HAL Id: jpa-00217631
https://hal.archives-ouvertes.fr/jpa-00217631
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.
CALCULATED ELECTRON-PHONON COUPLING
AND SUPERCONDUCTING TC OF TRANSITITION
METALS : Mo AND Pd
F. Pinski, P. Allen, W. Butler
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, suppliment au no 8, Tome 39, aotit 1978, page C6-472
CALCULATED ELECTRON-PHONON COUPLING AND SUPERCONDUCTING
TC
OF
TRANSITITION METALS:
M O AND Pdl
F.J. pinski', P.B. ~llen* and W.H. Butlerf
x ~ e p a r t m e n t o f P h y s i c s , S t a t e U n i v e r s i t y o f New Y o r k S t o n y B r o o k , New Y o r k 1 1 7 9 4 ; USA
i ' M e t a Z and C e r a m i c s D i v i s i o n , Oak R i d g e N a t i o n a l L a b o r a t o r y Oak R i d g e , T e n n e s s e e 3 7 8 3 0 , USA.
R6sumL.- Les paramztres
X
et cl 'F (W) du couplage electron-phonon sont calculLes pour MO et Pd. Lavaleur obtenue
X
= 0,40 est en assez bon accord avec la temperature de la transition supraconduc-trice du MO. La valeur obtenue pour Pd est
X
= 0,41Abstract.- Electron-phonon parameters
X
and a2F(w) are calculated for MO and Pd. The value X =0.40for MO agrees well with superconducting T ; for Pd
X
is 0.41.The electron-phonon coupling parameters of metals are conveniently summarized in the function a 2 ~
(n)
a2~~n)=~(~)kz,vl~k:1
I ' ~ ( Q - R ; - ~ ~ ) ~ ( E ~ ) ~ ( E ~ ~ ) I
PP' CG(E~~s(E~~). ( 1 )
From this function, the coupling constant X and
the value of Tc cah be computed, as is well known /I/. Realistic numerical evaluation of this func-
tion for d-band metals is complicated because (1)
v
the electron-phonon matrix clement Mkk, must be
evaluated ~sin~complicated wave functions, (2) the
double summation runs over the Fermi surface which has a complicated topology, and (3) realistic va-
v
lues of the phonon frequencies $-kl must be taken from experiment. The only d-band metal so far stu- died in realistic detail is Nb for which four inde- pendent calculations have recently been reported /2,3,4,5/. In this paper we report calculations for
MO and Pd which use (1) the rigid muffin-tin sche-
me /6/ for the matrix elements, (2) state-of-the-
art KKR band theory for the wave-functions and Fermi
surface topology, and (3) experimental phonon fre- quencies /7,8/ for both elements.
The method of calculation is the same as re-
ference 141. First the phonon linewidth y calcu-
Qv lated on a mesh of points (728 in the irreducible 1148th of the Brillouin zone for MO, and 1378 for
Pd). The linewidth calculation was done using a
ly uniformly distributed over the Fermi surface (1227 in the irreducible 1148th for MO, and 676 for Pd). Then the linewidths were summed to get
a2F(C2) using a formula due to ~llen/9/, and a
Gilat-Raubenheimer computer code :
a2F(n>=(2~~(0) )-l C (y /C? . ) 6 (Q-ilw)
.
Q
v Qv Q 0 (2)The results are shown in figures 1 and 2. Also
shown are the density of states F(Q) (which was
calculated simultaneously) and their ratio,a2(Q)
Finally, the coupling constants X = 0.40 for MO
and 0.41 for Pd were found by integration of c12F (Q)
.
dense mesh of electron states (k,k') approximate-
X~upported in part by U.S. National Science Foun- Fig.] : Upper curve : Phonon density of st tes
dation Grant f f DMR 76-82946. F(u) for No vs frequency w in units of 10'' cps
:Research sponsored by U.S.D.O.E. under contract (THz)
.
Middle curve : electron-phonon spectralwith Union Carbide Corporation. function a2F(w). Lower curve : their ration,a2 (U).
References
Pig.2 : Same as figure I, for Pd.
The result
X
= 0.40 for MO is in close agre-ement with the value 0.41 deduced empirically McMillan /10/. When our value for the band density
of states, N(0)=4.74 states/(spin Ryd at), is mul-
tiplied by I + l= 1.40, the result is 19 percent
higher than the specific heat Other parameters
Y '
we have calculated are 0 = 5.40 ~ v / A * ,<w2>V2 =280 K,
and tolog= 265 K, using the definitions of Allen
and Dynes /]l/. Using the approximate Tc equation /lO,Il/
Tc = (m1 /l.2)exp[-l.04(1+~)/(~-p~-.62~p*)] 08
our calculations give good agreement with experi- ment (0.92 K), when u X is chosen to be 0. I I
.
The resulth= 0.41 for Pd is harder to inter-
pret, as Pd is not known to be a superconducter.
When our value N(O)= 15.5 states/Ryd is enhanced
by 1.41, the result is somewhat smaller than expe-
rimental y. An additional enhancement due to spin
fluctuations /12/
XSF=
0.34 would account for thediscrepancy in and the absence of superconducti-
vity. Other parameters we have calculated are II
= 2.15 ~v/A', <w2> @ = 165 K, and tolog= 130 K.
If we neglect spin fluctuations and use eq.(3) with
l.1*=0.13, the calculated value of Tc is 0.3 K.
/l / Scalapino, D. J., in Superconductivity,
R.D. Parks, editor (M.Dekker,N.Y.,1969)
/2/ Harmon,B.N. and Sinha,S.K.,Phys.Rev.
B16
(1977) 3919.
/3/ Peter,M.,Ashkenazi,J.,and Dacorogna,M.,
t1elv.Phys.Acta
50,
(1977) 267./4/ Butler ,W.H. ,Smith,H.G., and Wakabayashi,N,
.
Phys.Rev.Lett.
2
(1977)1004/5/ Varma,C.N. and Weber,W. to be published.
/6/ Butler,W.H., Olson,J.J., Faulkner,J.S. and
Gyorffy,B.L.,Phys.Rev. (1976) 3823.
/7/ Woods,A.D.B. an Chen, S.H., Solid State Comun. 2 (1964) 233.
-
/S/ Miiler,A.P. and Brockhouse, B.N.,Can.J.Phys.g (1971) 704.
/9/ Allen, P.B., Phys Rev.
B5
(1972) 2577./l01 McMillan, W.L. Phys.Rev.
167
(1968) 331./11/ Allen,P.B. and Dynes, R.D.,Phys.Rev.B
12
(1975) 905.
/l2/Gladstone, G., Jensen, M.A. and Schrieffer,J.R.
