### Structural characteristics of 2D *α*-Mo_{2}C and *β-*Mo_{2}C crystal sheets

As reported previously^{6}, 2D *α*-Mo_{2}C crystals grown by CVD at a low flow rate of methane mainly have 6 kinds of regular shapes (hexagons, rectangles, triangles, etc.), indicating a typical characteristic of good crystallization. In contrast, the 2D Mo_{2}C sheets used in our study were grown by CVD with a high flow rate of methane and were mostly rectangular and hexagonal, with lateral sizes of 10–20 μm and thicknesses (*d*) of 5–30 nm. The thicknesses of the sheets were measured by atomic force microscopy, and representative images are shown in Supplementary Fig. S1. The superconducting properties of Mo_{2}C crystal sheets are highly stable under ambient conditions due to their excellent thermal and chemical stabilities.

High-resolution STEM was utilized to characterize the crystal structures of the Mo_{2}C sheets. Figure 1a, b shows bright-field TEM images of the Mo_{2}C crystal sheets of rectangular and hexagonal shapes, respectively, and Fig. 1c, d shows the corresponding SAED patterns. It can be seen that the rectangular- and hexagonal-shaped Mo_{2}C sheets have different crystal lattice symmetries. The rectangular-shaped sheets of Mo_{2}C are of the *α*-phase with an orthorhombic structure. As for the hexagonal-shaped sheets, they are of the *β*-phase with a hexagonal structure. Moreover, the *α*-Mo_{2}C sheet grows along the [100] direction, while the *β*-Mo_{2}C sheet grows along the [0001] direction, perpendicular to their surfaces. As highlighted by the yellow circles in the SAED pattern shown in Fig. 1c, superlattice diffraction spots are observed for the *α*-Mo_{2}C sheets. This feature has been proven to result from the ordered distribution of carbon atoms, which attract each other in the Mo octahedrons, leading to a regular distortion of the hexagonal close-packed Mo lattice^{7,8,24}. In contrast, there is no Mo lattice distortion in the *β*-Mo_{2}C sheets because the disordered distribution of carbon atoms mutually offsets the interactions^{7}. It is worth mentioning that the *α*-Mo_{2}C sheets can be transformed into *β*-Mo_{2}C by the electron beam irradiation-induced order-disorder transition^{7}. In other words, *β*-Mo_{2}C is a disordered counterpart of the ordered *α*-Mo_{2}C phase at high temperature^{25}.

Figure 1e, f shows atomic-resolution HAADF-STEM images of *α*-Mo_{2}C and *β*-Mo_{2}C sheets, respectively. As seen from the magnified images shown in the insets of Fig. 1e, f, there exist extra Mo lattice points (denoted by the green spheres) located at the centers of Mo hexagons, which is due to the stacking faults in the growth direction^{26}. In addition, it can be seen that the Mo atoms in the *α*-Mo_{2}C (along the [100] zone axis) and *β*-Mo_{2}C (along the [0001] zone axis) sheets have very similar atomic configurations. Actually, *α*-Mo_{2}C and *β*-Mo_{2}C are closely crystallographically related, namely, *a*_{α} = *c*_{β}, *b*_{α} = 2*b*_{β}, and *c*_{α} = (sqrt 3)*a*_{β}, as illustrated in Fig. 1g. Therefore, considering the almost identical Mo lattices, the main structural difference between *α*-Mo_{2}C and *β*-Mo_{2}C sheets is the different distributions of carbon atoms.

Another interesting feature is the presence of diagonal domain boundaries in hexagonal *β*-Mo_{2}C sheets, as shown in Fig. 1b. An atomic-resolution HAADF-STEM image near the boundary is shown in Fig. 1f, and the domain boundary is highlighted by the yellow dashed line. Interestingly, the lattice remains unchanged on the two sides of the boundary. The SAED patterns in each domain area are also studied (Supplementary Fig. S2), and the same SAED patterns in all areas indicate that the crystal structure is uniform throughout the whole crystal. It should be noted that diagonal domain boundaries have also been observed in hexagonal-shaped orthorhombic *α*-Mo_{2}C crystal sheets, but the lattice of the adjacent domain areas rotates by 60° in plane, which results in different SAED patterns in the neighboring domains^{15,24}. Moreover, the fringe contrast at the domain boundary in Fig. 1f indicates that there exists a translational strain of the Mo lattice at the domain boundary of *β*-Mo_{2}C sheets, which is also different from the feature of *α*-Mo_{2}C sheets^{15,24}.

### Hydrostatic pressure responses of the superconductivities in 2D *α*-Mo_{2}C and *β*-Mo_{2}C crystal sheets

With the distinct structural features, it will be interesting to study the transport property differences between the *α*-Mo_{2}C and *β*-Mo_{2}C sheets by tuning a thermodynamic variable—pressure. Figure 2a, b shows the temperature dependences of the sheet resistances (*R*_{s}) for a 23.7-nm-thick *α-*Mo_{2}C sheet (labeled *α-*1) and a 13.3-nm-thick *β-*Mo_{2}C sheet (labeled *β-*1) at various hydrostatic pressures. *R*_{s} is defined by *R*_{s} = *ρ*/*d* = *RW*/*L*, where *W* is the width and *L* is the length. Obviously different pressure effects on *T*_{c} were observed for *α-*Mo_{2}C and *β-*Mo_{2}C sheets, as the pressure dependences of *T*_{c} in Fig. 2c, d show. Here, the top and bottom lines of the error bars are the onset temperature *T*_{c,onset} and zero resistance temperature *T*_{c,zero}, respectively, and *T*_{c,onset}, *T*_{c}, and *T*_{c,zero} are defined by the resistance dropping to 90%, 50%, and 0.1% of the normal state resistance, respectively^{27,28}. For *α-*Mo_{2}C sheets, *T*_{c} first increases and then decreases with increasing pressure, presenting a dome-like shape. For *β-*Mo_{2}C sheets with *T*_{c} higher than that of *α*-Mo_{2}C sheets, a monotonic reduction in *T*_{c} under pressure with d*T*_{c}/d*P* = –0.12 K GPa^{–1} was obtained for *β-*1. Similar measurements were also performed for a 25.1-nm-thick *α-*Mo_{2}C sample (labeled *α-*2) and a 7.1-nm-thick *β-*Mo_{2}C sample (labeled *β-*2, d*T*_{c}/d*P* = –0.10 K GPa^{–1}), and consistent pressure effects were observed (Supplementary Fig. S3). In addition, we noted that the resistivity transitions in some of the samples, such as *α-*1 (Fig. 2a) and *β-*2 (Fig. S3b), show pronounced shoulders, which may be due to sample inhomogeneity^{29} (such as domain boundaries^{24}) and/or electrode contact quality.

Here, the hydrostatic pressure effect on the superconductivity of Mo_{2}C can be discussed in the BCS framework^{30,31}. Typically, the hydrostatic pressure dependences of *T*_{c} for BCS superconductors can be analyzed in terms of McMillan theory as follows^{32}:

$$T_{mathrm{c}} = frac{{Theta}_{D}}{1.45}exp left{frac{{-1.04(1 + lambda)}}{{lambda – mu^{*} (1 + 0.62lambda)}} right}.$$

(1)

Here, Θ_{D} is the Debye temperature, *λ* is the electron-phonon coupling parameter, and the Coulomb pseudopotential *μ** is equal to 0.1 and is insensitive to pressure^{21,33,34}. Equation (1) has successfully described the pressure-manipulated *T*_{c} in many systems, including elemental superconductors^{19}, transition-metal nitrides^{35}, MgB_{2}^{36}, LaH_{10}^{37}, *etc*^{34}. Based on Eq. (1), the following relationship can be obtained by the logarithmic volume derivative^{33}:

$$frac{{{mathrm{d}}ln T_{mathrm{c}}}}{{{mathrm{d}}ln V}} = – Bfrac{{{mathrm{d}}ln T_{mathrm{c}}}}{{{mathrm{d}}P}} = – gamma + Delta left{ {frac{{{mathrm{d}}ln eta }}{{{mathrm{d}}ln V}} + 2gamma } right},$$

(2)

where *V* is the sample volume and *B* is the bulk modulus parameter. (eta = N(E_{mathrm{F}})langle I^{2}rangle) is the product of the Fermi-level density of states *N*(*E*_{F}) with the average squared electronic matrix element (langle I^2rangle). (gamma {mathrm{ = }} – {mathrm{d}}ln langle omega rangle /{mathrm{d}}ln V) is the Grüneisen parameter, (langle omega rangle) is the mean phonon frequency, and (Delta = 1.04lambda left[ {1 + 0.38mu ^ast } right]left[ {lambda – mu ^ast left( {1 + 0.62lambda } right)} right]^{ – 2})^{ 38}. It can be seen that Eq. (2) contains the electron-phonon coupling, the density of states at the Fermi level, and the energy scale of the phonon excitations (Debye frequency). The terms (gamma {mathrm{ = }} – {mathrm{d}}ln langle omega rangle /{mathrm{d}}ln V) and ({mathrm{d}}ln eta /{mathrm{d}}ln V) represent the variations in the lattice and electronic characteristics with volume, respectively. It has been noted that the electronic term ({mathrm{d}}ln eta /{mathrm{d}}ln V) typically equals −1 for simple metal superconductors (*s*, *p* orbital electrons)^{39} and −3 to −4 for transition metal superconductors (*d* orbital electrons)^{21,33}. For Mo_{2}C, *N*(*E*_{F}) is mainly contributed by the 4*d* orbitals of the molybdenum atoms^{40}. According to Eq. (2), the negative electronic term ({mathrm{d}}ln eta /{mathrm{d}}ln V) tends to enhance *T*_{c} under pressure^{21}. However, for most BCS superconductors, the effect from the pressure-induced phonon stiffening (i.e., *γ* > 0) overcomes the effect related to the change in electronic properties, which leads to a ubiquitous decrease in *T*_{c}^{21}.

For *β-*Mo_{2}C sheets, the negative d*T*_{c}/d*P* = – 0.12 K GPa^{–1} means a positive (- B{mathrm{d}}ln T_{mathrm{c}}{mathrm{/d}}P) on the left side of Eq. (2). Using *T*_{c} = 4.33 K at ambient pressure, Θ_{D} = 590 K^{41}, and *μ** = 0.1, we obtained *λ* = 0.46 and Δ = 4.46 from Eq. (1). The electron–phonon coupling parameter *λ* = 0.46 is in agreement with the value *λ* = 0.5 obtained from first-principles density-functional theory calculations^{30}. Inserting these values, d*T*_{c}/d*P* = −0.12 K GPa^{–1}, ({mathrm{d}}ln eta /{mathrm{d}}ln V)= − 3 to − 4, and *B* ~ 290 GPa^{41}, into Eq. (2), we obtain *γ* = 2.7 to 3.3. The large and positive *γ* suggests significant stiffening of the lattice vibration spectrum under pressure, which results in the reduction in *T*_{c} in *β-*Mo_{2}C sheets.

For *α*-Mo_{2}C sheets, *T*_{c} first increases under low pressures and then decreases with further increases in pressure, leading to a distinct dome-like pressure dependence of *T*_{c}, which is obviously different from the decrease in *T*_{c} in *β-*Mo_{2}C sheets. Typically, an increase in *T*_{c} with pressure for a BCS superconductor occurs when the effect of the variation arising from the electronic characteristics with pressure overcomes the pressure-induced phonon stiffening effect. For example, in some BCS superconductors (e.g., Lu, Nb_{3}Ge, and NbSe_{2}), owing to the pressure-induced complex variation in electronic properties, *T*_{c} initially increases under low pressures^{21,42,43}. Similarly, for *α*-Mo_{2}C sheets, the increase in *T*_{c} under low pressures may also be because the complex variation in electronic properties overcomes the phonon stiffening. For the decrease in *T*_{c} under high pressures, the phonon stiffening effect may overcome the electronic effect, similar to that in *β-*Mo_{2}C sheets. In other words, the distinct pressure responses of the superconductivities in *α*– and *β-*Mo_{2}C suggest their different electronic and phononic properties under pressure. Compared with the always dominant lattice effect in *β-*Mo_{2}C sheets, both the lattice and electronic effects play an important role in *α*-Mo_{2}C sheets, which may be related to their different carbon atom distributions. The more significant influence of the electronic term in *α*-Mo_{2}C should be related to the ordered carbon atoms, while the strong phonon stiffening effect in *β-*Mo_{2}C with randomly distributed carbon atoms may be due to the less sensitive pressure response of its electronic characteristic.

### Upper critical fields *H*_{c2} of 2D *α*-Mo_{2}C and *β*-Mo_{2}C crystal sheets

To further understand the different superconducting characteristics of *α-*Mo_{2}C and *β-*Mo_{2}C sheets, their upper critical fields *H*_{c2}(*T*) (defined by the resistance dropping to 90% of the normal state resistance^{28}) were studied under various pressures. Based on the temperature-dependent resistances measured in various perpendicular magnetic fields under different pressures (Supplementary Fig. S4), the upper critical fields *H*_{c2}(*T*) of *α-*Mo_{2}C (sample *α*-1) and *β-*Mo_{2}C (sample *β*-1) sheets are shown in Fig. 3a. For 2D superconducting systems, *H*_{c2}(*T*) can be well fitted by the following relationship^{44,45}:

$$H_{{mathrm{c2}}}left( T right){mathrm{ = }}H_{{mathrm{c2}}}left( {mathrm{0}} right)left( {1 – T{mathrm{/}}T_{{mathrm{c,onset}}}} right)^{1 + alpha },$$

(3)

where *H*_{c2}(0) and *α* are the fitting parameters, and *α* ~0.34 and *α* ~0.04 are obtained for *α-*1 and *β-*1, respectively. Figure 3b shows the pressure dependences of *H*_{c2}(0) for *α*-1 and *β*-1 sheets with distinct nonmonotonic and monotonic curvatures, respectively. At ambient pressure, the *H*_{c2}(0) for *α-*Mo_{2}C sheets is 0.34 T, comparable with earlier reports^{6,13}, while the *H*_{c2}(0) of 5.55 T for *β-*Mo_{2}C sheets is ~16 times that for *α-*Mo_{2}C sheets. The initial slopes ({mathrm{ d}}H_{mathrm{c2}}/{mathrm{d}}T|_{T{mathrm{ = }}T{mathrm{c,onset}}}) for the *α*-Mo_{2}C and *β*-Mo_{2}C sheets are −0.06 and −1.11, respectively, and they are almost independent of pressure. Moreover, similar measurements were also performed for the *H*_{c2}(*T*) in *α-*Mo_{2}C (sample *α*-3, *d* = 18.9 nm) and *β-*Mo_{2}C (sample *β*-3, *d* = 15.6 nm) sheets in parallel magnetic fields, and much larger *H*_{c2}(0) in parallel magnetic fields than that in perpendicular magnetic fields can be observed, as expected for 2D superconductor systems (Supplementary Fig. S6).

The *α*-Mo_{2}C sheet with an ordered carbon atom distribution was considered to be a clean superconductor^{6}, while for *β-*Mo_{2}C, it should be treated as a dirty superconductor system due to its disordered carbon atom distribution. Typically, for a dirty superconductor, *H*_{c2} is highly related to the mean free path *l* and the coherence length *ξ* as follows^{46}:

$$H_{{mathrm{c2}}}left( 0 right) approx Phi _{mathrm{0}}/{mathrm{2}}pi xi l propto {mathrm{3}} times {mathrm{1}}0^{mathrm{4}}T_{mathrm{c}}/v_{mathrm{F}}l,$$

(4)

where (Phi _{mathrm{0}}) is the magnetic flux quantum and *v*_{F} is the Fermi velocity. The product *v*_{F}*l* is proportional to the electron diffusion coefficient, and a smaller *v*_{F}*l* typically indicates stronger disorder^{47,48}. According to Eq. (4), the *H*_{c2} of *β-*Mo_{2}C should be proportional to its *T*_{c} and inversely proportional to *v*_{F}*l*. Therefore, the dramatic enhancement of the *H*_{c2} in *β-*Mo_{2}C can be due to its higher *T*_{c} and smaller *v*_{F}*l* induced by the disordered carbon atom distribution. From the above discussions, it can be seen that the significant differences between *α*-Mo_{2}C and *β*-Mo_{2}C sheets should be related to the electronic structure differences, which are affected by the ordered and disordered carbon distributions. The detailed contributions of the ordered and disordered carbon atom distributions to the electronic structures of Mo_{2}C require further theoretical studies in the future.

### Relationships among the thickness, residual resistivity ratio, and superconductivity of samples

For 2D sheets, the sample thickness generally has a significant influence on the electronic transport properties^{49,50}. It is thus interesting to study the effects of thickness on the superconductivities of *α*-Mo_{2}C and *β*-Mo_{2}C sheets. Figure 4a, b shows the superconducting transitions of representative *α*-Mo_{2}C and *β*-Mo_{2}C crystal sheets with different sheet thicknesses, respectively. It can be seen that the values of *T*_{c} decrease with decreasing thickness for both *α*-Mo_{2}C and *β*-Mo_{2}C crystal sheets. Typically, in many superconducting systems, *T*_{c} is maximum (*T*_{c-max}) in the bulk sample and gradually decreases with decreasing sample thickness, and when the thickness is below a certain level (*d*_{c}), the sample will no longer be in the superconducting state^{51,52}. Based on the Ginzburg-Landau equations, Simonin proposed a relationship between *T*_{c} and the thickness as follows^{53,54}:

$$T_{rm{c}} = T_{{rm{c}}-{max}}left( {1-d_{rm{c}}/d} right).$$

(5)

Figure 4c shows the evolution of *T*_{c} as a function of the inverse of the thickness (1/*d*), and the experimental results can be fitted very well by Eq. (5) with *T*_{c-max} = 4.03 K and *d*_{c} = 3.3 nm (~ 7 unit cells) for *α*-Mo_{2}C sheets and *T*_{c-max} = 5.67 K and *d*_{c} = 2.8 nm (~ 6 unit cells) for *β*-Mo_{2}C sheets. The *T*_{c-max} values of *α*– and *β*-Mo_{2}C fall in the ranges of their bulk *T*_{c} of 4 K-12.2 K and 2.4 K-7.2 K^{8}, respectively, and the Mo_{2}C sheets may no longer be in a superconducting state when the sample thicknesses are below 6~7 unit cells^{53}. Furthermore, the *T*_{c} of *β*-Mo_{2}C sheets is ~1.6 K higher than that of *α*-Mo_{2}C sheets with similar thicknesses. The higher *T*_{c} of *β*-Mo_{2}C than that of *α*-Mo_{2}C may be due to the enhanced electron-phonon coupling related to the disordered carbon distribution in *β*-Mo_{2}C^{14} as well as the higher Debye temperature of *β*-Mo_{2}C (~590 K) than that of *α*-Mo_{2}C (~580 K)^{41}. To further illustrate the role of the carbon distribution in Mo_{2}C, e-beam irradiation was used to produce varying degrees of disorder in the carbon distribution^{7}, and the order-disorder transition from *α*-Mo_{2}C to *β-*Mo_{2}C results in increases in *T*_{c} and the sheet resistance (Supplementary Figs. S7 and S8), consistent with Fig. 4.

According to the BCS superconducting mechanism, a stronger electron–phonon coupling, related to the intrinsic phononic and electronic properties, will lead to a higher *T*_{c}, which can be reflected by the normal state properties. For example, in elemental superconductors, a relatively strong electron-phonon coupling often results in a relatively poor electronic conduction^{21,46}, and disorder-enhanced electron-phonon coupling accompanied by an increase in resistivity was also observed in other superconductors, such as Mo_{3}Ge^{55}. Thus, it will be interesting to study the normal state transport behaviors of *α*-Mo_{2}C and *β*-Mo_{2}C sheets. Figure 5a shows the temperature dependence of the normalized resistance *R*_{s}/*R*_{s-300K} from 1.9 K to 300 K for representative *α*-Mo_{2}C (*α*-4, *d* = 23.2 nm) and *β*-Mo_{2}C (*β*-4, *d* = 21.8 nm) sheets with similar thicknesses. A faster decrease in the normal-state resistance with decreasing temperature was observed in the *α*-Mo_{2}C sheets, which indicates a stronger metallicity in the *α*-Mo_{2}C.

To obtain a more comprehensive assessment of the normal state properties of *α*-Mo_{2}C and *β*-Mo_{2}C sheets, the thickness dependences of their residual resistance ratio (*RRR* = *R*_{s-300K}/*R*_{s-6K}) and sheet resistance at 10 K (*R*_{s-10K}) are shown in Fig. 5b, c, respectively. Compared with *α*-Mo_{2}C sheets, *β*-Mo_{2}C shows a much smaller *RRR* and a much higher *R*_{s-10K}. In particular, the thickness-dependent trends for *RRR* and *R*_{s-10K} are obviously different between *α*-Mo_{2}C and *β*-Mo_{2}C sheets. A thicker *α*-Mo_{2}C sheet possesses a higher *RRR* and a smaller *R*_{s-10K}, while a thicker *β-*Mo_{2}C sheet has a lower *RRR* and a larger *R*_{s-10K}. A larger *RRR* means less electronic scattering, which will lead to a better electrical conductivity, namely, a smaller sheet resistance^{56}. The electronic scattering in 2D sheet or film materials mainly results from two aspects: surface and/or interface scattering and intrinsic lattice scattering. In *α*-Mo_{2}C sheets, the intrinsic lattice scattering is relatively weak due to the ordered carbon atom distributions, and the surface scattering is dominant. Thus, for *α*-Mo_{2}C sheets, the importance of the surface scattering declines for thicker samples, which leads to an overall lower electronic scattering, a higher *RRR*, and a smaller sheet resistance. For *β-*Mo_{2}C sheets, the disordered carbon atom distributions will result in strong intrinsic lattice scattering. As a result, the overall stronger lattice scattering in a thicker *β*-Mo_{2}C sheet will lead to a smaller *RRR* and a larger sheet resistance.

Furthermore, the relationships between the *T*_{c} and *RRR* of *α*-Mo_{2}C and *β*-Mo_{2}C sheets plotted in Fig. 5d show different variation trends, and *β-*Mo_{2}C has a higher *T*_{c} with a smaller *RRR*, in contrast to *α*-Mo_{2}C. It is worth mentioning that a higher *T*_{c} usually accompanies a higher *RRR* due to the less scattering and higher sample quality^{49,56}, which is the case for *α-*Mo_{2}C sheets. In contrast, a higher *T*_{c} with a lower *RRR* was observed in the *β-*Mo_{2}C sheets. Considering the almost identical Mo lattices in *α*-Mo_{2}C and *β*-Mo_{2}C, it can be concluded that although the disordered distribution of carbon atoms increases the electronic scattering, it may promote the electron-phonon coupling and is beneficial for higher *T*_{c}, similar to Mo_{3}Ge^{55}.