[4]Om Prakash [6]Srinivasan Ramakrishnan 1]Department of Condensed Matter
Physics and Materials Science, Tata Institute of Fundamental Research, Mumbai,
400005, India 2]Department of Theoretical Physics, Tata Institute of
Fundamental Research, Mumbai, 400005, India 3]Department of Quantum Matter,
AdSE, Hiroshima University, Higashi-Hiroshima, 739-8530, Japan 4]Quantum
Computing Hardware, VTT Technical Research Centre of Finland, Espoo, 02150,
Finland 5]Amity Institute of Applied Sciences, Amity University Jharkhand,
Ranchi, 834002, India 6]Department of Physics, Indian Institute of Science
Education and Research, Pune, 411008, India

# Fragile electronic superconductivity in Bi Single crystal

Anil Kumar  Rajendra Loke  Arindam Pramanik  Rajdeep Sensarma  Sitaram
Ramakrishnan<EMAIL_ADDRESS>Biplab Bag  Arumugam Thamizhavel
<EMAIL_ADDRESS>[  [  [  [  [  [

###### Abstract

It was presumed that semimetal Bismuth (Bi) would not show superconductivity
(SC) even at ultra-low temperatures ($<$10 mK) due to its very low carrier
density ($\approx 3\times 10^{17}$cm-3). Recently, we have established bulk
superconductivity in ultra-pure (99.9999%) Bi single crystal at
$\mathrm{T_{C}=0.53}$ mK with an extrapolated upper critical field
$\mathrm{H_{C}(0)=5.2\mu}$T measured along the [$0001$] (trigonal)
-crystallographic direction prakash2017a . At very low concentrations of the
charge carriers, we are dealing with fragile Cooper pairs with an estimated
large coherence length $\mathrm{\xi_{GL}(0)\approx 96\mu}$m. We also stated
that one needs to go beyond the conventional electron-phonon coupling (BCS-
like) mechanism to understand the SC state in Bi. Bi is a compensated semi-
metal with electrons and holes as charge carriers. In order to find the charge
carriers responsible for the SC, we report the temperature dependence of the
anisotropic critical field along the [$01\bar{1}0$]
(bisectrix)-crystallographic direction and compared it with the earlier data
from measurements along the trigonal. Our theoretical analysis of the
anisotropy of critical fields suggests that the light electrons in the three
pockets of Bi bands are responsible for the SC and indicates that Bi is an
extremely weak type-II (close to type-I) superconductor. Finally, we review
the current theories proposed to explain the SC in Bi.

###### keywords:

Superconductivity, Low carrier density, ultra-low temperature

Although semimetal Bi has been studied for more than a century, it still draws
attention from both theorists and experimentalists who work in the frontier
areas of condensed matter research edelman1976a . Bismuth with the electronic
configuration ([Xe]- 4f15 5d10 6s2 6p3) crystallizes into a distorted
rhombohedral structure described by the space group $R\bar{3}m$ (166) with
$\bf{a}$ $=$ 4.538 Å and $\bf{c}$ $=$ 11.823 Å. The unit cell consists of two
pentavalent Bi atoms, giving rise to a Fermi surface (FS) comprising of four
ellipsoidal-shaped pockets: three electron and one hole pocket; making Bi
nearly charge compensated semimetal. These pockets account for a very small
($\approx 10^{-5}$) area of the FS resulting in a carrier density of
$\mathrm{n\approx p\approx 3\times 10^{17}/cm^{3}}$, small density of states
at FS $\mathrm{g(E_{F})\approx 4.2\times 10^{\textendash 6}/(eV-atom)}$ (see
figures 1-3 in SM). The low carrier density translates to a single charge
carrier shared by nearly 100,000 Bi atoms. This led to a presumption that, if
at all, superconductivity in bulk Bi would occur only at ultra-low
temperatures. Indeed in 2016 prakash2017a  studying a high purity (6N) single
crystal of Bi, some of us established superconductivity below
$\mathrm{T_{C}=0.53mK}$ with an estimated upper critical field of
$\mathrm{H_{C}(0)\approx 5.16\pm 0.06\mu T}$ along the [$0001$]
(trigonal)-crystallographic direction.

Extensive studies on the charge carriers have established that the electrons
in Bi have small effective masses ($\mathrm{m_{eff}\approx 0.001m_{e}}$) with
significant anisotropy ($\mathrm{m_{eff}(B\parallel binary)/m_{eff}(B\parallel
bisectix)\geq 200}$), while the holes are heavier ($\mathrm{m_{eff}\approx
0.07m_{e}}$) and have relatively isotropic effective mass (anisotropy $\approx
10$)Zhu2011 . To better understand the nature and possible mechanism of
superconductivity in Bi, we have studied the critical field anisotropy in
superconducting Bi using dc magnetization measurements. Here, we report
measurement of the temperature dependence of the upper critical field in a 6N
highly pure single crystal of Bi along the [$01\bar{1}0$]
(bisectrix)-crystallographic direction and present a comparison with the
critical field measured along the trigonal prakash2017a . Our theoretical
analysis of the electronic g-factor, together with the observed critical field
anisotropy suggest that the light electrons are primarily responsible for the
superconductivity in Bi.

To study the upper critical field along bisectrix, a Bi crystal sized
$2.1\times 0.3\times 0.2$ cm3 (from the same batch used in prakash2017a , RRR
$\geq 500$) was cut along the [$01\bar{1}0$] crystallographic direction (see
figure 1 in SM). The crystal was push-fitted to an annealed high-purity (5N)
silver (Ag) rod with fine Ag powder at the interface to enhance effective
surface contact area (see fig. [1]). The Ag rod was further pressed onto the
crystal to provide better thermal contact and threaded to the Cu nuclear stage
(NS), enabling cooling of the sample down to $\mathrm{T\approx 100\mu}$K. The
dc-magnetization measurement assembly consists of a magnetometer pick-up coil
(4 turns) and an excitation coil, both made up of superconducting niobium
(Nb). The pickup coil as well the excitation coil outgoing leads are twisted
in pairs and enclosed in Pb foil to minimise coupling to the external magnetic
fields and reduce noise in the measurements. The measurement assembly was
enclosed in a magnetic shielding consisting of high permeability material
Cryoperm-10 ($\mu_{r}\geq 10^{4}$ at 4.2K, SEKELS GmbH, Germany) and
superconducting lead (Pb) shields and attached to the mixing chamber (MC,
T$=7$mK) plate of the dilution fridge (DRS 1000, Leiden Cryogenics,
Netherlands). This magnetic shielding setup is tested to shield the sample
from the external magnetic fields of the order of $\mathrm{B_{ext}\approx
10mT}$ down to $\mathrm{B_{shielded}\lesssim 10nT}$, when there is no current
in the primary coil. The magnetometer coil is wound directly onto the single
crystal to maximize the filling fraction and connected to the input coil of
the dc-SQUID (Tristan Technologies, USA). Figure [1] shows the schematic
drawing of the measurement setup.

Figure 1: The schematic drawing of the measurement setup with magnetic
shielding assembly: (Right) The overview of the measurement setup consisting
of an excitation coil ($\mathrm{B/I\approx 0.04\mu T/\mu A}$) enclosed in
magnetic shields attached to the MC, Pt-NMR thermometer and samples mounted on
the NS plate (100 $\mu$K). (Middle) Actual measurement setup in the dilution
insert. The arrows mark different components shown in the drawing. (Left) Bi
single crystal oriented along the [$01\bar{1}0$] (bisectrix)-crystallographic
direction is attached to the silver rod. The measurement setup used here is
similar to the thermalization and measurement arrangements reported earlier
prakash2017a .

Apart from shielding external magnetic fields, the magnetic shields also
decrease the field produced by the enclosed primary coil. The excitation coil
was calibrated while enclosed in the magnetic shielding assembly at 4.2 K
using a single-axis magnetometer with the low field probe (Bartington
Instruments Ltd, England, $\mathrm{\pm 1nT}$ resolution) to accurately
determine the excitation magnetic fields used in the measurements. The pick-up
coil is connected to the dc-SQUID (Fig [1]), which in turn are connected to
the RF-amplifier fixed at the head of the cryostat at room temperature. The
RF-head is connected to the squid control unit which directly reads output in
volts. The dc$-$SQUID output has been calibrated at 4.2 K by measuring the
diamagnetic signal from classical superconductors, Nb and Pb. To calibrate the
SQUID output voltage with the diamagnetic susceptibility, we used a Rh metal
of the same dimension as Bi crystal and measured the jump in the SQUID output
voltage at the transition temperature with different excitation fields. We
used similar excitation and pick-up coil setups consisting of the magnetic
shields mentioned above for calibration (see prakash2017a ).

The experiment was carried out in a dilution refrigerator equipped with a
single Cu-adiabatic nuclear demagnetization stage. The Cu-nuclear stage was
first cooled by the dilution refrigerator down to 7 mK followed by
magnetization of the Cu-nuclear spins by applying a magnetic field of 9 T
using a superconducting magnet (Cryogenics, UK). The Cu-nuclear stage was
thermally connected to the mixing chamber using an Aluminium (Al)
superconducting thermal switch to facilitate isothermal magnetization. The
application of the 9 T magnetic field heats up the Cu-stage to nearly 40 mK
due to the heat of magnetization and we have to wait for nearly 36 hours to
cool down the magnetized Cu-stage to 10 mK. Subsequently, the Al thermal
switch is turned off to thermally disconnect the Cu-NS from the MC by removing
the current in the solenoid enclosing Al switch. A slow adiabatic
demagnetization of Cu nuclear spins over a period of 48 hours cools down the
NS to a base temperature of 100 $\mu$K. Slow demagnetization helps in
maintaining thermal equilibrium between the samples and NS as well as with the
NMR thermometer. We used 195Pt-NMR thermometer for the temperature
measurements below 10 mK during adiabatic demagnetization. The NMR thermometer
is calibrated against a Cerium Magnesium Nitrate (paramagnetic thermometer)
and a SQUID based noise thermometer (MAGNICON GmbH, Germany) at 10 mK. The
SQUID based noise thermometer can also measure temperatures down to 1mK and is
used along with the NMR thermometer below 10 mK. The details of the adiabatic
TIFR nuclear refrigerator with temperature measurement and calibration are
given in an earlier report naren201a .

The superconducting transition of the Bi in a excitation field of 0.2 $\mu$T
along the bisectrix is observed (see figure 6 in SM) below 0.53 mK in the form
of a sharp jump in the dc-susceptibility ($\mathrm{\chi}$). The
$\mathrm{T_{C}=0.53}$ mK is in agreement with the $\mathrm{T_{C}}$ observed
for field along the trigonal prakash2017a . The SC transition is suppressed to
lower temperatures with increasing excitation field. All measurements were
performed on the free end of the sample, away from the Ag rod joint, avoiding
any artefact due to the interface effects. The temperature dependence of the
critical fields along the trigonal and bisectrix is shown in fig.[2]. The data
fitted to $\mathrm{H_{C}(T)=H_{C}(0)[1-(T/T_{C})^{2}}$], to estimate the value
of the critical field at T$=0$ K. The critical field value along the bisectrix
estimated from the fit is $1.58\pm 0.07\mu$T. The resulting critical field
anisotropy is: $\mathrm{H_{C}^{tri}(0)/H_{C}^{bis}(0)=3.26\pm 0.18}$.

Figure 2: The temperature dependence of the critical fields $\mathrm{H_{C}}$
for field along the trigonal and bisectrix-crystallographic directions.

This critical field anisotropy presents both a challenge and possible window
into understanding the mechanism behind the suppression of superconductivity
due to the magnetic field in Bi. If Bi was a strong type II-SC, the anisotropy
can be explained in terms of the anisotropic superfluid stiffness of the
material (inherited from the anisotropic band dispersion of Bi). However, the
presence of a sharp jump in the magnetization, and nearly identical
diamagnetic susceptibility in the field cooled (FC) and zero field cooled
(ZFC) prakash2017a  precludes the possibility that Bi is a strong type II-SC.
This is not strange given that majority of elemental superconductors are
type-I superconductors. If we consider Bi as a type-I superconductor, the
critical field $\mathrm{H_{C}}$, is determined by equating the free energy of
the superconductor at zero magnetic field (field does not penetrate below
$\mathrm{H_{C}}$) with the free energy of the normal state at
$\mathrm{H_{C}}$(0). The anisotropy can then only arise from the dependence of
the free energy of the normal state on the direction of the applied field.
This free energy is given by the equation
$\mathrm{H^{2}_{C}/2\mu_{0}(1+\chi_{v}}$), where $\mathrm{\chi_{v}}$ is the
volume magnetic susceptibility of the material. For a strongly diamagnetic
material like Bi, $\mathrm{\chi_{v}\approx 10^{-4}}$ Fukuyama1970  the
anisotropic effects from $\chi$v can be neglected. Thus, it is hard to explain
an anisotropic factor of 3.26 in terms of $\mathrm{H_{C}}$. Additionally, a
simple estimate of $\mathrm{H_{C}}$, assuming an energy gap
$\Delta(0)=1.764\mathrm{k_{B}T_{C}}$ prakash2017a  with $\mathrm{T_{C}=0.53}$
mK and pairing of electrons with the density of states (per pocket per spin)
at the Fermi level $N^{e}_{0}=9.2\times 10^{18}$ eV-1 cm-3 ruhman2017a  leads
to an isotropic $\mathrm{H_{C}(0)=0.2\mu}$T which is much lower than the
values observed experimentally. A similarly low value of $\mathrm{H_{C}}$(0)
is obtained if one assumes hole pairing instead of electron pairing. If both
electrons and holes are paired, the estimated $\mathrm{H_{C}(0)=0.3\mu}$T is
still lower than the experimentally measured values.

The anisotropy of the critical field can be understood if one assumes that Bi
is a weakly type II superconductor and the transition is given by the Pauli
paramagnetic limit. In this case, one needs to compare the pairing energy gain
to the Zeeman energy cost paid by the electrons/holes to maintain similar
Fermi surfaces for the two spin components:
$g\mu_{B}\mathrm{H_{C}}=\frac{\Delta(0)}{\sqrt{2}}$tinkham1996 . To further
test this, we use the following model hamiltonian for each of the three
electron pockets to extract the anisotropic $g$ factor.

$H_{e}(\vec{k})=[\hbar v_{z}k_{z}\sigma^{z}+\hbar
v_{\perp}(k_{x}\sigma^{x}+k_{y}\sigma^{y})]s^{x}+\Delta_{bg}s^{z}$ (1)

where $\sigma^{i}$s and $s^{i}$s are the Pauli matrices in the spin and
orbital space, respectively. We use $v_{z}=6.6\times 10^{4}\,m/s$,
$v_{\perp}=8.1\times 10^{5}\,m/s$, and $\Delta_{bg}=7.5\,meV$ ruhman2017a .
Here z denotes the direction along the $\mathrm{\Gamma-L}$ line in the BZ,
which is also the long-axis of the electron pockets. Figure [3] shows the BZ
and the electron & holes pockets contributing to the FS of Bi. The electron
pockets have a tilt angle of $7^{\circ}$ with respect to the binary-bisectrix
plane.

Figure 3: (a) The first Brillouin zone and the Fermi surface of bismuth. (b)
Enlarged electron (blue) and hole (red) pockets. The electron pockets make
$\alpha\approx 7^{\circ}$ tilt angle with respect to binary-bisectrix plane.
The pocket e2 lies along the bisectrix direction.

To find the spin-splitting in the presence of an external magnetic field, we
block diagonalize wolff1964 ; zhu2011 the hamiltonian in the orbital space to
get the effective spin hamiltonian:

$H_{spin}=\frac{\hbar e}{\Delta_{bg}}v_{z}v_{\perp}H_{x}\sigma^{x}+\frac{\hbar
e}{\Delta_{bg}}v_{z}v_{\perp}H_{y}\sigma^{y}+\frac{\hbar
e}{\Delta_{bg}}v_{\perp}^{2}H_{z}\sigma^{z}$ (2)

where H is the external magnetic field. This allows us to extract the
effective $g$ factors for different electron pockets for H in three different
directions; trigonal, binary, and bisectrix (see fig.[3](b)), which are listed
in table [1]. These estimates (see SM for more details) are roughly consistent
with estimates from cyclotron resonances Zhu2011 .

Table 1: Calculated $g$ factor anisotropy of the three electron pockets in Bi pocket | trigonal | binary | bisectrix
---|---|---|---
e1 | 293 | 1716 | 1000
e2 | 293 | 162 | 1979
e3 | 293 | 1716 | 1000
|  |  |

We note that a larger value of $g$ in a particular direction implies a smaller
critical field $\mathrm{H_{C}}$(0) in that direction. From the table [1], we
see that the $g$ factor along the bisectrix direction is larger than the $g$
factor along the trigonal direction, and hence one would expect the critical
field along trigonal to be larger than the in-plane critical field along the
bisectrix, as seen in our experiments. Further, the ratio of
$\mathrm{H_{C}}$(0) along trigonal to that along the bisectrix is
$\mathrm{H_{C}^{tri}(0)/H_{C}^{bis}(0)=g_{bis}/g_{tri}\approx 3.41}$ (see
table [1]). This matches quantitatively with the anisotropy ratio of $\approx
3.26$ measured experimentally. Our individual estimates for
$\mathrm{H_{C}^{tri}(0)\approx 3.4\mu}$T and $\mathrm{H_{C}^{bis}(0)\approx
1\mu}$T are also close to the experimentally observed values. We also note
that the $g$ factor for the hole pocket is $\approx 63$ in the trigonal
direction and $\approx 0.8$ in the plane Zhu2011 . Thus a Pauli limit
calculation with the hole pocket would yield a larger in-plane upper critical
field compared to the critical field along the trigonal, contrary to the
experimental findings. Thus the anisotropy seen in the experiments strongly
points to the pairing of electrons driving superconductivity in Bi.

We now consider the models put forth to explain the observation of SC in Bi.
Earlier, Srinivasan, $et\hskip 2.84544ptal$. srinivasan1979a  used a
simplified two-band model for describing the longitudinal dielectric function
and found that the attractive interaction responsible for the instability of
the normal ground state comes not only from the exchange of lattice phonons,
but also from the electron-hole sound mode, provided the ratio of the averaged
hole to the electron mass, $\mathrm{m_{h}/m_{e}\neq 1}$. This condition is
easily satisfied in the case of Bi and they estimated the $\mathrm{T_{C}}$ to
be around 1 mK. However, so far there is no experimental evidence for
existence of the electron-hole sound (EHS) mode (a sound-like longitudinal
collective mode) in Bi. Another model proposed by Zaahel Mata-Pinzón $et\hskip
2.84544ptal$ pinzon2016a  numerically calculated electronic and vibrational
densities of states (eDOS and vDOS, respectively) of crystalline and amorphous
forms of Bi. The calculations showed that the eDOS of amorphous Bi (a-Bi) is
about $4\times$ larger than that of crystalline Bi (c-Bi) at the Fermi energy,
whereas, for the vDOS, the energy range of the amorphous is roughly the same
as the crystalline even though the actual shapes are quite different. Using
the experimental parameters obtained for a-Bi and employing a simple weak
coupling BCS mode, they gave an upper limit of 1.3 mK as the superconducting
transition for a pure Bi crystal. Furthermore, they suggested that the
electron-phonon coupling ($\lambda$) is larger in a-Bi as compared to c-Bi as
indicated by the $\lambda$ obtained via McMillan’s formula, $\lambda_{c}$ =
0.24 for c-Bi and while experiment and theory suggest $\lambda_{a}$ = 2.46 for
a-Bi. Therefore, with respect to the c-Bi, superconductivity in a-Bi is
enhanced by the higher values of $\lambda$ and eDOS at the Fermi energy.
Though this model is simple and predicts $\mathrm{T_{C}}$ of the Bi crystal
near to its actual value, use of the electron-phonon mechanism in the
formation of Cooper pairs is not justified due to the failure of the Migdal’s
adiabatic approximation migdal1958a  which is central to any electron-phonon
coupling scheme including the BCS-theory. This is due to the fact that the
Fermi energy (25 meV) and lattice energy (15 meV) of crystalline Bi are
comparable. We believe that the underlying mechanism for superconductivity in
c-Bi is entirely different from that of a-Bi. The mechanism for
superconductivity in the latter is due to electron-phonon interaction which is
firmly established.

Let us now look at the other models which suggest unconventional mechanisms
koley2017a ; baskaran2017a ; ruhman2017a ; tewari2018a ; ekkehard2018a  for
the observation of superconductivity in Bi. S. Koley et.al. koley2017a
suggested that fluctuating excitons could be the glue for superconductivity in
Bi. They showed that a two-fluid model composed of preformed and dynamically
fluctuating excitons coupled to a tiny number of carriers can provide a
unified understanding of the anomalous temperature dependence of the
resistivity below 1 K as well as superconductivity in Bi below 1 mK. Our
unpublished resistivity data (see fig.5 in the SM) clearly show deviation from
T2 dependence of the resistivity below 800 mK and ultimately shows a T5
behaviour down to 15 mK (in agreement with an earlier study Uher1977a ).
Koley’s model has proposed that resonant scattering involving a very low
density of renormalized carriers and the excitonic liquid drives logarithmic
enhancement of vertex corrections, boosting superconductivity in Bi. The model
also explains the temperature dependence of the normal state resistivity down
to 15 mK. However, at present it is not clear whether such a model can explain
the anisotropy observed in the critical fields. Bhaskaran baskaran2017a
suggested that a potential high $\mathrm{T_{C}}$ SC from the RVB mechanism is
lost in Bi. However, some superconducting fluctuations survive and the tiny
Fermi pockets are viewed as remnant evanescent Bogoliubov quasiparticles that
are responsible for the anomalous normal state. Multi-band character admits
possibility of PT violating chiral singlet superconductivity Bi. It remains to
be seen whether such a chiral superconductor exists and can explain the
observed anisotropy. Moreover, Bhaskaran’s model also suggests elements Sb and
As should also exhibit SC at low temperatures. We did not observe SC in single
crystals of both of Sb and As (unpublished) down to 0.1 mK in a field of
$0.4\mu$T. Ruhman and Lee ruhman2017a  have argued that conventional electron-
phonon coupling is too weak to be responsible for the binding of electrons
into Cooper pairs. They showed that Bi is the first material to exhibit
superconductivity driven by the retardation effects of Coulomb repulsion
alone. They claimed that SC of Bi at low carrier concentration arises only due
to the long-ranged interactions that are capable of causing such an
instability. In the absence of any experimental evidence of a critical point,
they investigated the more likely scenario in which the dynamically screened
Coulomb repulsion gives rise to an effective retarded attraction on the energy
scale of the longitudinal plasma oscillations. Further, they used an
approximate isotropic band structure and the random phase approximation for
the screened Coulomb interaction. Within these approximations, they found the
above-mentioned weak coupling instability. The transition temperature is
greatly enhanced by the existence of a heavy hole band which has a large mass
that allows for an enhancement of the static screening (Thomas-Fermi) without
enhancing the plasma frequency. They also showed that $\mathrm{T_{C}}$ is not
dramatically decreased when the acoustic plasma mode is absent. Therefore it
was concluded that acoustic plasmon does not contribute to attractive
interactions in Bi. They emphasized that the model works only in the s-wave
coupling and the scenario might change with higher angular momentum coupling
that might result in the SC of Bismuth. Our analysis suggests that electrons
(not holes) are responsible for SC in Bi and that needs to be reconciled with
ruhman2017a .

Tewari and Kapoor tewari2018a  have suggested that superconductivity of Bi
arises due to the electrons belonging to the three pockets. They claimed that
the electrons in Bi behave like a rare gas with inter-particle separation of
nearly $185\AA$. In such a dilute system, the peculiar oscillatory behavior of
the generalized electronic dielectric function at large distances can give
rise to an attractive interaction between two electrons. This model is valid
at extremely low temperatures for a very low density electron gas, so that
there is no electron–phonon interaction and the interaction amongst the
electrons can be expressed in terms of weak two-body potential characterized
by negative scattering length. The dilute nature of the electron gas is
crucial and necessary requirement (i.e. $\mathrm{k_{F}\times a\ll 1}$, where a
is the distance between atoms) to observe SC in Bi. We have calculated the
Sommerfeld coefficient($\gamma$) using the formula
$\mathrm{\gamma=\frac{\pi^{2}}{3}k_{B}^{2}g(E_{F})}$, where
$\mathrm{g(E_{F})}$ is the density of carriers at the Fermi level. Here the
carrier comprises both electrons and holes. A simple estimate of $\gamma$,
assuming the density of states (per pocket per spin) of electrons at the Fermi
level, $\mathrm{N_{0}^{e}=9.2\times 10^{18}eV^{-1}cm^{-3}}$ ruhman2017a , and
the density of states (per spin) of holes at the Fermi level,
$\mathrm{N_{0}^{h}=3.5\times 10^{19}eV^{-1}cm^{-3}}$ ruhman2017a , gives
$\gamma\approx 4\,\mu$J K-2 mol-1. This estimated value is quite close to the
experimentally obtained $\gamma\approx 5\,\mu$J K-2 mol-1 prakash2017a .
Therefore, a refinement of the model proposed by tewari2018a  is required to
understand the temperature dependence of the critical field. Finally, the
model proposed by Ekkehard Krüger ekkehard2018a  relies on the hypothesis that
the non-adiabatic Heisenberg model presents a mechanism of Cooper pair
formation generated by the strongly correlated atomic-like motion of the
electrons in narrow, roughly half-filled “superconducting bands” of special
symmetry. In this case, the formation of Cooper pairs is not only the result
of an attractive electron–electron interaction but is additionally the outcome
of quantum mechanical constraining forces. According to his assertion only
these constraining forces operating in superconducting bands may produce
eigenstates in which the electrons form Cooper pairs. He argued that both Bi
at atmospheric pressure and Bi at 122 Gpa have nearly half-filled narrow
electronic bands which may be responsible for superconductivity in both these
phases of Bi. However, this model does not provide any information on
superconducting properties of Bi apart from the estimation of
$\mathrm{T_{C}}$. It is clear that more theoretical work is required to
understand the superconducting properties of this fragile electronic SC of Bi.

## Acknowledgment

R.S. and A.P. acknowledge use of computational facilities at Dept. of
Theoretical Physics. R.S. acknowledges support from the Department of Atomic
Energy, Government of India, under Project Identification No. RTI 4002. R.S.
thanks Pratap Raychaudhuri, Unmesh Ghorai and Mursalin Islam for useful
discussions.

## References

  *   * (1) O. Prakash, A. Kumar, A. Thamizhavel, S. Ramakrishnan, Evidence for bulk superconductivity in pure bismuth single crystals at ambient pressure.  Science 355(6320), 52–55 (2017)
  * (2) V. Édel’man, Electrons in bismuth.  Advances in Physics 25(6), 555–613 (1976)
  * (3) Z. Zhu, B. Fauqué, Y. Fuseya, K. Behnia, Angle-resolved landau spectrum of electrons and holes in bismuth.  Phys. Rev. B 84, 115,137 (2011)
  * (4) H.R. Naren, R.S. Sannabhadti, A. Kumar, V. Arolkar, S. Ramakrishnan, Setting up of a microkelvin refrigerator facility at tifr, journal = AIP Conference Proceedings, volume = 1447, number = 1, pages = 503-504, year = 2012
  * (5) H. Fukuyama, R. Kubo, Interband effects on magnetic susceptibility. ii. diamagnetism of bismuth.  Journal of the Physical Society of Japan 28(3), 570–581 (1970).  10.1143/JPSJ.28.570
  * (6) J. Ruhman, P.A. Lee, Pairing from dynamically screened coulomb repulsion in bismuth.  Phys. Rev. B 96(235107) (2017)
  * (7) M. Tinkham, _Introduction to Superconductivity_ (Dover Publications, Mineola, New York, 1996)
  * (8) P. Wolff, Matrix elements and selection rules for the two-band model of bismuth.  Journal of Physics and Chemistry of Solids 25(10), 1057–1068 (1964)
  * (9) S. Srinivasan, P. Bhattacharyya, S.S. Jha, Superconducting transition temperature for semimetals like bismuth.  Pramana 13(2), 131–143 (1979)
  * (10) Z. Mata-Pinzón, A.A. Valladares, R.M. Valladares, A. Valladares, Superconductivity in bismuth. a new look at an old problem.  PLOS ONE 11, 1–20 (2016)
  * (11) A.B. Migdal, Zh. eksp. teor. fiz.  Soviet Physics JETP 34, 1438–1446 (1958)
  * (12) S. Koley, M.S. Laad, A. Taraphder, Dramatically enhanced superconductivity in elemental bismuth from excitonic fluctuation exchange.  Scientific Reports 7(10993) (2017)
  * (13) G. Baskaran, Theory of ultra low tc superconductivity in bismuth: Tip of an iceberg?  arXiv:1705.01506v1 [cond-mat.supr-con] (2017)
  * (14) S. Tewari, C. Kapoor, Non-phononic mechanism of superconductivity in pure semi-metallic bismuth single crystal at sub-milli kelvin.  Physics Letters A 382(42), 3131–3134 (2018)
  * (15) E. Krüger, Constraining forces stabilizing superconductivity in bismuth.  Symmetry 10(2) (2018)
  * (16) C. Uher, W.P. Pratt, High-precision, ultra low temperature resistivity measurements on bismuth.  Physical Review Letters 39(491) (1977)