Classifying Induced Superconductivity in Atomically Thin Dirac-Cone Materials

Abstract

Recently, Kayyalha et al. (Phys. Rev. Lett., 2019, 122, 047003) reported on the anomalous enhancement of the self-field critical currents (I_c(sf,T)) at low temperatures in Nb/BiSbTeSe₂-nanoribbon/Nb Josephson junctions. The enhancement was attributed to the low-energy Andreev-bound states arising from the winding of the electronic wave function around the circumference of the topological insulator BiSbTeSe₂ nanoribbon. It should be noted that identical enhancement in I_c(sf,T) and in the upper critical field (B_c2(T)) in approximately the same reduced temperatures, were reported by several research groups in atomically thin junctions based on a variety of Dirac-cone materials (DCM) earlier. The analysis shows that in all these S/DCM/S systems, the enhancement is due to a new superconducting band opening. Taking into account that several intrinsic superconductors also exhibit the effect of new superconducting band(s) opening when sample thickness becomes thinner than the out-of-plane coherence length (ξ_c(0)), we reaffirm our previous proposal that there is a new phenomenon of additional superconducting band(s) opening in atomically thin films.

1. Introduction

Intrinsic superconductors can be grouped into 32 classes under “conventional”, “possibly unconventional”, and “unconventional” categories, according to the mechanism believed to give rise to superconductivity [1]. One of the most widely used concepts to represent all 32 classes of superconductors was proposed by Uemura et al. [2,3]. The concept of the Uemura plot is based on the utilization of two fundamental temperatures of superconductors: one is the Fermi temperature (T_F) (X-axis), and the superconducting transition temperature (T_c) (Y-axis). In the most recently updated Uemura plot (Figure 1), it can be seen that elemental superconductors are located for wide range of T_c/T_F ≤ 0.001, while all unconventional superconductors, including both nearly-room-temperature superconductors of H₃S [4] and LaH₁₀ [5] (for which experimentally measured upper critical field data [6,7] was analyzed in Refs. [8,9]), are located within a narrow band of 0.01 ≤ T_c/T_F ≤ 0.05.

It should be mentioned that Hardy et al. [10] in 1993 (seven years after the discovery of high-temperature superconductivity in cuprates by Bednoltz and Mueller [11]) were the first to experimentally find that YBa₂Cu₂O_7-x has nodal superconducting gap. This experimental result was used to propose d-wave superconducting gap symmetry in HTS cuprates by Won and Maki [12]. It should be noted that several researchers and research groups (over last 33 years) have proposed different mechanisms for high-temperature superconductivity in cuprates, the first two-band BCS superconductor (MgB₂), pnictides, and hydrogen-rich superconductors, for which we refer the reader to original papers and comprehensive reviews [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31].

Despite some differences, all intrinsic superconductors can induce superconducting state in non-superconducting materials via the Holm-Meissner effect [36] (also designated as the proximity effect [37,38]). As direct consequence of this, non-dissipative transport current can flow throw the non-superconducting material at superconductor/non-superconductor/superconductor (S/N/S) junctions. The amplitude of this non-dissipative transport current at self-field conditions (when no external magnetic field is applied) (I_c (sf,T)) was given by Ambegaokar and Baratoff (AB) [39,40]:

$$I_c\left(sf,T\right)=\frac{\pi \mathsf{\Delta }\left(T\right)}{2e{R}_n}tanh\left(\frac{\mathsf{\Delta }\left(T\right)}{2k_{B}T}\right),$$

(1)

where Δ(T) is the temperature-dependent superconducting gap, e is the electron charge, normal-state tunneling resistance (R_n) is the normal-state tunneling resistance in the junction, and k_B is the Boltzmann constant.

Many interesting physical effects are expected if the non-superconducting part of the S/N/S junction is made of single-layer graphene (SLG) [41]; multiple-layer graphene (MLG) [42]; graphene-like materials [43]; and many other new 2D- and nano-DCMs, which are under on-going discovery/invention/exploration [44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73]. One interesting class of S/N/S junctions is the non-superconducting part of the device made of topological insulators (TI) [74,75,76,77,78,79,80,81]. Temperature-dependent self-field critical currents (I_c(sf,T)) in this class of junctions were first reported by Veldhorst et al. in Nb/Bi₂Te₃/Nb [52], and later by Kurter et al. in Nb/Bi₂Se₃/Nb [53], by Charpentier et al. in Al/Bi₂Te₃/Al [76], and by other research groups in different systems (extended reference list for studied S/TI/S junctions can be found in Refs. [79,80]).

Recently, Kayyalha et al. [82] have reported on anomalous enhancement of I_c(sf,T) at the Nb/BiSbTeSe₂-nanoribon/Nb junction at temperatures of T ≤ 0.25T_c. They confirmed the effect in all five studied junctions [82], for which TI parts were made of BiSbTeSe₂ flakes with thicknesses of 2b, which varied from 30 nm to 50 nm, and flakes widths of 2a, which varied from 266 nm to 390 nm. It should be noted that in all these S/TI/S junctions, BiSbTeSe₂-nanoribbons thicknesses and widths were smaller than the ground state superconducting coherence length (2b << 2a < ξ(0) ~ 600 nm) in these devices [82]. For one junction, made of wider BiSbTeSe₂-nanoribon (2a = 4 μm (Figure S4 of Supplementary Information of Ref. [82])), measurements were performed only at low temperatures (T < 2 K, which is about T < 0.2T_c (taking into account that Nb has T_c = 8.9–9.6 K [83])), and thus more experimental studies are required for this 4-μm wide Nb/BiSbTeSe₂-nanoribon/Nb junction to see the I_c(sf,T) enhancement.

It needs to be stressed that identical I_c(sf,T) enhancement (or, in another words, I_c(sf,T) upturn [46]) at approximately the same reduced temperature of T ≤ 0.25T_c in atomically-thin S/N/S junction was first reported by Calado et al. [46] in MoRe/SLG/MoRe junction in 2015. One year later, less prominent I_c(sf,T) enhancement (which wass, however, still very clearly visible in raw experimental data [84]) in nominally the same MoRe/SLG/MoRe junctions at T ≤ 0.25T_c was reported by Borzenets et al. [49]. Based on this, it would be incorrect to attribute the I_c(sf,T) enhancement at low reduced temperatures in Nb/BiSbTeSe₂-nanoribon/Nb [82] to unique property of S/TI/S junctions.

In addition, it is important to mention that Kurter et al. [53] were the first to report I_c(sf,T) enhancement at the S/TI/S junction at a reduced temperature of T ≤ 0.25T_c. At Nb/Bi₂Se₃/Nb junctions, Bi₂Se₃ flake had a thickness of 2b = 9 nm, and, thus, the condition of 2b < ξ_c(0) was also satisfied.

Overall, both S/TI/S [53,82], as S/SLG/S [46,49], studied junctions, for which the effect of the low-temperature I_c(sf,T) enhancement was observed to have non-superconducting parts thinner than the ground state out-of-plane coherence lengths, ξ_c(0). SLG thickness is 2b = 0.4–1.7 nm [84], and thus the condition of 2b << ξ_c(0) satisfies any SLG-based junctions.

It should be noted that several intrinsic superconductors exhibit multiple-band superconducting gapping [85,86] and the enhancement of the transition temperature [87,88,89,90,91,92,93] when the condition of 2b < ξ_c(0) [86] is satisfied. The first discovered material in this class of superconductors is atomically thin FeSe [88,89,90], in which a 13-fold increase (i.e., 100 K vs 7.5 K) is experimentally registered. Another milestone experimental finding in this field was reported by Liao et al. [43], who observed the effect of new superconducting band opening and T_c enhancement in few layers of stanene (which is the closest counterpart of graphene) by tuning the films’ thicknesses. To date, maximal T_c increase due to the effect [86] stands with another single-atomic layer superconductor, T_d-MoTe₂, for which Rhodes et al. [93] reported a 30-fold T_c increase when samples were thinneed down to a single atomic layer.

This paper reports the results of an analysis of I_c(sf,T) in Nb/BiSbTeSe₂-nanoribbon/Nb [82] and Nb/(Bi_0.06Sb_0.94)₂Te₃/Nb [94] junctions, and of the upper critical field (B_c2(T)) in Sn/single-layer graphene (SLG)/Sn junctions [95]. In the results, it is shown that a new superconducting band opening phenomenon in atomically thin superconductors, which we proposed earlier [85,86], has further experimental support.

2. Models Description

In [85], it was proposed to substitute Δ(T) in Equation (1) by analytical expression given by Gross et al. [96], as follows:

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

(2)

where Δ(0) is the ground-state amplitude of the superconducting band, ΔC/C is the relative jump in electronic specific heat at the T_c, and η = 2/3 for s-wave superconductors [96]. In result, T_c, ΔC/C, Δ(0), and R_n of the S/N/S junction can be deduced by fitting experimental of an I_c(sf,T) dataset to Equation (1) (full expression for Equation (1) is given in Ref. [85]).

In [85], it was shown that S/SLG/S and S/Bi₂Se₃/S junctions exhibit two-decoupled band superconducting state, for which, for general case of multiple-decoupled bands, I_c(sf,T) can be described by the following equation:

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

(3)

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.

It should be noted that multiple-band induced superconductivity in junctions should be detectable by any technique which is sensitive to additional bands crossing the Fermi surface, for instance multiple distinct gaps should be evident in the temperature-dependence of the upper critical field, B_c2(T), for which general equation is:

$$B_{c2}\left(T\right)={\sum }_{i=1}^N{B}_{c2,i}\left(T\right)\theta \left(T_{c,i}-T\right),$$

(4)

where, within each i-band, the upper critical field can be described by known model. In this paper, four B_c2(T) models were used to show that main result is model-independent. For instance, the following was used:

• Two-fluid Gorter-Casimir (GC) model [97,98], as follows:

  $$B_{c2}\left(T\right)={\sum }_{i=1}^N\left[B_{c2,i}\left(0\right)\left(1-{\left(\frac{T}{T_{c,i}}\right)}^2\right)\theta \left(T_{c,i}-T\right)\right]=\frac{{\varphi }_0}{2\pi }{\sum }_{i=1}^N\left[\frac{\theta \left(T_{c,i}-T\right)}{{\xi }_i^2\left(0\right)}\left(1-{\left(\frac{T}{T_{c,i}}\right)}^2\right)\right],$$

  (5)

  where φ₀ = 2.06810⁻¹⁵ Wb is flux quantum and ξ_i (0) is the ground state in-plane coherence length of the i--band. This model is widely used for single-band superconductors ranging from 3D near-room-temperature superconducting hydrides [4,7,8,9,99,100] to 2D superconductors [88,89,95,101].
• Jones-Hulm-Chandrasekhar (JHC) model [102], as follows:

  $$B_{c2}\left(T\right)=\frac{{\varphi }_0}{2\pi }{\sum }_{i=1}^N\frac{\theta \left(T_{c,i}-T\right)}{{\xi }_i^2\left(0\right)}\left(\frac{1-{\left(\frac{T}{T_{c,i}}\right)}^2}{1+{\left(\frac{T}{T_{c,i}}\right)}^2}\right),$$

  (6)
• Werthamer-Helfand-Hohenberg model [103,104], for which we use analytical expression given by Baumgartner et al. [105] (we will designate this model as B-WHH herein), as follows:

  $$B_{c2}\left(T\right)=\frac{{\varphi }_0}{2\pi }{\sum }_{i=1}^N\frac{\theta \left(T_{c,i}-T\right)}{{\xi }_i^2\left(0\right)}\left(\frac{\left(1-\frac{T}{T_{c,i}}\right)-0.153{\left(1-\frac{T}{T_{c,i}}\right)}^2-0.152{\left(1-\frac{T}{T_{c,i}}\right)}^4}{0.693}\right),$$

  (7)
• Gor’kov model [106], for which simple analytical expression was given by Jones et al. [102], as follows:

  $$B_{c2}\left(T\right)=\frac{{\varphi }_0}{2\pi }{\sum }_{i=1}^N\frac{\theta \left(T_{c,i}-T\right)}{{\xi }_i^2\left(0\right)}\left(\left(\frac{1.77-0.43{\left(\frac{T}{T_{c,i}}\right)}^2+0.07{\left(\frac{T}{T_{c,i}}\right)}^4}{1.77}\right)\left[1-{\left(\frac{T}{T_{c,i}}\right)}^2\right]\right),$$

  (8)

3. Results

3.1. Planar Sn/SLG/Sn Array

Superconductivity in planar graphene junctions varies by the change of the charge carrier density when it moves away from the Dirac point in the dispersion [46,47,51]. This change is usually controlled by the gate voltage (V_g) that applies to the junction. Han et al. [95] reported on a proximity-coupled array of Sn discs with diameter of 400 nm on SLG that were placed in a hexagonal lattice separated by 1 μm between disks centers.

In Figure 2 and Figure 3, we show reported B_c2(T) for Sn/SLG/Sn array by Han et al. [95] in their Figure 4 and Figure 5 at gate voltage of V_g = 30 V. We defined B_c2(T) by two criteria of R = 0.01 kΩ (Figure 2) and R = 0.2 kΩ (Figure 3). It can be seen that there is an obvious upturn in B_c2(T) at T ≤ 0.4T_c, independent of the upper critical field definition criterion. It should be noted that the upturn occurs at practically the same reduced temperature at which Borzenets et al. [49] observed the I_c(sf,T) enhancement in MoRe/SLG/MoRe junctions.

Accordingly, these B_c2(T) datasets were fitted to four two-band models (Equations (5)–(8)); they are shown in Figure 2 and Figure 3. Deduced parameters, including the ratio of transition temperatures for two bands, $\frac{T_{c2}}{T_{c1}}=0.32\pm 0.02$ for R = 0.01 kΩ criterion (Figure 2), and $\frac{T_{c2}}{T_{c1}}=0.38\pm 0.01$ for R = 0.2 kΩ criterion (Figure 3), agreed with each other despite the fact that experimental B_c2(T) data were processed by four different models.

It should be noted that experimental data of Han et al. [95] show that there is a third upturn in B_c2(T) that can be seen at lowest experimentally available temperatures of T < 0.1 K and applied fields of about B ~ 4.5 mT in Figure 4 and Figure 5 [95], if the criterion of R ~ 0.05 kΩ (for the B_c2(T) definition) are applied.

Despite the fact that authors [61] did not mention the presence of these two upturns in raw experimental B_c2(T) data and more detailed measurements of B_c2(T) requires to reveal more accurately the position and parameters for the third band, there is already enough experimental evidence that Sn/SLG/Sn array exhibits at least two-superconducting bands gapping, and thus the report of Han et al. [61] supports the idea that atomically thin films exhibit multiple-band superconducting gapping phenomenon [85,86].

3.2. Planar Nb/BiSbTeSe₂-Nanoribbon/Nb Junctions

Recently, Kayyalha et al. [82], in their Figure 2 and S1, reported I_c(sf,T) for five Nb/BiSbTeSe₂-nanopribbon/Nb junctions at different V_gs. The thickness of BiSbTeSe₂ flakes varied from 2b = 30 nm to 50 nm, and based on reported ξ(0) ~ 600 nm [82], the condition of 2b < ξ(0) [85,86] was satisfied for all junctions.

3.2.1. Nb/BiSbTeSe₂-Nanoribbon/Nb Junctions

In Figure 4, we show experimental I_c(sf,T) datasets for Sample 1 [82] reported for three gate voltages: V_g = −20 V (Figure 4a), 0 V (Figure 4b), and +45 V (Figure 4c). I_c(sf,T) fits to Equation (3) were performed for all parameters to be free, as experimental raw datasets were rich enough to carry out these sorts of fits.

Deduced R_ni, $\frac{T_{c1}}{T_{c2}}$, ΔC_i/C_i, Δ_i(0), and $\frac{2{\mathsf{\Delta }}_i\left(0\right)}{k_B{T}_{c,i}}$ for both superconducting bands as functions of V_g are shown in Figure 4d–f).

It needs to be stressed that within the range of uncertainties, deduced R_n1 values are well agree with directly measured values by Kayyalha et al. [82] (these values are reported in Figure 1a of Ref. [82]). Increasingly often, measured raw I_c(sf,T) data, and especially at high reduced temperatures, are required to reduce the uncertainty for R_n1 values.

Most notable outcome of our analysis is that, within uncertainty ranges, fundamental superconducting parameters for both bands, including the ratio of $\frac{T_{c2}}{T_{c1}}$, remain unchanged vs. gate voltage variation in the range from −20 V to 45 V. This means that two-band superconducting state in Nb/BiSbTeSe₂-nanoribbon/Nb junction is very robust and mostly independent from the change in V_g. This is an unexpected result, because there is generally accepted view that because V_g is determined the electronic state in 2D-systems in the normal state, it should also determine the superconducting state. However, performed analysis shows that this is not a case in general. As was already mentioned, there is a need for more frequent measurements of raw I_c(sf,T) data, which will allow one to reduce uncertainties for all deduced parameters.

3.2.2. Nb/BiSbTeSe₂-Nanoribbon/Nb Junction

In Figure 5a, we show experimental I_c(sf,T) dataset for Nb/BiSbTeSe₂-nanoribbon/Nb (Sample 3) reported by Kayyalha et al. [82].

Raw experimental I_c(sf,T) dataset for this sample was not reach enough at $T\ge 0.6$, and thus we cannot perform the fit to Equation (3) for all parameters to be free. To run the model (Equation (3)), we make the same model restriction, as we did in our previous work [85]:

$$\frac{\mathsf{\Delta }{C}_1}{C_1}=\frac{\mathsf{\Delta }{C}_2}{C_2}=\frac{\mathsf{\Delta }C}{C},$$

(9)

$$\frac{2{\mathsf{\Delta }}_1\left(0\right)}{k_B{T}_1}=\frac{2{\mathsf{\Delta }}_2\left(0\right)}{k_B{T}_2}=\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c},$$

(10)

i.e., we forced ΔC_i/C_i and $\frac{2{\mathsf{\Delta }}_i\left(0\right)}{k_B{T}_{c,i}}$ values to be the same for both bands. As a result, deduced R_ni, T_ci, $\frac{T_{c2}}{T_{c1}}~\frac{1}{4}$, ΔC/C, Δ_i(0) and $\frac{2\mathsf{\Delta }\left(0\right)}{k_B{T}_c}$ for this junction are very close to ones deduced for Sample 1 (Figure 4).

3.3. Planar MoRe/SLG/MoRe Junction

To demonstrate that findings in regard of Nb/BiSbTeSe₂-nanoribbon/Nb junctions are generic for a much wide range of atomically-thin DCM-based Josephson junctions, in Figure 5b we show raw I_c(sf,T) dataset and fit to the model (Equation (3)) for MoRe/SLG/MoRe reported by Calado et al. [46] for their Device A [46]. For the I_c(sf,T) fit for this device, we used the same parameters restrictions (Equations (9) and (10)), as for Nb/BiSbTeSe₂-nanoribbon/Nb Sample 3 [82].

In work [85], this I_c(sf,T) dataset for MoRe/SLG/MoRe Device A [46] was already analyzed. What was found in this paper was that there is remarkable and practically undistinguishable similarity between reduced I_c(sf,T) datasets and fits for Nb/BiSbTeSe₂-nanoribbon/Nb [48] and MoRe/SLG/MoRe [46] junctions (Figure 5). In an attempt to further extend atomically-thin S/DCM/S junctions, in next Section we analyze I_c(sf,T) data for Nb/(Bi_0.06Sb_0.94)₂Te₃-nanoribbon/Nb junction [94].

3.4. Planar Nb/(Bi_0.06Sb_0.94)₂Te₃-Nanoribbon/Nb Junction

In Figure 6, I_c(sf,T) in Nb/(Bi_0.06Sb_0.94)₂Te₃-nanoribbon/Nb reported by Schüffelgen et al. [94] are shown. TI nanoribbon has thickness of 2b = 10 nm, and, thus the condition of 2b < ξ(0) [85,86] is satisfied.

Due to the fact that the reported I_c(sf,T) dataset was not rich enough at high reduced temperatures, we restricted the model by utilizing Equations (9) and (10). Overall, fitted curves and all deduced parameters were very close to one reported by Borzenets et al. [49] for MoRe/SLG/MoRe junctions (which we processed and showed in our previous paper [85] in Figure 7).

4. Discussion

It should be noted that the idea of multiple-band superconductivity in bulk superconductors was proposed by Suhl et al. [107] in 1959, and it took more than forty years to discover the first two-band BCS superconductor (MgB₂) [108] and about fifty years to discover multiple-band iron-based superconductors in 2006 [109]. Interband scattering in these materials have been discussed in details elsewhere [24,110,111,112].

It needs to be mentioned that Shalnikov discussed the discovery of the T_c increase in thin films [113], who reported the effect for lead and tin thin films more than eighty years ago. Three-fold increase in the transition temperature of thin granular Al films was reported three decades later by Cohen and Abeles [114], and the superconductivity in granular Al films is still active scientific topic [115,116]; the discussion of this effect in intrinsic superconductors, however, is beyond the scope of this paper.

It needs to be stressed that Calado et al. [46] in 2015 emphasized the necessity for a new model to explain the upturn in I_c(sf,T) registered in their MoRe/SLG/MoRe junction (Device A) at $T~\frac{1}{4}{T}_c$ (which we show in Figure 5b), because this I_c(sf,T) enhancement was not possible to explain using either the Eilenberger model (which is used to describe clean S/N/S junctions) [117] or the Usadel model (which describes diffusive S/N/S junctions) [118].

Our explanation for this upturn [85], which is well aligned with the I_c(sf,T) upturn in natural atomically thin superconductors [86], is that this I_c(sf,T) enhancement is due to a new superconducting band opening phenomenon when sample dimensions become smaller than some critical value. For this critical value, we proposed to use [86] the out-of-plane coherence length, ξ_c(0), which is still, after expanding our analysis herein, a good choice for the criterion.

It should be pointed out that this new opening band phenomenon does not necessarily cause the increase in observed transition temperature in comparison with “bulk” material. For instance, in pure Nb films [119], this new “thin film” band has lower transition temperature in comparison with “bulk” band [86]. In these circumstances, the researchers are not able to explore the further creation of devices or films for new superconducting band.

Thus, in many atomically thin films, which in fact exhibit a new band opening phenomenon, this effect has not been registered yet, because there is no guarantee that something important/interesting can be observed at low reduced temperatures, well below “bulk” or observed T_c for given atomically thin film.

It should be noted that the effect of new superconducting band opening [86] in atomically thin films can be detected using any experimental techniques that are sensitive to additional band(s) crossing the Fermi surface. To date, most evident confirmations for the phenomenon are related to the I_c(sf,T) upturn [43,85,86] and B_c2(T) upturn [43]; however, other techniques also should be able to detect this.

In this regard, the observation of the I_c(sf,T) upturn reported by Li et al. [78] in their Figure 4a at T = 2.5 K in Nb/Cd₃As₂-nanowire/Nb junction should be mentioned. However, raw experimental I_c(sf,T) dataset [78] was limited by measurements at T < 3.5 K, and thus we are not able to perform the analysis for this very interesting atomically-narrow S/TI/S junction at the moment.

There are very interesting results reported by Sasaki et al. [120] and by Andersen et al. [121], who found that temperature-dependent B_c2(T), in nanostructures of topological insulators, cannot be explained by single-band WHH model [103,104]. However, reported, to date, raw experimental B_c2(T) datasets [119,120] are insufficient to perform two-band model fit to reveal the presence of additional band at low reduced temperatures in these structures.

We also need to mention an interesting research field of interfaced superconductivity [71,122,123,124,125], where, as was proposed earlier, the enhancement of the superconductivity is also due to new superconducting band opening [86]. However, the discussion of this interesting field is beyond the scope of this paper.

5. Conclusions

In this paper, an analysis of recently reported experimental data on induced superconducting state in atomically thin Dirac-cone films was performed. It was shown that the phenomenon of the new superconducting band opening in atomically thin films [85,86], when the film thickness becomes thinner than the ground state out-of-plane coherence length, ξ_c(0), can be extended to an induced superconducting state in atomically thin DCM, as one was established before for natural superconductors, i.e., pure Nb, exfoliated 2H-TaS₂, double-atomic layer FeSe, and a few layers of stanene [9].