University of Nigeria Research Publications

ONWUAGBA, Beniah Ndudim

Aut

hor

PG/Ph.D/80/851

Title

Local Field Correction and Short-Range Forces in

the Lattice Dynamics of d-Band Metals

Facu

lty

Physical Sciences

Dep

artm

ent

Physics and Astronomy

Dat

e

May, 1984

Sign

atur

e

LOCAL FIELD CORRECTION AND SHORT-RANGE FORCES

M THE LATTICE DYmAMICS OF +BAND METALS

RENIAH NDUDN ONWUAGBA

B.Sc., M. Phil' . Physics (~i~eria). r~l~h.~/80/851

A Thesis submitted to the Department of Physics m d

Astronomy in the Faculty of Physical Sciences in

fulfilment of the requirements for the degree of

OF THE

UiE53RSITY OF NIGERIA, MSUKKA.

I ' .. .i

MAY 1984

This i s t o certify that the work described

in th i s T F r - i s was carried out by

in the Department of Physics a& Astronorq,

'Jniveraity of Nigeria, Nsukka.

/-- .- d

0

A' '\

1 wish t o express my thanks t o Prof. A.O.E. Animalu

for h i s guhiance, advice and encouragaent in carrying out

t h i s work.

I would l i k e t o express my gratitude t o h i . B.A. O l i

for stimulating discussions. I am very graterul t o Dr. S.

Pal who took over the superviaion of my work from Prof.

A.O.E. flnimalu.

I gratefully acknowledge the financial assistance

promised by the Federal Republic of Nigeria.

Thanks t o my wife, Felicia Ngozi Onwuagba for her

patience and understanding during very trying periods.

Also, I would l ike t o e q r e s s my thanks t o my brothers

Nathaniel C. Onnagba and Christian I. Onwuagba for the i r

f inancieil support.

I a m grateful t o t h t entire s taff of the Computer , >

Centre, University 'ok' Nigeria, Nsuklra far providine me with

the necessasy fac i l5 t ics t o carry out my computations.

And f inal ly, thanks t o Mr. D.N. Ugwu for patiently typing

the manuscript, and Mr. P. ' ~eriie for p repar in~ my diagrams.

iii

Recent experildental and theoret ical studies o r t h e

electron-phonon in te rmt ion in Group VB &band metals vanadium,

niobium an? t an t~Jum which c r y s t d l i z e i n the body - centered

cubic phase, a c ~ i : t ha t t h e i r phonon spectra exhibit same soft

modes believed t o be due t o electronic band s t ructure e f fec ts

and loca l f i e l d corrections. These metals have received

considerable att.ention because of t n e occurrence of these soft

modes; t h e high superconducting t rans i t ion temperature

associeted with then! m d t h e i r compounds such ae vanadium

+in (v3 ~n); and metal insulator t rans i t ion i n vanadium

trioxi.de {V 0 ) m d vanadium n i t r ide (vN), 3 3

The purpose o t t h i s thes i s is t o formulate i n an

inhmmll-J consistent mmner the theory of the loca l field

corrections t o %he dynmical matrix fo r phonon dispersion

relat ion i n these d-band me tds and t o exhibit t h e appa:-ent

cancellation be.t;yew,. theee ooxect ions and repulsive short-

range forces, This w i l l be done i n t he t rans i t ion metal

model potent ia l (T?4MP) approximation of O l i and Animalu

(1916) which w i l l enable I us .. t o investigate t h e poss ib i l i ty of

i-epresent ing t h e loca l f i e l d correction associated with r-d

interactions as an a t t r ac t ive short-range interatomic force

having the same functional form i n r ea l space ae the Barn-Mayer

repulsion due t o the overlap of core electron wavef'unctions

centered on neighbouring ions.

In chapter 2 : we s h d 1 eive a gene~al introduction and

brief survey of the previous work in t h i s f ie ld. I n chapter

2, we sha l l present the m a t h e m a t i ~ ~ l tools based on the

formalism of l a t t i c e dynamics which w i l l de~ena on the

harr~onic and a d i a b ~ t i c qproxhat ions . I n the iy.Smmrk of

these approxbat ims, we sha l l characterize the interatomic

?orces arising from the direct short-rcige and l o n c r a n ~ e

i'orces, and the indirect ion-ion interaction virt t he

polarization f i e l d of the conduct ion electrons ( the electrcm-

@.xon interaction). I n chapter 3, we shaU use the pseudo-

p t e n t i s l approach t o describe t h e contribution f r o m the local

f ieie coTrection in the Transition-Metal Model Potential of

O l i and Anhalu (1976). Tnis w i l l enable us t~ investigate & ,, 4 ,, >.' 3 1s

the contribution t o t h e indirect interaction between ions

v i m the polarization f i e X of the vdence electron gas arising

from s-d interaction contribution t o interatomic pa i r

pot&tisl. In t h i s appYo~ch;' we sha l l adopt direct metho8

using actusl atomic potentials and charge clensittes via 8-8

interaction t o show how the a t t rac t ive short-range forces

V

associated with the loca l f i e ld correction may be d fined t o

have the same fXnctirma1 form in r ea l space as t he repulsive

short-range (~orn-Mayer) potential. In chapter 4, we sha.I.1

discuss the energy bands of trrmsition metals in order t o

display the 9c;-~.rrenoe of s-d hybridization and t o show haw

t h i s hybridization i s simulated in the framework of the

pseudopotential or model potential method. Logically, i n

order t o use the pseudopotential OY - d e l po te i~ t i a l derived

fron: the energy band theory for investigation of electron-

phonon interaction and other electron scattering processes,

it i s necesearj t o sake C ~ E ~ L F distinction between t he

V-matrix (used in Energy band theory) and the 'P-matrix

(required for scattering theory). Tkis dbtincfion will be

inc'rufied in c k p t e r 4. In chapter 5, we shall apply the

above theozy t o the specific examples of vanadium, niobium

ant? tantalum in order t o demonstrate how cancellation between

the local f ie ld4mrrea t ion md t he short-range Born-Eleyer

contribution t o these me-blA occur i n the phonon f'requmcies.

Ant1 i n chapter 6, we shal l obtain numerical results and

compsre them with expa~ipeti2al data. This w i l l be followed

by a summary and the atterldant conclwion.

- v i

TABLE OF CONITNS Page

CERTIFICATION . . . . . . i

ABSTRACT 0 .

TABLE OF CONI'm!Tf: . . . . . . iii

0. . v i

APPENDICES . . . . _ .. viii LIST OF FIGURES . . .. . ix

LIST OF TABLI:..S . . . . - 0 . x

1.1 Introduction . . . . - -. 1

1 . Obtlineof kesearchie+kodr~logy .. . 14

CKAPER. 2 : LrYiTICE DYNAMICS FORMALISM . . . . 17

2.5 Off-diagonal Ccmpozzent of Dielectric 14atris , , . . ' . C ' . I

. . 40

3.1 Introduction . . . . - a . 46

3.2 Pseudopotential Approach- -- . . . 47

3.3 Contribution from Local Field correction 52

3.4 Attractive Short-Range Forces via s-d interaction 5 5

vii Page

CHArmER 4: S-D HYBRIDIZATION IN ENERGY BAND AND mDEL RYlBlTIAL IN d-BAND METALS

Introduction . . . . - .. S-d Hybridization in Energy Band of Transition Metals

Resonance Model in d-band metals .. . . Generalization of the OPW-pseudopotential transfonnation to the d-band metals . . . . d-Band Model Potential . . . . Magnitude of the Orthogonalisation or Depletion Charge - . . . ,a.

The Theor, of T-riratrix Scattering . . .. T-matrix Form Factor . . . . Virtual Bound State Problem . . . .

CEK?TER 5 : APPLICATION TO WE PHONON FREQUENCIES OF d-BAND l@Tm 7-12

5.1 Introduction . . . . - .. 112

5.2 Demnstrz~ion of the Cancellation of Short-Range forces by the local field correction . . 114

5.3 Coulol~b;-c and Born-Mayer Contributions to the phonon frequencies . . ,. . . I . .:', .. . . - 0 . 120

5.4 Calculation of the Electronic Contribution and local Field Corrections to the Phonon frequencies . . 12 5

8 ) _ .- 6.1 Introduction . . . . - . . 137

6.2 Comparison of Calculate6 pilonor1 frequencies with experiment . . . . -_ . . 137

6.3 Sunnnary and Conclusion . . . . 152 References . . . . - .. 195 Biographical Note . . . . . . 200

APPENDICES

Page

A, Isnrmim of the Dielectric Matr'x . . .. 160

B. I.B.M, Computer Program for the Corngutation of the Born-Mayer Contributim . . .. 162

C. 1.B.M. Computer Progx rm for the computation of the Electronic Contribution and Total Phanon Frequencies excluding Local Field Correction . . 171

D. I .B.M. Computer h-ogrcm for the computation of thr? Total Phonon Fr~c~~encies including Local Field Correction. . . ..

LIST OF FIGURES - TITLE PAGE FIGURE

m i n t e r a c t i n g s and d bands . . . . Interacting and hybridized s and d bands for vanadium . . . . - .. Reduced Zone Scheme . . . . Extended Zone Scheme - . . . . Iiigh-lying resonant d-states situated well above the muffin-tin potential . . . . Comparison of the Heine and Abarenkov Model Potential % and the model wave function % with the true potentta.1 V and true wave function @.

Bo-and State . . . . . . Virtual EounG State - . . . . Ecc Brillouin Zone . . . . Phonon Dispersion Curves in Vanadium excluding calculated local f ie ld correction . . Phonon Pispersion Curves in Niobium excluding calculated local f ield correction . . P5.0non Dispersion Curves in Tantalum excluding cdaziated local f ie ld correction . . Phonon Dispersion Curves in Vanadium including calculated .local f ie ld correction . . Phmon Dispersion Curves in Niobium including calculated local f ield correction . . Phonon Dispersion Curves in Tantalum including calculated local f ield correction . .

TABLE

2.1 2.2

2.3 2.4

3.1 5.1 5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.13 6.1

6.2

6.3

6.4

6.5

6.6

LIST OF TABLES

TITLE

Repulsive (Born-Mayer) Contribution I I 19 I t

I I t 1 I 1

PAGE

a:' for vanadium 30 II I 1 niobium 31 ll II tantalum 32

Parameters for calculation of -1s ive (Born-Mayer) Potential . . . . . . 33

Attractive Born-Mayer Coefficients .. 62

Coulomb Frequencies w2 for bcc metals Electronic Contribution w i for vanadium Electronic Contribvtion 'I niobium

I t 19 " tantalum 131

Local Piold bmxtion wiC for vanadium I I t 1 1 I i 9 " niobium 19 I I I 1 I 1

" talialum Comparison of local field correction by long-range method w i t h our result by short-range method for vanadium . . . . . . 135 Model Potential Parameters . . . . 136

Parameters for Phanon Frequencies Calculation 136 Total Frequency excluding lacal field correction for 140 vanadhn . . . . . . - Total Frequency . I , I . . . . . + excJudkg I local field correction for niobium . . . . . . - 141

Tot a1 Frequency excluding local field correct ion for tantalum . . . . . . - 142

Total Frequency including local field correction for vanadium . . . . . . - 143

Total Frequnw kcludhg local field correction for niobium . . . . . . - 144

Total Frequency including local field correction for tmtalum . . . . . . 14 5

CHAPma 1 --- 1.1 rmTAODUCTION

The study of the subject-matter of l a t t i c e dynamics of

metals is of Fundamental importance for the investigation of

t he thermodynamics, e lectr ical and other physical properties

of metals. Lattice dynamics deals with small displacements

o r vibrations of atoms about t h e i r mean equilibrium position

in the crystal l a t t i ce . Lattice vibration involves a large

number of atoms interscting with one another through forces

which may be represented by springs connecting various

neighbouring atoms. n he quantized normal modes of vibration

are called phonons). The springs provide t he restoring

forces on the atoms when they are displaced from t h e i r

equilibrium positions. Born and Von-Karman (1912) used t h i s

model t o provide a mathematical theory of lattice vibrations,

and considerable development of the formalism of lattice

dynamics has since~theh'"e~6ltted based on various sophisticated

models of interatomic forces in the solids.

A general review of the l a t t i c e dynamics of solids can

be fomd in standard texts, such as Maradudin, Montroll, Weiss

and Ipatova (1971). For our present purpose, we shal l be interested

in the l a t t i c e dynamics of d-band metals, i n part icular the group

2

VB t ransi t ion metals vanadium, niobium and tantalum which

czystallize in the body-centpi-ed cxbic phase. These metsla

have receiv d considerable attention in recent years because

of the apparent connection between the occurrence of soft modes

in t h e i r phonon dispersion curves and the high superconducting

transi t ion temperature associated with them and their compounds

such as vanadium t i n (V ~ n ) and metal insulator t ransi t ion in 3

vanadium trioxide ( ~ ~ 0 ~ ) and vanadium ni t r ide (w). (weber

In the l a t t i c e dynamics of metals i n general, one has t o

deal separately with the metal ions, the valence o r conduction

electrons and the intersction between them. This is a

complicated tnany-body problem. To reduce it t o a t ractable form,

it is customary to i-make the adiabatic or Born-Oppenheimer

approximation, which makes use of the fac t tha t the ions move

so slovly compered with the conduction electrons tha t the system

can be decsupl.ed i n swh a way tha t the ions and the electrons , < , ."6. +' , '

can be treated separately tizl the lowest order while the

interaction betweem them i s fneluded as a perturbation. In t h i s

approximatior?, one can isolate the various terms tha t contribute , .- .*

t o the total. potenti51 eaergy for the motion of the ions, namely,

-the direct coulomb interaction and the exchange core (~orn-Mayer)

replsioa bct;mx ions, wid the inilircct ion-ion interaction via

3

the polarisation f i e l d of the conr lvr5c~ electrons (which i r

associated with the elettron-phonon interaction). fa t h i s

model of t b ~ interatomic potential energy, the frequencies of

l a t t i c e vibrations can be determined e i the r bg the force conrtsnt

method ( ~ o r n snd Huang 1954) or by the reciprooal latt ice method

(~ochran 1963), or by a combination of the two methods.

Generally, the force conatant method is adequate for short-range

forces while the reciprocal l a t t i c e method is required for

deeling with long range forrcee.

A careful investigation of the electm-phonon interaction

involves a detailed knowledge of the electronic energy baad

structure of the metal under consideration. In simple

(nontransition) metals, the energy bands ham the parabolic

shape of s-p bands and an ordinary free-electron d ie lec t r ic

screenkg theory is adequate f o r incorporating these band

structure effects i n the phonon dispersion relation. However,

i n the trru~aiLion netais a ,pe&r&~r free electron broad par8boLic .' , , , , A .I.?>. ,

s-band croc:ses and h y % r i d k s with a tight-binding nsrmw

d-band. For example, i n the 3d transition-series, the broad

s-band associated with tphc )+s atomic level crosses snd interacts 8 ) ._ _.

with the narrow 3d bandz, Tnis s-d hybridizing interaction and

the tight-binding character of the d-electrons give r i s e t o

l o c d :f.;cl.ci efI'%9-.s, vhic'n con?licat= the free-electron d ie lec t r ic

screening theory.

As is well known (Ekenreich and Cohen (1959). Alder (1962)

and Wiser (1963)), locd . f i e l d e f fec t s are associated with t h e

screening action of bound o r v i r t u a l bound electrons i n

c rys ta l l ine materials which involve the dynmicaJ. matrix f o r

phonon dispersion v i a the electron-phonon interact ion, o r what

i s the same thing, the interact ion between ions v i a t h e

polar isa t ion f i e l d of t h e conduction electrons. I n t h e l a t t i c e

dynamics of the t r ans i t i on m e t a l s characterized by the v i r t u a l

bound nature of t h e i r d-electrons, t he local f i e l d e f f ec t s

appear v ia t h e non-vanishing off-diagonKL components of t h e -1 A 2

inverse d i e l ec t r i c matrix E (g,gt) which determines the l i nea r

response of the conduction electrons t o a periodic c rys t a l

potent ia l i n a rn2tal.

Because of the c~mplexi ty of the l oca l f i e l d correction,

various approximate procedures have been developed f o r incor-

porating it i n the phonon spectrum calndekion in semiconductors ,<. . .*. ,,' , *

and t r ans i t i on metals, where such corrections are important.

Prakash m d Joshi (1970) in t h e i r calculation of t h e phonon

spectrum of nickel used non-interaction s and d-band scheme I, - .r

so t h a t only t he diagonal par t of t he d i e l ec t r i c matrix was

involved in the calculation. Van Vechten and Martin (1972) .A -A

approximated ~ ( ~ , g ' ) bv e finrite dimensiond (53 x 59) dimensional

5

matrix which was then invertod with the a id of the computer. In ord-

er of s i m p l i c i e t h i s conylicated prcblem, Hanke (1973a) showed

tha t by working with a tightbinding basis se t , only a f i n i t e

dimensional matrix need be inverted, the dimension of which is

determined by the number of overlapping tight-binding basis

h c t i o n s : he prcpoaed a factorization ansatz f o r the

d ie lec t r ic matrix which allows a complete inveq%dn procedure

leading t o a simple form of the loca l f i e l d corrections. Also

Hanke ( l973b) extended h is cFilcul~tions t o paramagnetic nickel

and palladium and included the diagonal and off-diagonal parts

of the d-d intraband t rens i t ions which give r i s e t o dipolar

model of l a t t i c e dynamics of t rans i t ion metals while

neglecting the diagonal and off-diagonal par ts of the d ie lec t r ic

m t r i x f o r d-s, s-d d-d interband transit ions.

Another factorization ansatz f o r the d i e l ec t r i c matrix was

proposed by Sinha e t al (1974) which was useful in the study

of l a t t i c e dynanics otl instdlators end se~jconductors. The

work in t h i s direction was fur ther extended by using 2nd order

Brilloxin-Wiper perturbation scheme by Bertoni e t al (1974).

The off-diapona elenents of the d ie lec t r ic matrix in the study

of l a t t i c e vibrations. and effective ion-ion potential , give

r i s e t o ncn-central forces between the ions znd sa t i s fy the 2

correct q + o l i m i t f o r thz l o ~ g i t u d i n d acoustic modes by

M f i l m e n t of acoustic sum rule. A few year l a t e r , Koichi and

Hisashi (1971) adopted the Lsedis t ic band models t o obtain the

microscopic d i e l ec t r i c clatrix for s i l i con and germanium i n the

random phase rzpproxima-i;ion (RPA) . Subsequently , Nizzoli (1977)

used a proper model Hmiltonian t o describe the covalent band

m d modified sg3 orDitals as basis s t a t e s t o evaluate the inverse

d ie lec t r ic tensor e i 0 ( d i n c rys ta l l ine t r igonal Se along

two symmetry directions. But recently, Sturm (1979) had shown

t h e generalization of the longitudinal d i e l ec t r i c function of

Ehrenreich and Cohen by Alder (1962) and Wiser (1963) to the

local f i e l d effect. Also, Inkson (1978), Littlewood (1979),

Van Camp e t al (1979), Oliveira and Inkson (1979) included

loca l f i e ld through the off-diagonal elements of the d ie lec t r ic a 4

matrix €(g,g1) and found the correction important i n the cubic

I V - V I compound, insulators and semiconductore.

I n t h i s thes i s we sha l l begin by using the pseudopotential

o r model pot^; t i a l me0'hOd''in ' t rea t in6 loca l f i e l d correction,

The model po t en t i a l method was first applied t o t rans i t ion

metals by A n ~ m l u (1973a), who called it the t rans i t ion metal

model potent ia l (TNMP) method beciuse it incorporates s-d

hybridizatior- h the resonmce model (Ziman (1965), Heine

(1907), Hubbard (19673 md hi-l;h e t z i l (1974). O l i and Animalu

(1976) achieved a i l inversion of the die lec t r ic matrh by applying

7

a pseudopotential transformation t o the t rue Bloch AvlcCions i n

the standard expression f o r t he d i e l ec t r i c metrix in t h e l i nea r

response theory. O l i (39%) extended t h i s t3eory t o thorium

and recently t o Zirconhn ( 0 l i 1981).

In practice, the evaluation of t he l oca l f i e l d correction

involves a complfca-ked t r i p l e s u m t i o n over an i n f i n i t e set

of reciprocal l a t t i c e vectors, and it is d i f f i c u l t t o aohieve

numerical r e su l t s consistent with the symmetry requirements

whenever t he summations are cut o f f after a f i n i t e number of

reciprocd. l a t t i c e vectors is included. To t h i s end, we wish

i n t h i s thes i s t o investigate the poss ib i l i ty of representing

the l oca l f i e ld correction associated with s-d in teract ions as

an a t t r ac t ive short-range i-nteratomic force having the same

functional fonn i n r e d space as the Born-Mayer repulsion due

t o t h e overlap of core-electron wavefunctions centered on

neighbouring ions. Onwuagba and Animalu (1981) have recently

reported t h ~ premble of t h i s theory. < ,< 4 - 6 >. , *

Experimentally, ine las t ic neutron s n ~ t t ering is currently

the most powerf'ul t oo l f o r the measurement of the phonon

dispersion re la t ion i n solids. 1t.h well known t h a t neutrons

are producec? by fissi.on of heavy nuclei , such as uranium-235.

Neutrons prod-uced by t h i s technique are f a s t and have a;n

average energy of abm% 2MeV. Hcwever, neutrons can be slowed

down by passing the beam through enough matter u n t i l the

6

beam i s i n thermal equilibrium with the molecular motions in

t he material. Then, such ~ e u t r o n s a re reearded ss slow o r

t h e m 1 neutrons with m- .~g ie s dk t r ibu ted about t he value of

O.025eV. The wave1eng:h of a neutron having t h i s energy is of

the order of 18, which means t ha t they a re l i a b l e t o diffraction

by crys ta l planes. However, neutrons serve as an excellent

probe f o r l a t t i c e dynamical studies of sol ids on account of the

f ac t t h a t neutrons in te rac t with both nuclei and magnetic, or

unpaired electrons, md t h a t slow neutrons have energies and

wavelengths t h a t match those of excitations and col lect ive

modes e.g. rotational energy leve ls i n molecules and acoustic

and optic phonons i n solids.

The most useful scat-bering processes are those involving

one phonon scat ter inz, Subject t o the ejcj~tence of coupling

mechanism, the s ingle ghonons produce s ignif icant sca t te r in8

and tile ~easurement of t h i s yields valuable and direct i n f o r

m s t ion on > k i l ~ ; l spegt~~rm.,, ,,The incoherent scat ter ing ~ i v e a

in format io~ fibout the frcqwncy distribti t ion functions, while

the one-phor,on coherent sccbtei-ing i s used for finding frequency

(v ) versus phuncn wave vector (G) dispersion relations. The , . .. one-phonon xllerent scat ter ing process o b e p energy and crys ta l

momentum conservation laws:

i i

KO - K' = q (1.2) -h

where Eo, E' a r e t he i n i t i d and f i n a l energies, c, Kt the

corresponding neutron wave vectors, v i s the frequency of a 9

4

l a t t i c e wave of wave vector q, m i s the neutron mass, and h n Q

is t he momentum transfer . Thus, f ron accurate measurement of .A 2

KO, K' and t h e energy lo s s of the scat tered neutron beam as a

function of t he sca t te r ing direction, t he phonon frequency (v)

versus phonon wave vector (i) dispersion r e l s t i on is determined.

Nakagawa and Woods (1963) applied t h e above technique using

the chalk River t r ip le-axis spectrometer i n i ts constant-Q mode

of operation rockho house 1960), f o r l a t t i c e waves t rave l l ing -1

along the symmetry direct ions $00.-j , '110 1 and. /'111] f o r ..I I. ... i

niobium. The i n i t i a l s l o ~ e s of t he dispersion curves a r e in

pood agreement with those calculated from the measured e l a s t i c

constants reported by Bolef (1961), and s t r i k i n ~ features

include : -;. ,,....;,.I: ,;. . , . , v i .'. .

( i ) The i 1 0 0 ~ longitudinal (L) and transverse (T) branches L . .

cross st (-$--)q = 0.7. The crossing over of the 2T

acoustical branches i n t h e absence of a synnetry

requirement i s p d c d & t o t h i s material.

( ii) me twa nondegenerate j'110 / T branches (designated by - T and T ~ ) in te rsec t a t (&)q = 0.3. 1

a ( i i i ) Anomalies occur in t h e 111 1 L branch a t (-i;;)q = 0.49

l

and pronounced chcnges in slope a t (*$)q = 0.47 and

f o r the T branch.

Analysis of t h e data on the basis of the Born-von Karman theory

(1912) indicates t ha t the effect ive interatomic forces are of

very long range nature, therefore, t he actual forces i n t h e

metal m e quite complic&tcd.

Bearing in mind the complicated nature of the interatomic

forces i n a metal, Woods (1964) determined the frequency-wave

vector dispersion re la t ion v(q) f o r l a t t i c e waves t rave l l ing i n

-1 ' -- - - I

the high symmetry directions (100 i, 110 1 a d 1 111 ; for tantalum . - .i -- L

at 296'~. The experiment was carr ied out using the t r i p l e

axis neutron spectrometer i n i ts constant Q mode of operation

m d with variable incident neutron e n e r a 0.0225eV. The

specimen consisted of three s ingle crysta ls each 5cm long and

aboxt l c m i n diameter. The (110) plane was perpendicular t o t he

cyl indr ical axis whcic4*,~as, aounted ver t ical ly . Each of t he

three crysta ls could be oriented indepencently and t h e i r

alignment was carr ied out by means of neutron deffraction. The

measured mosnic spreo,d of the (222) plane measured in t he I, - ..

pa ra l l e l posit ion against a germanium single c rys ta l was about

directions a r e obtained with many of the features s imilar t o

those previously observed f o r niobiwas. For example, the

I- - I , 100 i L and T branches a r e nbserved t o cross a t q = 0.7 and t h e -. -1

tendency f o r t h e [lo0 / m d [110 1 transverse branches t o deviation L

from norm1 dispersive behaviour is apparent. Hexever, these

fe3tures do not appear t o be a s pronounced i n tantslum as they

a re i n niobium. It should be noticed t h a t , unlike the case f o r

niobium, the r1103 T1 and T branches apparently do not L 2

in te rsec t i n tantalum. The reason f o r t h i s behaviour i s t h e t in

tantalum the e l a s t i c constant Cb4 is greater than (C -C )/2 11 12

whereas in niobium, Cb4 is l e s s than (cll-cl2)/2.

Colella and B a t t e m n ( 3 970) adopted n similar experimental

technique t o vanadium. Both authors measured t h e phonon

frequencies along the principal symmetry directions i n vanadium

by means of thermal diffuse-scattering of X-rays. The frequencies

corresponding t o a phonon wave vector are obtained from the

absolute intensi ty of the thermal sca t te r ing a t the appropriate

sac t te r ing vector. , , $ The . . t .. deqpil . exper imata l tzchnique employs

a proportional counter SPG-6 ( ~ e n e r a l ~ l e c t r i c ) operated a t

1870V, followed by a charge-sensitive preamplifier and a single-

channel analyser. The strong Wol vanadium fluoresence were reduced I / " A ..

t o a negligible l eve l with adequate f i l t e r s while t h e polishing

and annealing procedure t:as adopted t o eliminate sca t te r ing

associated with i m p p i t i e s . By t h i s technique, the dispersion

12

curves of vanad5um have been determined along the principal

symmetry directions I 100 ' j110 md 1 111 -1 . These dispersion 1-

curves have slopes a t the origin In agreement with the e l a s t i c

constants determined bj u l t r a s ~ n i c techniques. Within

experimental errors involved, the dispersion curves could be

adequately f i"vtd t o a Born-von Karman model with interactions

extended out t o seven neighbours.

Therefore in contract t o the other featureless dispersion

curves f o r simple bcc metals such as potassium, those f o r group

VB t ransi t ion metals vanadium, niobium, tantalum and t he i r compounds

were found t o exhibit a l o t of interesting s t r ~ c t u r e s . These

anomalous features were discussed on the basis of the Kohn

anomaly (powell e t a1 (1968), (1977) 1, t o which through the

Fed-surface nesting: s m a l l anomalies observed in simpler

metals had been attributed.

Previously it was thought

high-T superconductor TaC and C , -6. >. ,

tha t the pronounced anomalies for

the lack of the anomalies in the

nonsupercond~~ctor HfC seem t o support the conjecture tha t the

phonon anomalies were closely related t o superconductivity i n

general. But t h i s conjec'iiure is now found t o be more subtle. 1, - ..

For exmple, in sp i te of the low superconductivity t ransi t ion

temperature for I%, the metsl seems t o show anomalous dispersion

curves ( ~ e b e r , 1930)~ Nevertheless, a close relationship

between the momctlies and supercol?c?-wtivity does seem t o exis t ,

and one generally expects ?o 2ind some type of anomaly i n the

phonon diepers ion cilXwiTi:r of high- Tc ~uperconductors , although

a low-T superconductor xay also show rather strong anomalies. C

The anomalies and superconductivity should be t reated as the

resul t s of, o r differen5 manifestations of, the same electronic

property of the metal. Weber (1.980) has shown tha t the

relat ion between phono~ anomalies and ~ u p e ~ c o n d u e t i ~ i t y may

best be i l lus t ra ted by comparing the phonon linewidths.

Nevertheless, the important parameter which deternines the

t ransi t ion temperature Tc ie the electron-phonon coupling

constant A ( ~ c ~ i l l a n 1963). For simple metals, one can

rel iably estimate X by using pseudopotentials ( ~ i m n 1962,

Allen and Cohen 197'3 j . h t for the 8-band metals, we can use

McMillant a strong coirpling theory. In t h i s approach, we

hem tha t i n the rigid-ion, one electron approximation

2 1 = n(ey)<l (1.3)

where M i s t L e ~ tomic msss; a2> is the r c n o m l i z e d phonon

frequency; n(€ ) is the density of s t s t e s ?or one %ype of F 2

spin. a t the Fermi energy,, e ; n d <I > is the square of the I?

electron--i::ionori matrix element averaged over the F e d surface.

By determining A fi.m the experimentdy measured Tc, and

<a2> from neutrrjil scat-tering experiments, McMillan (1968) has

It 2 evaluated n(tzp) <I > f o r most of t he supe~y:onducting m e t s l s .

The most s t r i k ing r e su l t j u that while both II(E ) and <id> vary F 2 considerably from element t o elxment, n(cF)<l > remains

2 approximtely constant (,.- 7 e ~ / 8 ) f o r a l l t he bcc t r ans i t i on

metals.

Therefo-e, t he character is t ic feature of t rans i t ion metala

is t h a t t he electron-ion interact ion potent ia l v(=] hss a d

resonance above the muffin-tin zero. A s is well hnown, this

resonance l i e s near the F e d energy and is believed t o be

responsible f o r many of t h e most s t r i k ing properties of

t r ans i t i on metals. From t h i s p i n t of view, we s h a l l i n t h i s

present research c la r i fy the ro le played by such resonance i n

determining the strength of t h e electron-phonon interaction.

1 ,2 - OUTLINE OF RYKYJT;'-1_CR Mi3TH@JXILC!-C~

W e have a l r e a e s ta ted above t h a t t he objective of t h i s thesis

is t o explore a representation of the l oca l f i e l d correction as

an a t t r a c t i r e short-xanta;e force which tends t o cancel o r screen

t h e usual Born-Mayer r e p d s i v e short,-range force due t o the

overlap of the wavef'unctrions of electrons c e n t e r d on

neighbouring ions. To c lear t h e ground f o r t h i s investigation - .. we s h a l l hegjn by presenting the mathematical t oo l s based on the

formalism of Lat t ice djmamiks in chapter 2. This formalism w i l l

depend on the haxmo~.ic and adiabatic approximations. I n the

15

framework of these ~pproximstions, -.-I sha l l characterize the

interatomic forces a r i s inc fmm the direct short range and lone

range forces, and the indirect ion-ion interact ion v i a the

polarisation f i e l d of the conduction electrons ( the electron-

phonon interact ion).

The electron-phonon matrix element C. which is proportional % A

t o the screened pseudopotential o r model potent ia l form factor ,

~ ( q ) , depends on the electronic s t ructure of the solid. In the

simple (nearly-free-electron) metals, t h e screening of the

pseudopotential o r model potent ia l may be represented by a free-

electron d ie lec t r ic function. However, i n t r a n s i t ion (d-band)

metals, loca l field corrections a re called in to play: the

representation of the screening of the t r a n s i t ion metal

pseuc?opotential o r model potent ia l requires therefore t h e use

of a ctielectric matrix whose off-diaponal components characterize

the loca l f i e l d corrections.

I n chapter 3, w e , s h d . l . w o t he ~ceufiopotential approach t o

describe t h e contribution from the loca l f i e l d correction in

t h e Transition Metal Model Potential of O l i end Animalu (1976).

This w i l l ezable us t o investiqate , - .. the ccntribution t o the

indirect Lzteraction 'cetween ions v i a the polarizetion f i e l d of

the valence electron gas a r i s ing from s-d interaction contribution

t o interatomic ?a i r potential. In ih is approach, we sha l l adopt

16

direct method using actual atomic pt.entirzls and charge densit ies

v ia s-d intercction t o show how the a t t rac t ive short-range forces

associated with the loca l f i e l d correction nay be defined t o

have the same f'unctional f o m i n r e a l s p c e as the repulsive

short-range (~orn-~4ayer ) potent id.

In chapter 4, w e shall discuss t h e enerm bmds of t ransi-

t ion metals in order t o display the occurrence of s-d hybridiza-

t i on and t o show how t h i s hybridization is simulated in the

framework of the pseudopotential o r model potent ia l nethod.

Logically, in order t o use the pseudopotential o r nodel potential

derived from the energy band theory f o r the investigation of

electron-phonon interacticm m d other electron scat ter ing

processes, it i s necessary t o make a c lear d i s t inc t ion between

the V-matrix (used i n Ehergy band theory) and the T-matrix

(required Tor scattering theory). This dis t inct ion w i l l be

made i n chapter 4.

I n chapter 5, we shel1,apply the above theory t o the ,,, 4 . t . . 5

specif ic examples of vanadium, niobium and t ~ n t a l u m i n order t o

demonstrate how cancellation between the l oca l f i e l d correction

and the shor t - ran~e ( ~ o r n - ~ n ~ e r ) contribution of these metsls I) .> .-

occur il the phonon dispemion curves. An6 i n chapter 6 , we shall

obtain numerical resu l t s and compare them with experimental

data. This w i l l be followed by a summary and the attenfiant

conclusion.

LATTICE DYNAMICS FORMALISM

2.1 INTRODUCTION

In the previous chapter, we have presented the gene*

introduction t o the subject-matter of l a t t i c e dynamics and a

review of both the theoret ical and experimental background of

the aspects of l a t t i c e dynamics i n d-band metals. Also, we have

indicated i n chapter 1, the aspect of l oca l f i e l d correction

t h a t i s dealt with i n t h i s thesis . In order t o develop t h e

too ls f o r investigating the relationship between the loca l f i e l d

correction and short range forces i n d-band metals, we proceed

i n t h i s chapter t o formulate l a t t i c e dynamics from t h e f i r s t

principles.

I n section 2.2, we sWl derive the phonon dispersion

re la t ion i n the hamonic approximation: the dispersion

re la t ion w i l l depend on the interatomic forces through the

dynamical matrix. In ' iie'i.iidhli. 3, we sha l l discuss the short-

range (~orn-Mayer) par t clue t o overlap of core wave Punction

and the long-range ion-ion coulomb interaction part of the

interato&c forces. In secti'bn -2;f , we s h a l l describe how the

electronic Land s t ructure contribution due t o the indirect ion-

ion interaction v ia the polarisation f i e l d of the conduction

electrons involves the screening action of t he eas of conduotion

electrons through the electron-p5ona.n interact ion matrix

elements g s,AB which are proportional t o the screened pseudo-

potential o r model potentiel f o m factor viq). And i n section

2.5, we s h a l l show how the off-diagonel. part of d ie lec t r ic

matrix a r i s ing from the electronic contribution is determined

by the depletion hole associated with the nonlocal model

potential VM.

2.2 PHONON DISPERSION REZATION

We proceed i n t h i s section t o derive the dispersion

re la t ion for the l a t t i c e waves in the harmonic approximation.

For t h i s purpose only small displacements of t he atoms about

t h e i r mean equilibrium positions w i l l be considered. Also,

we s h e l l consider a Bravais l a t t i c e of atoms of chemical

valancs z crystal l iz ing in a cubic phase.

L e t the displacement of t he nth atom i n t he direction be

denoted by u.(n) where a = x,y,z. The kinet ic energy of the

nth atom i s gi-ven - ,< 4 . 6 , , . . ,,,, . .

U where M is the mass of the atom and va(n) = dt

is the velocity

Therefore the t o t a l kinet ic , ens& of 8 system of Il atoms i n

a crystal i? given by

19

In te rns of t he momentum, %(a) r: ??$(xi), Eq. (2.2) becomes

The t o t a l potent ia l energy of t h e c rys ta l as a function of

t h e instantaneous positions of all the ions is

where Rn - - Rno + u,(n), Rno being the equilibrium posit ion and

ua(n) t he displacement f r o m equilibrium. In t h e harmonic

approxhi t ion, we expand t h e potent ia l energy i n powers of the

displacements and re ta in terms up t o t he second order i n t he

displacements :

The subscript '0' means t h a t t he quantity is evaluated a t t h e

eqc.l:t.?'br-ium configuration. The f i r s t term i n t h e expansion is

t he equilibrium energy which is a constant and may be s e t equal

t o zero; the second term is zero b e c a ~ s e 3W/au (n) evaluated c ,< . .'. >>' , * , J

Q

at equilibrium posit ioa i s zero; and the t h i r d term which is

the second orOer o r harmonic term is characterized by the

non-zero force constants,

This represents t he force on the nth atom i n t h e a direct ion

when the mth atom is given a uni t displacement i n t h e B

direction. Thus,

The equation of motion of the n''''."' tiSox is therefore,

In order t o obtain the dispersion relation, we assume t he wave

in the crystal is a simple harmonic wave. Then, 8 travelling: I harmonic wave~bolution of the equation of motion is of the form

where

< = the l a t t i c e wave vector

s = longitudinal (L) or transverse (T T ) polarization 1' 2

inaex hS A

~ ~ ( q ) unit pclai-ieation vector with a = 1,2,3 being r,

2. = amplitude of t5e wave 9

Substituting f o r u (n) i n to the equation of motion (2.8) a

yields -,r ,, . , $ . ,.. ,

i ( q . ~ n o w t ) m s i( G.$.owt ) - ~ o ' ~ ; ( < ) e = z eB(<)e

no

MultLplying by c i ( c ) t!("p)eiq*Rno , eliminating the common factor -- .-

e -iwt, atld r.-oxe-.ging u:; g&t

21

Using the f ac t t ha t t he com?onent o f the polarization

vectors arc orthogonal, i.e.

where 6aX i s the Kronecker de l ta with the property t h a t

= 1, if u = X

= 0, otherwise

equation (2.10) may be rewritten i n the f o m

This can be rewrit ten as

where

i s the cfyn~mical matrix. It is the Fourier transform of the force

constant. Equation (2.13) is the dispersion re la t ion f o r the

l a t t i c e wave. The dl~p+s'/ZjTi re la t ion may be writ ten i n the

e x ~ l i c i t matrix form

Wow, because of t he 2er iodici ty of the c i ~ s t a l l a t t i c e , we have

t ha t

i.e. t he force constants a r e functions of the distances, between

the atoms concerned. Moreover,

C a2w = O (2.17) n aua(n)au8(m)

because i f all a tcss a re given equal displacement, the resul tant

force on any part icular atom, say t h e n t h atom is zero. Putting

the nth atom at the cr igin , we may express the dynmical matrix

in the form

and hence, bj vi r tue fir Equation :2.17), we get 2.

A a2w iq.R ~ ~ ~ ( d = - " (U (o)au (m) > o (1-e mo)

]do a e In practice, summation over the d i rec t l a t t i c e i n r e d . ,,,," .,.. ,'. > > ' . , . , . . J * , '

s p c e dces not converge r q j d l y f o r long r m c e forces. There-

fore it is frequently useful t o derive an a l te rna t ive

eq re s s ion fo r D (q) i n which the sumnation is carr ied out i n aB

, "- .- the reciprccal l a t t i c e space, a procedure introduced or iginal ly

by Cochran (1963). F w our p rposes we need cnly consider

potent ie l energy functions t ha t can be expressed a s s sum of

two-body effect ive pairwise potent ia ls i n the form (2.4) i u e u

Taking Fourier transform gives

and t he force constant becomes

Thus t he dynamical matrix by v i r t ue of Eqn. (2.19) becomes

Now, using the completeness re la t ions viz.

a where g is the reciprocal I.%ttice vector, then t h e dynamical

matrix is of t he form

Then, subst i tut ing Eq. (2.26) in Eq. (2.25), t he f i n a l form of

the dynamicel matrix becomes

!The choice between the expression (2.19) and (2.27) depends upon

the type of force actin8 between the ions. In the case of

short range forces, however, we can work i n reaJ space and the

form cf Eq. (2.19) is adequate. But fo r long range forces such

8s the ion-ion interaction of coulomb potent ia l o r the ion-ion

interaction v i a the polarisation f i e l d of conduction electrons,

Eq. (2.27) is most appropriate.

2.3 SHORT RANGE AND TGTG WTGE FORCES

In t h i s section, w? ~ r o c e c d t o describe the short range and

lone range parts of the interatomic forces discussed i n t he

previous section. I n a solid, t he vave functions of adjacent

4 - 6 .. . ,* a t o ~ i c cores overlap"-to s a x ext?nd, giving r i s e t o short

range repulsive interactions. These repulsive interactions are

s t ructure dependent and cannot be ignored i n the interatomic

forces because they are significant whsn compared with the Van

der Wads type of interaction.

Then, subst i tut ing Eq. (2.26) in Eq. (2.25), the f i n a l form of

the dynamical matrix becomes

The choice between the expression (2.19) and (2.27) depends upon

the type of force acting between the ions. In the case of

short range forces, however, we can work i n r ea l space and the

form cf Eq. (2.19) is adequate. But fo r long range forces such

as t he ion-ion interaction of coulomb potential o r the ion-ion

interaction v ia the polarisation f i e l d of conduction electrons,

Eq. (2.27) is most appropriate.

2.3 SHOW RANGE AND JEJG RANGE FORCES

In t h i s section, wz nrrocecd t o describe the short range and

lone ranee parts of the interatomic forces discussed i n the

previous section. I n a solid, t he xeve functions of adjacent

,,-; 4 -6. >.. 5 I + , . atonic cores overlap co sale extsnd, giving r i s e t o short

range repulsive interactions. These repulsive interactions a re

s t ructure dependent md cannot be ignored in the interatomic

forces because they a re s ignif icant whsn comyared with the Van

der Wads type of interaction.

According t o e l e c t r o s t a t i c theory, t h e in te rac t ion energy

o t two nonoverlapplng spher ical ly symmetric charge d i s t r ibu t ion

is e e /r where e and e are t h e t o t a l charges associated 1 2 12' 1 2

with t h e two d i s t r ibu t ions and r is t h e dis tance between t h e i r 12

centres. Similarly, t h e t o t a l e l e c t r o s t a t i c energy Ue of n

such charges of naenitude e. ( i = l , . . . ,n) 5.5 1

Ue = C Yi r.. pa i r s 1j

i n which t h e summation extents over a l l p a i r s o f charges, each

p a i r being considered once. This may be wr i t t en i n t h e form

'e xi = $ 1 (2.29)

i , j i j

where t h e summation i s now a double sum over a l l charges and

t h e superscr ip t prime indicates t h a t t h e cases i= j e r e t o be

excluded.

The repulsive force ' b e txzn ions i s very s m a l l u n t i l t h e

ions ecme in to contact, when t h i s force increases more rapidly

than t h e e lec t ros+,a t ic force. It 1~n.s e a r l i e r assumed t h a t , ,, . - 5 . ,.' , I *

t h e repulsive force2 beLt.?ea ions gave r i s e t o an interact5on

energy of t h e type

f o r t h e ent;ire c ry s t a l , where A and n a r e constants and r i s

t he distance between neares t unlike ions. However, i f we

assume t h a t only nearest neighbouring ions contribute t o Eq.

(2.30), t h i s term implies tha t ions repel each other with a

?+1 central force tha t varies as l!rL . But investigetLons carried out on interatomic forces on

t h e basis of quantum rnechrnics indicated that a repulsive term

of Eq. (2.30) was not consistently correct, although it may be

a f a i r approximation for a short range of r. Barn and Mqer

empirical intemtomic potential t o represent the interaction

tha t ar ises as a resul t of the mutual overlap between the

electron wavefunctions centred on neighbadring atoms. This

Born-Mayer potectial , as it i s called, is necesssrily a short

range potential and has the form

a t a separation R between two &tms A and b being empirical

Born and May~r found that they could take p as 0.345 x 10-~cm \ - --

for a l l types of ions i f they determined A from the equation

Here, A i s 2.25 x 10-l2 erg f o r all types of ion, b is another

f ixed constant, z and z. a re the valences of two interaction ions, i J

n. and n. a re t he numbers of valance electrons i n t he outer she l l s 1 J

of the ions Roi and R are t he ionic rad i i , and R . . i s t he 0 j 1.1

separation of the ions ( ~ e i t z 1940). The introduction of such

potent ia l was important in order t o account f o r the cohesive

energy of the ionic compounds of the a lka l i metals, such as NaCl,

i n which the Born-Mayer potent ia l , being repulsive, c o u n t e r

balances the tendency of a l a t t i c e b u i l t from an al ternat ing

+ array of posit ive ( ~ a ) and negative (cI-) ions t o collapse

under t h e i r mutual nearest-neighbour coulomb at t ract ion. Such

Born-Mayer potent ia ls were introduced i n the study of the

binding and vibrational energies of the pure a lka l i metals.

Benedek (1977) has studied the problem of core overlap

interactions i n metals and ta5ulated the Born-Mayer parameters

f o r three a l k a l i metals, namely, sodium ( ~ a ) , potassium (K)

and rubidium 1,3... (~b),. . bu; , not Tor Cesium (cs). He also comyared h i s

parameters t o those extracted from Gilbert 's work (see Gilbert

(1968)) f o r N a , K and Rb. Recently, Upadhyaya e t al (1980)

have calculated the contribution from the Born-Mayer type I, _ . I

repulsive interactions t o the s t ructure coefficient a f o r K and

Rb by using both Benedek and Gilbert parameters, Their resu l t s

show t h a t short-range ( ~ o r n - ~ a y e r ) interaction reduces the

binding energy more f o r t h c bcc phase than f o r the fcc and hcp phases.

Here, A i s 2.25 x 10-l2 erg f o r all types of ion, b i s another

fixed constant, z. and z are the valences of two in teract ion ions, 1 j

n. and n. a r e the numbers of valance electrons i n t he outer she l l s 1 J

of the ions Roi and R are t he ionic rad i i , 8nd R.. i s t he 03 1 J

separation of the ions (Seitz 1940). The introduction of such

potent ia l was important in order t o account f o r the cohesive

energy of the ionic compounds of the a lka l i metals, such as NaC1,

in which the Born-Mayer potent ia l , being repulsive, counter-

balances the tendency of a l a t t i c e b u i l t from an al ternat ing

+ array of posit ive (Na ) and negative (cI-) ions t o collapse

under t h e i r mutual nearest-neighbour coulomb at t ract ion. Such

Born-Mqer potent ia ls were introduced i n t he study of the

binding and vibrational energies of the pure a l k a l i metals.

Benedek (1977) has studied the problem of core overlap

interactions i n metals and tabulatec? the Born-Mayer parameters

fo r three a l k a l i metals, namely, sodium (Na) , potassium (K)

and rubidium I ( (~b)?. . . . bus , not fo r Cesium (cs). He also compared h i s

parameters t o those extracted from Gilbert 's work (see Gilbert

(1968)) f o r N a , K and Rb. Recently, Upadhyqa e t a1 (1980)

have calculated the contribution from the Born-Mayer type I, ..

repulsive interactions t o the s t ructure coefficient a f o r K and

Rb by using both Benedek and Gilbert parameters. Their r e su l t s

show t h a t short,-rang e ( ~ o r n - ~ a y e r ) in te rac t ion reduces the

binding energy more f o r t he bcc phase than f o r the fcc and hcp phases.

However, i n t he pure alkali metals and even i n t h e non-

t r an s i t i on polyvalent metals, the Born-Mayer contributions a r e

usual ly small compared with t he d i r ec t coulomb repulsion

between t he pos i t ive ions and i t s screening by t he valence

e lect ron gas. The reason f o r t he smallness of t he Born-Mayer

po t en t i a l in these simple (non-transition) metals lies..

i n t h e v a l i d i t y of t h e so-called small-core approximation

introduced t y Harrison (1966) i n connection with t h e ~ s e u d o -

p t e n t i a l nethod, According t o t h i s app-oxination, t h e

adjacent ion cores do not overlap i n these metals, s o t h a t the re

is p r ac t i c a l l y no d i r ec t in te rac t ion between t he ion cores

e x c e ~ t t h e i r coulomb repulsion, But t h i s approximation is not

expected t o hold f o r t he noble metals (CU, Ag and AU) and t he

t r a n s i t i o n metals, such as V, Ms and Ta, because t he d-shells

a re much too 1arg.e and the d-electrons associated with these

s h e l l s cannot be t r e a t ed a s conduction e lect rons , and addi t ional , ,< . q. >> , I d -

complication a r i s e s from t h e in te rac t ion of t h e d-electrons

with t he s - e l ec t rms ( c d l e d sd-hybridization i n t h e energy

band theory). 1, . .I

The expes s ion f o r t he contribution of short-range

po ten t ia l of t h e Born-Mayer type t o the Qmamical matrix i s

A

where 1 gn{ is the se t of direct lattice vedara, and R = R .e na n a

A

is the ath component of Rn. Because of t he exponential factor,

we sha l l sum over the f i r s t and a t most second nearest neighbours

i. e. d a Rn 5 ($)(+I. ~ 1 % I s ( ~ ) ( + 2 , O,OA ( 1 2% 0) and

Typical resul ts of our calculations for t he Born-Nayer potential

t o which we sha l l return l a t e r i n chapter 5 i s presented in

Tables 2.1, 2.2 and 2.3 f o r Vanadium, niobium and tantalum. The

parameters used i n the computer calculation of these numerical

values will be shown in Table 2.4.

< ,< . . c >. 3

TABLE 2.1

Repulsive (~orn-Mayer) Contribution o: ( i n units w2) far P

Vanadium

-L 2 .a q K 1 1 q = q(1,0,0) a = q(1,1,0) q = q ( l , l , l )

L T L T1 T2 L T

TABLE 2 .2

Repulsive (E3m-n-Mayer) Carribdim 0: (in units lom2 u2 for niobium P

33

TABLE 2.4

Parameters for Calculation of Repulsive (Born-~ayer) PotentiaL

Vanadium Niobium Tantalum

~ ( 1 ) z ~ ( 6 ) = p = constant ( A ) 0.345 0.345 0.345

~ ( 2 ) = A = constant (ergs) 2.2 5x10-l2 2 .25xl0-l2 2 . 2 5 ~ 1 6 ~ ~

~ ( 3 ) = a = l a t t i c e canstant (E) 8.01 3.30 3.31

~ ( 4 ) = ionic mass ( a ) 0 .84727~10-~~ 1.542401rl0-*~ 2 . 9 9 4 9 1 ~ 1 0 - ~ ~

~ ( 5 ) = vp = plasma freq. (BZ) 3?.189*10 2 5 . 5 5 9 ~ 1 0 ~ ~ 18.3422~10~~

Z2 E 2rg r equilibrium b

separation (8) 1.18 1.38 1.36

~ ( 1 ) = density (g/cm") 6.09 8.58 16.66

M(2) = Z = chemical valence 5+ 5+ 5*

~ l ( 3 ) = R = atomic volume

(atomic units) 93.90 121.30 121.33

Al(5) = ionic radius ( 2 ) 0.5900 0.6900 0.6800

The symbols B ( I ) , ~ ( 2 ) etc are the designation of the parameters

i n the coquter programne of Appendix B. ) < . +' . )>

However, apart from the short-range forces a r i s ing from

the core-core interactions, it i s a k o found tha t the long

range forces of

const i tute part

coulomb interaeilon of the form

of the core-core interactions. But i n a c rys ta l

many sets of neighbcurs interact s ignif icant ly . Therefore, we

need t o sum over many l a t t i c e s e t s t o get t he analogy of

the summatioLi

t o converge.

For t h i s reason, in dealing with any .:.:ong range potent ia l ,

it i s advantageous t o go over t o the reciprocal l a t t i c e space

by taking a Fourier tramform of the potent ia l viz:

The coulombic contribution t o the dynamical matrix can be t ,< .,*. >. * ,*

expressed i n tPe fbrm

Nuni.ericd values of th iv contribution w i l l be displayed i n

cha$ter 5 fo r vanadium, niobium and tantalum.

2.4 ELECTROWPI,. il 'ON INTERCICTION

In the previous section, we have accounted f o r t he short

range ( ~ o r n - ~ a y e r ) forces ar is ing from the overlap of core

wave functions and the long range (coulombic potent ia ls) due

t o ion-ion interaction. In t h i s section, we proceed t o

describe how the electronic band s t ructure contribution

involves the screening action of t he gas of conduction

electrons tLrough the electron-phonon interaction matrix

elements s,X' The electron-phonon interaction is essent ia l ly m a m i c a l

as revealed most s t r ikingly i n superconduct ing s ta tes . Also,

t he theory of the electron-phonon interaction, which is an

inport at element underlying many phenomena i n so l id s t a t e

physics, has a direct t e s t in the calculation of t he frequencies

of l a t t i c e vibrations i n metals. The study of t he ion-ion

interaction via the electrons requires the solution of the

many-electron Schrodfng'W edfiation. We consider a model of

essent ia l ly independent e l ec t rms influencing each other only

through a se l f consistent f i e l d which includes the Hartree

potential and a screened exohan& potential . Although the

effect ive interaction between two electrons, v~(; , ;~), which

occurs in the exchange term, i s known t o a f a i r degree of

sophistication, for convenience we s h a l l use the simple

"'" 36

Thomas-Fermi apyroximat ion. Its Fourier t ransfom i s

where k i s t he Thomas-Fermi screening parameter. S

As we are only interested in t h e term i n t he t o t a l energy

of second order i n the ionic displacements, we sha l l

perturbation expnsion in u of t he form R

use a

where

is the ionic displacement. However, it seems reasonable

t ha t the charge dis t r ibut ion of t h e con2uction electrons

within an ion core follows the ion without much change of

i ts shape and, therefore, it would not be r e a l i s t i c t o expand

a a a

t he t r u e wave function Q;(r,x) i n powers of uQ. We s h a l l

apply t h e perturbation theory t o t he redis t r ibut ion of t he

conduction electrons outside t h e core region, which concerns * * , . L , E . ?' * .,>

the smooth par t @;i;(r-x) ra ther Lhm t he t rue wave function. A

We simply t r e a t @;(r,x) as the wave function under the influence A

of the weak pseudopotentids ~ ( z - x ~ ) i n place of t he bore ion

potent ia ls ub(;GR) employing-t6e f a c t t h a t U is nonlocal, and

{$(;;) 1 cannot be exactly an orthonormal s e t , and t h a t

there should be a f rac t iona l fac tor i n t he density function,

because part of the charge has been taken out.

We can now expand e v e q quantity in powers of the a

ionic displacements u and denote the order of each term by e a superscript, omitting the zeroth order one. The f i r s t order

change, , H 1 T of the pseudo-Hamiltonian which .= includes the pseudo 20-tential and self-consistent f i e l d can

be interpreted as the effect ive elec+,ron-phonon scat ter ing

amplitude wil;l~in the framework of the Born-Oppenheimer method.

* It s a t i s f i e s an integral equation which i s discussed in some

de ta i l in e review a r t i c l e by Sham and Ziman (1963).

In the Eartree approximation, the solution is simply

a l i nea r screening of t he bare scat ter ing potential . The

screening matrix is given by

f o r all reciprocal l a t t i c e vectors and i' where

a -5

~ ( k ) bning the single electron energy and n(k) the occupation

number. Eq. (2.43) i s the same as what is vsually known as

the s t a t j c dielec-tYic d c t i o h i n the random phase approximat-

ion. However, the former rea l ly modifies the change of t he

ionic poJ~cnt ia l :-ather than the interaction between two point

charges, and exchange and correlation efrecta come i n t o play

different ly i n t he two cases.

Thus, we aim a t obtaining a solution neglecting the 2

k-dependence effect produced by tke exchange term. The exchange

term of Hubbard (1958) involves the replacement of t h e matrix 2 A

element vS (k--k) cP t he screened coalornb interaction, which 4 -L -L

sca t tc res k and c. in to k t end k t respectively, by

where i s some average of t h e F e d vector.

Our app-oximate solution w i l l then be of the same f o m in

the Hartree apgruxfmation, modieing only the coulomb inter- a A

action v(q+e) between two electrons in t he second t e r n on the

-L L ') a A

a - f(q+g) = 1 - 4 &It -- - * 2 2 2 (2.45) ( q + d +kF+kS

A .A

which tends t o one half f o r l a rge q + g. This upholds the

reasonable physical pj.cture t ha t t he short range par t of t he < ( < . * C . i *

inlerzcticm between ~ S . C Z ~ . T _ ' : I ~ ~ R c,f p r a l l e l spins is reduced by

the exclusion principle,,

The second order term of t h e t o t a l energy is eas i ly 1 - ..

shorn t o be

39 a

where V is t h e sum over all l a t t i c e vectors a of the pseudo- a a

potent ia ls u(r ,xa). The f i r s t term on the r igh t hand s ide is

due t o repeated one-phonon processes i n which a phonon is

annihilated producing an electron hole p ~ i r which recombines

emitting another ~'nonon. Before the enission of the second

phonon, the electron ~ i l d the hole are subjected t o the influence

of other electrons. The second term is due t o i n t r i n s i c two-

phonon processes i n vhfch an electron in t e r e s t s simultaneously

with two phono~s.

(1) By m c m z of our approximate solution f o r <4-., IH 142 , k

we can get tile atc.!ond order term of the t o t a l energy i n t he form

J2)(;) = ; Z U U P E@ . xl(;) -qr, qs' qs qs (2.47)

qss'

with t h e electronic contribution

-5

o(-~) being the F ~ u r i e ~ transform of the electron density. Each

t e r n j.n %h,? c o n t r i b u t i m ~ ~ & f e ~ W t h b repeated one-phonon

scattering pi-xesses is given ty

Hence, t h e electronic contribution can be writ ten i n t h e form

40 ; -

a A & -. .A

where w ~ ~ ( ~ , ~ ' ) is defined as the limit as q + 0 of waB(q+g,q+ge) A

except tihat when g = 0 , it is zero.

But the most general form of the electronic contribution

appropriate fo r a nonlocal pseudo- or model potential as given

by Hanke (1973asb) i s of the form

where the prime over the summation sign i n the second t e w implies

t ha t the t e r n (c = ;* = 0 ) ehould be excluded from the sumnation.

2 2 4 Ir. t h i s expressicn, o = ( h z e /MQ,) is the ion plasma P

frequency, M being the r ~ . a s , z the chemical valence and a the -.a - \ *

volume of the atom; ~(a .+g , q+gl) i e the normalized 0(0,0) = 1

energy-~?&ve-number characteristic metrix.

The chwacter is t ics t ,, ..(. of >. % The wave-vector dependent d ie lec t r ic

A f'unctbr. E l q + g, q + g' ) gci: CL! the electronic structure of

most crystrrl!.ine solids and can often be direct ly related t o

many experimentally observed que~nt.itias. Speci*ically, the 1 " _ .I

die lec t r ic j'cnction govzms tBke e lec t rmic contribution t o

the dyn&micalrnatrix, which determines the phonon spectrum.

The l a t t i c e i n s t a b i l i t i e s caused, for instance, by phonon

softening, charge density waves and spin density waves a re

presumably a l so related t o the sharp s t ructure observed i n the

d ie lec t r ic function or the closely related suscept ibi l i ty A 2. 3

function. The sitlgularity in c(q,q) a t q = 2% also play an

important rolc i n determining the Kbhn arlomalies in t h e phonon

spectrum of metals.

Nozieres and Pincs (1958), Ehrenreich and Cohen (1959)

were t:ie f i r s t t o d e r i v ~ expl ic i t expressions f o r the diagona3

component of the d ie lec t r ic matrix within the RDA. But t h e i r

treatment fa i led t o inclucie tke loca l 2icld corrections.

In t h i s section we p-mceed t o show t h a t the off-diazonal

component of d ie lec t r ic matrix ar is ing fmm the electronic

contribution t o the dynadcal matrix i s incorporated i n the

loca l f i e l d correction of d-band m e t a l s which we sha l l invoke

i n the next chapter. We have shom i n Eq. (2.51) t h a t t he

energyrave-ntanber.c&reuz0eristic matrix is absorbed i n the

electronic contribution and can be expressed in the form as given

by Hanke (19731,

where

(2.53)

n ( c ) being the occupation number of t he Bloch s t a t e Qi; with a

energy ~ ( k ) ; nnd

v(;++) = (4~e~/1:+;1~) l.-f(;+i) (2.54)

i s the Fourier transform of t he electron-electron interact ion

with modification f o r exchange and correla t ion b u i l t i n A A

through the frzc-i; ~ . - f ( ~ b - ~ ) . Consequently, Sham and Ziman (1963)

have shown t h a t Eq. (2.52) corresponds t o using t he exact

expression f o r t he

In order t o handle

d i e l ec t r i c matrix i n the RPA,

A d a.

x ; ti + 9. q + g v ) (2.55)

t he energy-wave-number charac te r i s t i c

conveniently, it has been reduced t o the standard expression c ,< . ,6. \.' , 1.r

currently used i n t he l i t e r a t u r e (see f o r example Hanke 1973)

and is of t h e form

where

i s the generalized suscept ibi l i ty matrix. 2 . a A A

We wisl- now t o transform the na t r ix element. %e"(q+g, q+g' )

ar-ising from t he d ie lec t r ic matrix of Eq. (2.55). Ey vi r tue of A

Bloch's themem i n the form $->fr) = e iL rt k I$;), we may write

A d a

e i (q+g*) . r @i = $+i+gt , so tha t

wnere we hwe used the fec t t h a t 3V /aE has the same properties M 2 as the projection operator (P = Ca(a> "1 = p ) of OPW pseudo- -

. p o t e n t i d theory so tha t formally ( l - ) ~ ~ / a ~ ) ' = 1-aVM/aE. .a

By transforming <$cle - i k ~ + g ) ~ r A d > I ~ k + ~ + g " s imilarly , and

subst i tut ing in Eq. (2.53) we f ind

Because

A27 A ,

we f ind tha t when we subs t i tu te f o r -,,(q+g, q+gf ) i n eq. (2.55) 2-

%g

and perform t.he summation over g", the l a s t -)rm i n Eq. (2.59)

may be replaced by

Sclkt i tut ing Eqn. (5.59) i n Eq. (2.55) we get the r e su l t

A -\

€*(q+g) bcmg tlia Lindimt! i ' i e l e ~ t r i c function. Zq. (2.61) A 2 & . A

ascer ta iw t.1hn-l; off-diegnn~.l psrt of c(q+g, q+gf) i s determined

c o r ~ l e t e l y by the depletion hole associated with the nonlocal

model potential V Row the correction t o the Linhard M'

expression due to the c?n-,lelion hols takes the form

In the simple metal qgG~.& ,potent ia l (sMMP) theory, the off- < , (

diagonal part of t h i s contribution i s t reated as s m a l l . while

t he diagonal part i s a3sclrbed i n the Lindhard expression A 2 .A .a

E , ( ~ + ~ ) by renormalizing the e l ec t r i c charge i n v(q+g), I ' 1 .I

through the substi tution

where

i s ?he magni-tude of the orthogonslization or depletion charge

which w e s h a l l dorive in chapter 4. Iil the m7'P theory, we

intend t o show t h a t the contribution of the d i e l ec t r i c matrix

to t he electronic corltribution does not cancel out (by symmetry)

f o r t h e transverse branches of the phonon spectrum of bcc crystals

as indicated by O l i and A n i d u (1976). This w i l l be shown

i n chapter 5.

f n t h e previous chapter, we have presented t h e mathematical

t o o l s f o r t h e l a t t i c e &~nwnics of d-band n e t a l s based on t h e

ad iaba t i c and harmonic ap~roximst ions . 1,Je hrzve a l s o described

t h e i n t e r s t o n i c forces a r i s i n q from ( a ) t h e d i r e c t coulomb

in te rac t ion between bare ions of 1 o n : ~ r a n y e nsture; ( b ) t h e

core-c?re exchange force due t o overlan betveen core wave

f'unctions centered on neighbourins ions of" short-ranee nature;

and ( c ) t h e i n d i r e c t in te rac t ion between ions v i a t h e

~ o l a r i s a t i o n f i e l d of t h e valence e lec t ron pas which depends on

t h e d i e l e c t r i c constant of t h e e lec t ron m s .

In t h i s c h a ~ t e r , we s h a l l inves t i ea te t h e contribution

t o ( c ) a r i s i n p from s-d in terac t ion. %is % r i l l be determined

by two methods, f",xs$l;vs. .,$he rec in roca l l a t t i c e method based

on p s e u d o p t e n t i a l s an6 yeudowave h c t i o n s and secondly, a

d i r e c t method usin? ac tua l a,tonic po ten t i a l s and charge densi t ies .

A s a ?relude, we s h a l l i n sec. 3.2 consider t h e >seudo- 1, " ..

- tent ial a p ~ r o a c h b:r i .nves t iv . t ing t h e d i ~ r o n a l nart of t h e 4 d

t o t a l in te rac t ion enerpr (?( F '. .. . ,nbT) ar i s i a tm Prom t h e screen in^ 1 ' -.

of t h e e l e c t r i c char{:e deasFty and electron-ion i n t e r a c t i o n

yo ten t i a l . "his v i l l enable us t o show how t h e d i e l e c t r i c

constant of the electron gas is called in to play, and the

circumstances under which ~ ( q , . . . ,%) represents a short-

range interatomic potential . In sec. 3.3, we ehal l investigate

the contribution f r o m local f i e l d correction by using the

reciprocal l a t t i c e method based on pseudopotentials and

pseudowave functions.

In order t o deal with the d i rec t method of using actual

atomic potent ia ls and charge densit ies, w e sha l l i n sec. 3.4

investigate the contribution from s-d interaction by considering

the following expression fo r the t o t a l electronic band s t ructure

energy i n a fixed configuration of the ions:

where V (G) is the potent ia l energy experience by an electron n

i n the nth band and p , (;) the electron density due t o the n

charge dist r ibut ion of e l e ~ i x m i in the n t th band. The contri-

bution from s-d hybridiztiLion w i l l be represented from t h i s

point of view by O (E . . . .%). We shal l show tha t i n a sd , &'. .'. ... .

certain approximation which w i l l be specified, 4 may be sd

represented by a short-range potential of t he Born-Mayer type.

As CL prelude, we proceed i n t h i s section t o consider

the diagonal part of the t o t s 1 interaction energy ar is ing from

the screening of the e l ec t r i c charge density and electron-ion

interaction potential . We write the t o t d interact ion in

the form (Animalu 1977a p. 7 3 ) *

- a

where p i ) is the electronic charge density a t r - Ri, 4 a

where V'( r R . ) i s the electron-ion interaction potent ia l a t J - A

r - R . The f i r s t term in Eq. (3.2) represents the d i rec t

coulomb interaction between ions, while the second term

represents the screening.

The bare interaction derived from ordinary coulomb

interact ion between an electron a t and a point ion a t takes

the form

Essentially, the physical idea associated with screening

by the valance electron gas is tha t i f s i s the d i e l ec t r i c

constant of the electron gas, then, the above p o t e n t i d is

modified t o the form 13

as i s c lass ica l e lectrosta t ics .

Considering the screening of the bare electron-ion 1 ..

potent ia l V by the d ie lec t r ic constant of the valence s C

electron gas, wc suppme t h a t the effect ive (screened)

potent icl ta.lces Lht! form

where, from c lass ica l e lec t ros ta t ics , t he screening po ten t ia l s A. -<

VSC rc la ted t o the density p(r-R) of t h e valence electrons

by Poisson's equation:

V'V- ac - ) = - 4ne26p(;-5)

dp(P5) being l;he change i n the electron density above 3r

below i t s mean value i n the crysta l . Such f luctuat ions i n

the l oca l electron density is due t o electron-electron

interact ions among the mobile valence electrons. But from the

point of view of qumtuin mechanics, we know t h a t p is the

p roba t i l i t y density function, a L A A

P ( ~ - R ) = 26 $FR)+~(=) k

where i n turn t h e electron wave flmction 9; is determined

by the solution of t he Schrodinger equation

.A 4

Here, we assume tha t the electrjn-ion potent ia l ~ ( r - R ) is

wea!c and we can t r e a t the above wave equation by perturbation

i &.- .- The Fourier transform of ~ ( P I ? ) is of t he form

If we subs t i tu te f o r $JR* and Jr i n th i s exproasion and r e t a in k k

t ~ m s of srder V a t most, then it i s readily seen t h a t .-. A

i verf(P) TffW &) 2 I 1 -

L? T j ; - T - A T- - T-' A - 1 (3.11) k I--. k+q k k-q -.

i:

(-q), t h i s reduces t o

where

Because we how p(;), t he Poisson equation (3.6) now gives an i

e w t i o n connecting V 9c (z) and p( q) ; viz ,

= 2K@1a0 defines t he Thomas-Fermi screening length,

And on eliminating vsC( ;) between t h i s and Eq. (3 .5) .

we f ind t he f i n a l re la t ion

where .A X2

4 9 ) = 1 + -- 2 x ( d 2 V 9

is the dielectr.Jc con:jtan-t of the f ree electron gas.

Typicdly, 57 i;he I'homas Fermi approximation (x=l) , t he

screened elec?;;.on-ion interact ion potent ia l has t he short-range 2 2. 2 -L -*

form (-e /I r-RI )exp( -XI ) , a t a separation ( ~ R I between

an electron and a pos i t ive ion, and

isterac.tLm is gi-ren by

where

the screcncd ion-ion

A e 2e 2 X2 $(R) = -- ... - s i n qR H K 2 4R dg (3.21)

0 q +x2

Crcrlshteyn and Pychi.k (3.465 g. 408) have shown in the Table c $ 3 . * 6 >. , * >+

of Ir~LegraLc that

Therefore, FA. (3.23.) tskhs thz final form

Thus

We observe t h a t t h e diagonal part of the screened ion-ion

in terect ion po ten t ia l i s represented by t'ne Yulrawa short-range

interatomic potent ia l .

3.3 CONTRIBUTION FROM I D C A L FIELD CORRECTION

We proceed t o show the contribution from the l o c a l f i e l d

correct ion i n the framework of t h e rec iprocal l a t t i c e method by

using t he i;s cudopotent i a l s and pseuclowave functions. This

approach w i l l require Fourier inversion of t he off-diagonal

part; of the e lect ronic contribution t o t h e 4vnamical matr ix in

t ? ~ transition-rnetnl model po ten t ia l (T-W) approximation.

Rn='-ally, we ,Ire t o determine t h e interatomic po ten t ia l @

by Fourier inversion of an expression l i k e Eq. (3.19), but with

G ( < ) replaced by a matrix of t he form

is t he generalized suscep t ib i l i ty matrix. Here, t h e d i e l e c t r i c

matrix is o f the form

(3.28) where

The inverse d i e l e c t r i c matrix &hated i n Appendix A has t h e form:

The d ie l ec t r i c niatrix E contains a free-electron contribution .A

c 0 ( ~ + ~ ) whl ch 1s diaganal 31 the r e c i p m c d l a t t i c e vectors g and d 2 a. g', m d rn off--diagonal (g # g ' ) par t , which givea r i s e t o the

loca l f i e l d effects i n the d i e l ec t r i c resporse through i t s

dependence on the localized electronic s t a t e s associated with

the pseudopotential projection operator BVM/OE.

Similarly, the energy-wave-number character is t ic G(q+g,q+gV)

associated with the d ie lec t r ic and inverse d ie lec t r ic matrices A d A 9

can be fur ther separated in to a diagonal part Go(q+g, q+g' ) 6- ', . s ,g

which represents the screening of the coulombic contribution and . a&.

an off-diagonal part Ggc(q+g, q+g1) which represents the local.

f i e l d correction, and tends t o cancel o r "screen" the repulsive

short-range contribh%i6nn; ' b p l i c i t l y ,

where

is t h e usual / .qression given by the f'ree-electron (~indhard)

screening theory and

is the contribution from the local f i e ld correction in which VM

is the transi-iion-metal model potential , and aVM/aE characterizes

the deplstfg:; hole whose dependence on enerpy and on core

electron wavefunctions is typically of the Harrison (1969) form

It is th i s dependence on core wavefunctions and the algebraic

sign of the local f i e l d correction tha t make it tend t o cancel

the repulsive short range ( ~ o r n - ~ a y e r ) force.

The diagonel component of the inverse dielectric matrix

has t h e coulombic form that provide8 the screening of the l o n g

range ( coulombic) contribution . Similarly, the off-diagonal

component of the inverse dielectr ic matrix which is associated ( < . . % *' , 1> . '

with the local f i e l d cor?ection tend t o cancel o r "screen"

the repulsive short-range ( ~ o r n - ~ a y e r ) potential.

In practice, evaluation of the local f i d d correction

f r o m the reciprocal l a t t i c e &etfiod involves a complicated

t r i p l e s m : ,".I t 7 J Z r RE fnf in i te se t of r e c i p m d . l a t t i c e

vectors ( 0 l i and ~nimalu 1976), and it i s d i f f icul t t o obtain

numerical resul ts consistent w i th the symetry requirements

whenever the sumnatioas are cut off after a f i n i t e number of

reciprocal l a t t i c e vectors i s i-cl nded.

To t h i s end, we sha l l in t h i s thesis adopt a direct method

using a c t u d ?,toxric 2otentials a112 charge densities via s-d

i:l",b:-nctic~~ .to p n r ~ ~ ~ e t e r i z e l o c d field correction as an

attractive short--r-.,nge ( ~ o r n - ~ a y e r ) force, Iiaving the same

functionel fora* i n real space as the Born-Mayer type. This

npproacb. ~ i . 1 1 be shown i n the next section.

3.4 ATEJLCTIVE SHORT-RANGE FORCXS VIA S-9 INTERACTION

We proceed in t h i s sec. t o use the d i rcc t method t o

investigate qualitatively under what circumstances s-d interact-

ion may be represented as a short-rmge potentiedl of Born-

Mayer type.

We shal l begin t h i s investigation by considering the

t o t a l interatomic potential energy of a t ransi t ion metal

containing N transition-metal ions of chemical valence

(% = Zg + zd) and a dietribution of s- and d-electrons with

charge density ( p =, ,pa.<+, P ~ ! whose i n t e r a ~ t i o n with the

positive ions is characterized by an electrowion potential

(V Vs + vd) is given by the expression (see, fo r example

, . = c ( z e ~ * + R. . i> j IJ

i

where R . . = The second term e represents the indirect 1J

ion-ion interaction via t he polarizstion f i e l d of the s- and

d-electrons, and may be rewritten as a sum of two diagonal

parts, and @id and two off-diagonal par ts , @id and @As, as follows:

where

(n,m = s ,d )

Thus, the off-6iagonal par t of the t o t a l energy ar i s ing

from the s-d interaction is of the form

where Ps(ri(i) is the electronic charge density of the a-electron A

b t Ri, while vd(;$ ) is the d-electron interaction potential j

a t 2 This gives the interaction potent ia l between the j

neikhbouring ( i and j atoms due t o an overlap of a f r ee s-

electron wave functfbd4'tif' tge ith aton and a bound o r v i r tua l

bound d-electron wave function of the j t h atom is the l a t t i c e .

Sincz the s-electron is nearly f ree , the charge density

associ&ted with z s-electron off. the ith atom m y be replaced by s

77 - whwe z is the chemical valence of t he s-electrons. Here,

s

Jlks has been approximated as a s.'mple plane wave so t h a t p is s

A

approximately a 6-f'unction centered on the atomic s i t e s Ri.

But f o r t he d-electron, it has the form:

NOW, p determines V via Poisson's equation i n the form d d

whose 20:~.ntion is qudLita.k.ively of t he form * A d

A a pd(rl-I+R.) v ~ ( ~ - R . ) = - 4ae j a

J ;I dr' (3.42)

irt-r + E.1 Then J

I n other wordss 4id is zg times a siqerposition of the atomic-

l i k e d-potentials. For our purposes we sha l l consider the

hydrogenic normalized radial and angular wavefunctions f o r the

3d band of the form ( ~ o l e s and Caplin 19.~6 p. I?'), which takes

the fonn , . > , '

Then 4 4 A I Z3 z r -2zdr/3ai

IJISd(r) i 2 = - . d d ; 4 e 2

(3 cos 0-112 (3.45) 81~x6 1 a:

where zd is the chemical valence of the d-electrons and ai is

the ionic radius.

Then Eq. (3.43) takes the form

A

where R = IRi-R i j

By subst i tut ing Eq. (3.45) i n to Eq. (3.461, we get

3 b ' 2 1 Z d "d O0 3 -2zdr/3ai 1

=.- 4ve -- . - mi -Ti Zs f r e drf ( 32 -112d2 . 31~x6 'i R. . -1

1J

But

Then Eq* (3.47) takes the form

I n t ~ g r a t i n g the above integral by par t s by using the re la t ion

j udt. - -? ,> - fvdu ,' ... (3.50) -2zdr/3ai

and nubstitc;.i,lg 1:=r3 an6 v = e , we get

--2zdr/3a. o Q) -2zdr/3ni 1 00 -2zdr/3ai -

f r e r e di. = 1 - 3 1 r e d r

2zdr R . . 22 r Rij d ( - - 1 1~ d R i j q- -1

dr 3ai d r 3ai

3ai - -22 R. ./3ai + - 9ai -2zdr/3ai = R e d 1 j

2zd 1 J r 3 e a; (3.51) 22d R .

lj Then

-22 r/3a. rn

9ai 2 .-?z r/3ai a; = 9ai r 2 e d 18ai - -

-1 r e d 2zdr 22d Rij *zd d (- - R.. 1 J

22d dr 3ai

2 27ai -2zdr/3ai =

t-- 54a: Q) -22 r/3ai

2 r e I + - 2 I re & d

4zd R.. 1 J bzd R . . 1 J

Px the r integration gives

3 l&ai -22 R. ./3a. 162a 3 OD -2zdr/3ai - - - ti.. 1. d 1 ~ 1, 1 e dr

82; lJ 8za 3 R~~ d 2 zdr -4- -) dr 3ai

3 162ai ., -22 R. . /3si 486ai 4 -2zdRi j/3c: d 1J = -- In. . l e

87,; IJ + 7 16zd

Cmbining Eqs. (3.51) - (3.53), we get

Of pMt~.cular intc,rgr;t6.ta.us i n t h i s thes i s is the f i r s t (zeroth 2

order) term i n powers of (R. . I which is of the Born-Mayer 1J

type, i.e.

A.

electron interaction pottxb?.al i n t h e f i r l d of an ion a t R . in J

the form

Thus, contribution t o the interatomic force due t o s-d

which is seen t o be an at t rac t ive Born-Mayer type potential ,

B a ~ l b .',' b : k ~ g the pert inec-t Born-Meyer coefficients defined

3 It i s remarkhble tha t , if we t r e a t the 3d electronic

s ta tes as virtual. bound in

Enhr radius of the v i r tua l

2 t h ~ z the atomic 4s . s t a t e ,

the solid, with the effective

bound 3d electronic she l l larger

then the effective nuclear charge

seen by :he v i r tua l bound 36 electrons w i l l be zd = 3, and

-10 -8 r - . - u 5 t n i n ( w i t h e = 4 . 8 ~ 1 0 esu; z m = 2 ; a i = 0 . 5 9 x 1 0 cm ( 3 4 \. 5

-8 -8 for vanadium, 0.69 x 10 cm for niobilzm and 0.68 x 10 cm

f o r taltalum east 1975)) tiie numerical values of the Born-

Mayer coefficients B and y which are displayed i n tab le 3.1, - ..

TABLE 3.1

ATTRACPIVE BORN-MEIYER COEFFICIETJTS

Metal Ionic Radii Born-Mayer Born-Mayer Coeff. (E) Coeff. (y)

Apart from the sign, these are of the same order of magnitude

8s the Born-Mayer parameters obtained empirically by Huntington

We now have a theory which can be used as a basis t o

explain the or igin of t he Born-Mayer potent ia l from the f i r s t

principle. We s h a l l use t h i s Born-Mayer (short-range) force t o

obtain local f i e l d corrections for the d-band metals vanadium,

niobium and tantalum. I n chapter 5 , our numerical r e su l t s c ,< 4 ,* >. 5

obtained by the use of short-range force w i l l . 3e compared

with the numerical resdt . ; evaluated by tile reciprocal l a t t i c e

method by O l i and Anim~lu (1936)for vanadium. Nevertheless,

we sha l l s-i ' ...I. use the ~r&si$on-~etal-~odel-potent ial ( W)

t o evaluate the r e s t of the contributions t o the phonon

frequencies which include the coulombic and electronic

contributions which we sha l l display in chapter 5.

S-D HYBRIDIZATION IN ENERGY BlWD AND MODEZ POTENTIAL I N d-BAND

4.1 INTRODUCTION

In the previous chapter, we have investigated the contri-

bution tc; -the indirect interaction between ions via the

polarization f i e l d of the valence electron gas. This enabled

us t o show tha t the.diagonal part of the screened electron-ion

and screened ion-ion interaction potent ia ls are ~epresented by

the Yuknwa short-range interatomic potential . In our detailed

derivation, we have investigated the contribution from the

locd- f i e l d correction in the framework of the reciprocel

l a t t i c e method by using the pseudopotentials and $seudowaVe

functions. I n t h i s approach, w e have shown tha t the pseudo-

potent ia l method i s use6 t o evaluate the off-diagonal component

of the inverse d ie lec t r ic roatrix which is associated with the . ,< . 6 +' , 14

loca l f i e l d correction, !lowever, t he complexi5y of the pseudo-

potent ia l approach i;i The computational analysis of the loca l

f i e l d correction demands an a l ternat ive technique. Thus, we

have parameterized loca l ' f idid correction associated with s-d

interaction as a short-rslge foroe having the same -functional

form i n real space as the '.<Jim-Mayer potential . Nevertheless,

our numerical calculations of the various contributions t o

the phonon frequencies w i l l s t i l l require pseudopotential approach.

ti4

To t h i s end, we proceed i n th i s c.h.$c~ t ~ t aeacribe how t h e

s-d hybridization i n energy band and model potent ia l i n d-band

metals a re used t o obtain numer ic~l r e su l t s i n the computation

of the phonon frequencies.

In order t o achieve our objectives, we s h a l l begin in sec.

4.2 by dzscribj-ug the s-d hybridimtion in Energy band by which

i s meant tP.3% a nearly f ree electron broad s-band crosses and

hybridizes with a tight-binding narrow d-band. In sec. 4.3, we

s h a l l show how s-d hybridization a r i ses fmm a resonance

interaction with = 2 component of the plane waves. I n

section 4.4, we s h a l l display the generalization of t h e O W -

pseudopotential transformation t o the &-band metals, This

w i l l enable ue t o appreciate t he fonnal s t ructure of the model

potent ia l fo r d-band metals in section 4.5. In section 4.6,

we s h a l l carefully derive the magnitude of t he okthogonali-

s a t ion o r depletion charge.

In order t o zmsetlc ps idopo ten t i a l o r motiel p o t e n t i 4

derived from the ene rg band theory f o r t he investigation of

electron-phonon interaction and other electron scat ter ing

processes, we shall introduce.:the T-matrix which is necessary

t o eliminate divergences from sca t te r ing off singular potentials.

Consequently, the T-matrix 3.s regular and well-behaved, whereas

the V-matrix is divergent. The formulation i n t he past reveal

t he s imi la r i ty of T-matrix and V-matrix i n t he Born approximation.

We wish t o show t h ~ t t h i s approxh&tioa is inadequate f o r the

d-band metals due t o the presence of the strong scat ter ing

resonances associated with the s-d kvbridization which is a

relevant feature of d-band metal electronic band structure.

In section 4.7, we sha l l carefully derive the theory of T-

matrix sc:.zttering in order t o es tabl ish an in tegra l equation

fo r t h e t r u t s i t i o n operator T. In section 4.8, we shal l

carefully determine the T-matrix form factor ~ ( q ) i n te rns of

the V-matrix form factor v(~). And in Sec. 4.9, t he v i r tua l

bound s t a t e problem w i l l be treated.

4.2 S-D HYBRIDIZATION I N ENERGY BAND OF TRANSITION METALS

In t h i s section we proceed t o deal with the occurrence of

s-d hybridization, by which is meant tha t a nearly free

electron broad s-band crosses and hybridizes with a t ight-

binding narmw 13-band. I n r ea l i t y the atomic 8-state is 5-fold

degenertxte but is s p l i t ii: a cubic crystal by t3e c rys ta l Field

in to two sub-beds, one. sr' ),,baing 3-fold degenerate and the c , P'J

other ( r ) being 2-fold asgclarate a t the ~ymmetry point r. 12

In order t o analyse s-d hybridization, the complex d-band may

be represented by a single l i n e crossing a parabolic s-band. 0 ., ..

For exmple, on considering the 3d transit ion-series metals,

t he broad 4s b a d approximated by a parabola is crossing and

interscting with the narrow 3d bands, and the simplified

(nm-interacting band) picture of energy bmds and the s-d

hybridization of vanadium are shoK.z i n Fig. 4.1.

Mathematically, s-d hybridizat ion i s incorporated i n energy

band theory through a sectilai: equation of the form

where 14 m d bCdd represent t he matrix elements of the s and S G

d banas respectively, while Ms-d and Md-s separately denote

the matrix element of the interaction.

The secular equation a r i s ing f r o m the enargy band theory

involves a one-electron Schrodinger equation

I

which t e l l s bow the % e r n 2 electron waves propagate inside

the crystal having the periodicity of the c rys ta l potent ia l

v(:) = v(; + ?) (4.2a) -.

where the vectors 2 aenote the app3-opriate l a t t i c e ~eCt0 rS of

the crys+.al, i, e.

and el' C2, !13 - l .ntr-~rs: R I ng, ag are l a t t i c e constants

(vectors ) ,

The nature of the eigenmlues Ei;: and eigenf'unctions $; 0 - .-

obtained from the solution of Eq, (3.1) fo r v ~ T ) = 0 (empty

l a t t i c e ) can be represented i n a number of ways since E; is 9

periodic in k i. e.

Non- interacting .

8 IL d bands Intoracting a hybridized

. s d bandr for vanadium

where g is a reciprocal l a t t i c e vector. We mag- represent Bi; r s A k (fig. 4.2) i n reduced zone scheme (as a multiple periodic

function) o r i n extended zone scheme (as an ordinary non-periodic

function). These are i l l u s t r a t ed i n the f r e e electron case

below where

By fo;: s i n g at tent ion on constant energy surfaces i n 3 dimensions,

we can rewrite Eq.

which is a sphere.

Fermi surface

(4.4) i n the form

But the surface of greatest interest is the

where $ = (3n2p)lI3, p being the electron density of fwe e l e c t h n .

In the context of onee lec t ron energy-band secular equ8tion

various ap~\maches ?,re used t o handle Eq. (4.1). Slater (1937)

first suggested the Augmented Plane Wave (APW) method by using

a potential that i s sphericalJy symmetrical &.ht each s r t d c ,<. . .<. ,7' 3 *

s i t e i n each atomi'.~ p?l.I. and constant i n the rest of t h e wit

cel l . Herring (1940) introduced the Orthogonalized Plane

Wave (oPW) method. Later Korrinp (19471, Kohn and Rest - , ) _ ..

(1954 ) used the Kbrringa-Kohn-Rostbker ( 5 ~ 5 ~ ) nethod otheruis.e

known as the Green's f'unction method. Finally, Phi l l ips and

Kle.inman (1.959) E-Iarrison (19631, Animalu and Heine (1965)

developed the pseodupotential method where the valence electron

Ra4uo.d Zona Scheme

I I

, ,, . .*. \:' ,

Fig. 4.2 (b)

Ext endad Zone Schema

appears t o see only a net weak pwbentjal. The efPective

electron wave M c t i o n corresponding t o the replacement of

the t rue potent ia l by the pseudopotential i n the t r u e

Schrodinger equation does not have all the rapid osc i l l a t ions

of the t r u e wave function but ra ther a pseudawave wave

function $.

h ' w our purpose, we s h a l l employ the APW method i n t he

t r a n s i t i ~ n metals involving the crossing of t he nearly-free-

electron (NFE) s-band and the tight-binding +band complex.

7 2 ~ - s e c u h r e q u ~ t i o n f o r one-electron eneriy ban& in t h e A P W

method (hucks 1967) ( i n uni ts such t h a t h = 2m = 1) is of

the. form

1 -- 7 4

where the AmJ pseudopotentoal form factor is defined fo r -. -1

arb i t ra ry s t a t e s , k and k t by

In these equat,ions, U ~ ( R . E ) and u~(R,E) = (duQ/dr) are,

respectively, t h e rad ia l w a v e &&ion and its d e r i v ~ t i v e

(evaluated a t r = R ) ; jQ is the spherical Bessel Punction;

PP( cos 8j;i(, ) is the Legendre's polynomial with argument 4

COS 0 4 kk' ' where 8-- i s the angle between k and E'; R is kk'

the radius of the muffin-tin potent ia l i n t he crysta l ; and Ro

is the unit c e l l volume, for a mvnoatomic crystal l a t t i ce .

The actual l a t t i c e potential appears implicit ly i n the

secular equation through the logarithmic derivative of the

radial wave b c t i o n determined by the radial-wave equation.

where v(;) is the muffin-tin potent ia l associated with a s ingle

l a t t i c e s l t e i n the crystal . By matching UE t o the corresponding

free-space wave outside each muffin-tin sphere, t he effect of

V :.# n hr simulated thmngh the ~ h . a e e s h i f t s (Ie G*,(E) defined

by the standard relat ion

where u2 = E, and je, ne are respectively, t he spherical Bessel

and N e m ~ n n ilmctions, w i t h derivatives, ~;(KR) = d,jQ/dr and

n i ( u ~ ) = dne/dr evaluated a t r = R.

Now, i n an "empty la t t ice" , a l l phase s h i f t s must vanish. ( 4 xi. " l I+.

Hosrever, we observe tha t making a l l Ga = 0 does not cause all

coefficients

t o vanish. If we make use of t he relat ion

then it may be ver i f ied tha t

where

cot Gi = cot G , - n e ( ~ I ? ) / j Q ( k ~ ) (4.14)

Consequently, i f we define the KKR form fac tor by the relat ion

Fmy = 4 p W - ( p W ) o (4.15)

then Zimm's secular equation ( Z i m 1965) f o r the KKR method

becomes

\-. . - .-I

i n whi ck

For R = 2, Z i m m (1965) noted tha t the tight-binding character

of t he or iginal atomic d s t a t e is preserved i n the c rys ta l and

postulated a d s t a t e resonant with a plane wave band. He noted

tha t , i n general, the phase s h i f t i n t he neighbourhood of such , , , . . I .<. . . I + ' .

a resonance behaves l i k e

tan G~ =aW E ~ - E

where En is t he energy of t he resonance and W is the width (i.e. I ' _ .I

t he reciprocal of t he decay life-time of t he s t a t e ) which depends

on the distance of Ed above the energy zero.

The implication of t h i s f ac t on the band s t ructure can be

demonstrated by considering a s ingle plane wave affected by only

73

the d-wave phase shiff; of this type. Nobr k = 0, we may

assume t h a t only one plane wave i s needed so tha t only the term d

with g, = &, = 0 appears in the secular determinant. Then

we f ind

By v i r tue of Eq.

G; =

where Ed, i s the

Subst i t u t i n ~ Eq.

(4.14), we may also write G; i n the form

point where Gi goes through n/2.

(4.20) into ( 4 . l9 ) and remrmging we get

1 where yaw defines a "hybridization gap" giving the degeneracy

resolved by s p l i t t i n g which occurs. i n E vs k re la t ion at the

poink where a very nsrrow d-band crosses the free-electron s

band. The d-band width ( A ) is given i n t h i s simple model by

the enerm difference ..$. +. . . ,z a

where Eo is the energy et which U' /U = 0. Typically, A is of

order 2y. Detailed theory by Lirna~u (1974) reveals t ha t i n the

t rans i t ion metal se r ies , A should depend l inear ly on the square

of the chemical v ~ l e n c e ( 2 ) of the transit ion-metal ions along

a row of the periodic table.

The attempt, by Prasad, Auluck and J3~uY. i (1.976) t o include

s-d hybridization in t he formil.ation of the d i e l ec t r i c screening

has the drawback t h a t only the diagonal contributions are

considered.

Mathematically, s-d hybridization i s incorporated i n

d i e l ec t r i c matrix through a secular equation of t he form

u k t= # - and E r e p r ~ s e n t 'the Intrabmd r w , ' ributions t o the I, s dd

dielee? r 2c m t r i x of tke s er,d d bands respectively, while

E ~ - ~ and E ~ - ~ ~fpfir9t~1.I.j)- (Im.-.l'e the interband contributions t o

the d i e l ec t r i c na t r ix of the interaction. The intraband

contribution involves t he unf i l l ed s and d bands. Therefore,

the off-diagonal csntributi.on t o t he dynamical matrix which

involves the local. f ie ld correcticns a r i ses f r o m t he d-s

intraband contrfiution t o the d ie l ec t r i c matrix. In t h i s

t hes i s , ve shall. Sn?l\5:8;e"t'he loca l f i e l d correction through the

off-diagom,!. c rn t r ibu t icns i n t he fornulation of the d i e l ec t r i c

screening vhich w i l l . ini-3lv e the inverse d i e l ec t r i c matrix

This folmii'ii~n w i l l be treated.& d e t a i l i n section 5.2.

4 . 3 ---- RESrOUKiCZ MODEL23 13-BAND METALS

In the pre-rlons nscl ion, we have sham t h a t t he a-d hybridi-

zation 1~ifC;s the degeneracy i n the eigenvalues of the s and d

bands of approximate symmetry at the point where t h e s and d

bands would i n t e r s ec t i n t h e absence of s-d in teract ion. In

t h i s section, we proceed t o point out how Aderson and McMillan

(1967) emphasized t h a t the hybridization emanates not s o much

from a t r u e s-d in te rac t ion mediated by a non spher ical

symmetrical po ten t ia l , as f'rom a resonance in te rac t ion with t h e

R - 2 com?onent of t he plane waves.

Fr,.--. sca t t e r ing theory, we note t h a t i f t he phase-shift,

Glt, passes through n/2 then t a n Ge would become i n f i n i t e , and

this i ccurs when t he r e is a resonance o r v i r t u a l l e v e l of

rn1.T': in-t in potential : For a t yp i ca l muffin-tin po t en t i a l of

we have 'try matching t h e solutions of t h e r a d i a l wave equation

9 . t I- = %, t h a t t he I.ogarithmic der ivat ive is given by

Put t ing C/B = t an .--

Cot GR 1 = I

I n the as-ymptotic region, tA?n can sxpnfied t o get

Expanding I$&x&(E) i n t he neighlourh~od of E the resonance re

Also, sf .ice &+l+vk(4) = 0, so tha t C t e ( ~ ) passes through

zero near E = Er, we get on subst i tut ing Eq. (4.28) in to Eq.

(4.26) t h a t Er-E

Cot GR = - i 1:

where

Whrn E - Er = f Jw, t he partial cross-section drops t o half i t s

va1-,-I?. Thus $W is the half-w?.d'ih of t he resonance. I n

particular, in t he neigilbou.?';.,cod of a d-state resonance, we

obtain

where E i s the resonancz energy, and W i s the reciprocal of the d

decay life-time of t he s ta te .

me d r a v e phase-shik in the expression (4.31) is of

primery in te res t t o us. 'Ine distingLshing Xeature of t h i s

reuoaance i n sc~t-Ler ing theoxy is tha t t he d s - t ~ t t e s which we were

thinkir.2 of i n t he %inding method as bound s t a t e s a r e

'(7 --

not bound s t a t e s a t dl, rather they are resonances o r v i r tua l

bound s ta tes . Basically, we have t o deal with a problem of

resonance located i n the middle of a f r ee electron band. This

implies t ha t in the standard ~ r ? ~ l f ln-t in approximat ion i n which

there is a f l a t potent ia l common t o the regions f e r from the

&tom, t he 6 - s ta tes turn out to be qui te f a r above t h i s f l a t

l eve l (see f ig . 4 . 3 ) .

4.4 GE3ERA1;IZATION OF THE OPW-PSEUDOPOTENTIAZ, TF?SFOFMATION

the formal s t ructure of

i n the resonance model,

t he model

we proceed

i n t h i s section t o exar ine c~:;.??vlly t he generalization of t h e

OPl+pseudopotential t rmafoimt t ion t o the d-band metals. .

In the simple m?tal theomy, we replace the one-electron

wave equation For a Bloch e k z t r o n i n a crystal ,

by a pseudowave equatiw ,.,,....,.+. . b2 (-zv2 + v om > = > (4 .33)

where VOW is understood t o be weaker than the t r u e potent ia l

V and the pseudowave function $z,, is - @?reral.ly .. a plane wave o r

a simple l i nea r combination of plane waves. The t ran~fonnat ion

t h a t re la tes the t rue Bloch f'unction @ t o t he pseudowave function

d = RESONANCES

Fig, 4 . 3

High - lying rasonon$ 4- 8tatas situated well above t h e .mu&fin-.tin potential.

f o r the class of pseudo potent id.^ and moue1 pwkentials of the

form

where l a >?1; nre the ion-core states. For, i f we tske the

pa r t i a l derivative of V with respect t o %, end subst i tute ow for aV,,/aP i n Eq. (14.34) we obtain the standard resul t , k

where

P = c l c > c cl (4.37) C I.

is the projection operhtor. [-$3" P) tha t orthogonalizes

$g to the ion-core stn;Lx ( c > . Eut i n d-band met-1s i n vh.ir;h +,fie 1 d > s t s t e s are now

complete set including the core s ta tes , the free-Son d s ta tes

Let N(;) be the difference between the true metal potential

then this extra potential appeRrs i n the hybridization

parmeter A defined by

fo r the class of pseudo potent id.^ and model p ~ e n t i a l s of the

form

where I GI > P~ w e the ioi~-.core s tates . For, i f we take the

pa r t i a l derivative of V with respect t o Ei;, end subst i tute om for av,,,,/a~~ k i n Eq. (h.34 ) we obtain the standard resul t ,

where

r . 2 is the projection operntor. :;J = P) tha t orthogonalims

($ t o the ion-core s t t i h ; . ~ lc > . But i n d-band met:-.ls i n ~rb.ich %he I d > s t a t e s are now

virb;ra.:l bound, Hei-risoa expm?f:? tb. ..o vave-Amction i n the o v e r

complete set including the core s ta tes , the free-ion d s ta tes

&a well as plane waves v i z : , t , , , . . t >P > , ' .

Let N(;) be the difference between the true metal potential

then this extra potential appears in the hybridization

parameter A defined by

Put t ing Ec,. (4.36) into Eq. (4.32) we get

To evtr-iuate a and a we multiply Eq. (4.41) on the l e f t by c d'

<c( and <dl respectively provided

and we obtain

I n R typical d-band metal, such as vanadium, c runs through t h e

ls22s22p63e'3p6 while d w s through t h e 3d3; % c h a r a c t a r i s t i z e ~

the ionized be2 free-electron s ta tes ; and A E 6V - <dld~ld> is

the hybridization potent ia l which generates t he d-band resonance

i n t h e energy band structure. Then the corresponding pseudowave

equation turns out t o be

where W is essent ia l ly the um%l 3PW-pseudopotentisl operator:

The s t r ik ing feature o f th i : , lxtension is the resonance o r

-1 bybridiaation term prn-,m*ional t o (E~-E) . To express

qual i ta t ively t he effec~; of t h i s term, we take the expectation

value of t he Schrodinger equation between plane waves t o get

where we have put

I where .\-re heve put

This is a quadratic in E with solution

and when this is plotted producns the gap in energy due to the

coupling of s and d states which is termed hybridization. The

occurrence of this s-d hybridization was demonstrated (see fig. a - 6 . \. ,

4.1) to occur in energr b ~ x h ~f d-bmd metals.

In order to handle the resonance term proportional to

(E~-E)- ' which gives rise to a singularity in energy near E 1 ' .. d'

we shall redefine the transition-metal-pseudopotential ( W)

to include this resonance tern. Thus, the real transition-

met al pseudopotent in].

and

But O i 1 ~ ~ l d Animlu (7-976) have shown tha t it is necessary t o

include the small imaginary part i.e. a resonance width i n order

t o remove the divergence occuring i n t h e pseudopotential. Thus,

and t h t corresponding wavefunction becomes

A Thus the new projection operator P 1 can be s p l i t in to a

simple part (P=) and a resonant p r t (P,) as follow^

and

then, because <dl A1 d, and <dl A1 d> = 0, and it is assumed t h a t A

is nearly constant over the core s t a t e s so tha t < d l ~ l d > = 0,

we have I, 1 .I

Thus

2 -- This i s the basic p r o p r t y (P = ?I j o f 9. val id pseudopo-t;entia,I.

transformation. It is on t h i s conceptual scheme t h a t we must

erect n nodel poteat?~ii . of the Haine-Abarenkov type f o r t h e

d- I l ~ i ~ d metals i n t he nexi; section.

4.5 d-BAND MODEL POTE2TTTAL

Iu this section, we sha l l carefully examine the appropriate

model. -3t.ent.ial lor t he d-band metals. The main feature of

the electronic s t ructure of d-band metals which has t o be

incorporate6 i n t he model potent ia l is the s-d hybridization

as discussed i n t he previous sections of t h i s chapter. Animalu

(1973b), Maclin and Animalu (1977) have shown how 8-d

hybridization may be s i m l a t e d i n t he framework of t he model

potent ia l method, s t a r t i n g fro13 Iiugmented Plane Wave (APW) and

Korringa-Koh-Rostoker--2imzn WCRZ methods of energy band

calculation. The model p t e n t i a l obtained in t h i s way is the

d-band model potentisl . Esscnt iaily , t he &--band model potent ia l , 4 6 ,. ,

has t h e mme form as the Eeine-Abarenkov model potent ia l , but

t he enez-gy-dependence of the R = 2 (resonant) term is qui te

different, being of t h e Corm A & ( E ) ~ ( E ~ - E ) - ~ ¶ near t h e d-band I, _ ..

resonant energy Ed.

The objective of .:-:I? model potent ia l however, is t o

replace the deep core potent ia l a t each ion s i t e i n e metal by

a weak model potent ia l constructed i n such a way t ha t the model

83

wave eq.ueAk3 I c . 4 ~ t o exactly the same eigen-due as the t rue

wave equakion for valence electrons in the m e t a l .

A model p o t e n t l s c m be clloscr i n n yariety of ways. A

simple choice that corresponds t o elimiilating a l l the radial

modes 2n the model wave f'unction, 4, i n the region O<r% so

as t o reproduce the phase s h i f t of the true potential (mod n) , is shown in Fig. 4.4. Outside the radius %, the t rue wave

function iy n.nd the model vave function 4 are identical. This

c h o i ~ l c ~ d s +,o the Heine-Absrenkov model potential ( ~ e i n e

and Abarc~kov 1964, Abarenkov and Hehe 1965, Animalu and

Heine 1965).

The Heir,%-Abarenkov model potential fo r a positive ion of

chemical valence z hes :.he form ( for each fi), i n atomic units,

In t h i s case, t h e m~d&i. radius, %, l i e s between the ion

core radius and the radius J P the sphere inscribed i n a Wigner-

Seitz c e l l for . , tb t .wy&al .lattice of the element, but i s

othervise ar3itraw. Round s ta tes of energy E and angular

momentum P, i n such a potential can be found analytically while

4 and the well-depth,, A ! are determined so as t o r~produca

the atomic spectroseo:~i~ term values of the atom o r ion.

When all mgnls-. .~oaentua s ta tes are taken into account,

the p o t e n t i d for a bare ion has the form

Fig. 4 . 4

Compasjson of the Heine and Aborenkov Model potential UMand the model wove funct ion with

M the t:ue. potential V and trqe y q v e funct ion yr.

where P is a projection operator t ha t ~ C K S ~ 2 - t the componknt E

of t he wave function with angular molilentm II.

The bare ion potent ia l % include uoc which is an

'orthogonalization correction' potent ia l due t o an orthogonaliza-

t i on o r depletion hole a r i s ing from t h e f ac t t ha t 1~1'1 = 1 $ 1 2 does not vanish inside the ion core. In order t o eliminate

the mgnitude of uoc, Heine and Abarenkov suggested tha t one

m y replace the'uniform electron density ze/Ro (z = valence of

ion, no = atomic volune) by a charge density z(l+a)e/Ro

everywhere, a being a correction, together with an extra

posi t ive charge ze/a spread uniformly over a sphere of radius

R equal t o t h a t of the ion core. C

The orthogonalization correction potent ia l then becomes

atomic un i t s )

I ) .> .. where

a eff

= 4(% E ortk.ogonalization charge Ra)

R, is the radius of a sphere of volme Ro. An addition correction

t o . t h e bare ion potent ia l is the so-called correl8tion correction

u described by a p o t e n t i d C C

\

i' -IEcl 9 ' < Rc ; .4 I

u ( r ) = A , CC i 0 , r > R c

i

The t o t a l bare ion potent ia l is then

I n order t o evaluate t he Fourier transform of the bare ion

potent ia l in a d-band metal, we put, following EIeine an4

Abarenkov (1964) and Animalu (1973)

so tha t 00

where i n the l a s t step, we have used the re la t ion OQ

C P p = l (4.67) a=o

The model potent ia l now takes the simple form

Eq. (4.68) is purely l o c d for r > % but non-local f o r

r < %, hence the matrix element of % between two plane waves I, ._ .+ - - A A % exp(ik.;) and dQ0 exp(izt.;) with E 1 = k+q, c~ now be

s p l i t i n to two parts as follows

where the loca l par t depends on only and is given by

- . . - \ while the non-local part depends on ?r, '7' t:nd and i s given

12 ( p1-c - -- cos 0 I j l (kr ) j l (k t r ) r2&

0

with

2&+1 hi-1 (x) = -y- q x ) - b-1 (XI for !Z > o

cor, 0 = ( k f 2 + k2 - q2)/2ktk .<, , . *,*.,. , ' .

By using t he re la t ion (%iff 1968 p. 86) of the form

L - we have f o r k. = kt

By u s k g the re la t ion

we have, f o r k

These a r e analyt ical expessions, involving only elementary

functions. Note t h a t i n these formulae t he AQ must be i n atomic

unj.1;~ ( i . e . ficuble rydber'gs ); " Theref ore, t h e f i n a l expression .A

f o r <hi%/k'> i s i n q+iwrgn.

The orthogmalizztion and correlation corrections, uoc and

u a re a lso loca l pc ten t ia l s and t h e i r contributions t o the cc'

-Z

l?~v.rier transfmm o the bare electron-ion interaction a r e a l so

For complctnass, we have t h a t t h e flcreened model potent ia l

i s given by

where t he l a s t term 1(7) represents the screening of the non-

l oca l (second) term 5.e.

2 2 pd .y~omkl o d we have used % B h p /2m*, m* being the bmd

effect ive mas8 including the e f fec t of electron-electron I "- .'

CouloTTih interactions. E( q) is the Hartree-Fock d i e l ec t r i c

function,

md i n t he Hubbard approximation

Apart from t he inclusion of the II =: 2 resonance t e r n ,

t he expressions a re the same as those for t h e simple metals.

The model pot%-Lial well-depths i n Eq.

where -:- 2 2 n k i2.2 P E = E F = - P 2m* Ed

(4.80) are expressed

f o r II = 0,1 (4.85)

(4.86)

4.6 MAGIJZTUDE OFc THE~.~~~~CX,_WP,L,~ZATI@N OR DEPLETION CHARGE

We proceed in t h i s section t o derive t h e magnitude L): t he

orthogona.Zjxation o r depletior, c2large i n terms of the model

potent ia l v,/(l3.f) and the,.inodel wave function $;. Shaw and . -.

Harrison (1967) appcached the problem of determining the

orthogonalization charge a from a more fundamental point of

view whereby a model potent ia l transformation replaces the

31

t rue Hartree equation f o r a s ingle ion:

li 2 (- , o2 + v(;)) $;(;) = E; +<(2) rL (4,881

v(;) being the t r u e ion core potent ia l plus the valence-electron

screening potent ia l , by a model wave equation

where V i s the bare ion model potent ia l , plus the same valence M ;i

electron screening potent ia l , as i n ~ ( r ) , The difference .A A

between the t rue e lec t r rn densiiy i$$r)$(r) m d t he "smooth" 4

part , ( ( I - ) , is defined by analogy with t he orthonogonr

1 ; : z d t i s 2 il(>L: of pseudopotential theory as a depletion hole :

The integral. i s over a ningle core volume % so tha t a

represents the depletion hole per electron a t a s ingle ion

s i t e , z being the valence o f t h e ion. To determine a, we use

(4.88) and (4.89) t o write

and

-+ - - A

Then we l e t k' - k = ;i and take q t o be small. On expanding 2.

the functions of <' around b, we obtain, t o lowest order i n t :

We Y W sub&lact (4.94) from (4.93) and integrate over the

volume 4 E W O L U A ~ one ion s i t e . Then on applying Green's ,I

theorem 50 t he terms on the left-hand side of t he resul t ing

equation, we see t h a t they contribute zero. since I& r $<

on the core boundary.

Thus we get the resu l t :

By v i r tue of (4.90) and (4.95) we have t h a t t h e depletion hole

is given in terms of t h e model po ten t ia l and t h e model

However, t h e depleticn 3ole a need not be t h e same a t every ion

s i t e but it can be s h m , i n t he lowest order of perturbation

93

theorfa t o be the sane a t every s i t e independent of ion

c~nf igu r&~ion . Our resu l t indicates t ha t a l oca l ( i .e. energy

independent) model poterkiul camot have a depletion hole.

But i n t he simplest approximation, we take t h e pseudowave A a

Punctim as a single plane wave, Q - ~ exp(ik . r ) , with no nixing 0

of waves by the periodic potent ia l i n t he solid. Then from

E q . (4.96) the Shaw-Harriaon d e ~ l e t i o n hole ( i n t he Haieree

approximation ) becomes Y

Now -;rc : :ze t h e f ac t t ha t only the non-local par t of t he model

potwtial . , V i'o-: r % dcpends expl ic i t ly on the energy, E, and 14

the approximation tha t Af = A2, f o r 2 > 2 t o obtain

are constants, we have

Shm and Harrison (1967) noted t h a t t he first cider estimate

of t he depletion hole (Eq. 4.99) a re PJWR~S too large. They

noted tha t i f we rewrits 1";~. 04.93) fo r t he depletion hole

using Eq. (4.91) end (4,921 as well as t he v i r i a l theorem,

we have

4.7 -p THEORY OF FMATRIX SCATTERING

1 n ' a metal, t he electrons--in t h e i r in te rna l motions are

alw~ys scattered by col l is ions with themselves and the l a t t i c e

s i t e s . Therefore i n tne standard treatment of e lectronic

properties of metals, -i,l,;! scat ter ing formalism is employed

extensively where t he l a t t i c e s i t e s in a perfect c rys ta l a re

regarded as the scat ter ing centres. The essence of a

scat ter ing problem l i e s in the determination of a f a r scat tered

f i e l d from a preswibed incident f ie ld . In t h i s section, we

sha l l carefully derive t h e theory of T-matrix sca t te r ing which

w i l l enable us t o es tabl ish an integral equation f o r t he

t r ans i t i on operctor. T.

Essentially, we s h a l l be interested i n solving

Schrodinger equation

h2 --v2J,+ql=EJI 2m A

f o r a sca t te r ing potent ia l ~ ( r ) ,

Rearranging t h i s equation we get

where we have put

A Thus we get an inhwa6'4&edhs equation with U( r ) as the source

A general solution of Eq. (4.105) is of t h e form

y -, ,;I a + % I _ ._ (4.107)

Since Eq. (4.198) is case or V) - 0, it represents a free-

pa r t i c l e o r a plane wave solution i.e.

But i n order t o solve Eq. (4.109) we use t h e Green function a

~ ( r , r ' ) technique which s a t i s f i e s the equation

A But s ince ~ ( r ) = , therefore, we have tha t

i2

which i s an integral equation with the unknown under the

integral sign. To obtain the Green function, we use an

integral representation t o the 6-function v i z .

i - and then assume a s imilar representation f o r ~ ( r , r ' ) i .e.

A d

G(F,;~) = !g(G)e iq . (r-r' a;i

Substituting Eq. (b .16) i n Eq. (5.13) we get 2 - A

A - iq. (r-rt 1 ~ ~ ( r , r ~ ) = - f e 2 d; k2 -

By changing t o sphe:.lcal coordinates with ' A -

(g - r f ) = R, and integrating out all the

then A a 1 G = -

4 i 1 r 2 ~ -ao k -q

4 i

lhis integrand has two poles < +k and q

polar axis along

angular dependence,

(4.120)

A

= -k. Using complex

variable method of integration, we sha l l choose the contour of -

integration which includes the pole +c and excludes the pole -k.

I n t h i s typical case, the contour is closed i n the upper half

plane and by the theory of residues

/f(z)dz = 21ri C residues (4.121)

we have

Since we require a solution t h a t tends t o f ( 0 )

. .

e i L F - where r

f (8) is the scattemingmnp3itude, then

98 7''"

Th-k gives the soL:.%ion of Eq. (1.105). Now in order t o

determine .%I abstract operator T, related t o V by

-u = 19, = W (4.124)

then the expl ici t srj.;r~.ion of Eq. (4.105) becomes

Y = (I - GT)Y, (4.125)

The determination of the operator T provides the solution '4'

everywhere witllin the scattering region. In order t o obtain the

far scattered f ields , only the values of matrix elements

<$acts w a ~ > on the she21 k ( the energy surface o r Fermi

surface in a metal.) are needed f o r description of the f a r

f ie ld .

We wish t o obtain the operator T we transform Eq. (4.123)

with an operator V, v i z

W = yvI, - VGyul (4.126)

or from Eq. (4.124) we have

w, = V$, - VGW (4.127)

Using Wa = W, we get

19, = vYl, - VGT$iwq * a (4.128)

Because of i ts validity for a l l $a in the Hilbert space

spanric: 17 $ , Eq. (4.128) becomes n

T =- V - VGT I e .. (4,129)

This is ~irc integrel equa$ion t o be solved, By simple

algebraic manipulation, we get from EQ. (1.129) the solution8

T = ( ~ - v G ) - ~ v (4.130)

T = V - VGV + VGVGV (4.131)

o r again

T = v ( ~ + G v ) - ~

This is an in f in i t e se r ies which is equivalent t o the

in tegra l in Eq. (4.129). The matrix elements of t h i s

t rans i t ion operator completely determine the probabi l i t ies of

t rans i t ions in the system. Thus the problem involving

p*rbkle scat ter ing can be reduced formally t o the solution

of an integral equation fo r the t rans i t ion operator T. As

already pointed out, t he T-matrix i s rela ted t o the V-matrix

used i n simple metals by the in tegra l Eqn (4.129) and both

reduce t o the same thing i n the Born approximation when we

throw away the higher order terms.

4.8 - T-MATRIX FORM FACTOR

I n the previous section, we have carefully established

an in tegra l equationfir the t rans i t ion operator T, we proceed L ,, 4 - 7 . ,:. , * $2 ' .

i n t h i s section t o solve the in tegra l equation in order t o

evaluate the T-matrix f o m factor ~ ( 2 ) i n terns of the V- 4

matrix form factor ~ ( g ) . I ) .- .+

If ve now use the one-particle Green's function in the

Lippaan-Schwinger form, viz:

of the T-matrix fo r a t rans i t ion on the energy surface can

be represented i n the form of an i n f i n i t e series

which corresponds t o the integral equation

which can be written in the momentum representation as

We can expand the potent ia l in p a r t i a l wave components

.. A

where v&( k' , k, E ~ ) is the p a r t i a l wave component o f the

potent ia l and 0 is the angle between the vectors E and c l . The various terms in Eq. (4.137) specify the interact ion i n

< , . ? < . > P ! * .P ~ .

s t a t e s with well-defined values of t he o rb i t a l angular

mmentum b. Similarly, the T-matrix and the scattering,

amplitude can be expanded i n pa r t iLd wave components in the I ' .< ..

form

and

A -r

where T (kt, k; E) represents the p a r t i a l t r ans i t i on matrix and 9"

f E denotes the p a r t i a l scat ter ing amplitude of the

entr-gy surface. Thus i n terms of the partial-wave amplitudes,

we can get from Eq. (4.136) the following equation f o r t he

p a r t i a l t rans i t ion matrix viz:

I n order t o move from the in tegra l equation i n momentum space t o

one i n energy, we use the f a c t t ha t

-3 3 ( 2 ~ ) Id p = ~ ! I ' ? ( E - ) ~ E ~ P

where N(E) i s the density of s t a t e s , and hence we have

By using complex analysis we can s p l i t up G(E;, EA) i n to the P

form

It is found tha t only the &function i n Eq. (4.143) contributes

t o t he i n t e g r d i n Eq. (4.142). %us t he integration of Eq.

( 4.142 ) yields the algebraic &&ion

A simple expression has been used f o r the density of s t a t e s viz

awl 4 is the r a t i o of the kth density of' s t a t e s t o the f r ee

eleckron density of s t a t e .

The scnt ter ing pertial amplitude fe(G' ,$;$) is connected

with the p a r t i a l t rans i t ion matrix l',(k',c;q) by the re la t ion

where v is the reduced mass. N s o , the p a r t i a l scat ter ing

amplitude on the energy surface can be expressed i n terms of

the phase s h i f t 6k(k) as

so tha t A - - is,

Tg(k' ,k;E) = - - s i n 6p

Similarly, the p a r t i a l V-matrix i n terms of phase s h i f t s is

where K = JE. In the special case of g = 1, we have t h a t

Eq. (4.147) and (4.148) sa t i s fy t he algebraic Eq. (4.144)

which gives < , < . , 5 . , . . , * > t ' .

1 i6, 1 i6, - jl s i n 6ke = - - t an 6, - i&- -- K tan 6 , e k K

i .e.

1 - s i n K

o r

s i n 6 i6k

ke = (l+ i h x ) t a n 6Q + s in 6

i 6 & 6& + s i n 6 e R

T h i s gives the re la t ion t h a t

(1 + iJFk) tan BE = 0

-L A *

Bo:: P, p a r t i a l mvc coupnent V (kt ,k;E) of t h e 2o ten t ia l i s R

This gives the model potent ia l of t he HAA - type i n the form

Ey using Eq. (4.151) and (4.152) i n the algebraic Eq. (4.144)

we can now re l a t e the T-matrix model potent ia l well depth h

A&(E) with t he V-matrix model potent ia l well depth A ~ ( E ) as h -1 h

A&E) = A&E) - i X Q A&E)A&E) A

and solving f o r Aks we get

= - A& -2 2 + &!L

+ All where

"ktt et resonance, i.. e. R = 2 we get

In t he s p i r i t of HAA model potential , we get

But g = 1, therefore, we have

By putt ing p = I' r, we get 'F

A t reeonance, i.e. R = 2 c ,< * *. 5. 5 la

In t h i s analysis, we have sham i n Eq. (4.154) t h a t i n t he T-

matrix formla t ion of the model potent ia l theory, there is a

s m a l l corj :xtion t o the ordinary V-matrix model potent ia l

well depth A ~ ( E ) which i s obtained i n t h e Born approximation.

Also t he T-matrix model potent ia l well depth A~(E) is

characterized bg the r ea l and imaginary parts . This o b s e m -

t i on enables us t o account f o r t he resonances which occur

i n the d-band metals. In order t o apply the above

treatment t o d-states, we shall c a r e m l y describe t he

hybridized v i r t ua l bomd s t a t e problem i n t he next section.

4.9 VIRTUAL BOUND STATE PROBLEM ------.--.

Considering the s-d hybridization i n t he resonance model,

we sha l l r eca l l that. t1.c bound s t a t e s are degenerate with

f r ee electron conduction band s ta tes . But i n t he in te rac t ing

model, new e j genst~tte:: ere formed by hybridization between

loca l and bound s ta tes . In the case of v i r t ua l bound s t a t e ,

Frf.edel and 3l.andin-I 61.959) ' have shown t h a t by analyzing t h e

plane wave conduction electrcn ~ t a t e s of angular momenL,uin R,

phase shirks can 3 e u,,ed ti, .:i-?cj'ibe many physical measurable

prcpv - ' - i e a , The phsse c,hifltS ere t he pttramcters t h a t contain

t he ~ f ~ e c t of tlie scat ter ing. They a r e posi t ive f o r a t t r ac t i ve

poten b i d s but nega.l;ive f o r repulsive potent ia ls . For t he

d-band metals, the d phase s h i f t ( R = 2 ) varies rapidly with

energy E between two m'litiples of K i n the resonance region

loG

i n order to accmodate appmximately the Z!(2%+1) which corresponds

t o 10 electronic: sl-dLes of its d she l l witliin t he v i r t u a l bound d

s t a t e created. The other phase s h i f t s f c r P 2 remain small

over t h i s energy range. Thus the excess density of s t ~ t e s

introduced has mah:r. d character peaked around Ed. Tine theom

of scsCtering by a muffin t i n potent ia l reveal t h a t at

resonance

-1 I w c*(E) = tan ~ f i p

which follows t h a t

and t h e scat ter ing amplitude becomes

while the t o t a l scat ter ing cross-section and t h e derivation

dG2/dE are given by

In & . typ ica l scat ter ing by a 3d l eve l , the addit ional density I I .+

of eie-tron s t a t e s associated with t h e virtual bound s t a t e s

m a expression ascertain* mat t h e density of s t a t e of 8

d-band metal, upon traversing the resonance exhibits very

prcmounced peeked and i s Lorentzian.

By using the re la t ions fo r T-matrix i n Eqs. ( 4 ~ 4 2 ) , (4.143) =d (4.144), we obtain t h a t B r R resonant s t a t e at

energy Er,

Thus fo r the d-band metals fo r instance, the resonance occurs

f o r t he R = 2 s t a t e s due t o the high-lying narrow d-band

hybridization strongly with the broad s-conduction bands. This

e f fec t i s noticed i n

e n e r a dependence of

depth $(E) (hLnicalu

t h e nodel potent ia l theory as a strong

the fonn ( E - E ~ ) - ~ i n the 2 = 2 well

1973b). For the R a 2 resonant s t a t e ,

the spectroscopic term values used i n deriving these model

potent ia l parameters f a l l in to the resonance region of t he

Ag, versus E curve m a *ab~fiOb supply complete information near

the resonance energy E t Ed. Hence, the important zesul t of

the TMMP is tha t t he R = 2 well c!opth has the form

This resu l t presents tke Ziff icul ty of i n f i n i t e solutions a t

resonance. For our purposes, we s h a l l take into account t he

f i n i t e width W of t he resonant s t a t e , and thereby modify the

well depth into the fonn

C (4.170) ( E~-E)+~ iw

which is a complex result f r o m the nSove T-mstrh formulation.

In the d-band metals we have that fo r J?, = 2, the contribution

t o the model pot-nt ia l well depth has the r ea l and imaginary

parts viz:

and

A

In general All i n t he T-matrix theory and Ak i n the V-matrix

theory are releted by Eq. (4.154) above. I n the d-band

resonance formulation by knimalu (1974) using Regge pole

theozy, the resonance well depth fo r energy band is of

u .< < - 6 ,?. , . 2 '

But i n the scat ter ing theory, the T-mtrix capable t o handle

the t r a n s p o ~ t properties, gives rise t o

Correspondingly, t he model potent ia l well depth for the

resonance i n the d-state of the d-band metals i s of a general

fonn

w e l l depth in to the fonn

C (4.170)

( E ~ - E ) + ~ ~ W

which is a complex result from the n5me Twatrix fornulation.

I n the d-band metals we have that for = 2, the contribution

t o the model potnntial well depth has the r ea l and imaginary

parts viz:

and

-. A

In general A$ i n the T-matrix theory and A& i n the V-matrix

theory are releted by Eq. (4.154) above, In the d-band

resonance formulation by finimalu (1974) using Regge pole

theory, the resonance well depth f o r energy band is of

the farm ax*w

A 2 ( ~ ) = Ed - E (4.173)

c , < ' , 5 ?' % 'J

But i n the scat ter ing theory, the T-mtrix capable t o handle

the transport properties, gives r ise t o

Correspondingly, t he model potent ia l well depth for the

resonance i n the d-stake of the d-band metals is of a general

fonn

reg where C is a constant. The expression for R d i f f e r s f r o m

the one given by the Born approximation due t o the presence

of the complex term J ~ w i n the denominator. This additional

term essent ia l ly eliminates Kohn anomaly ( ~ o h n 1959) i n the

form of singularity. The remvnl of t h i s s ingular i ty ensures

tha t the scat ter ing of a d-resonant potent ia l is regular and

well-behaved.

In t h i s section, we have used a different approach t o A

determine the model potential well depth $(E) i n the T-

matrix formulation as indicated i n Eqs. (4.171) and (4.172)

Our resu l t here involves a d-resonace expression whose energy

band parameters re ly on Kd and KF f o r t he evaluation of the A

m d e l potential well depth $(E). We have already obtained a

s imilar resu l t i n the previous section i n Eq. ( 4 -154 ) by using

an exact theory tha t involves a T-matrix integral equation.

Our r e su l t (~q. 4.15)+) in t h i s case does not depend on energy

band parameters end is a s t r ik ing feature of the exact theory. ,, 4 , 6 5. ,

Conclusively, it is remarkable tha t t he bound s t a t e is

of the form

! - .*

which is well represented by the sketch shown i n Fig. 4.5(s).

But the v i r tua l bound s t a t e includes the imaginary par t and is

of the form

This is represented by an oscillatory form as shown in Fig.

4-5(b).

F ign 4 . 5 (a) Bound State

Virtual

f l ~ - 4 . 5 (b)

Bound State

CHAPTER 5

APPLICATIOH TO THE FHONON FREQUENCIES OF d-BAND METALS

INTRODUCTION

In the preceeding chapter, we have discussed the energy

bands of d-band metals i n order t o display the occurrence of

s-d hybridization. We have also shown how t h i s hybridization

is simulated i n the framework of the pseudopotential o r

model potent ia l method. In order t o use the pseudopotential

o r model potential d e r i ~ e d from the energy band theory fo r

investigation of electron-phonon interaction and other electron

scat ter ing processes, we have made a c lear dis t inct ion between

the V-matrix (used in Energy band theory) and t h e T-matrix

(required f o r scat ter ing theory). I n t h i s chapter, we- wish

t o apply the above theoqr t o the specif ic examples of d-band

metals namely: vanadium, nrobium and tantalum, which a re of

grefit i ~ t e r e s t because of t h e i r high superconducting

trarisi t ion tc-iperature.

Experimental plionon &eWency measurements in bcc

Bril louin zone (see f ig . 5.1) &band metals vanadium, niobium

and tantalum show tha t t h e i r phonon dispersion curve exhibit

a number of ~ t r i k i n g pectflaritces, par t icular ly soft modes

and Kohn anomalies. It is 6;P interest to ascer ta in how much

of these pecular i t ies arise from the !l'MMP form fac tor ~ ( g ) -14-4-

end how much from the inverse d i e l ec t r i c matrix E (q+g,q+gl )

which has been the focus o f many of the exis t ing microscopic modele.

Fig 5-4

BRlLLOUlN ZONE FOR bcc STRUCTURE , . . '

The quantitative aspects

theory of l a t t i c e 4fnmics so

114

of the app2.ication of the general

far developed requires the calcul-

at ion of the loca l f i e l d correction which is represented as an

a t t r ac t ive short-range force and the calculation of t he various

contributions t o the d y n d c a l matrix. I n order t o achieve

these objectives, we sha l l i n section 5.2 demonstrate t h e

cancellation of t he repulsive short-range forces by the local

f i e l d correction. In section 5.3, we s h a l l calculate the

coulombic and Born-Mayer contributions t o the phonon

frequencies and i n section 5.4, we s h a l l display the calculations

of the electronic contribution and loca l f i e l d corrections t o

the phonon frequencies.

5.2 =NSTRATION 0-3 TEE CANCEUTION OF SHORT-WJGE FORCES BY

THE LOCAL FIELD CORRECTlON

W e proceed i n this section t o demonstrate expl ic i t ly the

cancellat~.cn o r scresning of t he short-range forces by the off-

dlagoilal ccmpnent cop. $.he, inverse d ie l ec t r i c matrix.

Tfia phonon dispersion re!.oti.on i n a metal having a Bravais

l a t t i c e s t ructure is determines by the secular equation

where 6 is %he phonon wave vector restricted t o the first

Bril louin tone; c8(;) i a the B conqponent (8 1.2.3) of the B unit polarization rector , with (longitudinal or transverse)

polariaation in&x a; M is the maas of t;he icm; aab

is the dynamical ( 3 x 3 ) matrix, consistifig cf t h e Coulombic

contribution 'JC the repulsive contribution D~ and the aR' a6

E electronic contribution DaR.

As is well-known, the dynasical matrix i n Eq. (5.2) i s

the Fourier transform of the respective p a i r r i s e interatomic

potentiel s, wC, # end 8. Here

r e p r e ~ e n t s the long-range "bare" ion coulomb gvtent ia l of -

chemical valence z , where Ri and 5. are the positions of the J

. th 1 m d j t h i o n s .

denotes the short-range ion-ion potential of the Born-Mayer type

where A and p are constants given by the volume and t he

compr+rJsibility data which represent a measure of the r ig id i ty

of th :.+.eraction "&d7'fi' thd equilibrium separation between 0

the pe i r of ions. Animalu (19778) has shown tha t the pa i r

potential a r i s ing from the t o t a l electronic (band-structure)

energy of the system may be wrieten i n the form:

where

and

Go and GQc being the diagonal and off-diagonal components of

the energy wave number character is t ic matrix Rznction:

The f i r s t term on the r ight represents t he diagonal non-local

pseudopotential par t while t he second t e r n incorporates both

t h e diagonal local pseudopotential par t and the off-diagonal

part.

!The t o t a l pa i r potent ia l is of t he form

$(R) = w~(R) + #(R) - W?R) o + ~ J R )

NOW, i f Go and GRc have the same s i p , then the

dq ( 5 . 9 )

screening

action of the electron gas may be well represented as p a r t i a l

C cancc3.1~tion of W + # by f + Sc. However, a r i s e s in Ilc

t he f r m m o r k of psoudapott?frtial theory f r o m the overlap of

core fmctionss it is apparent t h a t it ma;y have the same

spa t i a l dependence as 8 and therefore tends t o cancel 9. In

terms of t he associated dypamickl matrices,

and

t he two contributions a re seen t o have opposite sign, but the

determination of the effectiveness of the cancellation requires

expl ic i t numerical computation. Such a calculation was

performed by O l i and Animalu ( 1976) f o r vana.dium.

In order t o dewnstrate the screening o r cancellation of

the shofi-range (~orn-~ayer) contribution m d longrange

(~oulombic) contribution i n rea l space, w e have from Eq. (5.9)

t h a t the t o t a l interatomic pa i r potent ia l i s of the form

where 2 00 s in R

wE(R) a + $ c ( ~ ) , G(;) -y$- dq (5.13) 0

We have shown i n chapter 3 t ha t i n the Thomas-Fed approxima-

t ion, the diagonal and off-diagonal components of the energy

wave number charac te r i s t ics"m~t r fx f'unct ion takes the form

Thus Eq. ( 5.12 ) becomes

Transforming the above expression, we get ,

Q)

R Z Z ~ ? ? s i n ~q w~(R) = w'(R) + w (R) - (-Tr q (5.16) o q ( d + ~ * )

But we have also shown i n chapter 3 t ha t

Then Eq. (5.16) takes the form

By putting

we get

In t he above expression ( ~ q . 3.201, the t h i l Z term screens o r

cancels the long-rmge ( coulombic ) conti-ibntion , while t he

fourth and l a s t term screens o r cancels the repulsive short-

range ( ~ o r n - ~ a y e r ) contr<P,i:tion. It i s found tha t t he l a s t

term has some common features with the a t t r ac t ive short-range

( ~ o r n - ~ a y e r ) force described in chapter 3.

Thewyore, a carerul examination of t he r e su l t s of the

calculation t o be presented i n section 5.4 shows tha t in the

principal crystallographic direction, we nay simulate the

resu l t s by putting

f o r t he longitudinal branches, In t h i s expression ( ~ q . 5.21)

B represents the a t t r ac t ive short-range (~orn-Mayer ) potent ia l

coefficient as was indicated i n section 3.4, while A denotes

t h e repulsive short-range (~orn-Mayer) potent ia l

coeff ic ient whichaw&s emlier presented i n section 2.2. We

have incorporated exp(-aq/2n) i n Eq. (5.21) a s an empirical

damping factnr.

In order t o preservq th,e;-symmetry (crossover a t P and H

of longitudinal and t r a n v e r s e branches 1, we put

%c = - aw: exp(-aq/2~) (5.22)

fo r t he transverse branch, where the fac tor a represents the

normalized orthogonalisation charge which is characterized by

120

z (l+a) e , while exp( -aq/2n) i s a numerical dmping factor.

Thus the local f ield correction arising f r o m the

longitudinal branches is characterized by en at tract ive

short-range (~orn-&layer ) f ~ r c e . But in the transverse

branches, local f i e l d correction was found by adjustment to

arise from the model potential through the orthogonalisatfan

charge.

In chapter 3, we had used the 8-d interaction t o obtain

numerical values of the at t ract ive short-range om-~ayer)

-fficimts ss indicated i n Table 3.1. Apart from the sign,

our numerical results were found to be of the erne order of

magnitude as the repulsive short-range (Born- eyer) parametera

obtained empirically by Huntington ( 1953).

5.3 COULOMBXC AND BORN-MAYE3 CONTRIBIJTIONS TO THE PROMON

We proceed in t h i s section t o calculate the phonon

frequencies in unit of the ion plasma frequency w defined by P

The detailed analysis of the c~ l cu l a t i on of the coulombic '.

contribution ( Coulomb.' f requenw ) t o the phonon spectra in

the three principsl directions 1100+-\ , and '~11 of the - - -- -L .- - bcc Brillouin zone (see fig. 5.1) w i l l be presented.

The coulombic contribution t o the d y n d c a l matrix

diecussed in section 2.3 can now take the folrm

In t h i s expression,

and

In t he pr incipal directions, the dynamical matrix is

diagonal giving t he longitudinal and transverse modes as

follows :

The r e s u l t of t h i s c a l c u l ~ t i o n has been reported by various & , , 4 .6. ... '

authors e. g. Vosko e t al ( i965) . The resultimg coulomb

frequencies are tabulated i n Table 5.1.

TABLE 5.1

Coulomb Frequencies w2 (in units of w2 in each of the three C P

directions[loo 1 , L110 -,! ' ' and 111 , ( ~ w k o et al 1965) for bcc

The theory of the Born-Wer contribution t o the phonon

frequency has been given i n section .3. The de ta i l s of the cal-

culation of the Born-Mayer repulsive contributions a re cs follows:

The dynamical matrix has the form

Explicit ly, the dispersion relation ?

vectors i n the longitudinal (1) and two transverse (T1 and T*)

di rec t ions :

Eq. ( 5 . 3 2 ) v i t h &,L) t ha t

and

Thus - ' b -( 1; I-R~)/P-

n - d 1 1 1 - 2 = - 1 1--exp( iq .~ ) "L % + o L P e - - . n -.!

n

Since we are t o sum over complete she l l s of reciprocal l a t t i c e , -

vectors which include 'f& :?' -k , for each n in the imginary n9 n

parts i n these summation vanish. Each branch m a y be expressed

in unit of the ion p l a sm frequency by writing, e.g.

The results of the ca lcdat ion o f the Born-Mayer contribution t o

the phonon frequency had been displayed i n Section 2 .3 for vanadium,

niobium and tantalum in Tebles 2.1, 2 .2 and 2.3 respectively.

The computation was performed at the Computer Centre University o f

Nigeria, Nsulcka by usin@; the IBM 370. The program i s displayed in

Appendix B .

. < , ? . < $.,? , . a * '

We turn next t o the electronic contribution which will

include the local f i e l d correction. The dynamical matrix o f the

electronic contribution discussed in Section 3.3 is now of the form

In t h i s expression,

and

In the principal directions, the dynamical matrix is

diaganal giviug the longtitudinal and transverse modes as follows:

me program for computing th i s has two parts, namely: performing

the summn.tions over 368 reciprocal l a t t i c e vectors and calculation

of G(<) which depends on the model potenfial. The 368 reciprocal

lattice vectors are of the form:

The electronic contribution consists of the diagonal and off-

diagonal psrts. The dip4pbal part represents the usual electronic

contr<b u-tion srhi;? ?he off-diagonal par t represents the loca l

f i e l d correction.

We observe t ha t b c c a ~ s e o,f the complexity of the calculation

of the om-diagonal ;wrt representing the loca l field correction,

it i s difS.cult t o o3taiii numerical resu l t s consistent with the

synnetry requirements whenever the sumations are cut off af'ter a

f i n i t e number of reciprocal l a t t i c e vectors is included. Therefore, ., ,, . , c . C' 5 '

instead of ~Pop t ing the lonz-rmge method of eva1w.l i v y the loca l

f i e l d correction, we hhal l ir, t h i s thes i s obtain numerical values

of the local f i e l d correction by the short-range method. In our

calculation, +*he local fie18 sotkection is regarded as an attractive

short-range interatomic force laving the same functionaJ. form i n

real space as the Born-Payer repulsion due t o the overlap of core-

electron wavefunctions centered on neighbouring ions. The local

f i e ld correction of the longitudinal branches are simulated

by Eq. (5.21) while the tr=sveme branches assumed t o be zero

by O l i and Animalu (1976) ar2 found t o be non-zero and are

simulated by 6q. (5.22) .

The numerical resul ts fYoa the computation of the elect-

ronic contr? bution t o the phonon frequencies and the loca l f i e l d

corrections are presented i n Tables 5.2,'5.3, j.4, 5.5, 5.6, and

5.7 for vanadium, niobium and tantalum respectively. But i n

Table 5.8 we display the c m a r i s o n of our numerical resul ts of

the 1oca.l f i e ld correction by short-waves w i t ? the ea r l i e r

nmer ica l resul ts obtained by O l i and Animalu (1376) by the reciprocal l a t t i c e

method of long-waves. The model potential p e r a e t e r s used in our

calculation8 are s h m ~ in Table 5.9; and i n Table 5.10, we present

the other quatititier ;&?ch were used in the computation of the

phonon f'requencies. The computer program for the above elaborate , , 4 " 5 . . ' '

computation i s d i sp ley~d i n Appendices C and D.

Elect ronic Contribution w2 ( i n u n i t s u 2 ) for vanadium e P

TABLE 5.3 --,

Electronic Cortributicm w2 ( i n un i t s w 2 ) for niobium e P

TA3LE 5.4 3__

Electronic Contribution r? ( i n units u2) for tantalum e P

0.1 0.001 0.000 0. 003 0,000 0.000 0.003 0.000

0.2 0.001 0.00: 0.007 0.000 0.001 0.010 0.001

0.3 0.005 0.001 0.011 0.000 0.002 0.013 0.002

0.4 0.006 0.002 0.014 0.000 0.002 0.011 0.004

0.5 0.008 0.003 0.01S 0.000 0.002 0.007 0.006

0.6 0.008 0.003 0.003 0.008

0.7 0.007 0.004 0,001 0.007

0.8 0.007 0.004 0.003 0.006

0.9 0.006 o .oo4 0.005 0.005

1.0 0.0'35 0.004 $ 3 6 . , , 3 '

0.005 0.004

Local Fie ld Correction w2 fo r Niobium k

TABLE 5.8 .--..

Comparison of Local F ie ld Correction by the rtciprocal lattice

TABLE 5.9

~ D E L P O ~ I A L P- Bet . 6)

TABLE 5.10 .. PAEWEBBS FOR PHONON F R E Q U E N C ~ CALCULA&ON

Metals

Niobium

Model pot. well-depth f o r E = o

A.

Metals Density (p)

Vanadium I 6 . ow

Vanadium 1 3.250

F e d u a v e r 5 1.097

Tantalum 1 16.660

Correct inn Energy Ec

3.500

2.400

2.300

Model pot. well-depth forRs1

%

Niobium

Ion-Plasma Frepuency V

39 189

1. 900

1.007

2.900

2.250

2.250

Model pot well-depth forhl

A2

i Tantalum 1.850

18.342 '

z.600

2.000

2.000

Model p o t ) ~ t d e Chemical Effective Valence Mass

z ma

, I

radius

Rm

93.9ao

121.300

121.300

Volume a

lon-cord Effective radius

Rc

0.100 !LO00

orthogonalisstion charge aeff

l.OO0 1 1.115 1 0.031

5.W

5.000

0.100 1.000 0.038

1,000 0,038 0.100 -------

6.1 INTRODUCTION

I n the previous chapter, we have a p ~ ~ i e a our i m a l i s m of

l a t t i c e dynamics i n the calculation of the long-range caulombic

contriLution , the rep iLsive short-range ( ~ o r n - ~ a y e r ) contribution

and the electronic contribution t o the phonon ~ p e c t r a of vanadium,

niobium and tantalum. In t h i s chapter, we shal l combine these

contributions i n order t o obtain numerical resul ts for our specific

d-band metals.

I n section 6.2, we shal l cornpare the theoretical phonon dis-

persion curves w i t h experiment; and i n section 6.3 we sha l l give

a br ief summary and attendant conclusion.

6.2 COMPARISON OF CALCULATED PHONON FRE- WITH EXP-

In t h i s section, we proceed t o compare the theoreticd. phonon

dispersion curves with experiment, .On gathering resul ts fron the , & ,< . . ,6. ,a' 3 ' . I *

previous chapter, we have tha t the calculated phonon dispersion

relat ion is given formally by

But based on the tabulated values of we, the coulomb frequency

( ~ a b l e 5. I ) ; wr, the repulsive short-range (~orn-~ayer ) contri-

bution ( tables 2.1 - 2.3) ; we, the electronic contribution (Tables

5.2-j.4); and uk, t h e at t rzct ive short-range f m e ( loca l f i e l d

correction) Tab1,es 5.5-5.7) ; the calculated f'reguencies now take

t he form

i n uni ts of i012 E7..

Our nuiflei-icd r . suit f o r the t o t a l frequency excluding the

a t t r ac t ive shor t-range force is presented i n Tables (6.1-6.3) but - A- ....

the corresponding phonon dispersion e w e s sir the L100_) ,1110 1 - -! . -

r --*

and i - l l l j crystallographic directions a re displayed i n Figs.

(6.1-6.3) fo r vanadium, niobium and tantalum. In these figures,

we have compared our nunerical resu l t s with the experimental

points obtained by Colella and Batterman (1970), Nabaeawa and

Woods (1969, and Woods ( 19643 f u r vanadium, niobium a d tantalum

respectively.

A similar comparison is made while including the contribution

from the a t t rac t ive sh~~: tyrwge force, The inclusion of thi,:

contribution yields the t o t a l frequency given in Tables ( 6.4-6.6)

while the corresponding phonon dispersion curves i n the principal

crystallographic directions are , displayed _ _ in f igs . (6 ,4-6.6).

Here, we h e ~ c compred cur p2esent theory with the ea r l i e r

numerical calculations io Vuadium (01i and Anixalu 1976), niobium

(~n ima lu 1977) and tea:,-,: -9% (Ariimalu 1973a! .

Thus, inclusion of the local f i e l d correction which is well-

represented as an at t rac t ive short-range farce made a significant

improvement i n the ag eement between theory and experiment,

especially, i n the [ l l i j - and [ll~l branches. Ihe most s t r ik ing -1

agreement with experiment was the cross-over of the longitudinal r

and transveree branches i n the / 1111 direction a t the boundaries - 2

P and H, and the realination of the sound vel.ocity a t l aw reduced A r' -

MM vector (ii: of the tranverse branches in the 110 1 direction t,

=ising from the plot t ing of basically the coulombic contributian

??hi& ~bsorbs all o t k r contributions. Specif i c a w , we have

uiscd our present theory in niobium t o obtain for the first time in

t h i s approach the crossing of the t w o lower traverse branches

(TI and T ~ ) i n the 110 d i rec t im. However, in the Ll00i L 1 direction, our theory also gives the crossing of the longitudinal

and transverse branches for niobium and tantalum.

Tot& Frequency Y (in units of 1012 HZ) excluding the attractive

TABLE 6.2

Total Frequency d i n uni t s of lo1* Hz) eeluding the a t t r ac t i ve

short-range force (local f i e l d correction) i n niobium A A

(a/2n)q(lT1);=q(1,0,0) q=q(1,1,0) q = q(1,1,1)

L T L T1 T2 L T

' 8 .1 2.0918 1 . 0 1 2.7948 0.9905 1.3687 3.3022 1.4780

TABLE 6.3

Total Frequency d i n units of 10 l2 HZ) excluding the attractive

shortrange force (local field correction), i n Tantalum

W ~ E 6.4

Total Frequency in uni t s o f 1012 HZ) including the ~t tract ive short-

r a g e force (local f i e l d correction), i n vanadium

Total F'repuency in units of 1012 HZ) including the a t t r ac t ive

shortrange force (local f i e l d correction) i n niobium,

Total Frequency d i n units of 1012 ? z ) inc luding the attractive

short range force (local f i e l d correction) i n Tantalum.

F R E Q U E N C Y ( I O ~ ~ ~ H , )

Phonon Dirpmrdon Curve in Tantalum includinq calculated local f ield Correctim in thm 11 00], D lo]? and [I l d imt ion8 re~pectively exporimmntol L a - -- Animalu (1973b) present theory

6.3 SUMMARY AND CONCLUSIOIT

In t h i s section, we wish t o summasize what has been so

far achieved i n t h i s thesis and indicate directions fo r further

application of the theory.

I n this thesrs we have presented the formalism of the

l a t t i c e dynamics of d-band metals with a view t o identify the

effect of the loca l f i e l d correction and short-range forces on

the interatomic forces. In our formalism, we have used the stand-

ard adiabatic and h w i n i c approximations t o show tha t the

r';mamical matrix for phonon dispersion relat ion i n s Bravis

. i - ~ t t i e e is s p l i t , as i n the case of simple metals, i n to a sum

of three terms, namely, the coulombic contribution due t o the

long-range coulomb interaction between the bare ions of the

metal, a repulsive short-range ( ~ o r n - ~ a y e r ) contribution due t o

?lo overlap of the core-electron ravefunctions centered on neigh-

bowing ions, and the electronic contribution due t o t he indirect c $ 3 < 5 . <.' 8 *

ion-ion interaction v i a the polar isat ion f i e l d of the conduction

electrons. The electronic contribution involves, therefore the

screening action of the gas of conduction electrons through the I ' _ .I

electron-phonon interaction matrix elements, g3 which are pro- s,X'

portional t o the screened pseudopotential o r mo6el potent ia l i

form factor, ~ ( q ) .

Because of the v i r tua l bound character of the transition-metal

d-electrons, loca l f i e l d corrections a r i s e i n the conventional

screening calculation, as the off-diagonal components of the inverse

d ie lec t r ic matrix. In order t o show how the v i r tua l bound chare-

c t e r of d-electrons lead t o local f i e l d correction, we have used

the Fourier inversion of the off-diagonal par t of the electronic

contribution t o the dynarical mrttrix i n the transition-Metal

Model Potent ia l (TMMF') approximation of O l i and Animalu (1976) t o

show how the interatomic forces associated with the loca l f i e l d

correction nay be defined in real space. Due t o the complexity

of th is pseudopotential approach, we have adopted s d i rec t method

using actual atomic potentials end charge densi t ies via s-d inter-

action t o parameterize loca l f i e l d correction as an a t t rac t ive

short-range force, having the same functional form i n r ea l space as

the Born-Mzyer type. This apporach also enabled us t o formulate

a theory which can be ussd..ae la basis t o explain the origin of

the Born-Mayer potent ia l from the f i r s t principle.

The energy bands of d-band metals a r e incorporated i n t h i s

thesis i n order t o display the ocumrence . .- of s-d hybridization

and t o show how this hybridization i s simulated i n the f'rmework

of the pseudopotential o r model pokential meth3d. We have also

made a c lear d i ~ $ i n c t i o n between V-matrix and Fmatr ix iri order

t o use the pseudopotential or model potential derived from the

energy band theory for investigation of electronphonon inter-

action m d other electron sca t te r ing processes.

We have applied the above formalism t o Group VB bcc

t r ans i t i on m e t a l ~ vanadium, niobium and tantalum which have a

number of in te res t ; ng electronic properties such as high supel-

conducting t rans i t ion temperature. Our theoret ical r e su l t s for

these metals a re i n good agreement with experimental resu l t s .

The previous calculations by Pal (1974), O l i and Animalu (2976)

could not give symmetry agreement a t P i n the [111] direction

i n vanadium. Also, the calculations by V a m end Weber ( 19791,

Fielek (1980) in niobium, and the recent numerical calculation

using the pseudopotential approach by Sin& and Tripathi (1901)

i n tantalum, ha8 a l so f a i l ed t o get the required symmetry at same

P in the / 1111 d' ~ r e c t i o n . It i s remarkable t o note t ha t our i

numerical calculations.iaa&his thes i s achieved the symmetry require-

ment a t the zone boundaries P and H i n vanadium, niobium and

tantKlun. This j u s t i f i e s the representation of loca l f i e l d

correction as an a t t r ac t ive yhortArange . .- force associated with

s-d interaction.

I n practice evaluation of the l oca l f i e l d correction by the

reciprocal l a t t i c e method i n ~ l v e s a complicated t r i p l e summation

over an in f in i t e s e t of recipl-cel l a t t i c e vectors and it is

d i f f i cu l t t o obtain numerical refiults consistent with the

symmetry requirements whenever the summations are cut a f t e r

a f i n i t e numbel of reciprocal l a t t i c e vector is included, But

i n t h i s thes is , we hay.: evaluated local f i e ld correction by the

a t t rac t ive shor t - rq ;e force. Our approach in t h i s direction

is direct m d l e s s complicated. Comparison of the numerical

resul t s of local f i e ld correction in Table 5.8 for vanadium and

the phonon dispersion curves i n f igs (6.1-6.6) for vanadium,

niobium and tantalum by the reciprocal l a t t i c e method and use of

a t t rac t ive short-range force indicate tha t the use of a t t rac t ive

short-range form g i v e s . ~ be t te r agreement with experiment.

Thus, the ea r l i e r complicated procedure of evaluating local

f i e l d correction by the use of reciprocal l a t t i c e method can

now be replcced by the use of a t t rac t ive short-range force

which is be t te r and sirnglkr ,.to dmndle.

In t h i s thesis , our numerical resul ts have demonstrated

that the local f i e ld ccrrection which was ea r l i e r shown t o a r i se

P r ~ m the off-diagonal component of the inverse d ie lec t r ic matrix I - -.

CEUI d s o he ,-cpresented as an a t t rac t ive short-rage fbrce v ia

s-d interaction. This contribution cancels o r "screens" the

repulsive short-range ( ~ o r n - ~ a y e r ) force, Correspondingly, the

diagonal component of the inve-se die lec t r ic matrix which a r i ses

from the other par t of the electron-phonon interaction i n the

electronic contribntion cancels o r "screens" the long-range

(coulombic) force. Our numerical resu l t s show t ha t the canceller-

t ion or screening of the repulsive short-range ( ~ o r n - h y e r )

force by the a t t rac t ive short-range force i s such tha t the

effect ive short-range interatomic force turns out t o be a t t rac t ive

rather than repulsive i n vanadium, niobium and tantalum. This

observation is believed t o account fo r some peculiar shapes of

the major s o f t modes observed in- these metals.

We sha l l not conclude t h i s thesis without pointing out

some areas of possible application and future research. Having

s e t up an s-d intercction model for d-bad metals whereby the

loca l f i e l d correction is represented by an a t t rac t ive short-range

force, it i s noted tha t our successful application t o phonon

frequency calculations c a l l into, play a nufaber of other possible ,, .*6 ,. % 14-

applications.

Thus, an area of special concern is the superconductivity

of d-band metals and the i r compomds. There is a strong evidence I ' . .. - f i l

that the electron-phonon coupling constant, 1, and hence the

t rans i t ion temperature is very sensi t ive t o the phonon frequencies

i n the strong-coupling superconductors. A microscopic understanding

of the l a t t i c e dynamics i n the high superconducting t r ans i t i on

temperature materials is therefore of considerable importance

i n understanding c lear ly the factors which control superconducting

t rane i t ion tem2era.tu1-e i n these waterials.

The knowledge of the calculation of the interatom5.c forces

i n d-band metnls from microsoopic theory through the model poten-

t i a l theory, givcs the prospect of understanding the concept of

electron-phonon coupling mechanism. Such calculations require a

knowledge of both the band s t ruc ture and the eletron-phonon matrix

element. It i s of i n t e r e s t t o note t h a t information about electron-

pl?.onon coupling required t o understand the mgnitude of

superconducting t r ans i t i on temperature can be extracted d i r ec t l y

from measured phonon dispersion re la t ions empirically. The

remarkable pecu l ia r i t i es observed i n the phonon dispersion curves

in vmadiun, niobium and tantalum such as the softening of phonon

modes i n well-defined- ''reeion; of reciprocal space a re possible

sources of the l a t t i c e i n s t eb i l i t y which accompany changes i n

parameters essociated with high su?zrconductinfl t rans i t ion

temperature. . ..

The subject matter of t h i s t he s i s can be usefully applied i n

discuss ir,g s t r ~ c t u r a l phase t rculs i t ions . Many d-band metals and

t h e i r compounds undergo phase t r ans i t i on , It w i l l be of i n t e r e s t

t o extend the study of the s t ruc tura l phase t rans i t ion in vanadium

compounds: vanadium t i n ( v ~ s ~ ) , veradium tr ioxide (V 0 ), niobium 2 3 n i t r ide ( ~ b N), vsnadiun n i t r ide (VN) e t c which are of great

i n t e re s t due t o the i r high superconducting t rans i t ion tempereture.

In t h i s research, we have formulated the theory of the

a t t r ac t ive short-range ( ~ o r n - ~ a y e r ) force via s-d interaction

which is associated with the loca l f i e ld correction. This apporach

has been applied t o the simple closepacked body centered cubic

(bcc) structure. It is expected tha t t h i s research can be fur ther

extended t o the other simple closepacked structures such as the

face centered cubic ( fcc) and hex~gonal close packed (hcp)

structures.

We have used the pseudopotential theory t o generate the

Yukawa short-range force which is similar t o the existing force

i n nuclear physics. Therefore, it i s believed tha t t h i s new idea

should be the f i r s t stk$'%w&d~ the application of pseudopotential

theory t o nuclear physics, This s t r ik ing l i nk should -J* explored

i n de t a i l by experts in Solid s t a t e f ~ u c l e a r Physics.

In'conclvsion, we have r e p e n t e d local f i e l d correction as

an a t t rac t ive short-range component having the same functional

form in r e a l space as the Born-Mayer repulsion due t o overlap of

core electron wave functions centred on neighbouring ions, but of

an order of magnitude l a rge r t h ~ n the conventional Born-Meyer

potential , so t ha t the resul tant short-range interabomic potent ia l

is a t t rac t ive . We have provided a convenient method of including

locnl f i e l d co:*rection i n t he l a t t i c e dynamics of d-band metals.

Our procedure has greatly simplified the complicated method used

by O l i and Animalu (1976). The guarantee of symmetry 8t the

zone boundaries P ana H when our nmer i ca l resu l t s are compared

with experiments jus t i f ies the sound basis of our theory which

provides a c lear understanding of the interatomic forces associated

APPENDIX A ---- INVERSION O F 'lHE DTZLEC'IIHIC MATRIX

A basic problem i n the microscopic theory of screening is

t h e inversion of thrl dielectr ic matrix E'($+$, G+gt ) . We have sho

shown i n chapter 2 Sectiun 2.5 tha t the dielectr ic matrix is of

the form

which also take? th form

where

We can now carryY out a matrix expansion o f Eq. ( ~ 2 ) t o obtain the

This can be written in f'ull using Eqs . ( A ' ) and ~ 4 ) as

IBM Computer Program for the Computations of the

Born-Mayer Contribution t o tile I&namicctl Mstrix

i n bcc l ' ranci t ion Metals Vanadium, niobim and

tsntalm,

APPZm>IX C

rsM Computer Program for the Comp~Lationa of the

Electronic Contribution and T o t a l Phonon Frequencies

excludi:?g the Local field Correction i n the bcc

Transition Metals vanadium, niobium end tnntalm.

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APPENDIX D

IBM Computer P r o p m for the ~onycrtatione of the

Electronic Contribution, Total Phonon ~requencies and

Local. 1 l e l d Correction i n the bcc Transition

MOid~7.s 'Vewdu~, niob-hs-81ld tantalum.

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. . a

' ! . , ; - . ., i -! 7 - -

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BIOGRAPHICAL NOTE

BE3IIA.H NDUDIM ONWUAGBA hails from Nnakwe in I d d l i k c a l

Government Area of Anambra State. H e graduated B&. with Honour6

in Physics a t the V~!?vcrsity of' Nigeria, Nsukka in June 1975.

Innmediately af ter graduation, he served a year i n On& State under

the National Youth Service Corps ( A ,X.S .C. j scheme, when he became

the Senior Physics Mas ter et Doherty Memorial Gzrrnnnrtr Schol , Ijero-Ekiti. In August. 1976 he became the Senior Ph.;.sics Master

a t Federal Government G i r l s ' College Benin-City. In September

1978 he c&e t o Univprsity of Nigeria, Nsukka as the pioneer

reseasch student in Solid State Physics /Materials Science. But in

October ;YE0 he gr6duated the first M. Ph i l degree in Solid State

Fhysics/~mteria;~s Science p t t \ e Unjlversity of Nigeria, Hsukka*

fmmediately a f t e r graduation, he proceeded t o the Ph.D. degree

programs. His parents I& b Mrs ~ . b . Onwuagba are re t i red church

teachers. And he is married to Felicia Ngozi Oiwuagba and h a four , , . , ( +

children, Master Elochukwu Onwuagba, Mias Ijeoma Onwuagba,

Misa Chime Onwuagba, and Mies Ifeyinwa Onwuagba.