Temperature dependence of Liquid density and Magnetic field
entrance depth in high-temperature Superconductors
Dzhumanov Safarali
1
, Mayinova Umida
2
1
Institute of Nuclear Physics, Uzbek Academy of Sciences, 100214, Ulugbek, Tashkent, Uzbekistan
2
University of Tashkent for Applied Sciences, Gavhar Str. 1, Tashkent 100149, Uzbekistan
(djumanov)@inp.uz,
https://doi.org/10.5281/zenodo.10469033
Keywords:
High-
𝑇
𝐶
cuprate superconductors, Bosonic Cooper pairs, Three-dimensional Bose superfluid, Bose-liquid
superconductivity, Temperature-dependent magnetic field penetration depth and superfluid density, Unusual
exponential and power-law temperature dependences.
Abstract:
We show that the temperature-dependent superconducting order parameter and related superconducting
properties (in particular, the temperature dependences of specific heat, superfluid density and related London
penetration depth) of high- cuprates are fundamentally different from those of conventional superconductors
and cannot be understood within the existing theories based on the Bardeen-Cooper-Schrieffer (BCS)-type
condensation of weakly-bound Cooper pairs into a superfluid Fermi- liquid and on the usual Bose-Einstein
condensation (ВЕС) of bosonic Cooper pairs. We examine the validity of an alternative approach to the
unconventional superconductivity in high-
c
T
, cuprates and establish that these materials exhibiting a
-
like superconducting transition at the critical temperature
c
T
are similar to the superfluid
He
4
and are also
superfluid Bose systems. We argue that the doped high-
c
T
cuprates from underdoped to overdoped regime
are unconventional (bosonic) superconductors and the tightly-bound (polaronic) Cooper pairs in these polar
materials behave like composite bosons just like
He
4
atoms and condense into a Bose superfluid at
c
T
.
We
identify the superconducting order parameter
SC
in underdoped and optimally doped cuprates as the
coherence parameter
B
of bosonic Cooper pairs, which appears just below
c
T
and has a kink-like
temperature dependence near the characteristic temperature
T
T
c
.
INTRODUCTION
In conventional metals with weak electron-phonon
coupling, the Cooper pairing of electrons and Fermi-
liquid superconductivity occur simultaneously at the
critical temperature
𝑇
𝑐
of the superconducting
transition and these phenomena are well described by
the Bardeen-Cooper-Schrieffer (BCS) theory [1]. In
contrast,
in
doped
copper
oxide
(cuprate)
superconductors, the electron-phonon interaction is
strong
enough
and
the
origin
of
high-
𝑇
𝑐
superconductivity in these polar materials remains still
controversial [2, 3]. The temperature-dependent
superconducting
proporties
(in
particular,
the
temperature dependences of superfluid density and
related magnetic field penetration depth) of high-
𝑇
𝑐
cuprates are fundamentally different from those of
conventional
superconductors
and
cannot
be
understood within the existing theories based on the
BCS-type condensation of weakly-bound (fermionic)
Cooper pairs into a superfluid Fermi-liquid and on the
usual Bose-Einstein condensation (BEC) of tightly-
bound (bosonic) Cooper pairs. The magnetic field
penetration depth in superconductors is usually called
the
London
penetration
depth
𝜆
𝐿
[4]
and
temperaturedependent because, it depends on the
density
𝜌
𝑠
of superfluid carriers [3], which decreases
with increasing temperature
T
. The superfluid density
𝜌
𝑠
and London penetration depth
𝜆
𝐿
are the intrinsic
parameters of superconductors closely connected with
the mechanism of their superconductivity. The
temperature
dependences
of
𝜌
𝑠
and
𝜆
𝐿
in
superconductors are sensitive to the form of the
quasipartical excitation spectrum, which depends on
the fermionic or bosonic nature of Cooper pairs. In
particular, the London penetration depth
𝜆
𝐿
has an
exponential
temperature
dependence
in
the
superconductor if the quasiparticle excitation spectrum
has a finite energy gap, while it has a power-law
temperature dependence if the excitation spectrum of
quasiparticles is gapless in the superconductor.
For a long time, the different BCS-like theories of
Fermi-liquid superconductivity are often used to
describe the phenomenon of superconductivity in
doped high-
𝑇
𝑐
cuprates. However, the origins of many
anomalies in the superconducting properties of high-
𝑇
𝑐
cuprates have not been well understood yet within the
existing theories based on the BCS-like models or the
usual BEC models of an ideal Bose gas [3]. Actually, it
is found experimentally that the temperature
dependences of
𝜆
𝐿
in high-
𝑇
𝑐
cuprates at low
temperatures
𝑇
≲
0,4
𝑇
𝑐
and relatively high
temperatures 0,4
𝑇
𝑐
<
𝑇
<
𝑇
𝑐
do not follow the
predictions of BCS-like theories (see, e.g, Ref. [5]).
Some other experimental results for
𝜆
𝐿
in
Y
-based
cuprate superconductor
YBa
2
Cu
3
O
7-δ
(see Fig. 3 in Ref.
[6]) are also essentially at variance with the exponential
temperature dependence of
𝜆
𝐿
(
𝑇
) predicted by the BCS
theory. The doped high-
𝑇
𝑐
cuprates exhibiting a
𝜆
like
anomaly in the specific heat near
𝑇
𝑐
and other
anomalies (e. g, pronounced anomalies in
𝜌
𝑠
(
𝑇
) and
𝜆
𝐿
(
𝑇
) at a temperature somewhat below
𝑇
𝑐
or far below
𝑇
𝑐
) in the superconducting state are believed to be
unconventional
(bosonic)
superconductors
and
superfluid Bose systems. Therefore, the BCS-like
theories of Fermi-liquid superconductivity and the
usual BEC model of an ideal Bose gas are incapable of
describing the anomalous temperature dependences of
the superfluid density
𝜌
𝑠
(
𝑇
) and the London penetration
depth
𝜆
𝐿
(
𝑇
) in these unconventional superconductors.
In this work, we study the unusual temperature
dependences of the superfluid density
𝜌
𝑠
(
𝑇
) and
magnetic field penetration depth
𝜆
𝐿
(
𝑇
) in high-
𝑇
𝑐
cuprate superconductors
YBa
2
Cu
3
O
7-δ
. We examine the
validity of an alternative theory of high-
𝑇
𝑐
superconductivity in these materials based on the new
types of condensation of bosonic Cooper pairs into a
Bose superfluid and find out the origins of anomalies in
the temperature dependences of
𝜌
𝑠
(
𝑇
) and
𝜆
𝐿
(
𝑇
) in
𝑌𝐵𝑎
2
𝐶𝑢
3
𝑂
6.97
and
𝑌𝐵𝑎
2
𝐶𝑢
3
𝑂
7
. We demonstrate that the
distinctive exponential temperature dependence of
𝜆
𝐿
(
𝑇
) at temperatures somewhat below
𝑇
𝑐
and the
power-law temperature dependence of
𝜆
𝐿
(
𝑇
) at low
temperatures in the cuprate superconductors are direct
consequences of the theory of three-dimensional (3D)
Bose-liquid superconductivity. We argue that the doped
hole carriers in a polar crystal of the cuprates interact
with lattice vibrations (phonons) [7, 8] and they are
self-trapped just like self-trapping holes in ionic
crystals of alkali halides [9, 10]. In doped high-
𝑇
𝑐
cuprates the attractive hole-lattice interaction is strong
enough to overcome the Coulomb repulsion between
two self-trapped (polaronic) carriers [11] and the
Cooper pairing of such carriers results in the formation
of tightly-bound Cooper pairs [12], which behave like
bosons just like
𝐻𝑒
4
atoms and condense into a 3D Bose
superfluid [11, 13]. Therefore, we assume that the
Bose-liquid superconductivity is realized in high-
𝑇
𝑐
cuprates in which the quasiparticle excitations
characteristic of a superfluid Bose-liquid result into the
distinctive non-BCS temperature dependences of
𝜌
𝑠
(
𝑇
)
and
𝜆
𝐿
(
𝑇
).
According to the theory of the 3D Bose superfluid
[11], the quasiparticle excitation spectrum
𝐸
𝐵
(
𝑘
,
𝑇
) has
a finite energy gap
∆
𝑔
(𝑇) = √𝜇
𝐵
2
(𝑇) − ∆
𝐵
2
(𝑇)
when
the interboson coupling constant
𝛾
𝐵
is larger than a
certain critical value
𝛾
𝐵
∗
[11]. At
𝛾
𝐵
< 𝛾
𝐵
∗
the energy
gap in
𝐸
𝐵
(𝑘, 𝑇)
vanishes below a characteristic
temperature
𝑇
𝑐∗
, which is close to
𝑇
𝑐
(i.e.
𝑇
𝐶
∗
< 𝑇
𝐶
) at
𝛾
𝐵
≪ 1
and is much lower than
𝑇
𝑐
at
𝛾
𝐵
≲
1 [11]. The
gapped and gapless excitation spectra
𝐸
𝐵
(
𝑘
,
𝑇
) of a 3D
superfluid Bose-liquid result in the distinctive
exponential and power-law temperature dependences
of the superfluid density
𝜌
𝑠
(
𝑇
) and magnetic field
penetration depth
𝜆
𝐿
(
𝑇
) high-
𝑇
𝑐
cuprates.
We have shown that the theory of a 3
𝐷
Bose-liquid
superconductivity in high-
𝑇
𝑐
cuprates is better
consistent with the experimental data on the magnetic
field penetration depth and the superfluid density and
predicts the distinctive exponential and power-law
temperature dependences of
λ
𝐿
(𝑇)~1/√𝜌
𝑆
(𝑇)
in the
temperature ranges
𝑇
𝑐∗
<
𝑇
<
𝑇
𝑐
and
,
respectively. Thus, various experiments on the London
penetration depth
𝜆
𝐿
(
𝑇
) and the superfluid density
𝜌
𝑠
in
𝑌
-based high-
𝑇
𝑐
cuprate
superconductors [14,15] lend support for the validity of
the theory of 3
𝐷
Bose-liquid superconductivity.
EQUATIONS
The superfluid density
𝜌
𝑆
(𝑇)
and related London
penetration depth
𝜆
𝐿
(𝑇)
in high-
𝑇
𝐶
cuprates are other
key parameters of the superconducting state. The
temperature dependences of
𝜌
𝑆
(𝑇)
and
𝜆
𝐿
(𝑇)
in any
superconductor are sensitive to the form of the
quasiparticle excitation spectrum, which depends on the
fermionic or bosonic nature of superfluid carriers (e.g.,
Cooper
pairs).
In
unconventional
cuprate
superconductors, which are in the limit of bosonic
superconductors, these key superconducting parameters
are determined using the theory of the Bose superfluid.
One can assume that the interacting Bose gas in the
cuprate superconductors below
𝑇
𝐶
has two components.
The density of the superfluid component of such a Bose
gas is determined from the relation
𝜌
𝑆
(𝑇) = 𝜌
𝐵
− 𝜌
𝑛
(𝑇)
,
(1)
where
𝜌
𝑛
(𝑇)
is the density of the normal component of
an attracting Bose gas.
As is well known, the depth of the magnetic field
penetration into a superconductor depends on the
temperature
T
. According to the phenomenological
London theory, the magnetic field penetration depth is
determined from the expression
𝜆
𝐿
(𝑇) = √𝑚
∗
𝑐
2
4𝜋𝜌
𝑆
(𝑇)𝑒
∗2
⁄
,
(2)
where
𝜌
𝑛
(𝑇)
is the density of the effective mass and
charge of the superfluid carriers, respectively,
c
is the
light velocity. Then the expression for
𝜆
𝐿
(𝑇)
in the two
Bose-liquid model can be written as
𝜆
𝐿
(𝑇) = 𝜆
𝐿
(0) [1 −
𝜌
𝑛
(𝑇)
𝜌
𝐵
]
−1 2
⁄
,
(3)
where
𝜆
𝐿
(0) = (𝑚
𝐵
𝑐
2
4𝜋𝜌
𝐵
𝑒
∗2
)
⁄
1 2
⁄
= 2𝑒
is the
charge of Cooper pairs.
Taking into account that
𝜌
𝑆
(0) = 𝜌
𝐵
, the
expression (3) can be written in the form
𝜆
𝐿
2
(0)
𝜆
𝐿
2
(𝑇)
=
𝜌
𝐵
− 𝜌
𝑛
(𝑇)
𝜌
𝐵
=
𝜌
𝑆
(𝑇)
𝜌
𝑆
(0)
, (4)
The density of the normal component of a Bose-
liquid is determined from the expression [16]
𝜌
𝑛
(𝑇) = −
1
3𝑚
𝐵
∫
𝑑𝑛
𝐵
(𝑝)
𝑑𝐸
𝐵
(𝑝)
4𝜋𝑝
2
𝑑𝑝
(2𝜋ℏ)
3
(5)
where
𝑛
𝐵
(𝑝) = [exp (𝐸
𝐵
(𝑝) 𝑘
𝐵
⁄
𝑇) − 1]
−1
is the Bose
distribution function,
𝐸
𝐵
(𝑝) = √(𝜀(𝑝) + 𝜇̃
𝐵
)
2
− Δ
𝐵
2
,
𝜀(𝑝) = 𝑝
2
2𝑚
𝐵
⁄
is the kinetic energy of free bosons of
momentum
p
.
After performing the integration in Eq.(5), we can
write the expression for
𝜌
𝑆
(𝑇) 𝜌
𝑆
(0)
⁄
obtained from the
relation (3) as
𝜌
𝑆
(𝑇)
𝜌
𝑆
(0)
=
𝜆
𝐿
2
(0)
𝜆
𝐿
2
(𝑇)
= {1 −
√2𝑚
𝐵
3 2
⁄
3𝜋
2
ℏ
3
𝜌
𝐵
𝑘
𝐵
𝑇
×
∫
𝜀
3 2
⁄
exp (√(𝜀 + 𝜇̃
𝐵
(𝑇))
2
− Δ
𝐵
2
(𝑇) 𝑘
𝐵
𝑇)𝑑𝜀
⁄
[exp (√(𝜀 + 𝜇̃
𝐵
(𝑇))
2
−Δ
𝐵
2
(𝑇) 𝑘
𝐵
⁄
𝑇) − 1]
2
}
𝜉
𝐵𝐴
0
(6)
Now we compare the numerical recults for
𝜌
𝑆
(𝑇) 𝜌
𝑆
(0)
⁄
obtained using Eq. (6) with the
experimental data for
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
, which is identified
with the normalized superfluid density
𝜌
𝑆
(𝑇) 𝜌
𝑆
(0)
⁄
, in
high-
T
c
cuprate superconductor
YBa
2
Cu
3
O
6.97
[16]. As
can be seen in Fig.3, the fit of Eq.(6) to experimental
data is fairly good and the observed temperature
dependence of
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
in
YBa
2
Cu
3
O
6.97
are at
variance with the predictions of other models (see Fig.
9 in Ref. [16]) and the temperature- dependent radio
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
changes more slowly at low temperatures
than the high-temperature variation of this quanty.
Fig.3. Temperature dependence of
𝜌
𝑆
(𝑇) 𝜌
𝑆
(0)
⁄
or
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
(solid line) calculated by using Eq. (6)
with the fitting parameters
𝜌
𝐵
= 1.4 × 10
19
𝑐𝑚
−3
,
𝑚
𝑏
= 5.2𝑚
𝑒
,
𝜉
𝐵𝐴
= 0.05𝑒𝑉
and compared with the
experimental data for
YBa
2
Cu
3
O
6.97
[6].
A pronounced change in the observed slope of
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
occurs near the characteristic temperature
𝑇
𝑐
∗
≃ 0.4 𝑇
𝐶
well below
T
c
. Next we discuss the origin
of such anomalous change of the observed temperature
dependence of
𝜌
𝑆
(𝑇) 𝜌
𝑆
(0)
⁄
or
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
in
YBa
2
Cu
3
O
6.97
.
CONCLUSIONS
We have determined the quantity
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
,
which is identified with the normalized superfluid
density
𝜌
𝑆
(𝑇) 𝜌
𝑆
(0)
⁄
, and examined the distinctive
temperature dependences of
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
in
YBa
2
Cu
3
O
6.97
at low temperatures (below
𝑇
𝑐
∗
) and at
high temperatures (in the temperature range
𝑇
𝑐
∗
< 𝑇 <
𝑇
𝑐
). We have compared the obtained results for the
superfluid density
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
with the experimental
data on the temperature dependence of
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
[16] (see Fig.3), which were compared previously with
different inconsistent theories. Our close examination
of data on the temperature dependence of
𝜆
𝐿
2
(0) 𝜆
𝐿
2
⁄
(𝑇)
in
YBa
2
Cu
3
O
6.97.
ACKNOWLEDGMENTS
The research is supported in part by Grant No F-
FA -2021-443 of the Agency for Innovative
Development of the Republic of Uzbekistan.
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