HAL Id: jpa-00215647
https://hal.archives-ouvertes.fr/jpa-00215647
Submitted on 1 Jan 1974
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.
STRONG-COUPLING THEORY FOR DISORDERED SUPERCONDUCTORS
H. Takayama
To cite this version:
H. Takayama. STRONG-COUPLING THEORY FOR DISORDERED SUPERCONDUCTORS. Jour- nal de Physique Colloques, 1974, 35 (C4), pp.C4-299-C4-300. �10.1051/jphyscol:1974456�. �jpa- 00215647�
JOURNAL DE PHYSIQUE CoIloque C4, suppldment au no 5, Tome 35, Mai 1974, page C4-299
STRONG-COUPLING THEORY FOR DISORDERED SUPERCONDUCTORS
H. TAKAYAMA
Institut Max von Laue-Paul Langevin, 8046 Garching, Germany
R6sum6. - Nous Btudions l'interaction Blectron-phonon dans un metal desordonnk et en parti- culier la question, comment les traits caracteristiques de I'interaction sont modifies en l'absence de symetrie de translation. Dans l'etat supraconducteur, l'abaissement de symetrie entraine certaines modifications de 1'Bquation d'Eliashbgrg, qui est une Bquation de base pour les supraconducteurs dans le regime de couplage fort.
Abstract. - We study the electron-phonon interaction in a disordered metal, emphasizing on a problem how lack of a translational symmetry affects characteristic features of the interaction.
In the superconducting state the effect gives rise to modifications of the Eliashberg equation, which is a basic equation in the strong-coupling theory of superconductivity.
One of the typical examples of disordered metals is an amorphous film which is prepared by evaporation onto a cold substrate. These materials are particularly interesting since, by measuring tunneling current, we can obtain information about the product of the phonon density of state F(Q) and the absolute magni- tude of the electron-phonon interaction a2(Q) in these disordered systems [I]. In this communication we want to study the latter factor, i. e., the e!ectron- phonon interaction in a disordered metal, assuming its phononic property is already given.
For this purpose we expand as usual the electron- ion interaction He-' with respect to the ionic dis- placement operator S(X,) and write it down as
x 6 p p , - , + ,
s,
L 9 (1)where
b p -
C
e'k.xl - 4C
eik.xl +k - (2)
1 1
N is the ionic density c,f(c,) is the electron creation (annihilation) operator, v(k) is the electronic potential due to an indivisual ion, and e i is the polarization vector of phonons. In eq. (2), @ Q $- indicates the configuration average of a quantity Q with respect to equilibrium position of ions. The first and third terms are the parts proportional to
neity due to disorder, i. e., the inhomogeneous part of the interactions. Distinction of these two parts in the electron-phonon interaction has an importance in the present argument.
Given these electron-phonon interactions as well as the elastic scattering term due to disorder (the second term), we can calculate the electron self-energy parts straight-forwards, within the Migdal approximation (neglection of diagrams with crossing phonon pro- pagators) and within the so-called quasi-crystalline approximation (neglection of higher order correla- tions than the pair correlation of ionic distribution).
Effects of inhomogeneous part of the interactions are represented solely by the pair distribution function g2(k)
<
6pk 6p-k $.The Eliashberg equation, from which the strong- coupling theory of superconductivity is constructed, is a requirement of selfconsistency for the electron self-energy part expressed in the Nambu-Gorkov representation 121. It takes a form as
A
C(w) = oZ(w) - w - q(0)
GI ,
(3)where is the first Pauli spin operator, Z(o) is the renormalization function of the electron frequency, and q(w) is related to the superconducting gap function A(w) through A(o) = cp(w)/Z(w). The self-energy g(w) is estimated using the Green function which includes $(a) of eq. (3) itself. From the homo- geneous electron-phonon interaction (ordinary one), the self-energy shown in figure l a is obtained, and it is given by 121,
J --a
6
1
eik.'l % = Nbk,, ,
a1 n(Q)
+f
(o,)and we call them the homogeneous part of interactions. x
5-,
R + w + i o - w , The rest terms represent deviation from the homoge- X { N ~ I ) - I ( ~ I ) I:}
2 (4)Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974456
C4-300 H. TAKAYAMA
where terms in eq. (1). They are the processes of an inelastic
inhomogeneous scattering followed by an elastic R(o) = ~ e ' (
olJo2
- A2(o) , inhomogeneous scattering and vice versa. Sum of I(@) = Re(
d(o)/Jo2 - A2(o)) ,
these diagrams including other possible ways of and iz(0) and f ( o ) are the Bose and Fermi distri- combination gives rise to the contributionbution functions, respectively. The function a2(Q) F(Q) A
is called the Eliashberg function, and is given by C(s)(o) = -
m
where B~(q7 SZ) = - BA(q, - R, is the where the component proportional to 7, A as well as
of phonons. the Bose distribution function disappear.
From the above considerations we can see that in the study of disordered metals we need the following
- modification and reinterpretation of the ordinary
' . - A
'./
'7~~ Eliashberg eq. (3) to (5), which are the theoretical
( a ( u > ( c ; ( - 1
basis for the cc tunneling spectroscopy D, mentioned FIG. 1. - Electron self-energy parts. The solid, wavy and before. First, inclusion of eq. (8) breaks down the
broken lines are the electron Green function, the phonon Ordinary theoretical scheme, since it changes the
propagator and the pair distribution function g ~ ( k ) , respectively.
structure of the integral equations for Z(o) and The self-energy part shown in figure l b occurs from
the inhomogeneous electron-phonon interaction. It is given by the same functional form as eq. (4), but the corresponding Eliashberg function is now given by
a2(Q) F(Q) Ib =
To derive eq. (6'), we have introduced the electron elastic reIaxation time z by a simplest relation
where
{
dQor/
dR' is the integration with respect topcp(w). Secondly, even if we neglect this contribution, the reinterpretation of the Eliashberg function is of importance. Actually, when the Debye spectrum is substituted into B,(q, Q) at low frequencies, it is easily seen that a2(Q) F(Q)
I],
is proportional to Q2 as is the phonon density of states F(0) itself, whereas a2(Q) F(Q)1,
is proportional to SZ [3]. This indicates a change in a2(Q), i. e., the electron-phonon interaction in disordered metals. Relative importance of these correction terms is estimated from the ratio a2(Q) F(Q)la
to sr2(Q) F(Q)Ib.
It turns out that the corrections are dominant in the low frequency range given bywhere o, is the Debye frequency, q, the Debye wave number and 1 = v, z. Since q, 1 r 1
-
10' foramorphous metallic films measured, this range is certainly large. In the normal state (I(@) = 0 and R(o) = l), these correction terms give rise to the additional inelastic relaxation time of electrons z(o), or p' on the Fermi surface. Furthermore we obtain which has an identical expression to the one for an the self-energy corrections shown ,by figures lc and impure metal [4], within the approximation represented Id, from the combination of the second and fourth by eq. (7).
References
[I] KNORR, K. and BARTH, N., J. LOW Temp. Phys. 4 (1971) 469, and references cited in it.
[2] SCALAPINO, D. J., Supercondu~tivity edited by Parks, R. D., (Marcel Dekker Inc., New York) 1969, chap. 10.
[3] BERGMANN, G., Phys. Rev. B 3 (1971) 3797.
[4] TAKAYAMA, H., Z. Phys. 263 (1973) 329.
