of about the same dimensions. The magnitude
of the Meissner signal (the jump in the SQUID
voltage) observed for Bi is nearly the same as
the diamagnetic signal observed for super-
conducting Pb and Rh in the same excitation
field of 0.4 mT, suggesting that a large volume
fraction (bulk) of Bi crystal underwent the super-
conducting transition. The extrapolated critical
field at 0 K for Bi [HC(0) = 5.2 ± 0.1 m T] is similar
to the critical field for Rh ( 31), even though the
Fermi velocity vF, DOS at the Fermi level, and
carrier density in Bi are all very small compared
with those in Rh. The Fermi velocity of Bi was
calculated as vF ¼ ðℏ=meÞð3p2nÞ1= 3, where n is
the carrier density and ℏ is the Dirac constant
(also known as the reduced Planck constant).
Taking n ≈ 3 × 1017 cm– 3, we obtained vF = 2.4 ×
106 cm s–1 for Bi, which is two orders of magni-
tude smaller than the Fermi velocity in Rh.
To understand whether the superconductivity
in Bi is dirty or clean, we estimated the super-
conducting coherence length with the formula
x0 ¼ ℏvF=3:52kB TC, assuming the Bardeen-Cooper-
Schrieffer (BCS) framework (where kB is the
Boltzmann’s constant). We found x0 = 96 mm when
using the values of vF and TC for Bi. Because
the mean free path of Bi as estimated from the
resistivity measurements is ~300 mm at 4.2 K, Bi
can be classified as a clean type I superconduc-
tor. The BCS model also gives the relation BC(0)/
TC = (m0g/2VM)1/2, where BC is critical field, m0 is
the vacuum permeability, g is the electronic spe-
cific heat coefficient in the normal state, and VM
is the molar volume. Using the normal state param-
eters of Bi, we estimated this ratio to be equal
to 0.79 m T K–1, in contrast to the experimental
value of 9. 4 m T K–1, indicating the inapplicability
of the standard BCS theory ( 32). We roughly
estimated x0 by using the BCS formula so that
we could compare it with the mean free path;
the actual value of x0 might be different from
the value estimated above.
Superconductivity in metallic elements can
be understood from the BCS theory ( 32) and
its extensions, and the transition temperature
is given by TC = QDexp[–1/N(0)V], where QD,
N(0), and V are the Debye temperature, elec-
tronic DOS at EF, and phonon-mediated at-
tractive electron-electron interaction, respectively.
However, even though electron-phonon inter-
action occurs in Bi, the conventional BCS model
cannot be applied to Bi. Bi has a multivalley
band structure and small DOS at the Fermi
level. Studying the importance of the multi-
valley band structure in systems with low car-
rier densities, such as Bi, Cohen showed that
the attractive electron-electron interaction aris-
ing from the exchange of intravalley and inter-
valley phonons can be larger than the repulsive
Coulomb interaction in many-valley semicon-
ductors and semimetals, and it can cause these
materials to exhibit superconducting proper-
ties ( 33).
The Fermi energy EF ≈ 25 meV is comparable to the phonon energy ℏwD ≈ 12 meV in Bi,
where wD is the Debye frequency ( 34). The BCS
theory of superconductivity is formulated in the
so-called adiabatic limit, wD/EF ≪ 1. This assumption is clearly violated for Bi, given that wD/EF ≈
0.5. Many known superconductors, such as
Sr TiO3–d, fullerene (C60) compounds, and superconducting semiconductors, have EF ≤ ℏwD.
Several attempts have been made to extend
the BCS theory to account for superconductivity
in these systems in the nonadiabatic limit ( 35–40).
Some other theories on the mechanism of super-
conductivity, based on purely electronic correla-
tions, also exist ( 41, 42) but cannot be applied
to systems with low carrier densities like Bi.
The estimated TC for Bi based on the BCS theory
and its extensions is orders of magnitude smaller
(in the picokelvin range) than the observed TC of
In the nonadiabatic limit, wD/EF ≥ 1, Migdal’s
theorem breaks down and requires the inclusion of vertex renormalization and higher-order
diagrams in the self-consistent gap equation
( 37). The nonadiabatic effects produce strong
enhancement in TC with respect to the usual
Migdal-Eliashberg theory ( 43, 44). In particular,
( 38) generalized the many-body theory of superconductivity in a perturbative scheme with respect to the parameter lwD/EF, where l is the
electron-phonon coupling constant, by calculating the vertex correction function and self energy in the nonadiabatic limit. They found that
the vertex correction function behaves in a complex way with respect to the momentum q and
frequency w of the exchange phonon. Specifically, the vertex corrections are positive for small
values of q and can lead to strong enhancement
of TC, as compared with the usual BCS theory.
In this case, the superconducting transition
temperature is given by TC = 1.13QDe–1/l(1+l)–m*
(in the usual BCS theory, TC = 1.13QDe–1/l–m*).
Using the Coulomb screening m* = 0.105 ( 45) and
the observed TC = 0.53 mK in TC = 1.13QDe–1/l(1+l)–m*,
we obtained l = 0.16, suggesting rather weak
electron-phonon coupling in Bi. This value of
l is similar to but smaller than the value estimated for crystalline Bi in a recent simulation
study ( 46).
In systems with low carrier densities and
multivalley electronic structures (as is the case
for Bi), Cohen ( 46) showed that intervalley
electron-phonon interactions contribute considerably to the net attractive electronic potential. The intervalley scattering is associated
with large momentum transfer, whereas the
calculations ( 38) show that the enhancement
in TC caused by vertex corrections happens at
small q. Although the superconductivity in
Bi can be qualitatively explained by the model
in ( 38), new theoretical inputs are needed to
estimate the superconducting parameters in
systems with low carrier densities in the nonadiabatic limit.
REFERENCES AND NOTES
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54 6 JANUARY 2017 • VOL 355 ISSUE 6320 sciencemag.org SCIENCE
Fig. 2. Observation of superconductivity, the
Meissner effect, and the critical field HC(T)
in Bi single crystals. (A) The dc susceptibility
as a function of temperature [cn(T)] for samples
s1 and s2. A sharp drop in the susceptibility at
0.53 mK marks the transition into the superconducting state. FC, field-cooled; ZFC, zero field–
cooled. (B) cn(T ) at different magnetic fields. The
data corresponding to the 1.8-m T magnetic field
show the transition at 0.37 mK for both samples
s1 and s2. (C) Phase diagram of the critical magnetic field HC(T) versus critical temperature (TC) for
Bi. The data are fitted to HC(T) = HC(0)[1 – (T/TC)2],
and the extrapolated critical magnetic field value
is HC(0) = 5.2 ± 0.1 m T.