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

^{1}

^{2}

## Abstract

**:**

_{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

_{c}(sf,T) (i.e., when no external magnetic field applies), which is given by the following universal equation [2,3,4]:

_{c}(sf,T) in more than 100 superconductors, ranging from elemental Zn with T

_{c}= 0.65 K to highly-compressed H

_{3}S with ${T}_{c}\gtrsim 200\mathrm{K}$ [2,3,4], and samples dimensions from several Å to about 1 mm [5].

_{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]).

_{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.

_{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].

_{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].

_{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/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 I

_{c}(sf,T) enhancement in S/TI-nanoribbon/S junction at reduced temperature of T ≤ 0.25·T

_{c}.

_{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

_{c}(sf,T), in S/I/S junction was first given by Ambegaokar and Baratoff (AB) [7,8]:

_{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]:

_{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]:

_{c}is the most robust one.

_{2}Se

_{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, I

_{c}(sf,T), can be described by the following equation:

_{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].

_{c}(sf,T) in S/DCM/S junction at the conditions near the Dirac point can be described by the equation:

_{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:

_{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.

_{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

_{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

_{c}R

_{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.

_{n}= 241 ± 7 Ω is in a good agreement with experimental measured value for this junction [63].

_{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.

_{c}(sf,T) datasets should be reasonably dense to deduce parameters by AB model with small uncertainties.

#### 3.2. Planar Nb/BiSbTeSe_{2}-Nanoribbon/Nb Junctions

_{c}(sf,T) for five Nb/BiSbTeSe

_{2}-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.

_{i}/C

_{i}, Δ

_{i}(0), and $\frac{2\xb7{\mathsf{\Delta}}_{i}\left(0\right)}{{k}_{B}\xb7{T}_{c,i}}$, 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 $\frac{2\xb7{\mathsf{\Delta}}_{i}\left(0\right)}{{k}_{B}\xb7{T}_{c,i}}$, 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].

#### 3.3. Planar Nb/Bi_{2}Se_{3}/Nb Junction [56]

_{c}(sf,T), in Nb/Bi

_{2}Se

_{3}/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.

## 4. Discussion

_{c}(sf,T) in the S/N/S junctions is:

## 5. Conclusions

_{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.

## Funding

## Conflicts of Interest

## References

- Hirsch, J.E.; Maple, M.B.; Marsiglio, F. Superconducting materials classes: Introduction and overview. Physica C
**2015**, 514, 1–8. [Google Scholar] [CrossRef] [Green Version] - Talantsev, E.F.; Tallon, J.L. Universal self-field critical currents for thin-film superconductors. Nat. Commun.
**2015**, 6, 7820. [Google Scholar] [CrossRef] [PubMed] - Talantsev, E.F.; Crump, W.P.; Tallon, J.L. Thermodynamic parameters of single-or multi-band superconductors derived from self-field critical currents. Ann. Phys.
**2017**, 529, 1700197. [Google Scholar] [CrossRef] - Talantsev, E.F.; Crump, W.P. Weak-link criterion for pnictide and cuprate superconductors. Supercond. Sci. Technol.
**2018**, 31, 124001. [Google Scholar] [CrossRef] - Talantsev, E.F.; Crump, W.P.; Tallon, J.L. Universal scaling of the self-field critical current in superconductors: From sub-nanometre to millimetre size. Sci. Rep.
**2018**, 7, 10010. [Google Scholar] [CrossRef] [PubMed] - Holm, R.; Meissner, W. Messungen mit Hilfe von flüssigem Helium. XIII. Kontaktwiderstand zwischen Supraleitern und Nichtsupraleitern (Measurements using liquid helium. XIII. Contact resistance between superconductors and non-superconductors). Zeitschrift für Physik
**1932**, 74, 715–735. [Google Scholar] [CrossRef] - Ambegaokar, V.; Baratoff, A. Tunneling between superconductors. Phys. Rev. Lett.
**1963**, 10, 486–489. [Google Scholar] [CrossRef] - Ambegaokar, V.; Baratoff, A. Errata: Tunneling between superconductors. Phys. Rev. Lett.
**1963**, 11, 104. [Google Scholar] [CrossRef] - Josephson, B.D. Possible new effects in superconductive tunneling. Phys. Lett.
**1962**, 1, 251–253. [Google Scholar] [CrossRef] - Kulik, I.O.; Omel’yanchuk, A.N. Contribution to the microscopic theory of the Josephson effect in superconducting bridges. JETP Lett.
**1975**, 21, 96–97. [Google Scholar] - Kulik, I.O.; Omel’yanchuk, A.N. Properties of superconducting microbridges in the pure limit. Sov. J. Low Temp. Phys.
**1977**, 3, 459–462. [Google Scholar] - Kulik, I.; Omelyanchouk, A. The Josephson effect in superconducting constructions: Microscopic theory. J. Phys. Colloq.
**1978**, 39. [Google Scholar] [CrossRef] - Lee, G.H.; Lee, H.J. Proximity coupling in superconductor-graphene heterostructures. Rep. Prog. Phys.
**2018**, 81, 056502. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Park, J.; Lee, J.H.; Lee, G.-H.; Takane, Y.; Imura, K.-I.; Taniguchi, T.; Watanabe, K.; Lee, H.-J. Short ballistic Josephson coupling in planar graphene junctions with inhomogeneous carrier doping. Phys. Rev. Lett.
**2018**, 120, 077701. [Google Scholar] [CrossRef] - Takane, Y.; Imura, K.-I. Quasiclassical theory of the Josephson effect in ballistic graphene junctions. J. Phys. Soc. Jpn.
**2012**, 81, 094707. [Google Scholar] [CrossRef] - Strickland, N.M.; Long, N.J.; Talantsev, E.F.; Hoefakker, P.; Xia, J.A.; Rupich, M.W.; Zhang, W.; Li, X.; Kodenkandath, T.; Huang, Y. Nanoparticle additions for enhanced flux pinning in YBCO HTS films. Curr. Appl. Phys.
**2008**, 8, 372–375. [Google Scholar] [CrossRef] - Talantsev, E.F.; Strickland, N.M.; Hoefakker, P.; Xia, J.A.; Long, N.J. Critical current anisotropy for second generation HTS wires. Curr. Appl. Phys.
**2008**, 8, 388–390. [Google Scholar] [CrossRef] - Chepikov, V.; Mineev, N.; Degtyarenko, P.; Lee, S.; Petrykin, V.; Ovcharov, A.; Vasiliev, A.; Kaul, A.; Amelichev, V.; Kamenev, A.; et al. Introduction of BaSnO
_{3}and BaZrO_{3}artificial pinning centres into 2G HTS wires based on PLD-GdBCO films. Phase I of the industrial R&D programme at SuperOx. Supercond. Sci. Technol.**2017**, 30, 124001. [Google Scholar] - Paturi, P.; Malmivirta, M.; Hynninen, T.; Huhtinen, H. Angle dependent molecular dynamics simulation of flux pinning in YBCO superconductors with artificial pinning sites. J. Phys. Condens. Matter
**2018**, 30, 315902. [Google Scholar] [CrossRef] - Hänisch, J.; Iida, K.; Hühne, R.; Tarantini, C. Fe-based superconducting thin films—Preparation and tuning of superconducting properties. Supercond. Sci. Technol.
**2019**, 32, 093001. [Google Scholar] [CrossRef] - Qu, D.-X.; Teslich, N.E.; Dai, Z.; Chapline, G.F.; Schenkel, T.; Durham, S.R.; Dubois, J. Onset of a two-dimensional superconducting phase in a topological-insulator—Normal-metal Bi
_{1−x}Sb_{x}/Pt junction fabricated by ion-beam techniques. Phys. Rev. Lett.**2018**, 121, 037001. [Google Scholar] [CrossRef] [PubMed] - Li, C.-Z.; Li, C.; Wang, L.-X.; Wang, S.; Liao, Z.-M.; Brinkman, A.; Yu, D.-P. Bulk and surface states carried supercurrent in ballistic Nb-Dirac semimetal Cd
_{3}As_{2}nanowire-Nb junctions. Phys. Rev. B**2018**, 97, 115446. [Google Scholar] [CrossRef] - Reyren, N.; Thiel, S.; Caviglia, A.D.; Kourkoutis, L.F.; Hammerl, G.; Richter, C.; Schneider, C.W.; Kopp, T.; Rüetschi, A.-S.; Jaccard, D.; et al. Superconducting interfaces between insulating oxides. Science
**2007**, 317, 1196–1199. [Google Scholar] [CrossRef] [PubMed] - Gozar, A.; Logvenov, G.; Fitting Kourkoutis, L.; Bollinger, A.T.; Giannuzzi, L.A.; Muller, D.A.; Bozovic, I. High-temperature interface superconductivity between metallic and insulating copper oxides. Nature
**2008**, 455, 782–785. [Google Scholar] [CrossRef] [PubMed] - Di Castro, D.; Balestrino, G. Superconductivity in interacting interfaces of cuprate-based heterostructures. Supercond. Sci. Technol.
**2018**, 31, 073001. [Google Scholar] - Wang, Q.-Y.; Li, Z.; Zhang, W.-H.; Zhang, Z.-C.; Zhang, J.-S.; Li, W.; Ding, H.; OU, Y.-B.; Deng, P.; Chang, K.; et al. Interface-induced high-temperature superconductivity in single unit-cell FeSe films on SrTiO
_{3}. Chin. Phys. Lett.**2012**, 29, 037402. [Google Scholar] [CrossRef] - Zhang, W.H.; Sun, Y.; Zhang, J.; Li, F.; Guo, M.; Zhao, Y.; Zhang, H.; Peng, J.; Xing, Y.; Wang, H.; et al. Direct observation of high-temperature superconductivity in one-unit-cell FeSe films. Chin. Phys. Lett.
**2014**, 31, 017401. [Google Scholar] [CrossRef] - Ge, J.F.; Liu, Z.L.; Liu, C.; Gao, C.L.; Qian, D.; Xue, Q.K.; Liu, Y.; Jia, J.F. Superconductivity above 100 K in single-layer FeSe films on doped SrTiO
_{3}. Nat. Mater.**2015**, 14, 285–289. [Google Scholar] [CrossRef] - Zhang, H.M.; Sun, Y.; Li, W.; Peng, J.P.; Song, C.L.; Xing, Y.; Zhang, Q.; Guan, J.; Li, Z.; Zhao, Y.; et al. Detection of a superconducting phase in a two-atom layer of hexagonal Ga film grown on semiconducting GaN(0001). Phys. Rev. Lett.
**2015**, 114, 107003. [Google Scholar] [CrossRef] - Xing, Y.; Zhang, H.M.; Fu, H.L.; Liu, H.; Sun, Y.; Peng, J.P.; Wang, F.; Lin, X.; Ma, X.C.; Xue, Q.K.; et al. Quantum Griffiths singularity of superconductor-metal transition in Ga thin films. Science
**2015**, 350, 542–545. [Google Scholar] [CrossRef] - Navarro-Moratalla, E.; Island, J.O.; Mañas-Valero, S.; Pinilla-Cienfuegos, E.; Castellanos-Gomez, A.; Quereda, J.; Rubio-Bollinger, G.; Chirolli, L.; Silva-Guillén, J.A.; Agraït, N.; et al. Enhanced superconductivity in atomically thin TaS
_{2}. Nat. Commun.**2016**, 7, 11043. [Google Scholar] [CrossRef] [PubMed] - Yankowitz, M.; Chen, S.; Polshyn, H.; Watanabe, K.; Taniguchi, T.; Graf, D.; Young, A.F.; Dean, C.R. Tuning superconductivity in twisted bilayer graphene. Science
**2019**, 363, 1059–1064. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lucignano, P.; Alfè, D.; Cataudella, V.; Ninno, D.; Cantele, G. The crucial role of atomic corrugation on the flat bands and energy gaps of twisted bilayer graphene at the “magic angle” θ ∼ 1.08°. Phys. Rev. B
**2019**, 99, 195419. [Google Scholar] [CrossRef] - Fête, A.; Rossi, L.; Augieri, A.; Senatore, C. Ionic liquid gating of ultra-thin YBa2Cu3O7-x films. Appl. Phys. Lett.
**2016**, 109, 192601. [Google Scholar] [CrossRef] - Fête, A.; Senatore, C. Strong improvement of the transport characteristics of YBa
_{2}Cu_{3}O_{7−x}grain boundaries using ionic liquid gating. Sci. Rep.**2017**, 8, 17703. [Google Scholar] [CrossRef] - Paradiso, N.; Nguyen, A.-T.; Kloss, K.E.; Strunk, C. Phase slip lines in superconducting few-layer NbSe
_{2}crystals. 2D Mater.**2019**, 6, 025039. [Google Scholar] [CrossRef] - Guo, J.G.; Chen, X.; Jia, X.Y.; Zhang, Q.H.; Liu, N.; Lei, H.C.; Li, S.Y.; Gu, L.; Jin, S.F.; Chen, X.L.; et al. Quasi-two-dimensional superconductivity from dimerization of atomically ordered AuTe
_{2}Se_{4/3}cubes. Nat. Commun.**2017**, 8, 871. [Google Scholar] [CrossRef] - Pan, J.; Guo, C.; Song, C.; Lai, X.; Li, H.; Zhao, W.; Zhang, H.; Mu, G.; Bu, K.; Lin, T.; et al. Enhanced superconductivity in restacked TaS
_{2}nanosheets. J. Am. Chem. Soc.**2017**, 139, 4623. [Google Scholar] [CrossRef] - Ma, Y.; Pan, J.; Guo, C.; Zhang, X.; Wang, L.; Hu, T.; Mu, G.; Huang, F.; Xie, X. Unusual evolution of B
_{c2}and T_{c}with inclined fields in restacked TaS_{2}nanosheets. NPJ Quantum Mater.**2018**, 3, 34. [Google Scholar] [CrossRef] - Desrat, W.; Moret, M.; Briot, O.; Ngo, T.-H.; Piot, B.A.; Jabakhanji, B.; Gil, B. Superconducting Ga/GaSe layers grown by van der Waals epitaxy. Mater. Res. Express
**2018**, 5, 045901. [Google Scholar] [CrossRef] - Liu, C.; Lian, C.-S.; Liao, M.-H.; Wang, Y.; Zhong, Y.; Ding, C.; Li, W.; Song, C.-L.; He, K.; Ma, X.-C.; et al. Two-dimensional superconductivity and topological states in PdTe
_{2}thin films. Phys. Rev. Mater.**2018**, 2, 094001. [Google Scholar] [CrossRef] - Peng, J.; Yu, Z.; Wu, J.; Zhou, Y.; Guo, Y.; Li, Z.; Zhao, J.; Wu, C.; Xie, Y. Disorder enhanced superconductivity toward TaS
_{2}monolayer. ACS Nano**2018**, 12, 9461–9466. [Google Scholar] [CrossRef] [PubMed] - De La Barrera, S.C.; Sinko, M.R.; Gopalan, D.P.; Sivadas, N.; Seyler, K.L.; Watanabe, K.; Taniguchi, T.; Tsen, A.W.; Xu, X.; Xiao, D.; et al. Tuning Ising superconductivity with layer and spin-orbit coupling in two-dimensional transition-metal dichalcogenides. Nat. Commun.
**2018**, 9, 1427. [Google Scholar] [CrossRef] [PubMed] - Liao, M.; Zang, Y.; Guan, Z.; Li, H.; Gong, Y.; Zhu, K.; Hu, X.-P.; Zhang, D.; Xu, Y.; Wang, Y.-Y.; et al. Superconductivity in few-layer stanene. Nat. Phys.
**2018**, 14, 344–348. [Google Scholar] [CrossRef] - Wu, Y.; He, J.; Liu, J.; Xing, H.; Mao, Z.; Liu, Y. Dimensional reduction and ionic gating induced enhancement of superconductivity in atomically thin crystals of 2H-TaSe
_{2}. Nanotechnology**2019**, 30, 035702. [Google Scholar] [CrossRef] - Alidoust, M.; Willatzen, M.; Jauho, A.-P. Symmetry of superconducting correlations in displaced bilayers of graphene. Phys. Rev. B
**2019**, 99, 155413. [Google Scholar] [CrossRef] [Green Version] - Talantsev, E.F.; Mataira, R.C.; Crump, W.P. Classifying superconductivity in Moiré graphene superlattices. arXiv
**2019**, arXiv:1902.07410v2. [Google Scholar] - Rhodes, D.; Yuan, N.F.; Jung, Y.; Antony, A.; Wang, H.; Kim, B.; Chiu, Y.; Taniguchi, T.; Watanabe, K.; Barmak, K.; et al. Enhanced superconductivity in monolayer T
_{d}-MoTe_{2}with tilted Ising spin texture. arXiv**2019**, arXiv:1905.06508. [Google Scholar] - Yang, H.; Gao, Z.-Q.; Wang, F. Effect of defects in superconducting phase of twisted bilayer graphene. arXiv
**2019**, arXiv:1908.09555v2. [Google Scholar] - Talantsev, E.F.; Crump, W.P.; Island, J.O.; Xing, Y.; Sun, Y.; Wang, J.; Tallon, J.L. On the origin of critical temperature enhancement in atomically thin superconductors. 2D Mater.
**2017**, 4, 025072. [Google Scholar] [CrossRef] [Green Version] - Talantsev, E.F.; Crump, W.P.; Tallon, J.L. Two-band induced superconductivity in single-layer graphene and topological insulator bismuth selenide. Supercond. Sci. Technol.
**2018**, 31, 015011. [Google Scholar] [CrossRef] - Talantsev, E.F. Classifying induced superconductivity in atomically thin Dirac-cone materials. Condensed Matter
**2019**, 4, 83. [Google Scholar] [CrossRef] - Titov, M.; Beenakker, C.W.J. Josephson effect in ballistic graphene. Phys. Rev. B
**2006**, 74, 041401. [Google Scholar] [CrossRef] [Green Version] - Calado, V.E.; Goswami, S.; Nanda, G.; Diez, M.; Akhmerov, A.R.; Watanabe, K.; Taniguchi, T.; Klapwijk, T.M.; Vandersypen, L.M.K. Ballistic Josephson junctions in edge-contacted graphene. Nat. Nanotechnol.
**2015**, 10, 761–764. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Borzenets, I.V.; Amet, F.; Ke, C.T.; Draelos, A.W.; Wei, M.T.; Seredinski, A.; Watanabe, K.; Taniguchi, T.; Bomze, Y.; Yamamoto, M.; et al. Ballistic graphene Josephson junctions from the short to the long junction regimes. Phys. Rev. Lett.
**2016**, 117, 237002. [Google Scholar] [CrossRef] - Kayyalha, M.; Kargarian, M.; Kazakov, A.; Miotkowski, I.; Galitski, V.M.; Yakovenko, V.M.; Rokhinson, L.P.; Chen, Y.P. Anomalous low-temperature enhancement of supercurrent in topological-insulator nanoribbon Josephson junctions: Evidence for low-energy Andreev bound states. Phys. Rev. Lett.
**2019**, 122, 047003. [Google Scholar] [CrossRef] - Kurter, C.; Finck, A.D.K.; Hor, Y.S.; Van Harlingen, D.J. Evidence for an anomalous current–phase relation in topological insulator Josephson junctions. Nat. Commun.
**2015**, 6, 7130. [Google Scholar] [CrossRef] - Gross, F.; Chandrasekhar, B.S.; Einzel, D.; Andres, K.; Hirschfeld, P.J.; Ott, H.R.; Beuers, J.; Fisk, Z.; Smith, J.L. Anomalous temperature dependence of the magnetic field penetration depth in superconducting UBe
_{13}. Zeitschrift für Physik B Condensed Matter**1986**, 64, 175–188. [Google Scholar] [CrossRef] - Natterer, F.D.; Ha, J.; Baek, H.; Zhang, D.; Cullen, W.G.; Zhitenev, N.B.; Kuk, Y.; Stroscio, J.A. Scanning tunneling spectroscopy of proximity superconductivity in epitaxial multilayer graphene. Phys. Rev. B
**2016**, 93, 045406. [Google Scholar] [CrossRef] [Green Version] - Bardeen, J.; Cooper, L.N.; Schrieffer, J.R. Theory of Superconductivity. Phys. Rev.
**1957**, 108, 1175–1204. [Google Scholar] [CrossRef] [Green Version] - Dyson, F. A meeting with Enrico Fermi. Nature
**2004**, 427, 297. [Google Scholar] [CrossRef] [PubMed] - Piantadosi, S.T. One parameter is always enough. AIP Adv.
**2018**, 8, 095118. [Google Scholar] [CrossRef] - Jang, S.; Kim, E. Short ballistic Josephson coupling in micrometer-long tantalum/graphene/tantalum junction. Curr. Appl. Phys.
**2019**, 19, 436–439. [Google Scholar] [CrossRef] - Carbotte, J.P. Properties of boson-exchange superconductors. Rev. Mod. Phys.
**1990**, 62, 1027–1157. [Google Scholar] [CrossRef] - Nicol, E.J.; Carbotte, J.P. Properties of the superconducting state in a two-band model. Phys. Rev. B
**2005**, 71, 054501. [Google Scholar] [CrossRef] [Green Version] - Talantsev, E.F. Evaluation of a practical level of critical current densities in pnictides and recently discovered superconductors. Supercond. Sci. Technol.
**2019**, 32, 084007. [Google Scholar] [CrossRef] - Mackinnon, I.D.R.; Talbot, P.C.; Alarco, J.A. Phonon dispersion anomalies and superconductivity in metal substituted MgB
_{2}. Comput. Mater. Sci.**2017**, 130, 191–203. [Google Scholar] [CrossRef] - Alarco, J.A.; Talbot, P.C.; Mackinnon, I.D.R. Identification of superconductivity mechanisms and prediction of new materials using Density Functional Theory (DFT) calculations. J. Phys. Conf. Ser.
**2018**, 1143, 012028. [Google Scholar] [CrossRef]

**Figure 1.**Experimental I

_{c}(sf,T) for tantalum/graphene/tantalum (Ta/G/Ta) junction (Device 1) at gate voltage of V

_{g}= 10 V [63] and data fits to single-band ballistic model (Equation (7), Panel a) and single-band modified AB model (Equation (4), Panel b) (

**a**) Ballistic model. fit quality is R = 0.9948; (

**b**) modified AB model [51,52] fit quality is R = 0.9980.

**Figure 2.**Experimental I

_{c}(sf,T) for Ta/G/Ta junction (Device 1) at gate voltage of V

_{g}= 10 V [63] and data fits to two-band ballistic model (Equation (8), Panel a) and two-band modified AB model (Equation (5), Panel b). (

**a**) Ballistic model, fit quality is R = 0.9978; (

**b**) modified AB model [51,52]. Derived parameters: R

_{n1}= 429 ± 184 Ω, R

_{n2}= 603 ± 209 Ω, fit quality is R = 0.9994.

Parameter | TB Model | AB Model |
---|---|---|

T_{c} (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)/k_{B}·T_{c} | 22.7 ± 0.3 | 2.1 ± 0.1 |

Parameter | TB Model | AB Model |
---|---|---|

T_{c1} (K) | 1.052 ± 0.001 | 1.053 ± 0.003 |

T_{c2} (K) | 0.61 ± 0.02 | 0.63 ± 0.03 |

ΔC_{1}/C_{1} | 17.1 ± 0.6 | 2.2 ± 0.8 |

ΔC_{2}/C_{2} | 2.9 ± 3.8 | 1.1 ± 0.9 |

2·Δ_{1}(0)/k_{B}·T_{c1} | 21 ± 1 | 3.0 ± 0.9 |

2·Δ_{2}(0)/k_{B}·T_{c2} | 3 ± 1 | 1.9 ± 0.3 |

Parameter | TB Model | AB Model |
---|---|---|

T_{c1} (K) | 1.76 ± 0.01 | 1.74 ± 0.04 |

T_{c2} (K) | 0.236 ± 0.003 | 0.31 ± 0.02 |

ΔC_{1}/C_{1} | 0.019 ± 0.03 | 0.84 ± 0.18 |

ΔC_{2}/C_{2} | 1.8 ± 0.3 | 0.19 ± 0.07 |

2·Δ_{1}(0)/k_{B}·T_{c1} | 0.83 ± 0.04 | 2.5 ± 0.5 |

2·Δ_{2}(0)/k_{B}·T_{c2} | 10.0 ± 0.3 | 2.85 ± 0.70 |

Parameter | TB Model | AB Model |
---|---|---|

T_{c1} (K) | 2.10 ± 0.01 | 2.07 ± 0.03 |

T_{c2} (K) | 0.252 ± 0.005 | 0.33 ± 0.02 |

ΔC_{1}/C_{1} | 0.014 ± 0.001 | 0.6 ± 0.2 |

ΔC_{2}/C_{2} | 1.5 ± 0.2 | 0.20 ± 0.06 |

2·Δ_{1}(0)/k_{B}·T_{c1} | 0.94 ± 0.04 | 1.6 ± 0.2 |

2·Δ_{2}(0)/k_{B}·T_{c2} | 9.5 ± 0.3 | 3.1 ± 0.7 |

Parameter | TB Model | AB Model |
---|---|---|

T_{c1} (K) | 2.21 ± 0.01 | 2.19 ± 0.03 |

T_{c2} (K) | 0.274 ± 0.006 | 0.34 ± 0.01 |

ΔC_{1}/C_{1} | 0.027 ± 0.002 | 0.6 ± 0.1 |

ΔC_{2}/C_{2} | 3.4 ± 0.4 | 0.30 ± 0.08 |

2·Δ_{1}(0)/k_{B}·T_{c1} | 1.22 ± 0.01 | 1.9 ± 0.2 |

2·Δ_{2}(0)/k_{B}·T_{c2} | 12.3 ± 0.5 | 3.1 ± 0.7 |

Parameter | TB Model | AB Model |

T_{c1} (K) | 1.55 ± 0.02 | 1.73 ± 0.05 |

T_{c2} (K) | 0.51 ± 0.03 | 0.51 ± 0.03 |

ΔC_{1}/C_{1} | 4.0 ± 0.5 | 0.22 ± 0.06 |

ΔC_{2}/C_{2} | 15 ± 7 | 0.26 ± 0.05 |

2·Δ_{1}(0)/k_{B}·T_{c1} | 22 ± 5 | 2.1 ± 0.8 |

2·Δ_{2}(0)/k_{B}·T_{c2} | 15 ± 7 | 2.2 ± 0.8 |

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**MDPI and ACS Style**

Talantsev, E.F.
DC Self-Field Critical Current in Superconductor/Dirac-Cone Material/Superconductor Junctions. *Nanomaterials* **2019**, *9*, 1554.
https://doi.org/10.3390/nano9111554

**AMA Style**

Talantsev EF.
DC Self-Field Critical Current in Superconductor/Dirac-Cone Material/Superconductor Junctions. *Nanomaterials*. 2019; 9(11):1554.
https://doi.org/10.3390/nano9111554

**Chicago/Turabian Style**

Talantsev, Evgueni F.
2019. "DC Self-Field Critical Current in Superconductor/Dirac-Cone Material/Superconductor Junctions" *Nanomaterials* 9, no. 11: 1554.
https://doi.org/10.3390/nano9111554