Structural characteristics of 2D α-Mo2C and β-Mo2C crystal sheets

As reported previously6, 2D α-Mo2C 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 Mo2C 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 Mo2C 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 Mo2C sheets. Figure 1a, b shows bright-field TEM images of the Mo2C 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 Mo2C sheets have different crystal lattice symmetries. The rectangular-shaped sheets of Mo2C 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-Mo2C sheet grows along the [100] direction, while the β-Mo2C 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-Mo2C 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 lattice7,8,24. In contrast, there is no Mo lattice distortion in the β-Mo2C sheets because the disordered distribution of carbon atoms mutually offsets the interactions7. It is worth mentioning that the α-Mo2C sheets can be transformed into β-Mo2C by the electron beam irradiation-induced order-disorder transition7. In other words, β-Mo2C is a disordered counterpart of the ordered α-Mo2C phase at high temperature25.

Fig. 1: Crystal structures of Mo2C crystal sheets.

BF-TEM images of (a) rectangular-shaped α-Mo2C and (b) hexagonal-shaped β-Mo2C sheets. Corresponding SAED patterns for (c) α-Mo2C and (d) β-Mo2C sheets. Corresponding HAADF-STEM images for (e) α-Mo2C and (f) β-Mo2C sheets. The domain boundary is depicted by the dashed yellow line in (f). g Unit cell illustrations of orthorhombic α-Mo2C and hexagonal β-Mo2C.

Figure 1e, f shows atomic-resolution HAADF-STEM images of α-Mo2C and β-Mo2C 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 direction26. In addition, it can be seen that the Mo atoms in the α-Mo2C (along the [100] zone axis) and β-Mo2C (along the [0001] zone axis) sheets have very similar atomic configurations. Actually, α-Mo2C and β-Mo2C are closely crystallographically related, namely, aα = cβ, bα = 2bβ, and cα = (sqrt 3)aβ, as illustrated in Fig. 1g. Therefore, considering the almost identical Mo lattices, the main structural difference between α-Mo2C and β-Mo2C sheets is the different distributions of carbon atoms.

Another interesting feature is the presence of diagonal domain boundaries in hexagonal β-Mo2C 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-Mo2C crystal sheets, but the lattice of the adjacent domain areas rotates by 60° in plane, which results in different SAED patterns in the neighboring domains15,24. 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-Mo2C sheets, which is also different from the feature of α-Mo2C sheets15,24.

Hydrostatic pressure responses of the superconductivities in 2D α-Mo2C and β-Mo2C crystal sheets

With the distinct structural features, it will be interesting to study the transport property differences between the α-Mo2C and β-Mo2C sheets by tuning a thermodynamic variable—pressure. Figure 2a, b shows the temperature dependences of the sheet resistances (Rs) for a 23.7-nm-thick α-Mo2C sheet (labeled α-1) and a 13.3-nm-thick β-Mo2C sheet (labeled β-1) at various hydrostatic pressures. Rs is defined by Rs = ρ/d = RW/L, where W is the width and L is the length. Obviously different pressure effects on Tc were observed for α-Mo2C and β-Mo2C sheets, as the pressure dependences of Tc in Fig. 2c, d show. Here, the top and bottom lines of the error bars are the onset temperature Tc,onset and zero resistance temperature Tc,zero, respectively, and Tc,onset, Tc, and Tc,zero are defined by the resistance dropping to 90%, 50%, and 0.1% of the normal state resistance, respectively27,28. For α-Mo2C sheets, Tc first increases and then decreases with increasing pressure, presenting a dome-like shape. For β-Mo2C sheets with Tc higher than that of α-Mo2C sheets, a monotonic reduction in Tc under pressure with dTc/dP = –0.12 K GPa–1 was obtained for β-1. Similar measurements were also performed for a 25.1-nm-thick α-Mo2C sample (labeled α-2) and a 7.1-nm-thick β-Mo2C sample (labeled β-2, dTc/dP = –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 inhomogeneity29 (such as domain boundaries24) and/or electrode contact quality.

Fig. 2: Pressure effects on the Tc of α-Mo2C and β-Mo2C sheets.

Temperature dependences of sheet resistances for (a) α-Mo2C (α-1) and (b) β-Mo2C (β-1) sheets under various pressures. Insets are optical images of the corresponding Mo2C samples with electrodes. Pressure dependences of Tc for (c) α-Mo2C and (d) β-Mo2C sheets.

Here, the hydrostatic pressure effect on the superconductivity of Mo2C can be discussed in the BCS framework30,31. Typically, the hydrostatic pressure dependences of Tc for BCS superconductors can be analyzed in terms of McMillan theory as follows32:

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


Here, ΘD 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 pressure21,33,34. Equation (1) has successfully described the pressure-manipulated Tc in many systems, including elemental superconductors19, transition-metal nitrides35, MgB236, LaH1037, etc34. Based on Eq. (1), the following relationship can be obtained by the logarithmic volume derivative33:

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


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(EF) 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 Mo2C, N(EF) is mainly contributed by the 4d orbitals of the molybdenum atoms40. According to Eq. (2), the negative electronic term ({mathrm{d}}ln eta /{mathrm{d}}ln V) tends to enhance Tc under pressure21. 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 Tc21.

For β-Mo2C sheets, the negative dTc/dP = – 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 Tc = 4.33 K at ambient pressure, ΘD = 590 K41, 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 calculations30. Inserting these values, dTc/dP = −0.12 K GPa–1, ({mathrm{d}}ln eta /{mathrm{d}}ln V)= − 3 to − 4, and B ~ 290 GPa41, 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 Tc in β-Mo2C sheets.

For α-Mo2C sheets, Tc first increases under low pressures and then decreases with further increases in pressure, leading to a distinct dome-like pressure dependence of Tc, which is obviously different from the decrease in Tc in β-Mo2C sheets. Typically, an increase in Tc 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, Nb3Ge, and NbSe2), owing to the pressure-induced complex variation in electronic properties, Tc initially increases under low pressures21,42,43. Similarly, for α-Mo2C sheets, the increase in Tc under low pressures may also be because the complex variation in electronic properties overcomes the phonon stiffening. For the decrease in Tc under high pressures, the phonon stiffening effect may overcome the electronic effect, similar to that in β-Mo2C sheets. In other words, the distinct pressure responses of the superconductivities in α– and β-Mo2C suggest their different electronic and phononic properties under pressure. Compared with the always dominant lattice effect in β-Mo2C sheets, both the lattice and electronic effects play an important role in α-Mo2C sheets, which may be related to their different carbon atom distributions. The more significant influence of the electronic term in α-Mo2C should be related to the ordered carbon atoms, while the strong phonon stiffening effect in β-Mo2C with randomly distributed carbon atoms may be due to the less sensitive pressure response of its electronic characteristic.

Upper critical fields Hc2 of 2D α-Mo2C and β-Mo2C crystal sheets

To further understand the different superconducting characteristics of α-Mo2C and β-Mo2C sheets, their upper critical fields Hc2(T) (defined by the resistance dropping to 90% of the normal state resistance28) 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 Hc2(T) of α-Mo2C (sample α-1) and β-Mo2C (sample β-1) sheets are shown in Fig. 3a. For 2D superconducting systems, Hc2(T) can be well fitted by the following relationship44,45:

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


where Hc2(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 Hc2(0) for α-1 and β-1 sheets with distinct nonmonotonic and monotonic curvatures, respectively. At ambient pressure, the Hc2(0) for α-Mo2C sheets is 0.34 T, comparable with earlier reports6,13, while the Hc2(0) of 5.55 T for β-Mo2C sheets is ~16 times that for α-Mo2C sheets. The initial slopes ({mathrm{ d}}H_{mathrm{c2}}/{mathrm{d}}T|_{T{mathrm{ = }}T{mathrm{c,onset}}}) for the α-Mo2C and β-Mo2C sheets are −0.06 and −1.11, respectively, and they are almost independent of pressure. Moreover, similar measurements were also performed for the Hc2(T) in α-Mo2C (sample α-3, d = 18.9 nm) and β-Mo2C (sample β-3, d = 15.6 nm) sheets in parallel magnetic fields, and much larger Hc2(0) in parallel magnetic fields than that in perpendicular magnetic fields can be observed, as expected for 2D superconductor systems (Supplementary Fig. S6).

Fig. 3: Upper critical fields Hc2 of α-Mo2C (α-1) and β-Mo2C (β-1) sheets.

a Hc2 as a function of temperature under various pressures. The solid lines are the fits with Eq. (3). b Hc2(0) as a function of pressure, and the solid lines are guides for the eye.

The α-Mo2C sheet with an ordered carbon atom distribution was considered to be a clean superconductor6, while for β-Mo2C, it should be treated as a dirty superconductor system due to its disordered carbon atom distribution. Typically, for a dirty superconductor, Hc2 is highly related to the mean free path l and the coherence length ξ as follows46:

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


where (Phi _{mathrm{0}}) is the magnetic flux quantum and vF is the Fermi velocity. The product vFl is proportional to the electron diffusion coefficient, and a smaller vFl typically indicates stronger disorder47,48. According to Eq. (4), the Hc2 of β-Mo2C should be proportional to its Tc and inversely proportional to vFl. Therefore, the dramatic enhancement of the Hc2 in β-Mo2C can be due to its higher Tc and smaller vFl induced by the disordered carbon atom distribution. From the above discussions, it can be seen that the significant differences between α-Mo2C and β-Mo2C 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 Mo2C 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 properties49,50. It is thus interesting to study the effects of thickness on the superconductivities of α-Mo2C and β-Mo2C sheets. Figure 4a, b shows the superconducting transitions of representative α-Mo2C and β-Mo2C crystal sheets with different sheet thicknesses, respectively. It can be seen that the values of Tc decrease with decreasing thickness for both α-Mo2C and β-Mo2C crystal sheets. Typically, in many superconducting systems, Tc is maximum (Tc-max) in the bulk sample and gradually decreases with decreasing sample thickness, and when the thickness is below a certain level (dc), the sample will no longer be in the superconducting state51,52. Based on the Ginzburg-Landau equations, Simonin proposed a relationship between Tc and the thickness as follows53,54:

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


Figure 4c shows the evolution of Tc as a function of the inverse of the thickness (1/d), and the experimental results can be fitted very well by Eq. (5) with Tc-max = 4.03 K and dc = 3.3 nm (~ 7 unit cells) for α-Mo2C sheets and Tc-max = 5.67 K and dc = 2.8 nm (~ 6 unit cells) for β-Mo2C sheets. The Tc-max values of α– and β-Mo2C fall in the ranges of their bulk Tc of 4 K-12.2 K and 2.4 K-7.2 K8, respectively, and the Mo2C sheets may no longer be in a superconducting state when the sample thicknesses are below 6~7 unit cells53. Furthermore, the Tc of β-Mo2C sheets is ~1.6 K higher than that of α-Mo2C sheets with similar thicknesses. The higher Tc of β-Mo2C than that of α-Mo2C may be due to the enhanced electron-phonon coupling related to the disordered carbon distribution in β-Mo2C14 as well as the higher Debye temperature of β-Mo2C (~590 K) than that of α-Mo2C (~580 K)41. To further illustrate the role of the carbon distribution in Mo2C, e-beam irradiation was used to produce varying degrees of disorder in the carbon distribution7, and the order-disorder transition from α-Mo2C to β-Mo2C results in increases in Tc and the sheet resistance (Supplementary Figs. S7 and S8), consistent with Fig. 4.

Fig. 4: Superconducting transitions of α-Mo2C and β-Mo2C sheets.

Temperature-dependent resistivities of (a) α-Mo2C and (b) β-Mo2C sheets with different thicknesses. c Thickness dependences of the Tc of α-Mo2C and β-Mo2C sheets. The dashed lines are the fitting results obtained using Eq. (5).

According to the BCS superconducting mechanism, a stronger electron–phonon coupling, related to the intrinsic phononic and electronic properties, will lead to a higher Tc, 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 conduction21,46, and disorder-enhanced electron-phonon coupling accompanied by an increase in resistivity was also observed in other superconductors, such as Mo3Ge55. Thus, it will be interesting to study the normal state transport behaviors of α-Mo2C and β-Mo2C sheets. Figure 5a shows the temperature dependence of the normalized resistance Rs/Rs-300K from 1.9 K to 300 K for representative α-Mo2C (α-4, d = 23.2 nm) and β-Mo2C (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-Mo2C sheets, which indicates a stronger metallicity in the α-Mo2C.

Fig. 5: Electrical transport properties of α-Mo2C and β-Mo2C sheets.

a Temperature dependences of normalized resistances Rs/Rs-300K of α-Mo2C (23.2 nm) and β-Mo2C (21.8 nm) sheets from 1.9 K −300 K. b Thickness dependence of the residual resistance ratio (RRR = Rs-300K/Rs-6K). c Thickness dependence of the sheet resistance at 10 K (Rs-10K). d Tc vs residual resistance ratio RRR. The dashed lines in (b-d) are guides for the eye.

To obtain a more comprehensive assessment of the normal state properties of α-Mo2C and β-Mo2C sheets, the thickness dependences of their residual resistance ratio (RRR = Rs-300K/Rs-6K) and sheet resistance at 10 K (Rs-10K) are shown in Fig. 5b, c, respectively. Compared with α-Mo2C sheets, β-Mo2C shows a much smaller RRR and a much higher Rs-10K. In particular, the thickness-dependent trends for RRR and Rs-10K are obviously different between α-Mo2C and β-Mo2C sheets. A thicker α-Mo2C sheet possesses a higher RRR and a smaller Rs-10K, while a thicker β-Mo2C sheet has a lower RRR and a larger Rs-10K. A larger RRR means less electronic scattering, which will lead to a better electrical conductivity, namely, a smaller sheet resistance56. 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-Mo2C sheets, the intrinsic lattice scattering is relatively weak due to the ordered carbon atom distributions, and the surface scattering is dominant. Thus, for α-Mo2C 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-Mo2C 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-Mo2C sheet will lead to a smaller RRR and a larger sheet resistance.

Furthermore, the relationships between the Tc and RRR of α-Mo2C and β-Mo2C sheets plotted in Fig. 5d show different variation trends, and β-Mo2C has a higher Tc with a smaller RRR, in contrast to α-Mo2C. It is worth mentioning that a higher Tc usually accompanies a higher RRR due to the less scattering and higher sample quality49,56, which is the case for α-Mo2C sheets. In contrast, a higher Tc with a lower RRR was observed in the β-Mo2C sheets. Considering the almost identical Mo lattices in α-Mo2C and β-Mo2C, 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 Tc, similar to Mo3Ge55.

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