rev. 111, 648 (1960).

J. Bardeen, Phys. Rev. 53, 688 (1937).

M. Bailyn, Phys. Rev. 111, 974 (1960).

33-

34.7
' Benjamin, New York, 1964), P. 234.

35-

37-

39.

4o.
41.
42.
43.

141

 

L. Nordheim, Ann. Phys. 9, 607 (1951).
D. Pines, Elementary Excitations in Solids (w. A.

a. n. Mattuck, A Guide _tg Fe¥gan Diagrams in the
w-Body' Problem (NcGraw- k 60., Lo'r'fion,
)3 e 5:; e '

Electrons distort a small region of the lattice about
their position. It is this virtual phonon cloud which
is felt by another electron that forms the attractive
mechanism. Normal coulomb repulsion is, of course,
still present. '

The reason that g' is indicated instead of g' is that
the screened electron-phonon coupling constant g will
be sought self-consistently instead of the bare ion

coupling g.

A. R. Edmonds, ular M0mentum.in.Quantum Mechanics
(Princeton Uhivers y Press,PFinceEon, NewJersey,
1957). P- 5")-

?.9gglat and L. J. Raubenheimer, Phys. Rev. 144, 390
1 .

D. Pines, Phys. Rev. fig, 626, (1953).
L. Y. L. Shen, Phys. Rev. Letters 24, 1104 (1970).
A. n. S. Woods, Phys. Rev. 15, A781 (1964). '

S. H. Chen and s. N. Brockhouse, Solid State Communi-
cations g, 73 (1964).

     

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