Two-band superconductivity in Pb from ab initio calculations
A. Floris,
1
A. Sanna,
2
S. Massidda,
2
and E. K. U. Gross
1
1
Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany
2
INFM SLACS, Sardinian Laboratory for Computational Materials Science and Dipartimento di Scienze Fisiche,
Università degli Studi di Cagliari, S.P. Monserrato-Sestu km 0.700, I-09124 Monserrato (Cagliari), Italy
Received 31 October 2006; published 9 February 2007
We perform first-principles calculations of the band and k-point resolved superconducting gap of Pb in the
framework of the density functional theory for superconductors. Without any adjustable parameter or assuption
different from s-wave symmetry, we find two different values of the gap on the two sheets of the Fermi surface,
which can be related to the different electron-phonon couplings characterizing the electronic states in the
corresponding bands. These, in turn, derive from the different orbital character of the electronic states. We also
find some intraband gap anisotropy in each Fermi surface sheet. Our calculated gap, critical temperature and
total anisotropy of the gap are in good agreement with tunneling experiments. We estimate an 8% enhance-
ment of T
c
coming from the gap anisotropy. However, the experimentally found T
3
temperature dependence of
the specific heat cannot be found within our assumed anisotropic s-wave gap symmetry.
DOI: 10.1103/PhysRevB.75.054508 PACS numbers: 74.25.Jb, 74.25.Kc, 74.70.Ad
I. INTRODUCTION
The possibility of multiband superconductivity was
discussed
1
already a few years after the formulation of the
Bardeen-Cooper-Schrieffer BCS theory
2
and later reported
in several experimental papers.
3–5
Moreover, the dependence
of the superconducting SC gap on the direction in k-space
was analyzed for different materials, like Pb Refs. 6–8 and
Sn.
9
The recent discovery of two-gap superconductivity
TGSC in MgB
2
brought this problem back to the attention
of the scientific community. Theoretical and experimental
investigations on MgB
2
have emphasized how important the
presence of multiple gaps can be to enhance the critical tem-
perature T
c
of a specific material. Theoretical estimates have
in fact shown that retaining the information about the ,
band dependence of the electron-phonon interaction in MgB
2
greatly enhances the calculated T
c
, relative to an averaged,
single-band, calculation.
10–13
Of crucial importance is the
question: How and when does multigap superconductivity
increase T
c
? In this context, it is very interesting to investi-
gate the gap anisotropy in different materials showing this
peculiarity. Pb is such a material: the unconventional aspects
of its superconductivity, relative to BCS, have been dis-
cussed in several papers.
6–8,14,15,30–32
In particular, two sepa-
rated SC gaps have been found experimentally
14,15
and in-
vestigated theoretically.
6,7
Very recently, an approach to superconductivity, based on
density functional theory SCDFT
16,17
has been able, in a
completely parameter-free fashion, to describe successfully
the superconducting properties of several materials, ranging
from weak and intermediate to the strong coupling
regime
13,16,17
and from ambient to high pressure
conditions.
18,19
Unlike within Eliashberg theory, the Cou-
lomb interaction is included in the calculation on the same
footing as the electron-phonon interaction. This proved to be
very important in MgB
2
, where a band resolved treatment of
the Coulomb repulsion resulted to be crucial to reproduce the
T
c
of this material from first principles.
13
These consider-
ations show that a theory with a predictive character as
SCDFT, retaining the full band dependence and anisotropy
of the SC gap, can prove to be very useful to study TGSC.
In this work, we use a fully n , k-resolved formalism
within SCDFT and calculate from first principles the SC
properties. Although a generalization to general symmetry is
possible, our present formulation still assumes an anisotropic
s symmetry for the order parameter. Within this approach all
the features of the SC gap emerge naturally, without any a
priori, material specific, physical model and without adjust-
ing any parameter. Our results confirm the experimental find-
ing that Pb is a two band superconductor. The calculated
gaps, their overall anisotropy, and the T
c
result in good
agreement with experiments reported in Refs. 14 and 15 and
with previous calculations.
6,7
A T
c
enhancement of 8% can
be directly related to a gap anisotropy |
2
-
1
| /
iso
20%,
much smaller than that of MgB
2
. Here
1
,
2
are the aver-
aged gap inside each band and
iso
is the average gap result-
ing from an isotropic interaction. The ratios 2
1,2
/ K
B
T
c
are
in very good agreement with the corresponding experimental
values. Moreover, anisotropy slightly changes the tempera-
ture dependence of the electronic specific heat, but cannot
reconcile the calculations with the T
3
dependence reported
experimentally Ref. 31. The existence of the two gaps and
their intraband anisotropy are related to the corresponding
anisotropy of the electron-phonon e-ph coupling in the two
bands crossing the Fermi level, which strongly correlates
with the s and d character of the bands.
The paper is organized as follows: In Sec. II we summa-
rize the main features of the SCDFT. In Sec. III we describe
the computational details of our calculation. Our main results
are presented and analyzed in Sec. IV, and finally in Sec. V,
we draw our conclusions.
II. DENSITY FUNCTIONAL THEORY FOR
SUPERCONDUCTORS
The central result of the SCDFT is the generalized gap
equation
PHYSICAL REVIEW B 75, 054508 2007
1098-0121/2007/755/0545086 ©2007 The American Physical Society 054508-1