Pis’ma v ZhETF, vol. 114, iss. 8, pp. 551 - 552
© 2021 October 25
Reentrant orbital effect against superconductivity in the
quasi-two-dimensional superconductor NbS2
A. G. Lebed1)
Department of Physics, University of Arizona, 1118 E. 4-th Street, Tucson, AZ 85721, USA
L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117334 Moscow, Russia
Submitted 2 September 2021
Resubmitted 22 September 2021
Accepted 22
September 2021
DOI: 10.31857/S1234567821200088
It is well known that superconductivity at zero
any orbital effect against superconductivity. The aim
temperature is usually destroyed in any superconduc-
of our paper is to show that these happen due to the
tor by either the upper orbital critical magnetic field,
reentrant nature of the quantum effects of electron mo-
Hc2(0), or the so-called Clogston paramagnetic limit-
tion in a parallel magnetic field, theoretically predicted
ing field, Hp [1]. These are due to the fact that, in
and considered in [12-17]. To this end, we derive the
the traditional singlet Cooper pair, the electrons pos-
so-called gap equation, determining the upper critical
sess opposite momenta and opposite spins. By present
field in slightly inclined magnetic field, which directly
moment, there are also known several superconduct-
takes into account quantum effects of electron motion
ing phases, which can exist above Hc2(0) and Hp. In-
in a parallel magnetic field. The physical origin of the
deed, the paramagnetic limit, Hp, can be absent for
above mentioned quantum effects [8,12] is related to
some triplet superconductors (see, for example, UTe2
the Bragg reflections from the Brillouin zone boundaries
[2-5]). Alternatively, for singlet superconductivity, the
during electron motion in a parallel magnetic field. To
superconducting phase can exceed the Clogston limit by
compare the obtained results with the existing exper-
creating the non-homogeneous Fulde-Ferrell-Larkin-
imental data, we derive the gap equation both for a
Ovchinnikov (FFLO or LOFF phase) [6, 7]. On the other
strictly parallel magnetic field and for a magnetic field
hand, if Hc2(0) tries to destroy superconductivity, then
with some perpendicular component. The latter is de-
quantum effects of electron motion in a magnetic field
rived, for the best of our knowledge, for the first time.
can, in principle, restore it as the Reentrant Supercon-
We use comparison of these equations with experimental
ducting (RS) phase [8-17]. Although there are numer-
data [18] to extract the so-called GL coherence lengths
ous experimental results, confirming the existence of the
and in-plane Fermi velocity. These allow us to show that,
FFLO phase in several Q2D superconductors, there ex-
indeed, in the magnetic fields range, H ≃ 15 T , quan-
ist only a few experimental works [2-5], where the pre-
tum effects are very strong and completely suppress the
sumably RS phase revives in ultrahigh magnetic fields
orbital effect against superconductivity. As a result, the
due to quantum effects of electron motion in a magnetic
FFLO phase appears with the transition temperature
field in one compound - UTe2. On the other hand, the
value like for a pure 2D superconductor, which satisfies
above mentioned unique RS phenomenon has been the-
the experimental situation in NbS2 [18]. In our opinion,
oretically predicted for a variety of Fermi surfaces: for
this is the first firm demonstration of a reentrant nature
Q1D [8-10], for isotropic 3D [11], and for Q2D super-
of the orbital effect against superconductivity [8-17].
conductors [12-17].
Below, we consider a Q2D conductor with the follow-
Recently, the FFLO phase has been found by Lortz
ing electron spectrum, which is an isotropic one within
and collaborators in the Q2D compound NbS2 in a par-
the conducting plane:
allel magnetic filed [18]. The peculiarity of this work
(p2x + p2y)
p2F
is that at relatively low magnetic fields (i.e., in the
ǫ(p) =
- 2t⊥ cos(pzd), t⊥ ≪ ǫF =
,
(1)
2m
2m
Ginzburg-Landau (GL) area [1]) the orbital effect of
the field partially destroys superconductivity but, at
where m is the in-plane electron mass, t⊥ is the integral
high magnetic fields, everything looks like there is no
of the overlapping of electron wave functions in a per-
pendicular to the conducting planes direction; ǫF and
1)e-mail: lebed@arizona.edu
pF are the Fermi energy and Fermi momentum, respec-
Письма в ЖЭТФ том 114 вып. 7 - 8
2021
551
552
A. G. Lebed
tively; ℏ ≡ 1. Let us consider slightly inclined with re-
This is an excerpt of the article
“Reentrant
spect to the conducting planes magnetic field,
orbital effect against superconductivity in the quasi-
”. Full text
two-dimensional superconductor NbS2
H = (0,H∥,H⊥),
(2)
of the paper is published in JETP Letters journal.
DOI: 10.1134/S0021364021200017
since the experiments in [18] are done both for the par-
allel and the slightly inclined fields. For our calculations,
it is convenient to choose the following gauge, where the
vector-potential of the magnetic field (2) depends only
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∫π
∫
dφ
2πT[x1
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=U
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−π 2π
|x-x1|
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2πT |x-x1|
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vF | cos φ| sinh
| cos φ|
vF
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{
[
]
[
]}
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×J0
sinω∥(x-x1)2v
sinω∥(x+x1)
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F
2vF
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[
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[
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2
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2µB H(x-x1)
× cosω⊥pF sinφ(x2-x
cos
Δ(x1),
(4)
D. Braithwaite, G. Lapertot, M. Nardone, A. Zitouni,
vF
cos(φ)
vF cos(φ)
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d is a cut-off dis-
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HFFLO - H∗FFLO
l2⊥
∫ π dφ
∫ ∞ dz
(2014).
=2
HFFLO
d2
-π
2π
0
z
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]
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ω∥z cos φ
sin2
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4vF
(2µBHz
(2µBHz cosφ),(5)
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H∗FFLO - HFFLO
l2⊥
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= -0.2
(6)
ZhETF 111, 833 (2020)].
HFFLO
d2
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Finally, taking into account that, at H ≃ 15 T , l⊥/d ≃
18.
C.-w. Cho, J. Lyu, C. Y. Ng, J. J. He, T. A. Abdel-
≃ 0.27, we find that the relative change of the critical
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field of the appearance of the FFLO phase is very small:
accepted; preprint arXiv: 2011.04880 (2020).
H∗FFLO - HFFLO
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A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshinskii,
= -0.015.
(7)
HFFLO
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The author is thankful to N. N. Bagmet (Lebed),
Rolf Lortz, and V. P. Mineev for useful discussions.
Письма в ЖЭТФ том 114 вып. 7 - 8
2021