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Submitted on 1 Jan 1978
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FUNCTIONAL DERIVATIVE OF Tc WITH N(ϵ)
S. Lie, J. Daams, J. Carbotte
To cite this version:
JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-468
FUNCTIONAL DERIVATIVE OF Tc WITH N ( e )+
S.G. Lie, J.M. Daams and J . P . Carbotte
Physios Department, McMaster University, Hamilton L8S 4M1 Ontario, Canada.
Résumé.- La dérivée fonctionnelle de la température critique en fonction de la dépendance en énergie de la densité d'états électroniques est calculée avec les équations de Eliashberg. Pour plusieurs supraconducteurs nous trouvons que la dérivée fonctionnelle décroît rapidement avec l'énergie mesurée du niveau de Fermi sur une échelle de quelques meV.
Abstract.- The functional derivative of the critical temperature with energy dependence in the elec-tronic density of states has been computed from the full Eliashberg equations. For the several super-conductors considered it is found to be a rapidly decreasing function of energy away from the Fermi energy on the scale of meV.
Sharp structure in the density of electron states(N(e)) as a function of energy has been sug-gested as a possible reason for the large T values observed in the A15 compounds/1/. To study this pro-blem we have computed the functional derivative 6T /6N(e) from the full Eliashberg gap equations/2/, suitably generalized, so as to include approximately /3/ an energy dependent N(e). They are
W =0) + TTT £A(to -iy )(signoi )N(|5 |) (1)
n n m n m m ' m
A((D ) „
A(oo ) = TTT 2{A(w -a) )-uK} — N(|5 |) (2) n m n m i ~ i . m
K«
+p
with T the temperature, u) the Matsubara frequencies ito = i(2n-l)irT , n - 0, ±1, ±2,... (3) n and with CO A(to ) = 2 dm to a2(o))F(c)) (4) n I a/+ aiz
S(|ul> - f d e - i ^ L I(£L.
(5
)
nThe kernels a2(io)F(0)) and \i in these equations are available for a large number of materials from tun-neling measurements/4/. Finally, p is the pair brea-king parameter and
6T 6p <dp'
£-= / — (6) 6H(e) 6N(e) ^dljT
To compute (6) we use in(l)-(5) a constant density of states to which a delta function of infi-nitesimal weight is added at a specified energy e. Results for N(0) x 6T /<5N(e) as a function of e/T
c c are given in figure 1 for various materials. All
curves show the same characteristic rapid decrease Research supported by the National Research Coun-cil of Canada. 120 v o 0 5 jl 8 0
- \ A -oa> V S > ^
\ A. -025 60 \ *Bl -0301 ' • ' • , . \ 50 100 150 200 250 ° 0 5 10 15 20 25%
&T « c Fig. 1 : Functional derxvative x N(0) as as . 6 N ( e )
function of — , e=0 is the Fermi surface : Ta, Sn, Tl, -.-.-In, -..-..- Nb,
Hg, lower Pb, lower Nb3Sn. The insert shows the negative part of the curves, the three uppermost curves are Nb3Sn, Pb and Hg.
as a function of energy away from the Fermi surface. Reference to table I shows that, in all cases consi-dered, it has dropped to half its maximum value for ^ 8(T ) . Beyond 20, structure in N(e) will clearly not be very influential on T . At yet higher ener-gies ST /6N(e) can be negative as shown in the in-sert on figure 1. While this region is not numeri-cally very significant it is sufficient to explain the maximum in T found by Horsch and Rietschel/3/.
c
Additional calculations based on different shapes for N(e) leave the functional derivatives al-most unchanged in shape.
Finally we note that the area under the func-tional derivative curve gives c i.e.
dN(0)
