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of the peaked conductivity and its decaying tendency un-
der the application of the potential gradient will strongly
support the superconducting multiphase.
IV.
LINEAR AND NONLINEAR
PARACONDUCTIVITY IN THE IN-PLANE
MAGNETIC FIELD
In this section, we study the linear and nonlinear para-
conductivity of the system in the in-plane magnetic field
with and without the potential gradient.
8
(a)
0
0.5
1
1.5
2
2.5
0
0.2
0.4
0.6
0.8
1
H/Tc0
T/Tc0
(b)
0
0.01
0.02
0.03
0.04
0.05
0.06
0
0.5
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] |
2
2.5
0
0.2
0.4
0.6
0.8
1
H/Tc0
T/Tc0
(b)
0
0.01
0.02
0.03
0.04
0.05
0.06
0
0.5
1
1.5
2
2.5
q
H/Tc0
(c)
0.0
0.5
1.0
1.5
2.0
0
0.5
1
1.5
2
2.5
σxx/σ0
H/Tc0
(d)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx
(e)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx
(f)
0.00
0.02
0.04
0.06
0.08
0.10
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx
FIG. 6. (a) The superconducting transition line of the sys- | [
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] |
0.10
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx
FIG. 6. (a) The superconducting transition line of the sys-
tem in the in-plane magnetic field. (b) The magnetic field
dependence of Cooper pair’s momentum along the transition
line. Superconducting state changes from the BCS state to
the FFLO state near H/Tc0 ∼ 1.4. (c) The linear paracon-
ductivity. (d-f) Momentum dependence of αµ for qy = 0. (d),
(e), and (f) correspond to H = 0.00, 0.35, and 0.70, respec-
tively.
A.
The system with inversion symmetry | [
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] |
tively.
A.
The system with inversion symmetry
In the beginning, we consider the system without the
potential gradient. In this case, the system has inversion
symmetry, and therefore the nonlinear paraconductivity
vanishes.
We calculate the superconducting transition
line and the linear paraconductivity in the same way as
Sec. III.
As understood from Figs. 6(a) and 6(b), the super-
conducting phase diagram has the BCS phase in the low
magnetic field and the FFLO phase in the high magnetic | [
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] |
magnetic field and the FFLO phase in the high magnetic
field [76]: The Cooper-pair momentum becomes finite
for the magnetic field H ≳ 1.4Tc0. Furthermore, the lin-
ear paraconductivity σxx
1s has a peak near the tricritical
point as shown in Fig. 6(c). The origin of this peak is at-
tributed to the shape of αµ(q). We show the momentum
dependence of αµ(q) for the system in the BCS phase
(H = 0.00) and FFLO phase (H = 0.70) and near the
tricritical point (H = 0.35), in Figs. 6(d), 6(f) and 6(e), | [
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] |
tricritical point (H = 0.35), in Figs. 6(d), 6(f) and 6(e),
respectively.
It should be noted that the contribution
to the paraconductivity formula Eq. (21) mainly comes
from the region where αµ(q) is vanishingly small. Near
the tricritical point, we can see from Fig. 6(e) that the
area of small αµ(q) is much larger than that in Figs. 6(d)
and 6(f). This naturally leads to the peak of paraconduc-
tivity. In other words, the peaked paraconductivity near
the tricritical point may be interpreted as coming from | [
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] |
the tricritical point may be interpreted as coming from
the degeneracy of the BCS and FFLO states, in analogy
with the case in the perpendicular magnetic field. The
result obtained here is consistent with the previous study
which pointed out the enhanced linear paraconductivity
with the anomalous temperature scaling near the tricriti-
cal point [70, 71]. This result implies that the observation
of the linear paraconductivity can be used as a probe of
FFLO superconductors.
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] |
of the linear paraconductivity can be used as a probe of
FFLO superconductors.
B.
The system without inversion symmetry
In this section, we apply the potential gradients v =
0.01 and v = 0.10, which are smaller and larger than
the transition temperature Tc0 ∼ 0.0255, and correspond
to weakly and strongly noncentrosymmetric supercon-
ductors, respectively.
We consider not only the linear
paraconductivity but also the nonlinear paraconductiv-
ity, which is allowed due to the broken inversion sym- | [
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] |
ity, which is allowed due to the broken inversion sym-
metry and C2z rotation symmetry. As an indicator of
the nonreciprocity, in addition to σxxx
2s , we focus on the
quantity
ηxxx = σxxx
2s
(σxx
1s )2 ,
(33)
to study the nonreciprocal charge transport. This quan-
tity is convergent as approaching the transition temper-
ature, in contrast to σ1s and σ2s. It has the dimension of
the inverse of the current density and, therefore, repre-
sents the typical value of the current density where non- | [
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] |
sents the typical value of the current density where non-
reciprocity becomes visible [74].
We show the calculation results in Fig. 7. First, we
discuss the difference of the superconducting phase dia-
grams in the presence and absence of the inversion sym-
metry, by comparing Figs. 6 and 7.
By switching on
the potential gradient v, due to the inversion symmetry
breaking, the BCS-FFLO transition in Fig. 6 begins to
change to the crossover of the so-called helical supercon- | [
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] |
change to the crossover of the so-called helical supercon-
ductivity [74, 80–90]. Indeed, the Cooper-pair momen-
tum takes finite values in the low-field regime before it
takes large values in the high magnetic field, as shown in
Figs. 7(b) and 7(g). For clarity, we call the crossover in
weakly and strongly noncentrosymmetric cases as BCS-
FFLO-like crossover and helical crossover, respectively.
When the inversion symmetry breaking is weak (v =
0.01), the behavior of the linear paraconductivity in the | [
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] |
0.01), the behavior of the linear paraconductivity in the
moderate magnetic field is similar to that near the BCS-
PDW transition. Indeed, we can see that the linear para-
conductivity has a peak in the BCS-FFLO-like crossover
region [Fig. 7(c)] as well as near the BCS-FFLO transi-
tion point [Fig. 6(c)]. By contrast, Fig. 7(h) shows that
9
0
0.01
0.02
0.03
0.04
0.05
0.06
0
0.5
1
1.5
2
2.5
q
H/Tc0
0
0.5
1
1.5
2
2.5
0
0.2
0.4
0.6
0.8
1
H/Tc0
T/Tc0
0
0.01
0.02
0.03
0.04
0.05 | [
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0
0.5
1
1.5
2
2.5
0
0.2
0.4
0.6
0.8
1
H/Tc0
T/Tc0
0
0.01
0.02
0.03
0.04
0.05
0.06
0
0.5
1
1.5
2
2.5
q
H/Tc0
0
0.5
1
1.5
2
2.5
0
0.2
0.4
0.6
0.8
1
H/Tc0
T/Tc0
0.0
0.2
0.4
0.6
0.8
1.0
0
0.5
1
1.5
2
2.5
σxx/σ0
H/Tc0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.5
1
1.5
2
2.5
ηxxx
H/Tc0
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
0
0.5
1
1.5
2
2.5
σxxx
H/Tc0
×10!
×10"
0.0
0.2
0.4
0.6
0.8
1.0
0
0.5
1
1.5
2
2.5
σxx/σ0
H/Tc0
0.0
10.0
20.0
30.0
40.0
50.0
0
0.5
1
1.5
2 | [
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] |
0.6
0.8
1.0
0
0.5
1
1.5
2
2.5
σxx/σ0
H/Tc0
0.0
10.0
20.0
30.0
40.0
50.0
0
0.5
1
1.5
2
2.5
σxxx
H/Tc0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0
0.5
1
1.5
2
2.5
ηxxx
H/Tc0
×10!
×10"
(a)
(b)
(d)
(c)
(f)
(e)
(g)
(i)
(h)
(j)
FIG. 7. The superconducting transition line [(a),(f)], Cooper-pair momentum [(b),(g)], linear [(c),(h)] and nonlinear [(d),(i)]
paraconductivities, and nonreciprocity [(e),(j)] of the system in the in-plane magnetic field and potential gradient. (a-e) show | [
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] |
the results of v = 0.01 and (f-j) are the results of v = 0.10.
the peak vanishes in the strongly noncentrosymmetric
system, i.e., in the helical crossover regime. These are
natural results because the peaked linear paraconductiv-
ity signals the degenerate superconducting states and the
degeneracy is sufficiently lifted in the strongly noncen-
trosymmetric case in Fig. 7(h).
Interestingly, the behavior of the nonlinear paracon-
ductivity is significantly different from the linear para-
conductivity. | [
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] |
ductivity is significantly different from the linear para-
conductivity.
We can see in Figs. 7(d) and 7(i) that
there is a peak in the nonlinear paraconductivity in both
weakly and strongly noncentrosymmetric systems with
comparable strengths of σxxx
2s
and ηxxx. The obtained
peak in strongly noncentrosymmetric superconductors is
consistent with the results of Ref. [74], which studies the
enhancement of nonlinear paraconductivity associated
with the helical crossover in a strongly noncentrosym- | [
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] |
with the helical crossover in a strongly noncentrosym-
metric mono-layer Rashba model. Our results indicate
that the enhanced nonlinear paraconductivity associated
with the evolution of finite-momentum superconductivity
also exists in the weakly noncentrosymmetric regime.
To understand the behavior of the nonlinear paracon-
ductivity, we show in Figs. 8 and
9 the momentum
dependence of αµ(q) with q = (qx, 0) in the moderate
and strong magnetic fields. Here, µ specifies the min-
imum eigenvalue of ˆα(q). | [
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] |
and strong magnetic fields. Here, µ specifies the min-
imum eigenvalue of ˆα(q).
These figures clearly show
the realization of finite-momentum superconductivity in
the in-plane magnetic field.
The results of αµ(q) for
v = 0.01 is similar to those of the FFLO state shown in
Figs. 6(e) and 6(f), reflecting the weak inversion breaking
(v ≪ Tc0), while the strongly noncentrosymmetric case
(v ≫ Tc0) in Figs. 8(c) and 8(d) shows a large devia-
tion. We also show in Figs. 9(a) and 9(b) the enlarged | [
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] |
tion. We also show in Figs. 9(a) and 9(b) the enlarged
figures of Figs. 8(a) and 8(c), respectively, to consider the
main contribution to the paraconductivity coming from
the region αµ(q) ∼ 0. When seen in this energy scale,
(a)
0.00
0.02
0.04
0.06
0.08
0.10
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx
(b)
0.00
0.02
0.04
0.06
0.08
0.10
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx
(c)
0.00
0.02
0.04
0.06
0.08
0.10
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx
(d)
0.00
0.02
0.04
0.06
0.08
0.10
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx | [
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] |
-0.05
0
0.05
0.1
αµ(q)
qx
(d)
0.00
0.02
0.04
0.06
0.08
0.10
-0.1
-0.05
0
0.05
0.1
αµ(q)
qx
FIG. 8. The momentum dependence of αµ(q). (a) and (b)
are the case of v = 0.01, while (c) and (d) are the case of
v = 0.10.
(a) and (c) are for the moderate magnetic field
H = 0.036 and H = 0.039, respectively, where the nonlinear
paraconductivity shows the peak. (b) and (d) are obtained in
the high magnetic field, H = 0.070.
the structure of αµ(q) for v = 0.01 is essentially the same | [
-0.003949437290430069,
0.07190430909395218,
0.7556686997413635,
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0.3211800754070282,
0.10300413519144058,
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0.08865801244974136,
0.004288224969059229,
0.06317642331123352,
0.637161374092102,
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0.6890101432800293,
0.22982332110404968,
1.04521906375885,
0.4616715610027313,
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0.40809711813926697,
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] |
the high magnetic field, H = 0.070.
the structure of αµ(q) for v = 0.01 is essentially the same
as that for v = 0.1. Actually, the values of αµ(q) at the
local minima q ∼ ±0.02, dubbed α±, are not degenerate
due to the inversion symmetry breaking v = 0.01 ̸= 0,
and satisfy α− − α+ ≫ α+ ≳ 0. This means that the α+
valley mainly contributes to the nonlinear paraconductiv-
ity in the same way as the strongly noncentrosymmetric
case. Thus, at the temperatures considered here, the sys-
10
(a)
0.000
0.001
0.002 | [
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] |
case. Thus, at the temperatures considered here, the sys-
10
(a)
0.000
0.001
0.002
0.003
0.004
0.005
-0.02
0
0.02
αµ(q)
qx
(b)
0.000
0.001
0.002
0.003
0.004
0.005
-0.02
0
0.02
αµ(q)
qx
FIG. 9. Enlarged figures of Figs. 8(a) and 8(c) for small αµ(q).
The panels (a) and (b) correspond to Figs. 8(a) and 8(c),
respectively.
tem is sufficiently noncentrosymmetric for the nonlinear
paraconductivity even though v ≪ Tc0.
This explains
the appearance of the peaked nonlinear paraconductiv-
ity in Fig. 7(d). | [
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] |
This explains
the appearance of the peaked nonlinear paraconductiv-
ity in Fig. 7(d).
We expect that the peak width will
decrease when v is decreased from v = 0.01 with fix-
ing the reduced temperature, and eventually will vanish
when α− − α+ ∼ α+ is always satisfied along the tran-
sition line. This is consistent with σxxx
2s
= 0 at v = 0 as
enforced by the inversion symmetry.
The important point is that a small potential gradi-
ent v ≪ Tc0 is sufficient to obtain the sizable nonlin- | [
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] |
ent v ≪ Tc0 is sufficient to obtain the sizable nonlin-
ear paraconductivity. Such a small v does not usually
modify superconducting properties. In this case, on the
other hand, each valley of the FFLO state shown e.g., in
Fig. 6(f), has the intrinsic nonreciprocity even without v:
The αµ(q) dispersion has a (cubic) asymmetry with re-
spect to each local minimum, which has been identified as
the origin of the nonreciprocal charge transport in non-
centrosymmetric superconductors [62, 72–74]. The appli- | [
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] |
centrosymmetric superconductors [62, 72–74]. The appli-
cation of the potential gradient lifts the degeneracy of the
two valleys, and one of the valleys selectively contributes
to the nonlinear paraconductivity, picking up the intrin-
sic nonreciprocity of finite-momentum Cooper pairs. A
small potential gradient inevitably exists in most thin
film superconductors owing to substrate effects.
Since
the other contributions to the nonlinear paraconductiv-
ity unrelated to the FFLO state, if any, would be small | [
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] |
ity unrelated to the FFLO state, if any, would be small
when v ≪ Tc0, the observation of the peaked nonlinear
paraconductivity in intrinsically centrosymmetric super-
conductors under a small potential gradient serves as the
strong evidence for the FFLO state.
V.
SUMMARY AND CONCLUSION
In this study, we have investigated the linear and non-
linear paraconductivity in the bilayer system under the
perpendicular and in-plane magnetic fields. We have re-
vealed that the linear paraconductivity has a peak ac- | [
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] |
vealed that the linear paraconductivity has a peak ac-
companied by the BCS-PDW transition in the system
with a perpendicular magnetic field. The peak of linear
paraconductivity tends to vanish by applying the poten-
tial gradient, signaling the lifted degeneracy of the BCS
and PDW states. Similar results of the linear paracon-
ductivity are obtained for the in-plane magnetic field. In
this case, the degeneracy of the BCS and FFLO states
gives rise to the peak structure of the linear paraconduc- | [
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] |
gives rise to the peak structure of the linear paraconduc-
tivity, which is also smeared out by applying the strong
potential gradient. We have also studied the nonlinear
paraconductivity of the system in the in-plane magnetic
field. In contrast to the linear paraconductivity, a peak
appears robustly in both the weakly and strongly noncen-
trosymmetric cases, i.e., both near the BCS-FFLO-like
crossover and the helical crossover regions.
Since we neglected the orbital magnetic field in this | [
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] |
crossover and the helical crossover regions.
Since we neglected the orbital magnetic field in this
paper, our results, particularly those in the perpendic-
ular magnetic field, apply to the superconductors with
large Maki parameters. Even when the orbital magnetic
field can not be neglected, the qualitative results such
as the peaked linear paraconductivity are expected to
be unaffected.
The peaked nonlinear paraconductivity
in finite-momentum superconductors is also expected to | [
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] |
The peaked nonlinear paraconductivity
in finite-momentum superconductors is also expected to
survive to some extent in the orbital magnetic field [74],
leaving a quantitative study as a future issue.
Based on the results obtained in this paper, we propose
to use linear and nonlinear paraconductivity to probe
multiphase and/or finite-momentum superconductors. In
particular, this proposal is expected to be useful in thin
film superconductors because the thermodynamic mea-
surements are challenging there. | [
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] |
film superconductors because the thermodynamic mea-
surements are challenging there.
The linear and non-
linear paraconductivity have different advantages and
disadvantages.
The linear paraconductivity can gener-
ally be used to probe degenerate superconducting states,
through the detection of the peak structure, while it
seems not sensitive to the detailed nature of the supercon-
ducting states. The nonlinear paraconductivity can only
be used by breaking the inversion symmetry. However, | [
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] |
be used by breaking the inversion symmetry. However,
once the symmetry allows a finite response, it can pro-
vide strong evidence for the finite-momentum supercon-
ductivity, including (nearly-)FFLO and helical supercon-
ducting states. A sizable response is obtained by lifting
the degeneracy of the finite-momentum Cooper pairs. It
is known that there are a variety of finite-momentum su-
perconducting states such as the layer-decoupled state or
the orbital FFLO state [59, 91–93], and pair-density-wave | [
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] |
the orbital FFLO state [59, 91–93], and pair-density-wave
state in strongly correlated superconductors [94] (refer-
ring to a different state from the that studied in this pa-
per, accompanying the in-plane modulation of the order
parameter). It is an interesting future issue to quantita-
tively study the nonreciprocal charge transport of such
finite momentum superconductors.
ACKNOWLEDGMENTS
T.M. thank fruitful discussion with Sho Ishikawa.
This
work
was
supported
by
JSPS
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] |
T.M. thank fruitful discussion with Sho Ishikawa.
This
work
was
supported
by
JSPS
KAKENHI
No. JP21K13880, No. JP22H01181, No. JP22H04933,
No. JP22H04476, No. JP23K17353, No. JP23K22452,
No. JP24H00007, and No. JP24H01662.
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] |
Khim,
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F.
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R. Cardoso-Gil, U. Stockert, A. P. Mackenzie, D. F.
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transition within the superconducting state of CeRh2As2,
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Frontiers
in
Electronic
Materials
2,
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] |
tivity of the Two-Phase superconductor CeRh2As2,
Frontiers
in
Electronic
Materials
2,
10.3389/fe-
mat.2022.880579 (2022).
[8] J. F. Landaeta, P. Khanenko, D. C. Cavanagh, C. Geibel,
S. Khim, S. Mishra, I. Sheikin, P. M. R. Brydon,
D. F. Agterberg, M. Brando, and E. Hassinger, Field-
Angle dependence reveals Odd-Parity superconductivity
in CeRh2As2, Phys. Rev. X 12, 031001 (2022).
[9] S. Ogata, S. Kitagawa, K. Kinjo, K. Ishida, M. Brando,
E. Hassinger, C. Geibel, and S. Khim, Parity transition of | [
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] |
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] |
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0.9032615423202515,
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0.1150580495595932,
-0.3312169015407562
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14
Appendix A: Derivation of GL factor ˆα | [
-0.07654113322496414,
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] |
yond, Annu. Rev. Condens. Matter Phys. 11, 231 (2020).
14
Appendix A: Derivation of GL factor ˆα
In this section, we demonstrate the derivation of GL
factor ˆα from GL free energy,
F = − 1
β tr log e−β ˆ
H.
(A1)
Here, we can rewrite the Hamiltonian as
ˆH = 1
2
�
k
Ψ(k)† ˆHBdG(k)Ψ(k) + 2
�
k
ξ(k) +
�
m,k
|∆|2
U
(A2)
with the BdG Hamiltoninan ˆHBdG,
ˆHN(k) =
�HN(k)
0
0
−HN(−k)T
�
,
(A3)
∆ = ∆1φ1 + ∆2φ2
(A4)
ˆ∆ =
� 0
∆
Ơ
0
�
,
(A5)
ˆHBdG(k) = ˆHN(k) + ˆ∆,
(A6)
and the Nambu spinor,
Ψ(k)† = (c†
k↑1, c† | [
-0.2530110478401184,
-0.06963111460208893,
0.6692748665809631,
-0.2635977566242218,
-0.37665241956710815,
0.11064068228006363,
0.005884315352886915,
-0.8275172114372253,
-0.15845893323421478,
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] |
ˆ∆ =
� 0
∆
Ơ
0
�
,
(A5)
ˆHBdG(k) = ˆHN(k) + ˆ∆,
(A6)
and the Nambu spinor,
Ψ(k)† = (c†
k↑1, c†
k↓1, c†
k↑2, c†
k↓2, c−k,↑1, c−k↓1, c−k,↑2, c−k↓2).
(A7)
Then, the GL free energy is
F = − 1
2β
�
k
tr log
�
1 + e−βEn(k)�
+ 2
�
k
ξ(k) +
�
m,k
|∆|2
U
(A8)
= F|∆=0 + δF,
(A9)
where En(k) represents eigenvalues of ˆHBdG(k). By ex-
panding δF by ∆, we can obtain the GL factor ˆα,
δF/V =
�
m
|∆|2
U
+
1
2βV
�
k,ωn
trN
�
1
iωn − HN(k)∆(k)
1
iωn + HN(−k)T ∆(k)†
�
+ O
�
∆4�
=
�
m
|∆m|2
U
+ 1
2V
�
k,n,m
Fnm(k)
�
∆∗ | [
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0.7090108394622803,
0.21238769590854645,
-0.10835309326648712,
-0.19947156310081482,
-0.11244558542966843,
0.1913835108280182,
0.0038787678349763155,
-0.10054636746644974,
-0.8073616027832031,
-0.13423451781272888,
-0.02790035679936409,
-0.01667080633342266,
0.5194553732872009,
0.39250215888023376,
-0.5898369550704956,
-0.11713680624961853,
-0.49091583490371704,
0.11528702825307846,
-0.7643277049064636,
0.21471597254276276,
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0.1319153755903244,
-0.10568752884864807,
-0.2120349258184433,
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0.158390611410141,
-0.22536106407642365,
-0.08069899678230286,
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0.018413953483104706,
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0.2739531397819519,
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-0.17939762771129608,
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0.5763238668441772,
0.1926284283399582,
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0.4477291703224182,
0.5010755062103271,
-0.15855436027050018,
-0.26286646723747253,
0.22250740230083466,
0.3054334819316864,
-0.2523081600666046,
0.20903722941875458,
-0.5673450827598572,
0.7369982004165649,
0.049931205809116364,
0.2276257425546646,
-0.16638904809951782,
0.18883635103702545,
0.6776583790779114,
0.07920593023300171,
0.3959563672542572,
-0.7653043270111084,
-0.2460331916809082,
-0.2786273658275604,
0.700636625289917,
-0.27406013011932373,
-0.5258989334106445,
0.4266010522842407,
-0.21201099455356598,
0.13036365807056427,
0.4277157485485077,
0.041466619819402695,
0.4897335171699524,
-0.032480061054229736,
0.3547075092792511,
-0.7504565119743347,
0.24886251986026764,
-0.3452965319156647,
0.002415642375126481,
-0.36576977372169495,
-0.640027642250061,
0.15296947956085205,
-0.2997787594795227,
0.5878103375434875,
-0.2201775312423706,
-0.26420125365257263,
0.06674475222826004,
-0.3488268554210663,
-0.08846856653690338,
-0.1578861027956009,
-0.26565200090408325,
-1.1056411266326904,
-0.773453950881958,
-0.12422990053892136,
0.8914591670036316,
0.09163301438093185,
-0.1008530929684639,
-0.08942529559135437,
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-0.4316127598285675,
0.36146605014801025,
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0.22135622799396515,
0.30801868438720703,
0.0515836738049984,
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-0.22442908585071564,
0.6210564374923706,
-0.055268652737140656,
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0.4576490521430969,
-0.28415197134017944,
-0.4865645170211792,
0.1135827973484993,
0.21211759746074677,
0.1628403663635254,
0.41650518774986267,
-0.06738771498203278,
0.8294968605041504,
0.9229596853256226,
-0.3066382110118866,
-0.6102238893508911,
0.3468232750892639,
-0.5569194555282593,
0.3210602104663849,
0.0022539715282619,
0.7308663725852966,
0.4750840365886688,
0.46877145767211914,
-0.3402845859527588,
0.14393272995948792,
0.19509141147136688,
0.3619438111782074,
-0.14848534762859344,
0.030295727774500847,
-0.10352544486522675,
0.192750483751297,
0.7086693644523621,
0.5125240683555603,
-0.29511886835098267,
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0.0054091508500278,
0.2725306451320648,
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0.2274709790945053,
0.6123514175415039,
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0.0006620723288506269,
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0.06470462679862976,
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0.3539499342441559,
0.4684530794620514,
0.4162794351577759,
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-0.18163996934890747,
0.21910688281059265,
-0.15863443911075592,
-0.0008516800589859486,
-0.24261140823364258,
0.41276904940605164,
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-0.26399508118629456,
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0.0057061780244112015,
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0.206138476729393,
0.8432262539863586,
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0.481288880109787,
0.08060561865568161,
-0.6487627625465393,
0.8060501217842102,
0.9680539965629578,
0.266366183757782,
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0.029307056218385696,
0.3362767696380615,
0.9580951929092407,
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0.2209128886461258,
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0.8021929264068604,
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0.84989333152771,
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0.027008656412363052,
0.44303688406944275,
0.17505136132240295,
0.3336549401283264,
0.23962071537971497,
-0.08417952060699463,
-0.38873597979545593,
-0.23045185208320618,
-0.875909686088562,
-0.507962703704834,
0.331277072429657,
0.19372768700122833,
0.13561421632766724,
-0.2840607762336731,
-0.015453596599400043,
0.27916863560676575,
0.08717433363199234,
0.4779910147190094,
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0.02465609647333622,
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0.64162278175354,
-0.8419366478919983,
0.06094716861844063,
0.1448173075914383,
0.4927634596824646,
-0.09328009933233261,
-0.45884931087493896,
-0.3885820209980011,
-0.14439748227596283,
-0.39224621653556824,
0.9934356212615967,
0.28821536898612976,
0.0834212377667427,
0.041707172989845276,
-0.6922287344932556,
-0.10862723737955093,
0.5076273679733276,
0.2560722827911377,
0.00969100184738636,
-0.38073164224624634,
0.05071580782532692,
0.047129008919000626,
0.31543776392936707,
-0.18150927126407623,
-0.6816028356552124,
0.1205596998333931,
0.03971956670284271
] |
trN
�
1
iωn − HN(k)∆(k)
1
iωn + HN(−k)T ∆(k)†
�
+ O
�
∆4�
=
�
m
|∆m|2
U
+ 1
2V
�
k,n,m
Fnm(k)
�
∆∗
1∆1|⟨un(k)| φ1 |u∗
m(−k)⟩|2 + ∆∗
2∆2|⟨un(k)| φ2 |u∗
m(−k)⟩|2
+ ∆∗
1∆2 ⟨u∗
m(−k)| φ†
1 |un(k)⟩ ⟨un(k)| φ2 |u∗
m(−k)⟩ +∆∗
2∆1 ⟨u∗
m(−k)| φ†
2 |un(k)⟩ ⟨un(k)| φ1 |u∗
m(−k)⟩
�
+ O
�
∆4�
=
�∆∗
1 ∆∗
2
�
ˆα
�
∆1
∆2
�
+ O
�
∆4�
(A10)
with Fmn(k) in Eq. (12). Then we obtain the expression
of ˆα given in Eq. (11).
We prove that σBCS−PDW vanishes in the system in the | [
-0.47475555539131165,
-0.28482699394226074,
0.678063154220581,
-0.17002134025096893,
-0.09396917372941971,
0.18878506124019623,
0.2899059057235718,
-0.9919725656509399,
-0.1746899038553238,
-0.12659460306167603,
-0.14900624752044678,
0.6734760999679565,
-0.447512686252594,
0.6427035331726074,
0.6344319581985474,
0.6475386619567871,
0.7633091807365417,
0.05498458817601204,
-0.6042765974998474,
-0.8389165997505188,
0.12392130494117737,
0.07217934727668762,
0.7747407555580139,
0.08179237693548203,
0.3995671272277832,
0.18139588832855225,
0.20262360572814941,
-0.23333632946014404,
-0.058423273265361786,
0.5175333023071289,
-0.054520174860954285,
0.25086334347724915,
-0.6981928944587708,
-0.13431572914123535,
0.15628857910633087,
0.055367324501276016,
0.11286263167858124,
0.329893559217453,
0.07308925688266754,
-0.49992483854293823,
-0.10882242023944855,
-0.21444717049598694,
0.14789514243602753,
0.4989076554775238,
-0.22860974073410034,
-0.20145976543426514,
-1.0691968202590942,
0.16490650177001953,
-0.2361220419406891,
-0.5872433185577393,
-0.3708435595035553,
0.4240000545978546,
0.3986826539039612,
0.05664511024951935,
-0.5091161727905273,
0.025915047153830528,
-0.6234243512153625,
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0.1592719703912735,
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0.6182459592819214,
0.5699492692947388,
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0.47792571783065796,
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0.6075472831726074,
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0.10091910511255264,
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0.07776223123073578,
0.5758597254753113,
0.13304509222507477,
-0.18760459125041962,
0.2714967429637909,
0.05627397075295448,
0.558508574962616,
-0.32010146975517273,
-0.21758495271205902,
0.33219286799430847,
0.24717335402965546,
0.06370101869106293,
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0.6602204442024231,
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0.43571221828460693,
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0.3466585874557495,
0.6301170587539673,
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0.47620889544487,
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0.15821070969104767,
0.21214191615581512,
0.05953909829258919,
0.5138757824897766,
-0.4702407717704773,
-0.010033771395683289,
0.3364817500114441,
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0.26324179768562317,
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-1.0934778451919556,
0.1580592691898346,
-0.16339845955371857,
-0.4836554527282715,
-0.049972113221883774,
-0.15795399248600006,
-0.050921790301799774,
-0.5245298743247986,
0.1553669422864914,
0.26955896615982056,
0.1888664960861206,
0.3934832215309143,
0.16382859647274017,
0.4456460475921631,
0.481606125831604,
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0.5148035287857056,
0.7370724081993103,
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0.24172227084636688,
1.2894949913024902,
0.16110859811306,
0.13446711003780365,
-0.11977265775203705,
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-0.26812124252319336,
0.058939650654792786,
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0.38664019107818604,
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0.11079691350460052,
0.22449104487895966,
-0.3884579539299011,
0.04249342158436775,
0.12813134491443634,
0.3337007462978363,
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0.00006101978215156123,
0.11583580821752548,
-0.35913118720054626,
0.11629711091518402,
-0.2413344532251358,
-0.26745927333831787,
-0.09459636360406876,
-0.07293029874563217,
-0.44697749614715576,
-0.15221022069454193,
0.14730621874332428,
0.3011888265609741,
-0.15317673981189728,
-0.21347275376319885,
0.7203870415687561,
-0.5811724662780762,
0.21444132924079895,
-0.4916219413280487,
0.38640090823173523,
0.03854403272271156,
-0.08700603991746902,
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0.07542990148067474,
0.3060952425003052,
-0.5459553003311157,
0.12967891991138458,
0.13765455782413483,
0.20360060036182404,
0.059839747846126556,
-0.36608555912971497,
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-0.8966830372810364,
0.5833109021186829,
0.19023481011390686,
0.8032675981521606,
0.4077164828777313,
0.121111199259758,
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0.7250997424125671,
-0.8534950017929077,
1.4948233366012573,
-0.27731990814208984,
-0.5257905125617981,
0.5851281881332397,
0.10443850606679916,
-0.038376033306121826,
-0.06481210887432098,
0.23163777589797974,
0.07754989713430405,
-0.10224432498216629,
-0.48641011118888855,
0.41035956144332886,
0.00970219075679779,
-0.03595602884888649,
-0.682220458984375,
-0.11838475614786148,
0.25143587589263916,
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-0.18972857296466827,
-0.587432861328125,
0.3179848790168762,
-0.7840853333473206,
0.2291848361492157,
-0.56602942943573,
0.3640082776546478,
0.17201170325279236,
0.8707034587860107,
-0.24653680622577667,
0.681491494178772,
0.5442283749580383,
-0.2083393782377243,
0.5263265371322632,
0.23813775181770325,
-0.09929826855659485,
0.5457450747489929,
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-0.3988437056541443,
-0.09650787711143494,
0.0312868095934391,
0.13734014332294464,
0.36454543471336365,
-0.1625012457370758,
0.11010748893022537,
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0.24968154728412628,
0.7482489347457886,
-0.6794888377189636,
0.6217176914215088,
0.10459297150373459,
0.1046878769993782,
-0.21802359819412231,
0.2345692664384842,
-0.31676381826400757,
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-0.18802717328071594,
0.3474244177341461,
0.08963563293218613,
-0.7045029401779175,
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0.20529605448246002,
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0.1286965012550354,
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0.08822201192378998,
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-1.1624395847320557,
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1.4319688081741333,
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0.42827802896499634,
0.8818473219871521,
1.00801420211792,
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0.12461341917514801,
0.01728551648557186,
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-0.5696496367454529,
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0.3092639744281769,
0.2959098517894745,
-0.8388096690177917,
0.09709477424621582,
0.633176863193512,
-0.06781909614801407,
0.08073477447032928,
0.25273627042770386,
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0.11913898587226868,
0.034481700509786606,
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1.0375319719314575,
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0.003434646874666214,
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0.22429881989955902,
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0.20159457623958588,
0.8427914977073669,
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] |
of ˆα given in Eq. (11).
We prove that σBCS−PDW vanishes in the system in the
perpendicular magnetic field. In this case, the system has
both C2z symmetry and Mz symmetry,
ˆα(q) = ˆα(−q),
(A11)
σx ˆα(q)σ†
x = ˆα(q).
(A12)
Thus, we can express ˆα(q) by,
ˆα(q) =
�
a(q) b(q)
b(q) a(q)
�
,
(A13)
with real numbers a(q) and b(q). Accordingly, eigenvec-
tors of ˆα are given by
|BCS⟩ =
1
√
2
�
1
1
�
,
(A14)
|PDW⟩ =
1
√
2
�
1
−1
�
,
(A15)
regardless of q. This forces σBCS−PDW to vanish, since
⟨BCS| ji |PDW⟩ = 1
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] |
1
√
2
�
1
−1
�
,
(A15)
regardless of q. This forces σBCS−PDW to vanish, since
⟨BCS| ji |PDW⟩ = 1
2
�1 1�
∂i
�
a(q) b(q)
b(q) a(q)
��
1
−1
�
= 1
2∂i(a(q) + b(q) − a(q) − b(q))
= 0.
(A16)
Thus, the linear paraconductivity is given by summing
up the BCS and PDW contributions, σBCS and σPDW, in
the system in the perpendicular magnetic field. | [
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Progress in Developing Highly Efficient p-Type TCOs for
Transparent Electronics: A Comprehensive Review
Jarnail Singh1 and Pankaj Bhardwaj2,*
1Department of Mechanical Engineering, University Centre for Research and Development,
Chandigarh University, Mohali 140413, Punjab, India
2Department of Material Science and Engineering, National Institute of Technology Hamirpur-
177005
*E-mail address: [email protected]
ORCID ID: 0000-0001-8341-525X
Abstract | [
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] |
177005
*E-mail address: [email protected]
ORCID ID: 0000-0001-8341-525X
Abstract
Transparent conducting oxides (TCOs) represent a remarkable class of materials that possess
both excellent electrical conductivity and high optical transparency, which are typically
considered mutually exclusive in traditional materials. This unique combination of properties
makes TCOs highly desirable for various applications, particularly in optoelectronic devices such | [
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] |
makes TCOs highly desirable for various applications, particularly in optoelectronic devices such
as transparent electrodes for displays, solar cells, and touchscreens. However, developing high-
performance TCOs, especially p-type materials, has been a significant challenge. In conventional
materials, achieving both high electrical conductivity and optical transparency is difficult.
Because materials with wide optical band gap (≥ 3.1 eV) are transparent in the visible region but | [
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] |
Because materials with wide optical band gap (≥ 3.1 eV) are transparent in the visible region but
lack electrical conductivity, while conductive metals are opaque. Hence, the only way to induce
both properties in a single material is to create non-stoichiometry and/or defects. By introducing
shallow defects near the conduction band for n-type materials and the valence band for p-type
materials, it is possible to enhance the conductivity of the material at room temperature. | [
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materials, it is possible to enhance the conductivity of the material at room temperature.
However, developing efficient p-type TCOs has been particularly difficult due to the localized
nature of the valence band derived from O 2p orbitals and the challenges associated with shallow
acceptors, resulting in large effective mass of holes. While commercially available TCOs are
predominantly n-type, such as Sn:In2O3, Al:ZnO, and F:SnO2, the development of efficient p- | [
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] |
predominantly n-type, such as Sn:In2O3, Al:ZnO, and F:SnO2, the development of efficient p-
type TCOs lags. In this review, we have discussed the origin of p-type conductivity in TCOs, and
the difficulties encountered in developing efficient p-type materials. We have also demonstrated
the fundamental materials physics of p-type TCOs, including electronic structures, doping, defect
properties, and optical properties. A range of deposition techniques has been adopted to prepare | [
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] |
properties, and optical properties. A range of deposition techniques has been adopted to prepare
TCO films, and this review provides a detailed discussion of these techniques and their relative
deposition parameters. Overall, we have presented an up-to-date and comprehensive review of
different p-type transparent conducting oxide thin films, providing insights into ongoing research
and potential future directions in this field.
1. Introduction | [
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] |
and potential future directions in this field.
1. Introduction
Transparent conducting oxide (TCO) materials hold excellent potential in a wide variety of
applications, including thin-film solar cells, flat panel display, liquid crystal display (LCDs),
light-emitting diodes (LEDs), thin-film transistors (TFTs), smart windows, invisible circuits,
flexible and transparent devices [1]. For instance, transparent display devices with great refresh | [
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] |
flexible and transparent devices [1]. For instance, transparent display devices with great refresh
frequency and excellent resolution or pixel density can be realized by employing highly effective
TCO materials. Furthermore, energy-efficient displays that operate in typical lighting
atmospheres can be fabricated using TCO materials. TCOs may allow the removal of panel
backlight (that accounts for power consumption around 90% in current displays) [2,3]. However, | [
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backlight (that accounts for power consumption around 90% in current displays) [2,3]. However,
TCOs acquire a wide range of optoelectronic applications, but very modest work has been done
on the development of p-type TCOs. The p-type TCOs with comparable performance to their n-
type counterparts are necessary for the fabrication of active electronic devices based on
transparent p-n junction. But the p-type TCOs were missing in literature for a long time until | [
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transparent p-n junction. But the p-type TCOs were missing in literature for a long time until
Kawazoe and co-workers (from Tokyo Institute of Technology, Japan) in 1997 came up with an
article in which p-type conductivity is reported in the thin film form of CuAlO2 [4]. This
discovery helped other researchers to realize that the development of p-type TCOs is not an
unfeasible task and unlocked a new field of “invisible electronics” or “transparent
electronics”. | [
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] |
electronics”.
However, it is worth discussing that the first p-type semi-transparent conducting
material in the form of nickel oxide (NiO) was reported by Sato et al. in 1993 [5]. They have
deposited a thin film of NiO at various O2 content by using RF magnetron sputtering technique.
A very low value of resistivity of 0.14 Ω-cm with a carrier concentration of 1.3×1019 cm-3 was
observed in pristine NiO thin film at substrate temperature and oxygen partial pressure of 200 ⁰C | [
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] |
observed in pristine NiO thin film at substrate temperature and oxygen partial pressure of 200 ⁰C
and 1.33 Pa, respectively. Hall measurement results demonstrated the p-type conductivity in NiO
thin film. But transparency was not upto the mark with an average value of 40% in the visible
region for 110 nm thin film of NiO. They have also fabricated the p-i-n diode on a glass
substrate, consisting of p-NiO/i-NiO/i-ZnO/n-ZnO and observed that optical transparency further | [
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] |
substrate, consisting of p-NiO/i-NiO/i-ZnO/n-ZnO and observed that optical transparency further
narrows down to 20%, which cannot be considered for superior optoelectronics devices-based
applications. Still, this report has turned out to be a landmark in the field of “invisible
electronics” and optoelectronic technology development.
The research for developing p-type TCOs has been accelerated when Kawazoe et al. | [
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] |
The research for developing p-type TCOs has been accelerated when Kawazoe et al.
demonstrated the significant hole conductivity at room temperature [4]. Copper aluminium oxide
(CuAlO2) thin film having a thickness of 500 nm was deposited on (0001) sapphire substrate
using pulsed laser deposition (PLD) technique. As-deposited CuAlO2 thin film illustrated a wide
optical band of 3.5 eV. The presence of Al3+ ions in between Cu+ ions reduced the cross-linking | [
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] |
optical band of 3.5 eV. The presence of Al3+ ions in between Cu+ ions reduced the cross-linking
of d10 electrons of Cu+ network along the c-axis. This may be responsible for the larger optical
band gap of CuAlO2. The electrical conductivity of 1 S/cm was achieved in CuAlO2 thin film.
The positive value of Hall coefficient and Seebeck coefficient confirmed the p-type nature of
CuAlO2 thin film. Excellent Hall mobility of 10.4 cm2/Vs (like that of ITO) is responsible for the | [
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] |
CuAlO2 thin film. Excellent Hall mobility of 10.4 cm2/Vs (like that of ITO) is responsible for the
significant conductivity value of p-type CuAlO2. However, its conductivity value is still lower
than the highly effective n-type TCOs. Nevertheless, these results demonstrated that both
properties' simultaneous existence in p-type TCOs is not an invincible goal. After this report,
intensive research has been conducted to explore other materials as p-type TCOs. Here, we have | [
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