DC Self-Field Critical Current in Superconductor/Dirac-Cone Material/Superconductor Junctions

Abstract

Recently, several research groups have reported on anomalous enhancement of the self-field critical currents, I_c(sf,T), at low temperatures in superconductor/Dirac-cone material/superconductor (S/DCM/S) junctions. Some papers attributed the enhancement to the low-energy Andreev bound states arising from winding of the electronic wave function around DCM. In this paper, I_c(sf,T) in S/DCM/S junctions have been analyzed by two approaches: modified Ambegaokar-Baratoff and ballistic Titov-Beenakker models. It is shown that the ballistic model, which is traditionally considered to be a basic model to describe I_c(sf,T) in S/DCM/S junctions, is an inadequate tool to analyze experimental data from these type of junctions, while Ambegaokar-Baratoff model, which is generally considered to be a model for I_c(sf,T) in superconductor/insulator/superconductor junctions, provides good experimental data description. Thus, there is a need to develop a new model for self-field critical currents in S/DCM/S systems.

1. Introduction

Intrinsic superconductors [1] of rectangular cross-section (with width 2a and thickness 2b) exhibit non-dissipative temperature dependent transport self-field critical current, I_c(sf,T) (i.e., when no external magnetic field applies), which is given by the following universal equation [2,3,4]:

$$\begin{gathered} I_c\left(\mathrm{sf},T\right)=\frac{{\varphi }_0}{\pi ·{\mu }_0}·\left[\frac{ln\left(1+\sqrt{2}·{\kappa }_c\left(T\right)\right)}{{\lambda }_{ab}^3\left(T\right)}·\left(\frac{{\lambda }_c\left(T\right)}{b}·\tanh \left(\frac{b}{{\lambda }_c\left(T\right)}\right)\right)+ \\ \frac{ln\left(1+\sqrt{2}·\gamma \left(T\right)·{\kappa }_c\left(T\right)\right)}{\sqrt{\gamma \left(T\right)}·{\lambda }_{ab}^3\left(T\right)}\left(\frac{{\lambda }_{ab}\left(T\right)}{a}\tanh \left(\frac{a}{{\lambda }_{ab}\left(T\right)}\right)\right)\right]·\left(a·b\right), \end{gathered}$$

(1)

where T is sample temperature, ${\varphi }_0=2.067\times {10}^{-15} Wb$ is the magnetic flux quantum, ${\mu }_0=4·\pi \times {10}^{-7} H/m$ is the magnetic permeability of free space, ${\lambda }_{ab}\left(T\right)$ and ${\lambda }_c\left(T\right)$ are the in-plane and out-of-plane London penetration depths respectively, ${\kappa }_c\left(T\right)={\lambda }_{ab}\left(T\right)/{\xi }_{ab}\left(T\right)$, ${\xi }_{ab}\left(T\right)$ is the in-plane coherence length, and $\gamma \left(T\right)={\lambda }_c\left(T\right)/{\lambda }_{ab}\left(T\right)$ is the electron mass anisotropy. It has been shown in previous research that Equation (1) quantitatively and accurately describes I_c(sf,T) in more than 100 superconductors, ranging from elemental Zn with T_c = 0.65 K to highly-compressed H₃S with $T_c\gtrsim 200 \mathrm{K}$ [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, I_c(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 I_c(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, l_e, and the superconducting correlation length, ξ_s. These length scales classify whether the junction is in short (L ≪ ξ_s) or long (i.e., L ≫ ξ_s) regime and ballistic (L ≪ l_e) or diffusive (L ≫ l_e) limit, respectively.

For about one decade, the KO-1 model was considered to be the primary model to describe I_c(sf,T) in superconductor/graphene/superconductor (S/G/S) junctions (a detailed review of different models for I_c(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 I_c(sf,T) data [14]. A detailed discussion of all models, including a model by Takane and Imura [15], which was proposed to describe I_c(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 I_c(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 I_c(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 I_c(sf,T) enhancement at a reduced temperature of T ≤ 0.25·T_c. For instance, the enhancement in atomically-thin MoRe/single layer graphene (SLG)/MoRe junction was first reported by Calado et al. [54]. Raw experimental I_c(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·T_c. Based on this, the I_c(sf,T) enhancement at low reduced temperatures in Nb/BiSbTeSe₂-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 I_c(sf,T) enhancement in S/TI-nanoribbon/S junction at reduced temperature of T ≤ 0.25·T_c.

As a result of the performed I_c(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, I_c(sf,T), in S/I/S junction was first given by Ambegaokar and Baratoff (AB) [7,8]:

$$I_c\left(\mathrm{sf},T\right)=\frac{\pi ·\mathsf{\Delta }\left(T\right)}{2·e·R_n}·\tanh \left(\frac{\mathsf{\Delta }\left(T\right)}{2·k_B·T}\right),$$

(2)

where ∆(T) is the temperature-dependent superconducting gap, e is the electron charge, R_n is the normal-state tunneling resistance in the junction, and k_B 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]:

$$\mathsf{\Delta }\left(T\right)=\mathsf{\Delta }\left(0\right)·\tanh \left(\frac{\pi ·k_B·T_c}{\mathsf{\Delta }\left(0\right)}·\sqrt{\eta ·\left(\frac{\mathsf{\Delta }C}{C}\right)·\left(\frac{T_c}{T}-1\right)}\right),$$

(3)

where Δ(0) is the ground-state amplitude of the superconducting band, ΔC/C is the relative jump in electronic specific heat at the transition temperature, T_c, and η = 2/3 for s-wave superconductors [56]. In the result, T_c, ΔC/C, Δ(0), and normal-state tunneling resistance, R_n, of the S/I/S junction, or in the more general case of S/N/S junction, can be deduced by fitting experimental I_c(sf,T) datasets to Equation (2), for which the full expression is [51]:

$$\begin{gathered} I_c\left(\mathrm{sf},T\right)=\frac{\pi ·\mathsf{\Delta }\left(0\right)·\tanh \left(\frac{\pi ·k_B·T_c}{\mathsf{\Delta }\left(0\right)}·\sqrt{\eta ·\left(\frac{\mathsf{\Delta }C}{C}\right)·\left(\frac{T_c}{T}-1\right)}\right)}{2·e·R_n}· \\ \tanh \left(\frac{\mathsf{\Delta }\left(0\right)·\tanh \left(\frac{\pi ·k_B·T_c}{\mathsf{\Delta }\left(0\right)}·\sqrt{\eta ·\left(\frac{\mathsf{\Delta }C}{C}\right)·\left(\frac{T_c}{T}-1\right)}\right)}{2·k_B·T}\right), \end{gathered}$$

(4)

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, ΔC/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], ΔC/C is the fastest decaying parameter of the superconducting state in S/N/S junctions, over the junction length, L, while T_c is the most robust one.

In References [51,52], it was shown that S/SLG/S and S/Bi₂Se₃/S junctions exhibit two-decoupled band superconducting state. Thus, for the general case of N-decoupled bands, the temperature-dependent self-field critical current, I_c(sf,T), can be described by the following equation:

$$I_c\left(\mathrm{sf},T\right)={{\displaystyle \sum }}_{i=1}^N\frac{\pi ·{\mathsf{\Delta }}_i\left(T\right)}{2·e·R_{n,i}}·\theta \left(T_{c,i}-T\right)·\tanh \left(\frac{{\mathsf{\Delta }}_i\left(T\right)}{2·k_B·T}\right),$$

(5)

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 T_c,i, ΔC_i/C_i, Δ_i(0), and R_n,i. Equation (5) was also used to analyze experimental I_c(sf,T) data for several S/DCM/S junctions [60].

Titov and Beenakker [53] proposed that I_c(sf,T) in S/DCM/S junction at the conditions near the Dirac point can be described by the equation:

$$I_c\left(\mathrm{sf},T\right)=1.33·\frac{e·\mathsf{\Delta }\left(T\right)}{\hslash }·\frac{W}{\pi ·L},$$

(6)

where W is the junction width. In this paper, analytical equation for the gap (Equation (3) [57]) is substituted in Equation (6):

$$I_c\left(\mathrm{sf},T\right)=1.33·\frac{e·\mathsf{\Delta }\left(0\right)·\tanh \left(\frac{\pi ·k_B·T_c}{\mathsf{\Delta }\left(0\right)}·\sqrt{\eta ·\left(\frac{\mathsf{\Delta }C}{C}\right)·\left(\frac{T_c}{T}-1\right)}\right)}{\hslash }·\frac{W}{\pi ·L},$$

(7)

with the purpose to deduce T_c, ΔC/C, and Δ(0) values in the S/DCM/S junctions from the fit of experimental I_c(sf,T) datasets to Equation (7). For a general case of N-decoupled bands, temperature-dependent self-field critical current I_c(sf,T) in S/DCM/S junctions can be described by the following equation:

$$I_c\left(\mathrm{sf},T\right)=1.33·\frac{e}{\pi ·\hslash }·\frac{W}{L}·{{\displaystyle \sum }}_{i=1}^N{\mathsf{\Delta }}_i\left(T\right)·\theta \left(T_{c,i}-T\right),$$

(8)

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 T_c, ΔC/C, Δ(0)) to fit to the experimental I_c(sf,T) dataset. However, as we demonstrate below, the ballistic model (Equation (6) [53]) is not the most correct model to describe I_c(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., T_c, ΔC/C, Δ(0), to describe superconducting state in S/N/S junction and R_n 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 I_c(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.

3. Results

3.1. Micrometer-Long Tantalum/Graphene/Tantalum (Ta/G/Ta) Junction

Jang and Kim [63] reported experimental I_c(sf,T) datasets and fit to KO-1 model (in their Figure 2d [63]) for micrometer long ballistic Ta/G/Ta junctions. The I_c(sf,T) fit to KO-1 model (Figure 2d [63]) and deduced parameters are in disagreement with experimental values based on I_cR_n product. In Figure 1, we show I_c(sf,T) datasets for Device 1 [63] (recorded at gate voltage V_g = 10 V) and fits to single-band ballistic model, Equation (7) (in Figure 1a) and single-band modified AB model Equation (4) (Figure 1b). Device 1 has W = 6 µm, L = 1 µm, and ξ_s = 16 µm [63]. This means that the ballistic limit of L << ξ_s is satisfied for these junctions.

Results of fits to both models are presented in

Table 1. Deduced parameters for tantalum/graphene/tantalum (Ta/G/Ta) junction from fit to single-band Titov and Beenakker (TB) [53] and Ambegaokar and Baratoff (AB) [7,8] models.

Parameter	TB Model	AB Model
Tc (K)	1.052 ± 0.002	1.06 ± 0.01
ΔC/C	17.7 ± 0.6	1.15 ± 0.07
Δ(0) (meV)	1.03 ± 0.01	0.095 ± 0.002
2·Δ(0)/kB·Tc	22.7 ± 0.3	2.1 ± 0.1

.

Deduced parameters from the fit to ballistic model (Equation (7)) in Figure 1a are in remarkable disagreement with any physical-backgrounded expectations, i.e., the ratio of $\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}=22.7$ (which should be comparable with s-wave BCS weak coupling limit of $\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}=3.53$) and $\frac{\mathsf{\Delta }C}{C}=17.7$ (which should be comparable with s-wave BCS weak coupling limit of $\frac{\mathsf{\Delta }C}{C}=1.43$).

It needs to be noted that the highest experimental value for phonon-mediated superconductors of $\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}\approx 5$ was measured for lead- and bismuth-based alloys [64,65], and the deduced value by the ballistic model $\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}\approx 23$ does not have a physical interpretation.

In contract, the fit to Equation (4) reveals superconducting parameters in expected ranges of $\frac{2·\mathsf{\Delta }\left(0\right)}{k_B·T_c}=2.1\pm 0.1$ and $\frac{\mathsf{\Delta }C}{C}=1.15\pm 0.07$, i.e., these parameters are slightly suppressed from s-wave BCS weak-coupling limits as expected [52,60]. It should also be noted that free-fitting parameter R_n = 241 ± 7 Ω is in a good agreement with experimental measured value for this junction [63].

It can be seen (Figure 1), that there is an upturn in experimental I_c(sf,T) at T ~ 0.65 K, which is a manifestation of the second superconducting band opening in this atomically thin S/N/S junction [51,52]. Thus, the experimental I_c(sf,T) dataset was fitted to two-band models (Equations (8) and (5)). Results of these fits are shown in Figure 2 and deduced parameters are in

Table 2. Deduced parameters for tantalum/graphene/tantalum (Ta/G/Ta) junction at V_g = 10 V from fit to two-band Titov and Beenakker (TB) [53] and Ambegaokar and Baratoff (AB) [7,8] models.

Parameter	TB Model	AB Model
Tc1 (K)	1.052 ± 0.001	1.053 ± 0.003
Tc2 (K)	0.61 ± 0.02	0.63 ± 0.03
ΔC1/C1	17.1 ± 0.6	2.2 ± 0.8
ΔC2/C2	2.9 ± 3.8	1.1 ± 0.9
2·Δ1(0)/kB·Tc1	21 ± 1	3.0 ± 0.9
2·Δ2(0)/kB·Tc2	3 ± 1	1.9 ± 0.3

.

The fit reveals a large disagreement of parameters deduced by ballistic model with expected values within frames for BCS theory. In contrast with this, deduced parameters by modified AB model [51,52] are within weak-coupling limits of BCS. As shown in Reference [51], raw experimental I_c(sf,T) datasets should be reasonably dense to deduce parameters by AB model with small uncertainties.

3.2. Planar Nb/BiSbTeSe₂-Nanoribbon/Nb Junctions

Kayyalha et al. [56] reported I_c(sf,T) for five Nb/BiSbTeSe₂-nanopribbon/Nb junctions at different gate voltage, V_g. In this paper I_c(sf,T) datasets for Sample 1 at V_g = −20 V, 0 V and 45 V [56] were analyzed by two-band models (Equations (5) and (8)), because it was already shown in Reference [60] that these junctions exhibit two-band superconducting state. In Figure 3 experimental I_c(sf,T) dataset [56] and fits are shown. For this junction, L = 40 nm [56] and ξ_s = 640 nm [56]; thus, the ballistic regime, L << ξ_s, is well satisfied.

Despite the fact that fits to both models have a similar quality, deduced parameters of the superconducting state (

Table 3. Deduced parameters for Nb/BiSbTeSe₂-nanoribbon/Nb junction (Sample 1 [56]) at V_g = −20 V from fit to two-band Titov and Beenakker (TB) [53] and Ambegaokar and Baratoff (AB) [7,8] models.

Parameter	TB Model	AB Model
Tc1 (K)	1.76 ± 0.01	1.74 ± 0.04
Tc2 (K)	0.236 ± 0.003	0.31 ± 0.02
ΔC1/C1	0.019 ± 0.03	0.84 ± 0.18
ΔC2/C2	1.8 ± 0.3	0.19 ± 0.07
2·Δ1(0)/kB·Tc1	0.83 ± 0.04	2.5 ± 0.5
2·Δ2(0)/kB·Tc2	10.0 ± 0.3	2.85 ± 0.70

), i.e., ΔC_i/C_i, Δ_i(0), and $\frac{2·{\mathsf{\Delta }}_i\left(0\right)}{k_B·T_{c,i}}$, for the case of the ballistic models (Figure 3a), similar to the case of Ta/G/Ta junction (Figure 1 and Figure 2), are remarkably different from values expected from BCS theory. Additionally, there are two orders of magnitude difference between deduced ΔC_i/C_i for two bands for the same sample, and one order of magnitude for $\frac{2·{\mathsf{\Delta }}_i\left(0\right)}{k_B·T_{c,i}}$, which is unavoidable evidence that the ballistic model needs to be reexamined. In contrast with this, the fit to the modified AB model [51] (Figure 3b) reveals deduced parameters, including R_ni values, in the expected ranges. It should be noted that full analysis (within the modified AB model [52]) of I_c(sf,T) datasets in junctions reported by Kayyalha et al. [56] can be found elsewhere [60].

In Figure 4, experimental I_c(sf,T) dataset [56] and fits to two models for Sample 1 at gate voltage V_g = 0 V also demonstrate that the ballistic model is an inadequate tool to analyze experimental data in S/DCM/S junctions (deduced parameters are given in

Table 4. Deduced parameters for for Nb/BiSbTeSe₂-nanoribbon/Nb junction (Sample 1 [56]) at V_g = 0 V from fit to two-band Titov and Beenakker (TB) [53] and Ambegaokar and Baratoff (AB) [7,8] models.

Parameter	TB Model	AB Model
Tc1 (K)	2.10 ± 0.01	2.07 ± 0.03
Tc2 (K)	0.252 ± 0.005	0.33 ± 0.02
ΔC1/C1	0.014 ± 0.001	0.6 ± 0.2
ΔC2/C2	1.5 ± 0.2	0.20 ± 0.06
2·Δ1(0)/kB·Tc1	0.94 ± 0.04	1.6 ± 0.2
2·Δ2(0)/kB·Tc2	9.5 ± 0.3	3.1 ± 0.7

).

The same conclusion can be made for Sample 1 at V_g = 45 V (Figure 5 and

Table 5. Deduced parameters for for Nb/BiSbTeSe₂-nanoribbon/Nb junction (Sample 1 [56]) at V_g = 45 V from fit to two-band Titov and Beenakker (TB) [53] and Ambegaokar and Baratoff (AB) [7,8] models.

Parameter	TB Model	AB Model
Tc1 (K)	2.21 ± 0.01	2.19 ± 0.03
Tc2 (K)	0.274 ± 0.006	0.34 ± 0.01
ΔC1/C1	0.027 ± 0.002	0.6 ± 0.1
ΔC2/C2	3.4 ± 0.4	0.30 ± 0.08
2·Δ1(0)/kB·Tc1	1.22 ± 0.01	1.9 ± 0.2
2·Δ2(0)/kB·Tc2	12.3 ± 0.5	3.1 ± 0.7

).

3.3. Planar Nb/Bi₂Se₃/Nb Junction [56]

In Figure 6, temperature-dependent self-field critical currents, I_c(sf,T), in Nb/Bi₂Se₃/Nb (W = 1000 nm, L = 100 nm) reported by Kurter et al. [57] is shown. For this junction, 300 nm < ξ_s < 1,000 nm [57], and thus, the ballistic regime condition, L << ξ_s, is well satisfied.

There is a large difference between experimental data and the fit to ballistic model (Figure 6 and

Table 6. Deduced parameters for for Nb/Bi₂Se₃/Nb junction [57] from fit to two-band Titov and Beenakker (TB) [53] and Ambegaokar and Baratoff (AB) [7,8] models.

Parameter	TB Model	AB Model
Tc1 (K)	1.55 ± 0.02	1.73 ± 0.05
Tc2 (K)	0.51 ± 0.03	0.51 ± 0.03
ΔC1/C1	4.0 ± 0.5	0.22 ± 0.06
ΔC2/C2	15 ± 7	0.26 ± 0.05
2·Δ1(0)/kB·Tc1	22 ± 5	2.1 ± 0.8
2·Δ2(0)/kB·Tc2	15 ± 7	2.2 ± 0.8

). In addition, deduced parameters from the ballistic model fit have no physical interpretation. The fit to the modified AB model reveals parameters in the expected ranges (Figure 6).

There is a large difference between experimental data and the fit to ballistic model (Figure 6 and Table 6). In addition, deduced parameters from ballistic model fit have no any physical interpretation. The fit to modified AB model reveals parameters in expected ranges (Figure 6 and Table 6).

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 I_c(sf,T) in the S/N/S junctions is:

$$I_c\left(\mathrm{sf},T\right)=\underset{\phi }{\max }\left(I\left(\phi ,\mathrm{sf},T\right)\right)$$

(9)

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:

$$I_c\left(\mathrm{sf},T\right)=\underset{L\to 0}{\lim }\left(1.33·\frac{e·\mathsf{\Delta }\left(T\right)}{\hslash }·\frac{W}{\pi ·L}\right)\propto \underset{L\to 0}{\lim }\left(\frac{1}{L}\right)\to \infty .$$

(10)

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)):

$$\begin{gathered} I_c\left(\mathrm{sf},T\right)=\underset{L\to 0}{\lim }\left(1.33·\frac{e·\mathsf{\Delta }\left(T\right)}{\hslash }·\frac{W}{\pi ·L}\right)=\frac{{\varphi }_0}{\pi ·{\mu }_0}·\left[\frac{ln\left(1+\sqrt{2}·{\kappa }_c\left(T\right)\right)}{{\lambda }_{ab}^3\left(T\right)}·\left(\frac{{\lambda }_c\left(T\right)}{b}·\tanh \left(\frac{b}{{\lambda }_c\left(T\right)}\right)\right)+ \\ \frac{ln\left(1+\sqrt{2}·\gamma \left(T\right)·{\kappa }_c\left(T\right)\right)}{\sqrt{\gamma \left(T\right)}·{\lambda }_{ab}^3\left(T\right)}\left(\frac{{\lambda }_{ab}\left(T\right)}{a}\tanh \left(\frac{a}{{\lambda }_{ab}\left(T\right)}\right)\right)\right]·\left(a·b\right). \end{gathered}$$

(11)

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.

5. Conclusions

In this paper, I_c(sf,T) data for S/DCM/S junctions were analyzed by applying two models: the ballistic and the modified Ambegaokar-Baratoff model. It was shown that the ballistic model [10,11,12,53] cannot describe the self-field critical currents in S/DCM/S junctions. In conclusion, the ballistic model should be reexamined in terms of its applicability to describe dissipation-free self-field transport current in S/DCM/S junctions.