Pis’ma v ZhETF, vol. 115, iss. 6, pp. 392 - 393
© 2022
March 25
A chiral triplet quasi-two-dimensional superconductor in a parallel
magnetic field
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, 17334 Moscow, Russia
Submitted 6 February 2022
Resubmitted 21 February 2022
Accepted 21
February 2022
DOI: 10.31857/S1234567822060076
Since discovery of superconductivity in the quasi-
it with the experimental one, 0.45-0.5 [17-19]. To make
two-dimensional (Q2D) conductor Sr2RuO4 [1], it has
our argument against the chiral triplet scenario to be
been intensively investigated for more than 25 years (for
firm, below we calculate the above mentioned ratio ex-
reviews, see [2,3]). Some analogy of this Q2D super-
actly for the in-plane isotropic chiral triplet supercon-
conductor with the superfluid3He was recognized from
ductor with d vector order parameter [4, 20],
the beginning and the existence of a chiral triplet su-
d = z Δ0 (kx ± iky),
(1)
perconducting phase in the Sr2RuO4 was suggested [4].
This scenario of superconducting pairing was supported
and obtain even stronger inconsistency,
by the observations of no change of the Knight shift
between normal and superconducting phases [5,6] and
H∥(0) = 0.815 |dHGL∥/dT|T=Tc Tc,
(2)
breaking of the time reversal symmetry in the super-
conducting phase [7,8]. On the other hand, there were
with the experimental values [17-19], where, to the best
arguments against the chiral triplet superconductivity
of our knowledge, Eq.(2) is the first time obtained in the
scenario, which were almost ignored that time by sci-
Letter. The second our goal is to suggest one more test
entific community. One of the first argument was the
for chiral triplet superconductivity, which may already
paramagnetic limitation of the parallel upper critical
exist in the slightly in-plane anisotropic Q2D triplet su-
magnetic field in Sr2RuO4 [9, 10]. In addition, the pre-
perconductor UTe2 [21] and, as we hope, will be discov-
dicted in the chiral triplet scenario edge currents were
ered in some other Q2D compounds in the future.
not found in the Sr2RuO4 [11,12] but were found ze-
Let us consider a layered superconductor with the
ros of superconducting gap on Q2D Fermi surface (FS)
following in-plane isotropic Q2D electron spectrum:
[13, 14], which is against the fully gaped chiral triplet
scenario [4]. Recently, the strongest experimental argu-
ǫ(p) = ǫ(px, py) - 2t⊥ cos(pzc∗), t⊥ ≪ ǫF ,
(3)
ment against the triplet scenario of superconductivity
where
in Sr2RuO4 was published [15], where strong drop of
the Knight shift in superconducting state of the above
(p2x + p2y)
p2F
ǫ(px, py) =
,
ǫF =
(4)
mentioned material was experimentally discovered.
2m
2m
As seen from the above discussion, the situation
[In Equations (3) and (4), t⊥ is the integral of overlap-
with the chiral triplet scenario of superconductivity
ping of electron wave functions in a perpendicular to
in Sr2RuO4 is still rather controversial. The goal of
the conducting planes direction, m is the in-plane elec-
our Letter is two-fold. First, we improve and make
tron mass, ǫF and pF are the Fermi energy and Fermi
our pioneering argument [9] in favor of singlet su-
momentum, respectively; ℏ ≡ 1.] In a parallel magnetic
perconductivity in Sr2RuO4 to be firm. The point
field, which is applied along x axis
is that in [9] (see also recent [16]) we calculated
the ratio H∥(0)/(|dHGL∥/dT |T=Tc Tc)
= 0.75, where
H = (H,0,0),
(5)
|dHGL∥/dT |T=Tc is the so-called Ginzburg-Landau (GL)
slope, for s-wave Q2D superconductivity and compared
it is convenient to choose the vector potential of the field
in the form:
1)e-mail: lebed@arizona.edu
A = (0,0,Hy).
(6)
392
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2022
A chiral triplet quasi-two-dimensional superconductor in a parallel magnetic field
393
Using the Matsubara’s Green functions technique
1.
Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki,
[22], it is possible to derive the following so-called lin-
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conductor (1):
∫2π
∫∞
3.
C. Kallin, Rep. Progr. Phys. 75, 042501 (2012).
Δ(φ, y) =
g cos(φ - φ1)
×
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π
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×
(
)×
2πT |y-y1|
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vF | sin φ1| sinh
| sin φ1|
vF
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[
]
(1998).
2t⊥ωc
×J0
(y2 - y21) Δ(φ1, y1),
(7)
v2F | sin φ1|
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where g is the effective electron coupling constant, d is
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the cut-off distance, J0(...) is the zero order Bessel func-
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the position on the cylindrical FS (4), where φ and φ1
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J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and
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9.
A. G. Lebed and N. Hayashi, Physica C 341-348, 1677
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[16] to the Gor’kov’s Eq. (7) we find the so-called GL
slope for parallel upper critical magnetic field:
10.
K. Machida, JPS Conf. Proc. 30, 011038 (2020).
(
)
[
√
]
11.
P. G. Bjrnsson, Y. Maeno, M. E. Huber, and
φ0
8
2π2cT2c
K. A. Moler, Phys. Rev. B 72, 167002 (2005).
HGL∥(T) =
τ =
τ,
(8)
2πξ∥ξ⊥
7ζ(3)evF t⊥c∗
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More complicated problem is to solve Eq. (7) at
13.
E. Hassinger, P. Bourgeois-Hope, H. Taniguchi, S. Ren
T = 0 and, thus, to find the upper critical magnetic
de Cotret, G. Grissonnanche, M. S. Anwar, Y. Maeno,
field at zero temperature, H∥(0). This is possible to do
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only by means of numerical calculations. Here, we sum-
011032 (2017).
marize procedure of the numerical solution of Eq. (7)
14.
S. Kittaka, J. Phys. Soc. Jpn. 87, 093703 (2018).
and obtain the following new result:
15.
A. Pustogow, Y. Luo, A. Chronister, Y.-S. Su,
D. Sokolov, F. Jerzembeck, A. P. MacKenzie,
cT2c
H∥(0) = 10.78
(9)
C. W. Hicks, N. Kikugawa, S. Radhu, E. D. Bauer, and
evF t⊥c∗
S. E. Brown, Nature 574, 72 (2019).
Note that solution for the superconducting gap, Δ(y) of
16.
A. G. Lebed, JETP Lett. 110,
173
(2019)
[Pis’ma
Eq. (7) is not of an exponential shape and changes its
v ZhETF 110, 163 (2019)].
sign several times in space, in contrast to the 3D case.
17.
Z. Q. Mao, Y. Maeno, S. NishiZaki, T. Akima, and
Using Eqs. (8) and (9), it is possible to obtain Eq. (2).
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As we already mentioned, in the candidate for
18.
S. Kittaka, T. Nakamura, Y. Aono, S. Yonezawa,
the chiral triplet in-plane isotropic superconductivity,
K. Ishida, and Y. Maeno, Phys. Rev. B 80, 147514
(2009).
Sr2RuO4, the corresponding experimental coefficients
19.
S. Yonezawa, T. Kajikawa, and Y. Maeno, Phys. Rev.
[17-19] are almost two times smaller than the calculated
Lett. 110, 077008 (2013).
in this Letter (2), which is a strong argument against
20.
V. P. Mineev and K. V. Samokhin, Introduction to
the chiral triplet scenario.
Unconventional Superconductivity, Gordon and Breach
The author is thankful to N. N. Bagmet (Lebed) for
Science Publisher, Sydney, Australia (1999).
useful discussions.
21.
L. Jiao, S. Howard, S. Ran, Z. Wang, J. O. Rodriguez,
This is an excerpt of the article
“A chi-
M. Sigrist, Z. Wang, N. P. Butch, and V. Madhavan,
ral
triplet
quasi-two-dimensional superconduc-
Nature 579, 523 (2020).
tor in a parallel magnetic field”. Full text of
22.
A. A. Abrikosov, L. P. Gor’kov, and I. E. Dzyaloshin-
the paper is published in JETP Letters journal.
skii, Methods of Quantum Field Theory in Statistical
DOI: 10.1134/S0021364022200267
Mechanics, Dover, N.Y. (1963).
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2022
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