Role of the intraatomic coulomb correlations on the energy of cohesion in narrow band metals

HAL Id: jpa-00208773

https://hal.archives-ouvertes.fr/jpa-00208773

Submitted on 1 Jan 1978

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Role of the intraatomic coulomb correlations on the energy of cohesion in narrow band metals

F. Kajzar, J. Friedel

To cite this version:

F. Kajzar, J. Friedel. Role of the intraatomic coulomb correlations on the energy of cohesion in narrow band metals. Journal de Physique, 1978, 39 (4), pp.397-405. �10.1051/jphys:01978003904039700�.

�jpa-00208773�

ROLE OF THE INTRAATOMIC COULOMB CORRELATIONS

ON THE ENERGY OF COHESION IN NARROW BAND METALS

F. KAJZAR

(*)

^{and J. FRIEDEL}

(**)

DPh-G/PSRM, CEN-Saclay,

^{B.P. n°}

2,

91190 Gif sur

Yvette,

^France

(Reçu

^{le 24 octobre}

1977, accepté

^{le 20 décembre}

1977)

Résumé. ²⁰¹⁴ L’énergie de cohésion d’une bande s étroite est calculée dans diverses approximations (Hubbard, CPA, Gutzwiller)

développées

^au second ordre par rapport ^{au terme} de Coulomb U/w.

Les résultats sont comparés ^{à ceux fournis} par la méthode de

perturbation

^en liaisons fortes à un

site. Les deux premiers ^termes (de ^{bande et} de Hartree Fock) ^sont identiques ^{dans toutes ces} approxi-

mations. Le terme suivant dans le développement, ^{dû aux} corrélations, donne des résultats un peu différents dans les diverses approximations. Une amélioration possible ^est discutée. Ces calculs sont faits _pour une bande symétrique ^et rectangulaire ^de largeur ^{w; mais ils ne} dépendent pas beau- coup de la forme de la bande.

Abstract. ²⁰¹⁴ The energy of cohesion for a narrow s band is computed

using

^various approxi-

mations (Hubbard, CPA, Gutzwiller) developed ^to second order in the Coulomb correlation ratio

U/w. The results are compared with that obtained by ^the

perturbation

method in the

single

^site tight binding approximation, ^{The two first terms} (band and Hartree Fock) ^are identical in all these approxi-

mations. The next term in the development, ^{due to} correlations, gives slightly different results in the various approximations. ^A possible

improvement

is discussed. Computations ^{are carried out for a} rectangular ^and symmetrical band of width _w, but are shown not to be sensitive to the form of the band.

Classification

Physics ^Abstracts

61. SOL

1. Introduction. ^-

Recently

^some papers have dealt with the role of the intraatomic Coulomb cor-

relations on the energy of cohesion in narrow band metals. In one of them

(Kajzar

^{and Mizia}

[1]),

^the

energy of cohesion was

computed

in the Coherent Potential

Approximation (CPA)

^{and the} appearence of a

lowering

^{in this} energy observed for 3d transition metals near a half filled band

(Cr,

^{Mn, see Gschnei-}

der

[2])

^was

interpreted

^as

being

^{due to the} strong Coulomb correlations, whereas for the other series

(4d, 5d)

^such

lowering

is less marked. Numerical calculations have been

performed

^{in two limits :} strong Coulomb correlations

(U > w)

^{and weak Coulomb}

corrélations (U « w),

^{where U is the} energy of Cou- lomb

repulsion

^{between two electrons on the same}

site with

opposite spins

and w is the band width.

Similar conclusions were also reached

by Sayers [3]

who

computed

^the energy of cohesion in Gutzwiller’s

approximation

for values of

U/w

^{between 0 and 2.}

Friedel and

Sayers [4]

calculated the _energy of cohesion,

developing

^it in powers of

U/w,

^{in Gutzwil-}

ler’s

approximation

up ^to second order.

They

^included

the d band fivefold

degeneracy, spin

^orbit

coupling

and

exchange

corrections

arising

from the fact that free atoms are

magnetic.

Their conclusion was that the

lowering

^{in the} energy of cohesion near a hall filled band is due to the

special stability

^{of free} magne- tic atoms with half filled shells

(Cr, Mn) ;

^{it is not due}

to the

strong

^Coulomb correlations, ^{if Coulomb} interactions between electrons of different d orbitals

are

properly

^taken into account ; the ^{term in}

U/w

^then

only produces

^a

general lowering

^of

cohesion ;

the têrm in

(U/w)2 produces

^a small increase of cohe- sion near to half filled d

bands, responsible

^{for anoma-}

lies in surface tension

[4],

^{lattice constants and atomic}

volumes

[5].

In this _paper, we compare the _energy of cohesion

computed

in the first Hubbard

approximation (FHA),

in CPA and in Gutzwiller’s

approximation

^{with the}

values obtained from the

perturbation

method in the

single

^site

approximation,

up ^to second order in

U/w.

^{We use for this}

comparison

^a

simple rectangular

and

symmetrical

^{s band. We check in one of the}

approximations

that the results are not very sensitive (*) On leave of absence from Institute of Physics and Nuclear

Techniques, University ^of Mining ^and Metallurgy, al. Mickiewicza

30, 30-059 Krakow, ^Poland.

(**) Physique des Solides ^- Laboratoire associé au CNRS - Université Paris Sud, 91405 Orsay (France).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01978003904039700

398

to the form of the band. Extension to d bands would be

straightforward

^{in the}

approximations

^used

by

Friedel and

Sayers [4].

^{Thus their} conclusions are not

dependent

^{on the use} of Gutzwiller’s

approximation.

2. Basic formulae. ^- 2.1 GUTWILLER’S APPROXI- MATION. ^- In Gutzwiller’s

approximation [6],

^the

energy of cohesion _per atom for an s band in a para-

magnetic ^metal, using

^a

rectangular symmetrical

band

(cf. Sayers [3],

Friedel and

Sayers [4])

^is

^equal

^to

where n is the number of electrons per ^atom.

2.2 PERTURBATION METHOD IN SINGLE SITE APPROXIMATION. ^- The band term

t

^{w and}

the Hartree Fock correction

- 2 U n 2

^are

obtained. .

in the usual way

[4,

^7,

8]. They

^{are the same as in}

equation (1).

The second order term can be obtained

taking

^the

Fock functions and

projecting

^{them on the real one}

electron states

where

t/lki(rj)

^{is a Bloch} function with _energy

Ei, k,

^and

k3

^{have one}

spin direction, k2

^and

k4

^{the other.}

Expanding

^{the Bloch} functions into atomic wave functions with the

tight binding approximation

where

f (ri - ^R)

is the atomic

(s)

function centred on site _n, and if

e2/r12 only

^{acts within one}

site, expression (2)

can be written

In the

single

^site

approximation [9]

^where

only

^{the term}

Rn

^{= 0 is} considered in the summation _{over n,} one

obtains

For the

rectangular density

^{of states function}

this is

where

Thus the _energy of cohesion can be written as follows

F(n)

^{has the}

following properties :

These values are very close to those for the

correspond- ing

^{term in} Gutzwiller’s

approximation (formu-

la

(1)) :

^:

with

’

The lower limit of

F(n)

determined from the condi- tion :

E3

⁺

E4 - El - E2

^{2 W is}

equal

^{to one}

half of

G(n).

As the

single

^site

approximation neglects

^some

positive

^terms in the contribution in

U2/W

^{to cohesive}

energy,

F(n) gives

^a lower limit to the exact value of the second order coefficient. Gutzwiller’s

approxima- tion,

^{which is a} variational method based on the

single

site

approximation, gives

^an upper limit to the

single

site term. The

comparison

^{above shows} that Gutzwil- ler’s

approximation gives

^a

^good approximation

^{to the}

single

^{site term.}

2.3 FIRST HUBBARD APPROXIMATION. ^- The one

electron Hamiltonian of Hubbard

[10]

^{for an s band in}

single

^site

approximation

^{has the}

following

^{form :}

where

tij

^{are the} elements of the

scattering

^{T matrix.}

a:’, aj,,

^are

respectively

creation and annihilation

operators

^{of one} electron in site

i( j)

^with

spin

^Q.

U energy of Coulomb

repulsion

^{as before.}

ni

⁼

a,-, ai,,

^{is the} operator ^{of the} number of elec- trons with

spin

^{Q on site i.}

Eo

^{is the one} electron atomic _energy level.

In the first Hubbard

approximation (with

^an appro-

priate

choice of the

origin

^of

energy),

^{the coherent}

potential E O(z)

^{has the}

following

^form

[10]

where

n-,, = ni-,, >

^is the average number of electrons of

spin -

^Q.

For the

rectangular

^and

symmetrical density

^of

states function for uncorrelated electrons

the

corresponding

Slater-Koster functions has the

following

^{form :}

The Slater-Koster function for the system ^{with corre-} lations is obtained in the usual _{way :}

and the

corresponding density

^{of states function ,}

where the limits of

nonvanishing density

^{of states are}

determined from the

following

^{set of}

equations

which

gives

where

400

The first term in

ô’ (Eq. (8))

^which

gives

^{the shift of}

the energy of electrons due to the Coulomb correla- tions is

just

the Hartree-Fock energy ^{as we}

expected.

It is seen that even at very weak Coulomb correla- tions a band

splitting

^{d = U -}

(be

⁺

b2)

^occurs.

The energy of cohesion

(Friedel [8])

^is

given by

^the

difference between the

energies

^{of electrons in the}

atomic and solid states

where n ^- number of electrons in the band

Eo

^- energy of electrons in the atomic state.

The correction term CT corrects for the fact that the sum of one electron

energies

^{counts twice the}

électron-électron

interactions.

Choosing

^the

origin

of energy in such ^a way ^{as to}

get Eo

^{= 0 the} energy of cohesion will be

given by following

^formula

where with

ôe given by

eq.

(20).

^The

Fermi energy

Fp

^{will be}

always

ⁱⁿ the first band if we use the

particles

^holes symmetry ^{and can be deter-} mined from the condition that the total number of electrons

(holes)

^{in the} band is conserved

This condition

together

^with first of eq.

(19)

^and

(20)

yields

^from eq.

(22).

^,

or in the

paramagnetic

^state

(na

⁼ n - a ⁼

n/2)

^{for two electrons in the band}

In formula

(24)

^{we have taken one half in the sum over} interaction _{energy in} order to avoid

çounting

^the energy of electron-electron collisions twice.

. 2.4 COHERENT POTENTIAL APPROXIMATION. ^- In CPA the coherent

potential Eff(z)

is obtained from the condition that the average value of the T matrix vanishes. This

yields

^the

following equation for E’ff(z) (Soven [11], Velicky et

^al.

[12])

This

approximation applied

^{to metals is}

equivalent

^to the Hubbard

alloy analogy [13] (see

e.g.

Velicky

et al.

[12J, Fukuyama

and Ehrenreich

[14],

^Brouers and Ducastelle

[15]).

^{For a metal}

(cf. ^Kajzar

^{and Mizia}

^{[ 1 ],} Velicky et

^al.

[12],

^Mizia

[17])

^the

probabilities Pia

ⁱⁿ eq.

(26)

^are

equal

^to

and the centres of

gravity

of the bands

From eq.

(14)

^{we obtain}

following

solutions for the coherent

potential Z "(z)

which for

U/w

¹

yield

From these ^two solutions

depending

^on

F«(z), only

^the first verifies the well known condition :

and has a

physical meaning.

For the

rectangular density

^{of states}

po(E) (Eq. (14))

the Slater-Koster function has the form

(see Eq. (16))

where

The band boundaries are determined now

by

^{the condition Im}

IO’(z)

^{= 0 which}

yields

and the

density

^{of states is}

given by

At the bottom of the band :

where we have retained

only

^{first terms in}

U/w.

With U w a lower limit of the

logarithm

_g

(in (

^absolute

value) )

^{is In}

^3,

^thus

Fa a( (EÎ) cn - i)

ln 3. This would

w

give

^{for the} energy of cohesion calculated from _eq.

(22) using

^condition

(23)

or in the

paramagnetic

^state

The

expressions (35)

^and

(36)

for the cohesive _energy differ from the

corresponding

^{ones in FHA}

(Eqs. (24)

and

(25)) by

^{a factor}

equal

^to

2(1 - n

^{+ ln}

3) present

^{in the third term}

(proportional

^to

U2/w2).

^{Thus this term}

takes lower values than the

corresponding

^{one in FHA} and is characterized

by

^{a different}

dependence

^{on n,}

similar to that of

perturbation

and Gutzwiller’s terms

(see Fig. 1).

^{However we} have taken a lowest estimation of

F(f(Ef) (in

^absolute

value).

^With

decreasing

^{value of}

U, ) r F(f(Ef) increases

^{and this term becomes}

larger.

^{In the}

limit u --+

0 1 F(f(Ef)) --+

^{oo but the}

product UF(f{Ef)

^{tends to zero}

assuring

the convergence of the

expansion

^for

,E’ff (z) (formula (30)).

3. Discussion. ^- In all the

approximations

^used

^here,

^the

development

of the cohesive energy with

U/w

can be written as follows

402

FIG. 1. ^- A(n) for different approximations ^{for a} rectangular ^band (heavy line-FHA, dotted line-Gutzwiller’s approximation, light

line-CPA, ^broken line-perturbation).

The band term

n(2 - n)/4

and the Hartree-Fock term -

n’14 U/w

^are

identical,

^as

expected.

Figure

1 compares the variation of

A(n)

^{with the}

filling

of the band. It is seen that Gutzwiller’s _appro- ximation and

perturbation give

very similar behaviour

as was shown in formulae

(10)

^and

(11).

^{CPA also}

gives

^a similar value for

A(n)

^{but the values are about}

twice as

large.

^In

reality

^the différence is more

impor-

tant because of the _presence of

Fu(Ei)

in formula

(30),

which increases with

decreasing

^{U. For the semi-}

circular band

(cf. Appendix)

^{where that}

problem

does not appear this term is 1.5 times greater ^{than for a}

rectangular

^{band. FHA}

approximation gives

^values

for

A(n)

greater ^than Gutzwiller’s

approximation

and

perturbation (about

four times for n ⁼

1).

^Also

FHA’s terms show a different

dependence

^of

A(n)

^{on n,}

than the other

approximations.

^{The first is a}

poly-

nomial of odd order

in n,

^while the other is

éven.

As a

result,

the cohesive _energy, shown in

figure 2, depends slightly

^{on the}

approximation

^used

(the

differences

being

^{small at} low values of

U).

FIG. 2. ^- Energy of cohesion for different approximations ^{for a} rectangular ^{band and} U/w ^{= 0.5} (heavy line-FHA, dotted line- CPA, light line-Gutzwiller’s approximation, ^broken line-pertur-

bation).

’All

the

approximations

^{used are one site}

approxima-

tions, i.e. assume non conservation of momentum in electron-electron

collisions,

^or

again,

^each

moving

electron scattered

by

the other electrons considered

to be

fixed,

^{as if their} masses were infinite. As

empha-

sized

above,

^{the exact one site second order} pertur- bation

gives

^{a lower limit to} the cohesive _energy.

The Gutzwiller

method,

^{which is a} variational

method, gives correctly

^a somewhat smaller cohesive energy than the exact second order

perturbation.

^In contrast, the Hubbard and CPA

approximation give

^cohesive

energies

^{which are}

definitely larger

^{than the exact}

second order

perturbation.

This last result is due to the fact that the Hubbard and CPA

procedures

^{are not} variational

approxima-

tions on the _energy. Moreover

FHA,

^as

usually applied,

shows a lack of

selfconsistency

^{which can} be corrected for. Such a correction

strongly

reduces the discre- pancy with the exact second order

perturbation

^result.

4. Self consistent Hubbard

procédure.

^{- Because of}

correlation,

^{the coherent}

potential depends ^on

^the

energy, ^{as seen} from formulae

(16).

^{However in the}

usual

procedure,

^this

change

alters the real part ^of

F,,,(z)

^without

altering

^its

imaginary

part. ^{Thus the}

Kramers-Kronig dispersion

^{relations are not} pre- served.

It is in fact clear that the

change

^with energy of the coherent

potential

alters the

density of

^{states as well}

as the limits of the

band, by changing

the scale of energy.

To each uncorrelated one electron state with _energy E

corresponds

^a correlated state with _{energy E«}

given by

To take into account the

change

^of energy

scale,

one should then write for the cohesive _energy

(with Eo = 0) :

where the summation is ^over the uncorrelated ener-

gies .E,

^{up to} the uncorrelated Fermi level

EF given by

and

E"(E)

^as

given by (38)

^{are the actual one electron}

energies

^{of the} correlated system.

In the first Hubbard

approximation (13),

eq.

(38)

gives

The solution with the minus

sign gives

^{rise to the elec-}

trons band the solution

with the

plus sign

^to the holes band ^{For a}

paramagnetic

metal with the

symmetrical

and

rectangular

^band

(14),

^this

gives

^{for n}

1,

^thus

with

This somewhat

complex expression simplifies

^{in two limits :}

a) n «

^{1 : a}

development

ⁱⁿ

U/EF

^{is valid. It}

gives

For n - 0, the second order term is close to that in Gutzwiller’s

approximation.

b) n

^{= 1 : a}

development

ⁱⁿ

U/w gives

The second order term is thus

replaced by

^a

logarith-

mic term, which however ^takes values similar to those of the two first

approximations

^{for 0.1}

U/w

^0.5.

The cohesive _energy is

symmetrical for n ;

^1.

Thus the _energy of cohesion calculated in this modified FHA

procedure, taking

^{into account}

changes

in the

density

of states, is closer ^to Gutzwiller’s and

perturbation

^{for 0.1}

U/w

^0.5.

Appendix.

^{- For} the semi-circular

density

^{of states}

for a

noninteracting

system

the

corresponding

Slater-Koster function is

For the system ^with correlations

(cf.

^Section

II)

In FHA

developing

^{z -}

IO"(z) given by

eq.

(13)

^into

powers series of

Ulw

^{we obtain}

Contrary

^{to the case with}

rectangular

^band up ^to second order of

U/w,

^{the band is not}

split

^{and the}

limits of the

nonvanishing density

^{of states are deter-}

, mined from the condition

analogous

^to eq.

(18)

which

gives

and

where

The lower limit

Ei

^is

exactly

^{the same as that for rec-}

tangular

^band

(formula (19)).

From eqs.

(A. 6)

^and

(A. 7)

one can see that the centre of

gravity

^{of the band with}

spin J

is shifted

by

the value

of t5!.

404

Proceeding

^{in the same manner as before we obtain}

for the energy of cohesion the

following expression

where

is the band term.

In the

paramagnetic

^{state eq.}

^{(A. 8)} gives

This differs from the

corresponding

^{one for a rectan-}

gular

^band

only

^{in the band term which for a semi-}

circular band

(see Fig. 3)

^{is somewhat smaller and in}

the third term. In the last term the difference is also very

small ; Efl

takes the values from 0 for n ⁼ 0 to 0.212 for n ⁼ 1

(for

^a

rectangular

^{band the corres-}

ponding

value for n ⁼ 1 is

0.25).

^{Because of the}

nearly parabolic

behavior of

El

^on number of elec- trons n

filling

^{the band}

(cf. Fig. 3),

the behaviour of

FIG. 3. ^- The band term for a rectangular ^band (heavy line) ^and semi circular band (light line).

this term on n will be also

nearly parabolic

^{as in the}

case of Gutzwiller’s

approximation

and in the method of

perturbation

^for

rectangular band,

but with values

nearly

^{two times}

greater (for n

⁼

1).

In

CPA, only

^{the first solution}

for Eo(z) (Eq. (30)) gives

^{non zero}

density

^{of states. Thus} the band is also translated and not

split. Using

this solution we obtain

where

As before we get ^{for the} energy of cohesion

or in the

paramagnetic

^state

where

Efl

^{is as} before the band term.

Expression (A. 13)

differs from the

corresponding

one

(Eq. (A. 10))

^{in FHA}

^only

^{in the third term pro-}

portional

^to

U’lw’

^{which is 1.5 times} greater ^{in the} last case. The difference between this term and the

corresponding

^{one for a}

rectangular

^band

(Eq. (36))

is more

important (by

^a factor of about 1.5 for n ⁼

1).

This difference arises

mainly

^{from the} fact that we

FIG. 4. ^- Energy of cohésion for U/w ⁼ 0.2 -calculated by ^the

method of perturbation ^{with a} rectangular ^band (heavy line) ^and

in FHA (light line) ^{and CPA} (dotted line) using ^a semi-circular band.

have taken

(in absolute)

the lowest value for

FQ(El).

This value

depends

^{however on U} and n and should be greater. The difference in the band term

(similar

as in

FHA)

^{is small}

(see Fig. 3).

In

figure

^{4 the} energy of cohesion calculated for a

semi-circular band in FHA and in CPA for

U/w

^{= 0.2}

is shown and

compared

with that obtained

by

^the

method of

perturbation

^{for the same} value of Cou-

lomb correlation _energy. The behaviour of

Fe(yt)

is

nearly

^{the same. The} differences are most

important

for n ⁼ 1

(but

^{do not exceed 15}

%)

and arise from the band term which is greater ^{for the}

rectangular

^band.

Acknowledgements.

^- One of the autors

(FK)

would like to thank Dr. Ducastelle for useful discus- sions.

References

[1] KAJZAR, F. and MIZIA, J., ^J. Phys. F, ^Metal Phys. ⁷ (1977)

1115.

[2] GSCHNEIDNER, K. A. Jr, Solid. State Phys. ¹⁶ (1964) ^275.

[3] SAYERS, C. M., J. Phys. F, Metal Phys. ⁷ (1977) ^1157.

[4] FRIEDEL, ^{J. and} SAYERS, ^C. M., ^J. Physique ³⁸ (1977) ^697.

[5] FRIEDEL, ^{J. and} SAYERS, ^C. M., ^J. Physique ^{Lett. 38} (1977) L-263.

[6] GUTZWELLER, ^M. C., Phys. ^{Rev. 137} (1965) ^1726.

[7] FRIEDEL, J., ^J. Physique ^{Radium 16} (1955) ^829.

[8] FRIEDEL, J., The Physics of Metals I, Electrons, ^{ed. J. M. Ziman} (Cambridge University ^Press, Cambridge) ^1969.

[9] HODGES, L., EHRENREICH, ^{H. and} LANG, ^N. D., Phys. ^Rev.

152 (1966) ^525.

[10] HUBBARD, J., ^{Proc. R.} Soc. A 276 (1963) ^238.

[11] SOVEN, P., Phys. ^{Rev. 156} (1967) ^809.

[12] VELICKY, B., KIRKPATRICK, ^{S. and} EHRENREICH, H., Phys.

Rev. 175 (1968) ^747.

[13] HUBBARD, J., Proc. R. Soc. A 281 (1964) ^401.

[14] FUKUYAMA, ^{H. and} EHRENREICH, H., Phys. ^{Rev. B 7} (1973)

3266.

[15] BROUERS, F. and DUCASTELLE, F., J. Physique ³⁶ (1975) ^851.

[16] VEDYAEV, A. V., KONDORSKI, E. J. and MIZIA, J., Phys. ^Stat.

Sol. (b) ⁷² (1975) ^205.

[17] MIZIA, J., Phys. Stat. Sol. (b) ⁷⁴ (1976) ^461.