in describing neutron stars ( 4)
(see the figure, top panel). As
an effective theory of metallic
superconductivity, BCS theory
gives a satisfactory account of
physical properties. Even when
it is found near another competing instability and even in
the presence of a strong correlation among electrons, the superconducting state has been
found to be a condensate of
Cooper pairs, sometimes with
However, despite its glorious history, and unusually for
a celebrated scientific theory,
BCS theory has been effectively
impotent in predicting new
pursuit that has remained an
empirical enterprise. A mystery surrounds the attractive
interaction leading to the formation of Cooper pairs. In the
case of common superconductors, exchanging phonons between electrons can generate
such an attraction. The vicinity of a competing order offers
other pairing possibilities such
as those involving the spin of
electrons. It is fair to say that
while the existence of Cooper
pairs has been demonstrated
beyond a reasonable doubt,
save for the simplest superconductors, the identification of
the binding glue remains a difficult, and open, question.
The discovery of ultralow-temperature superconductivity in bismuth provides a new
challenge. The lattice structure
has modified the familiar electron to complex Bloch waves
beyond recognition (see the figure, bottom panel). Each mobile charge carrier
in bismuth occupies a volume containing
105 atoms. Half of the carriers are hole-like
and the other half, electron-like. The bare
mass of electron-like carriers is a thousandth of the bare electron mass along one
orientation. The spin and the momentum
are locked to each other, and therefore
one cannot separate the spin and the spatial components of the wave function. The
quantum numbers defining each fermion
are thus quite different from those of electrons in a vacuum.
What drives pairing in the case of bis-
muth? One may think that the phonon-me-
diated scenario of pairing would have no
difficulty in explaining superconductivity
at such a low temperature. After all, amor-
phous bismuth and pressurized bismuth
(both more atomically packed than the
rhombohedral crystal found at ambient
pressure) are ordinary superconductors
with a decent critical temperature. Why
should one care about the 0.5-mK instabil-
ity of this strangely dilute metallic system?
The caveat comes from an important de-
tail in BCS theory. An infinitesimal attrac-
tion between fermions would destabilize
the Fermi sea, as long as it only restricts
itself to a narrow thickness around the
Fermi level ( 5). In almost all superconduc-
tors, this restriction is readily provided by
the fact that the Debye temperature (the
typical energy of phonons) is much smaller
than the Fermi energy. But in bismuth, the
two have comparable magni-
tudes. The formation of Coo-
per pairs by an infinitesimal
attraction is intimately con-
nected to a fundamental prob-
lem in quantum mechanics ( 6).
In two dimensions, a potential
well would generate a bound
state no matter how shallow it
is. This is not the case in three
dimensions, where bonding
requires a minimum depth.
The hierarchy between Debye
and Fermi energies is required
to transform pairing from a
three-dimensional problem to
a two-dimensional one ( 7). In
doped SrTiO3, superconduc-
tivity survives even when the
Fermi temperature becomes
an order of magnitude lower
than the Debye temperature
( 8). This motivated theorists ( 7,
9) to consider plasmons, col-
lective excitations of the elec-
tronic system, as substitutes
for phonons. The plasmon
energy can become lower than
the Fermi energy in a dilute su-
perconductor such as bismuth.
By becoming the newest
member of the select club of
superconducting elements ( 10),
bismuth raises another intrigu-
ing question. Antimony and
arsenic share the same rhom-
bohedral crystal structure, but
remain outside this club ( 10). Is
bismuth the only column-V su-
perconducting element despite
its much lower carrier concen-
tration? If so, why? Is it be-
cause of its stronger spin-orbit
coupling or its specific Fermi
surface? Or is there an entire
family of superconductors to be
discovered by cooling arsenic and antimony
below 15 mK, the lowest temperature at-
tained in previous studies ( 11)? j
1. O. Prakash et al., Science 355, 52 (2017).
2. Y. Fuseya et al., J. Phys. Soc. Jpn. 84, 012001 (2015).
3. D. Vollhardt, P. Wölfle, The Superfluid Phases of Helium 3
(Taylor & Francis, London, 1990).
4. D. Pines, M. A. Alpar, Nature 316, 27 (1985).
5. P. G. De Gennes, Superconductivity of Metals and Alloys
6. K. Yang, M. de Llano, Am. J. Phys . 57, 85 (1989).
7. Y. Takada, J. Phys. Soc. Jpn. 49, 1267 (1980).
8. X. Lin et al., Phys. Rev. X3, 021002 (2013).
9. J. Ruhman, P. A. Lee, https://arxiv.org/abs/1605.01737
10. M. Debessai etal., J.Phys.Conf.Ser. 215, 012034 (2010).
11. C. Uher, T. Morelli, J.Phys.FMet.Phys. 16, L103 (1986).
λ F 100 nm
1018 1020 1022 1024 1037
Fermion concentration (cm- 3)
Across many length scales
( Top) Pairing is ubiquitous in fermion systems with a variety of fermion densities.
(Bottom) Sketch of electron-like carriers and the crystal lattice in bismuth. The
trigonal axis is perpendicular to the page. The central atom and its third neighbors
lie in a common plane, sandwiched between planes containing first and second
neighbors. Fermions in bismuth are anisotropic Bloch waves in three distinct
flavors and extending over many atomic distances.