Theoretical calculation of Tc for lead

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Theoretical calculation of Tc for lead

A.D. Zdetsis, E.N. Economou, D.A. Papaconstantopoulos

To cite this version:

A.D.

Zdetsis,

E.N.

Economou,

D.A.

Papaconstantopoulos.

Theoretical

calculation

of

Tc for lead.

Journal de Physique Lettres,

Edp sciences,

1979,

40 (12),

pp.253-255.

�10.1051/jphyslet:019790040012025300�. �jpa-00231619�

L-253

Theoretical calculation

of

_Tc

for lead

_{(*) (**)}

A. D.

_Zdetsis,

E. N. Economou

_(***)

N.R.C. _{Demokritos, Aghia Paraskevi, Attikis,} Greece

and D. A.

_{Papaconstantopoulos}

Naval Research _{Laboratory, Washington, D.C.,} U.S.A.

(Re~u le 10 aout 1978, revise le 2 avril _{1979, accepte} le 24 _{avril 1979)}

Résumé. 2014 Nous calculons la

température de transition

_{supraconductrice}

du Pb avec la théorie de

_Gaspari

et

Gyorffy, en utilisant nos calculs ab initio de structure de bande. La valeur calculée ainsi est la moitié de la valeur

expérimentale.

Cet écart est attribué à

_{l’approximation}

du

_{potentiel muffin}

tin

_rigide.

Pour tenir _compte de cette

approximation

nous

_présentons

une méthode

_simple,

sans _{paramètre ajustable.} La nouvelle valeur calculée de la

température de transition est en accord excellent avec la valeur

_{expérimentale.}

Abstract. 2014

Using

our ab initio band structure results, we calculate the transition _temperature of Pb

_according

to the

_{Gaspari-Gyorffy}

_theory. The calculated value is half the

_experimental

one. The

_discrepancy

is attributed

to the

_rigid

muffin tin

_{approximation.}

A

_simple

method with no

_adjustable

_parameters is

_developed

to account for this

_{approximation.}

The new transition _temperature is in excellent _agreement with

_experiment.

LE JOURNAL DE PHYSIQUE - LETTRES _TOME

40, 15 JUIN 1979,

Classification

Physics Abstracts

74.55 - 74.70G - 74.90

Following

the work of McMillan

_[1],

which relates in a

_practical

_{way the}

superconducting

transition

temperature

T~

to

_{microscopic quantities,}

a

_large

amount of research has been

_{reported along}

those lines.

In

_particular,

the

_theory

of

_Gaspari

and

_Gyorffy

_[2,

3],

which we

_employ

here,

was

_{designed especially}

for transition metals.

_However,

as we shall illustrate

below,

this

_theory

_properly

corrected can

_give

_very

good

estimates of the

_{electron-phonon}

interaction ’1

for normal metals like lead as well. The _parameter ~,

which

_together

with the _average

_phonon

_frequency

w2

~, [1, 4]

defines

_7~,

can be

_expressed

in the

Gas-pari-Gyorffy

theory

in terms of

_partial

densities of

states and

_phase

shifts

readily

available from tradi-tional band structure calculations.

In the first _stage of this work we have

determined,

within the

_Rigid

Muffin Tin

_{Approximation}

_(RMTA),

the

_{quantity ’1 using}

our self-consistent ab initio band calculation. The latter includes relativistic

effects,

such as the mass

velocity

and Darwin

corrections,

---(*) Partly supported by NSF _grant No. DMR 76-19458 at the

University of Virginia.

(**) This _paper was _presented at the LT 15 Conference as a

post-deadline paper.

(***) On leave from the Dept. of Physics, University of _Virginia,

Charlottesville, Virginia 22901, U.S.A.

which for a

_heavy

metal such as Pb are

important

_{[5, 6].}

The

_spin

orbit interaction has been

neglected

because,

although

it is _very

_important

for the structure of the Fermi

surface,

is not

_expected

to be _very

_significant

for the

_{electron-phonon}

interaction,

and

consequen-tly

for

_7~,

which is our

_primary

concern here. Details of our band structure

_calculation,

_together

with

comparisons

with

_previous

calculations and

experi-mental data are

_published

elsewhere

_[7].

The total and

_partial

densities of _{states nt}

_{and nl}

Fig. 1. - Electronic

density of states (DOS) of Pb and its s-,

p-and d-like components.

L-254 JOURNAL DE _{PHYSIQUE -} LETTRES

respectively,

which are calculated

_by

our modified

QUAD

method

_[8],

are shown in

_figure

1. The same

quantities

evaluated at the Fermi level

_EF,

_together

with the

_scattering

_phase

shift

_bl,

the mass

enhance-ment A and the free scatterer

_partial

densities of states

nfl)

are listed in table I. We have calculated

_7~

accord-ing

to the Allen and

_Dynes

equation

[4]

with

_M

_{c~2 ~}

the

_phononic

contribution to

_A,

taken from their _paper. The Coulomb

_{potential /1*}

was evaluated

_by

the Benne-mann and Garland formula

[9].

The

experimental

value of

_A,

measured

_through

_{tunnelling experiments}

(see

refs.

_{[4, 10])}

is twice as

large

as our ab initio value calculated here.

This

_discrepancy

must be due to the inherent appro-ximations of the

_{Gaspari-Gyorffy}

(GG) theory;

namely

the

_spherical,

the local and the

_rigid

muffin tin

_{approximation [2,}

_3,

_{11 ].}

The first two

approxima-tions are not

_expected

to be

_important

for lead. The

spherical approximation

is known to be well satisfied for cubic

_crystals

and I =

0, 1,

2

[12].

The local

_{approximation}

for lead is also

_expected

to be well

_obeyed

due to the

_large

contribution of

Umklapp

electron-phonon

scattering [11, 13-15].

It has been demonstrated

_{[11-13, 14]}

that the local

approximation

is better satisfied for

_large

_Umklapp

scattering.

Furthermore,

the

_validity

of the local

approximation

can be also verified from the

similarity

of the

_phonon

_density

of states curve,

F(w)

and the

a2(c~)

F(co)

profile

[11].

The

_Eliashberg

function

_{OC2 ((o)}

_F(c~)

defined in

ref.

_{[1] has}

been determined

_{experimentally}

_by

super-conducting tunnelling

measurements

_(see

refs.

_[4,10]).

Thus,

the

_discrepancy

between the observed and calculated values of À and

_Tc

must be

_mainly

attributed

to the

_rigid

muffin tin

_{approximation}

_(RMTA).

The

validity

of RMTA

_depends

on the

importance

of the

tails of the self consistent

_potential

about the

_edge

of the muffin-tin

_sphere.

For transition

metals,

where

the

_scattering

of electrons is dominated

_by

the atomic

potential

inside the muffin tin

_sphere,

this

approxima-tion is

_expected

to be

_quite

_good.

For

_lead,

_however, one must _go

_beyond

this

approxi-mation

_{by adding}

a correction term to the RMTA

potential.

On

_physical

_grounds

this correction term

Vc(r)

is

_expected

to have a _strong influence near the muffin-tin

radius,

rMT, and decrease

rapidly

away from the muffin-tin

_sphere.

Therefore a

_plausible

choice for

_V~(r)

is

where qTF is the inverse of the Thomas-Fermi

screen-ing length

and

_Vo

is an as yet unspecified parameter which denotes the

_depth

of the correction

_potential.

To determine

_Vo

without _any resort to

_experimental

information,

we have

_adapted

the

equivalent

RMTA

electron-phonon pseudopotential

of Lee and Heine

_[16]

which has been

_expressed

in terms of the same quan-tities that appear in the GG formula

_[11].

This

pseudo-potential

in _{q-space has} the form

_[11]

_]

where

_kF

is the free electron radius of the Fermi

sphere

and

_P,

is the

_{Legendre polynomial}

of order I.

The rest of the

_symbols

have been defined before in the

calculation of fl.

The sine term in the above

_expression

and

equiva-lently

in the GG formula was obtained

_by

evaluating

the radial matrix element of the derivative of the RMTA

_potential

_[3].

_Obviously

the addition of the

potential

(1)

into the RMTA

_potential

will result in

replacing

the sine terms above

_by

Table I. - Calculated densities

of

states

_{(total n~}

and

_{partial n,)}

at the Fermi level and other basic

_quantities

entering

the

_{formula for Tr.}

Notation

_according

to the text and

_{references [1-4].}

L-255 THEORETICAL CALCULATION OF _T, FOR LEAD

The

_{terms AV,,,,,,}

_being

the radial matrix elements of the derivative of

_(1),

are

_expressed

in terms of

_Vo.

The

_{depth Vo}

can be determined then

by

demanding

that the new

_{pseudopotential}

Vep(q)

has the correct

limit

_[17,

_{18] : -}

_{Z/2 nt(EF)}

as _q

_approaches

zero. In

_figure

2 we have

_plotted

both

_VRMTA(q)

and

Yep(q)

as a function of

_ql2

_kF.

The

_deficiency

of the

RMTA can be seen from the fact that the RMTA

pseudopotential

reaches about half the correct value

as q

approaches

zero.

The new renormalized values of

n*,

~* and

_7~*

are listed in table I. The _agreement with the

experi-mentally

determined value of A is excellent.

Good _{agreement with}

_experiment

have been obtain-ed before

_[11,

15,

19-22] using empirical

pseudopo-tentials. In the _{present approach} the

_{pseudopotential}

was determined from our first

_principles

band

struc-Fig. 2. - The RMTA (broken lines) and the corrected,

_Vep(q),

(solid lines) pseudopotentials for lead.

ture calculation and its

limiting

value _{as ~ -~}

0,

without _any resort to

_experimental

data.

References

[1] MCMILLAN, W. L., Phys. Rev. 167 ₍₁₉₆₈₎ 331.

[2] GASPARI, G. D. and _GYORFFY, B. L., Phys. Rev. Lett. 29

(1972) 801.

[3] EVANS, R., GASPARI, G. D. and GYORFFY, B. L., J. Phys.

F 3 (1973) 39.

[4] ALLEN, P. B. and _DYNES, R. C., Phys. Rev. B 12 ₍₁₉₇₅₎ 905.

[5] MATTHEISS, L. F., Phys. Rev. 151 (1966) 450.

[6] KOELLING, D. D. and HARMON, B. N., J. _Phys. C 4 (1971) 2064.

[7] PAPACONSTANTOPOULOS, D. A., ZDETSIS, A. D. and

ECONO-MOU, E. N., Solid State Commun. 27 ₍₁₉₇₈₎ 1189.

[8] MUELLER, F. M., GARLAND, J. W., COHEN, M. H. and

BENNE-MANN, K. H., Ann. _Phys. 67 ₍₁₉₇₁₎ 19.

[9] BENNEMANN, K. H. and _GARLAND, J. W., AIP Conf. Proc. 4 (1972) 103.

[10] GRIMVAL, G., Phys. Scr. 14 _{(1976) 63 ;} and references therein.

[11] RIETSCHEL, H., Z. Phys. B 30 (1978) 271.

[12] JOHN, W., J. Phys. F. Metal Phys. 3 (1973) L 233.

[13] RIETSCHEL, H., Z. _Phys. B 22 ₍₁₉₇₅₎ 133.

[14] WINTER, H. and RIES, G., Z. _Phys. B 24 (1976) 279.

[15] YORK, E. D. and _{BARDASIS, A.,} Lett. Nuovo Cimento 2 ₍₁₉₇₁₎ 1228.

[16] LEE, M. J. G. and HEINE, V., Phys. Rev. B 5 ₍₁₉₇₂₎ 3839.

[17] ZIMAN, J. _M., Electrons and Phonons _(Clarendon Press, Oxford)

1967.

[18] HEINE, V., NOZIERES, P. and _WILKINS, J. _W., Philos. _Mag. 13 (1966) 741.

[19] ALLEN, P. B. and COHEN, M. L., Phys. Rev. 177 ₍₁₉₆₉₎ 704.

[20] CARBOTTE, J. P. and _DYNES, R. C., Phys. Rev. 172 (1968) 476.

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Can. J. Phys. 47 (1969) 1107.

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