Moiré Superconductivity and the Roeser-Huber Formula

As shown previously, a relation between the superconducting transition temperature 1 and some characteristic distance in the crystal lattice holds, which enables the calculation of 2 the superconducting transition temperature, Tc, based only on the knowledge of the electronic 3 configuration and of some details of the crystallographic structure. This relation was found to 4 apply for a large number of superconductors, including the high-temperature superconductors, 5 the iron-based materials, alkali fullerides, metallic alloys, and element superconductors. When 6 applying this scheme called Roeser-Huber formula to Moiré-type superconductivity, i.e., magic7 angle twisted bi-layer graphene (tBLG) and bi-layer WSe2, we find that the calculated transition 8 temperatures for tBLG are always higher than the available experimental data, e.g., for the magic 9 angle 1.1◦, we find Tc ≈ 4.2–6.7 K. Now, the question arises why the calculation produces larger 10 Tc’s. Two possible scenarios may answer this question: (1) The given problem for experimentalists 11 is the fact that for electric measurements always substrates/caps are required to arrange the electric 12 contacts. When now discussing superconductivity in atomically thin objects, also these layers may 13 play a role forming the Moiré patterns. The consequence of such substrate-induced super-Moiré 14 patterns is that the resulting Moiré pattern always will show a larger cell size, and thus, a lower Tc 15 of the final structure will result. (2) A correction factor to the Roeser-Huber formalism may be 16 required to account for the low charge carrier density of the tBLG. Here, we test both scenarios and 17 find that the introduction of a correction factor η enables a proper calculation of Tc, reproducing 18 the experimental data. We find that η depends exponentially on the value of Tc. 19


Introduction
Moiré superconductivity, which was first demonstrated experimentally in 2018, 23 involves creating large, periodic superstructures in 2D materials as compared to the 24 atomic scale. The first sample belonging to this new family of superconductors was 25 found when stacking two graphene layers together with a small misalignment angle, 26 Θ ∼ 1.1 • , called also the first magic angle. This graphene stack is called twisted bilayer 27 graphene or abbreviated tBLG [1,2]. The misalignment between the two graphene layers 28 It is important to note here that Moiré patterns can be formed also in cases when 48 different types of 2D-layered materials are stacked together, with or without angular 49 misalignment, or between a 2D layer and a substrate [17,18]. As result, the resulting 50 Moiré lattice parameter, a M , may be considerably larger than the original atomic unit 51 cells of any ingredient. Several details of the mathematics of Moiré patterns were 52 already presented in Refs. [19][20][21][22]. Thus, the stacking of various 2D-layered materials 53 offers a versatile new way to control superconductivity in layered 2D-systems ("Moiré-54 superconductors"), the full potential of which has been barely explored yet [23]. Thus, 55 to further investigate this field and unleash more possibilities to find new materials with 56 higher T c 's, a simple calculation procedure which can be included in machine-learning 57 approaches, see, e.g., Refs. [24][25][26][27][28], is extremely useful. knowledge about the electronic configuration [29][30][31][32][33][34][35][36]. All this information may be found 66 in existing databases. Using the Roeser-Huber formula, the T c of several superconducting 67 materials could be calculated with only a small error margin, and recently, the approach 68 was employed to predict T c of metallic hydrogen with different crystal lattices [37]. In  In the present contribution, we present the application of the Roeser-Huber equation 76 to Moiré superconductivity using the existing literature data for comparison. The basic idea behind this approach is the view of the resisitive transition to the superconducting state as a resonance effect between the superconducting charge carrier wave, λ cc , and a characteristic length, x = λ cc /2, in the sample. The details of this were already discussed in Refs. [29,30,35]. The Roeser-Huber-equation, originally obtained for high-T c superconductors, is written as where h is the Planck constant, k B the Boltzmann constant, x the characteristic atomic  Thus, n for tBLG is taken to be n = 1 as the two graphene layers at the magic angle give 85 together one superconducting unit. A system corresponding to n 0 = 2 would be then a 86   stack of two 2D layers like h-BN-tBLG-h-BN-tBLG-h-BN, where the two tBLG layers are   87 separated by a h-BN layer. As charge carrier mass, we assume in a first approximation 88 M L = 2m e , corresponding to a Cooper pair. 89 An energy, ∆ (0) , can be introduced via which may correspond to the pairing energy of the superconductor. So we can write Using eq. (2) and regrouping of the terms leads finally to Now, the formalism described above requires only minor adapations to the case of tBLG Refs. [38][39][40]. In the literature, T c is often derived often from 50% of the normal-state   In Ref. [9], also a tri-layer structure was presented with the top and bottom layers tilted 114 by ±5 • with respect to the center layer. This situation is depicted in Fig. 1b.

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The lattice parameter of graphene is a G 0 = 0.246 nm, and the one of WSe 2 is a WSe 2 0 = 0.353 nm [43]. Then, the possible Moiré patterns of two identical layers at an angle Θ have a periodicity according to    Table 1  authors [1][2][3][4][5][6][7] together with data of a graphene tri-layer [9], the data of WSe 2 -stabilized 150 tBLG [10] and the data obtained on twisted WSe 2 bi-layers [11]. Listed are the tilt angle to a T c of 6.714 K, which is even higher and unrealistic. There are two possible scenarios to explain this outcome.

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In case of a stack of h-BN with graphene, there is a misfit between the two lattices, so the resulting superlattice can be described as [2,20] where δ denotes the lattice mismatch between h-BN and graphene (1.8 %) and Φ is the  Table 1. Table giving the experimental data of T c , the angles and the resulting characteristic length, x, the calculated energy ∆ (0) and T c (calc) using the Roeser-Huber equation (eq. 1 with n =1 and M L = 2m e . The energy ∆ * (0) and the transition temperature T * c (calc) are calculated using the correction factor η. Furthermore, the sample names of the original publication and the references are given. The T c marked by † is the value claimed by the authors from a two-step transition. Our T c determined from their data is T c = 0.32 K.
‡ This value gives the zero resistance. Stars (*) mark the WSe 2 T c -data from the experiments of An et al. [11], where the T c values given are determined by us. ( ⊗ ) as given by the authors for R = 0 Ω. (**) indicates T c determined via a 50% normal-state resistance criterion.   Thus, we introduce a correction factor, η, to the charge carrier mass M L in eq. (1) by writing: The situation η = 2 will then correspond to our initial value of 2. Now, we come 201 back to Table 1. The energy ∆ * (0) and the corresponding T * c (calc) were obtained by 202 introducing the correction factor η to the Roeser-Huber equation, which is listed as well.

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The parameter η was obtained by adapting the calculation procedure manually to the  and also off the fit in Fig. 3.
214 Figure 3 plots the resulting values for η as function of temperature. The dashed green line indicates the bottom value of η = 2, which corresponds to the case of HTSc materials. The lower the measured transition temperature, the larger the parameter η. Fitting the data with an exponential decay of the type we obtain a quite good correlation with the parameters A 1 = 14.17, x 0 = 0.6 and t 1 = 0.766 as shown in Fig. 3. The tBLG/WSe 2 -data fall below this fit line, and the TLG and WSe 2 are located above it. Furthermore, the values for η are only in a small range between 2 and 22, which is equal to the narrow window for the tBLG samples in the Uemura plot (T c as a function of the Fermi temperature, T F = E F /k B with E F denoting the Fermi energy) in a line below the HTSc samples [1,8]. As T F is directly linked to the Fermi velocity, v F , via and there is the effective mass, m * , and the density of the charge carriers, n, directly involved. 215 We also must note here that η depends not only on the calculated value of T c at a given 216 misorientation angle Θ, but also on the charge carrier density. Thus, the parameter 217 η determined here should contain all this information, which will then also enable to 218 judge via the value of m * (m * < 0.1m e ) [8], if a material can be a superconductor or not.

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The newer experimental reports also present superconductivity measured at different The case of bi-layer WSe 2 [11] is more complicated to be solved. The first problem 226 in the case of WSe 2 is the value for n 0 to be taken in the calculations. If a monolayer 227 WSe 2 is superconducting itself, n 0 must be taken as 2. If only the product from two 228 misaligned WSe 2 layers is superconducting, we would have n 0 = 1 like for tBLG. A 229 first glance on Table 1gives the idea that n 0 = 2 could be correct, but as seen from the 230 combined WSe 2 -tBLG-data from Arora et al. [10], we can consider n 0 = 1 to be the more 231 realistic case. Thus, we have listed both cases in Table 1 to give some predictions of T c for 232 the WSe 2 system. As seen from Fig. 1c, the larger lattice parameter of WSe 2 will lead to 233 slightly larger a M for a given angle Θ, and thus, the resulting values for T c are higher as 234 compared to tBLG, which is also observed experimentally [11]. The main problem is now 235 that the experiments of Ref. [11] do not convincingly demonstrate superconductivity 236 in this system as compared to the tBLG data, where much more detailed information  Table 1    Funding: This work is part of the SUPERFOAM international project funded by ANR and DFG under the references ANR-17-CE05-0030 and DFG-ANR Ko2323-10, respectively.

Conflicts of Interest:
The authors declare no conflict of interest.