1. Introduction
Intrinsic superconductors [
1] of rectangular cross-section (with width 2
a and thickness 2
b) exhibit non-dissipative temperature dependent transport self-field critical current,
Ic(sf,
T) (i.e., when no external magnetic field applies), which is given by the following universal equation [
2,
3,
4]:
where
T is sample temperature,
is the magnetic flux quantum,
is the magnetic permeability of free space,
and
are the in-plane and out-of-plane London penetration depths respectively,
,
is the in-plane coherence length, and
is the electron mass anisotropy. It has been shown in previous research that Equation (1) quantitatively and accurately describes
Ic(sf,
T) in more than 100 superconductors, ranging from elemental Zn with
Tc = 0.65 K to highly-compressed H
3S with
[
2,
3,
4], and samples dimensions from several Å to about 1 mm [
5].
All intrinsic superconductors [
1] can induce a superconducting state in non-superconducting materials by the Holm-Meissner effect [
6]. However, a universal equation for non-dissipative self-field critical transport current,
Ic(sf,
T), in superconductor/non-superconductor/superconductor junctions is still unknown. Ambegaokar and Baratoff (AB) [
7,
8] were the first who proposed an equation for
Ic(sf,
T) in superconductor/insulator/superconductor (S/I/S) systems [
9]. Later, Kulik and Omel’yanchuk (KO) [
10,
11,
12] proposed two models for different types of superconductor/normal conductor/superconductor junctions (which are known as KO-1 [
10] and KO-2 [
11]).
In general, superconductor/normal metal/superconductor (S/N/S) junctions are classified by the comparison of the device length (L) to two characteristic length scales of the junction, which are the mean free path of the charge carriers, le, and the superconducting correlation length, ξs. These length scales classify whether the junction is in short (L ≪ ξs) or long (i.e., L ≫ ξs) regime and ballistic (L ≪ le) or diffusive (L ≫ le) limit, respectively.
For about one decade, the KO-1 model was considered to be the primary model to describe
Ic(sf,
T) in superconductor/graphene/superconductor (S/G/S) junctions (a detailed review of different models for
Ic(sf,
T) in S/G/S junctions was given by Lee and Lee [
13]). However, recent technological progress in fabricating high-quality S/G/S junctions demonstrates a large difference between the KO-1 model and experimental
Ic(sf,
T) data [
14]. A detailed discussion of all models, including a model by Takane and Imura [
15], which was proposed to describe
Ic(sf,
T) in superconductor/Dirac-cone material/superconductor (S/DCM/S) junctions, is given by Lee and Lee [
13].
It should be noted that a universal quantitatively accurate equation for critical currents at the applied magnetic field,
B, is unknown to date for intrinsic superconductors [
16,
17,
18,
19,
20] and for Josephson junctions [
13,
21,
22]. However, the discussion of these important problems, as well as the discussion of interface superconductivity [
23,
24,
25] and generic case of two-dimensional (2D) superconductivity [
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36,
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47,
48,
49,
50], is beyond the scope of this paper.
The primary task for this work is to show that
Ic(sf,
T), in a variety of S/DCM/S junctions in the ballistic regime, cannot be described by the KO-based model. To prove this, experimental
Ic(sf,
T) datasets in S/DCM/S junctions were analyzed by two models: the modified Ambegaokar-Baratoff model [
51,
52] and ballistic Titov-Beenakker model [
53].
It needs to be noted that some S/DCM/S junctions show the
Ic(sf,
T) enhancement at a reduced temperature of
T ≤ 0.25·
Tc. For instance, the enhancement in atomically-thin MoRe/single layer graphene (SLG)/MoRe junction was first reported by Calado et al. [
54]. Raw experimental
Ic(sf,
T) data reported by Borzenets et al. [
55] in nominally the same MoRe/SLG/MoRe junctions also shows the enhancement at
T ≤ 0.25·
Tc. Based on this, the
Ic(sf,
T) enhancement at low reduced temperatures in Nb/BiSbTeSe
2-nanoribon/Nb reported by Kayyalha et al. [
56] cannot be considered as a unique property of superconductor/topological insulator/superconductor (S/TI/S) junctions, but is rather the demonstration of a general feature of S/DCM/S devices and atomically thin superconducting systems. Additionally, it is important to mention that Kurter et al. [
57] were the first who reported
Ic(sf,
T) enhancement in S/TI-nanoribbon/S junction at reduced temperature of
T ≤ 0.25·
Tc.
As a result of the performed Ic(sf,T) analysis in this paper, it is shown that a new model is needed to describe dissipation-free transport currents in S/DCM/S junctions.
2. Models Description
The amplitude of dissipation-free transport current,
Ic(sf,
T), in S/I/S junction was first given by Ambegaokar and Baratoff (AB) [
7,
8]:
where ∆(
T) is the temperature-dependent superconducting gap,
e is the electron charge,
Rn is the normal-state tunneling resistance in the junction, and
kB is the Boltzmann constant. In one research [
51], it was proposed to substitute ∆(
T) in Equation (2) by the analytical expression given by Gross et al. [
58]:
where Δ(0) is the ground-state amplitude of the superconducting band, Δ
C/
C is the relative jump in electronic specific heat at the transition temperature,
Tc, and
η = 2/3 for
s-wave superconductors [
56]. In the result,
Tc, Δ
C/
C, Δ(0), and normal-state tunneling resistance,
Rn, of the S/I/S junction, or in the more general case of S/N/S junction, can be deduced by fitting experimental
Ic(sf,
T) datasets to Equation (2), for which the full expression is [
51]:
It should be noted that direct experiments performed by Natterer et al. [
59] showed that the superconducting gap does exist in graphene, which is in proximity contact with superconducting electrodes. The gap amplitude, Δ(
T), has a characteristic decaying length [
59], which is the expected behavior from primary idea of the proximity effect [
6]. As a direct consequence, clear physical meaning remains for the relative jump in electronic specific heat at the transition temperature, Δ
C/
C, due to this parameter is an essential thermodynamic consequence for the appearance of the superconducting energy gap, Δ(
T). As was shown in another study [
51], Δ
C/
C is the fastest decaying parameter of the superconducting state in S/N/S junctions, over the junction length,
L, while
Tc is the most robust one.
In References [
51,
52], it was shown that S/SLG/S and S/Bi
2Se
3/S junctions exhibit two-decoupled band superconducting state. Thus, for the general case of
N-decoupled bands, the temperature-dependent self-field critical current,
Ic(sf,
T), can be described by the following equation:
where the subscript
i refers to the
i-band,
θ(
x) is the Heaviside step function, and each band has its own independent parameters of
Tc,i, Δ
Ci/
Ci, Δ
i(0), and
Rn,i. Equation (5) was also used to analyze experimental
Ic(sf,
T) data for several S/DCM/S junctions [
60].
Titov and Beenakker [
53] proposed that
Ic(sf,
T) in S/DCM/S junction at the conditions near the Dirac point can be described by the equation:
where
W is the junction width. In this paper, analytical equation for the gap (Equation (3) [
57]) is substituted in Equation (6):
with the purpose to deduce
Tc, Δ
C/
C, and Δ(0) values in the S/DCM/S junctions from the fit of experimental
Ic(sf,
T) datasets to Equation (7). For a general case of
N-decoupled bands, temperature-dependent self-field critical current
Ic(sf,
T) in S/DCM/S junctions can be described by the following equation:
Based on a fact that
W and
L can be measured with very high accuracies, Equation (7) has the minimal ever proposed number of free-fitting parameters (which are
Tc, Δ
C/
C, Δ(0)) to fit to the experimental
Ic(sf,
T) dataset. However, as we demonstrate below, the ballistic model (Equation (6) [
53]) is not the most correct model to describe
Ic(sf,
T) in S/DCM/S junctions. It should be noted that Equation (4) utilizes the same minimal set of parameters within the Bardeen-Cooper-Schrieffer (BCS) theory [
60], i.e.,
Tc, Δ
C/
C, Δ(0), to describe superconducting state in S/N/S junction and
Rn as a free-fitting parameter to describe the junction.
It should be stressed that a good reason must be presented for requiring a more complex model than is needed to adequately explain the experimental data [
61,
62].
In the next section, Equations (4), (5), (7), and (8) will be applied to fit experimental
Ic(sf,
T) datasets for a variety of S/DCM/S junctions with the purpose to reveal the primary superconducting parameters of these systems and by comparison deduced parameters with weak-coupling s-wave BCS limits we show that the modified Ambegaokar and Baratoff model (Equations (4) and (5)) [
51,
52] describes the superconducting state in S/DCM/S junctions with higher accuracy.
4. Discussion
One of the most important questions that can be discussed herein is as follows: what is the origin for such dramatic incapability of ballistic model to analyze the self-field critical currents in S/DCM/S junctions? From the author’s point of view, the origin is the primary concept of the KO theory, in that
Ic(sf,
T) in the S/N/S junctions is:
where
φ is the phase difference between two superconducting electrodes of the junction. Despite this assumption is a fundamental conceptual point of the KO theory, there are no physically background or experimental confirmations that this assumption should be a true. In fact, the analysis of experimental data by a model within this assumption (we presented herein) shows that Equation (9) is in remarkably large disagreement with experiment.
One of the simplest ways to show that Equation (9) is incorrect is to note that when the length of the junction,
L, goes to zero, Equation (6) shows:
Herein, the simplest available function [
53] that was proposed for the S/DCM/S junction in the Equation (9) was chosen as an example. However, other proposed functions for Equation (9) (for which we refer the reader to Reference [
12]) have identical unresolved problem, because, as this was shown for about 100 weak-link superconductors [
2,
3,
4,
5,
66], the limit should be (Equation (1)):
This means that the primary dissipation mechanism, which governs DC transport current limit in S/N/S, is not yet revealed. However, as we show herein, it is irrelevant to achieving values within the primary concept of KO theory, Equation (9). It should be mentioned that the Density Functional Theory (DFT) calculations [
67,
68] are currently unexplored powerful techniques, which can be used to reveal dissipation mechanism in S/DCM/S junctions.