Introduction

Novel two-dimensional (2D) materials exhibit unparalleled and highly tailorable physical and chemical properties, making them appealing for both fundamental and technological reasons1,2. With a rich diversity of phases, including superconducting (SC)3,4,5,6,7,8,9, ferroelectric10,11, (anti)ferromagnetic12,13, charge density wave (CDW)14,15, topological insulator16, and excitonic insulator17, all reachable by tuning temperature, pressure or doping, it is of great interest to understand the phase diagrams of 2D materials.

Molybdenum disulfide (MoS2) is a quasi-2D layered semiconducting transition-metal dichalcogenide (TMD) well known for its exceptional optical properties and strong excitonic features18,19. When doped with excess electrons, MoS2 exhibits a complex, yet largely unexplored phase diagram, including various structural phases and many-body features such as multi-valley superconductivity3,20,21, Holstein polarons22,23,24,25, and CDWs14,26,27,28. The ground state of the MoS2 monolayer is the 1 × 1 H phase, with a characteristic graphene-like hexagonal structure. Mechanically and chemically exfoliated MoS2 samples differ in structure. While the former results in the 1 × 1 H structure, in the latter, multiple phases coexist29,30,31,32,33,34, such as 1T and 1T′ phases with various periodicities28,35, mostly due to doping-induced structural modifications. Recently, a CDW phase with 2 × 2 and \(2\sqrt{3}\times 2\sqrt{3}\) reconstructions was shown to exist in potassium-intercalated MoS214. Intriguingly, the SC transition temperature Tc in electron-doped MoS2 follows a characteristic dome structure3,36, resembling the phase diagrams of other unconventional correlated materials such as high-Tc copper oxides37, iron-based pnictides and chalcogenides38, SrTiO339, 1T-TiSe24, and Weyl semimetals WTe240 or MoTe241. Even though the SC mechanism in MoS2 is well studied for lower carrier concentrations and is understood in terms of multi-valley phonon-mediated pairing20,27, the origin of the dome structure is still unresolved. Along with the various coexisting many-body interactions, the presence of polarons, CDWs, and the H to 1T′ structural phase transition all inherently change the structure of MoS2, such that reconstructing a doping-dependent SC dome in MoS2 is theoretically challenging. For the aforesaid systems hosting a SC dome, several mechanisms leading to a dome formation have been discussed in the literature, including the doping-induced Lifshitz transition in the Fermi surface20, the parametrization of McMillan’s formula for Tc26, anharmonic damping close to the phase transitions42, soft-phonon-mediated pairing43,44,45,46, and dynamical electron-electron interaction47. Interestingly, the impact of a doping-induced change of structure on Tc has not yet been investigated in MoS2, even though CDW and SC phases seem to coexist. It is not clear whether CDW supports or suppresses an SC phase15,48,49,50,51,52, but both phases employ the same conduction electrons. As the CDW phase is suppressed, the electronic density of states (DOS) at the Fermi level increases. Consequently, the SC phase is able to receive more electrons for an SC condensate. In general, this results in a simultaneous reduction of a CDW Tc and an increase of a SC Tc4,53,54,55, and it could provide a valid explanation for the SC dome in the CDW systems such as MoS2.

Theoretical endeavors, which include electron-phonon coupling (EPC) calculations of SC Tc in the 1 × 1 H-MoS2 phase within the isotropic Migdal–Eliashberg formalism56, McMillan’s formula57, or the Allen–Dynes formula58, lead to large EPC constants and Tc’s20,26,59. These results could be improved by including anharmonic effects, which would lower the Tc and delay the onset of CDW60, while the anisotropic Migdal–Eliashberg formalism might be important for MoS2 and low-dimensional systems61. These effects were recently taken into account, while the doping mechanism was included in a field-effect configuration62, which resulted in the accurate prediction of a SC Tc, and a suppression of the EPC constant by an order of magnitude. However, in that study, the SC dome was not found, and the existence of CDW in the experimental doping range was ruled out.

Here, we conduct a comprehensive first-principles investigation of the doping-dependent phase diagram of MoS2 and find various stable phases across a large doping range. Structural changes of MoS2 lead to a dome structure of the SC Tc and encompass the experimental findings of a SC dome and CDW phase. We perform nonadiabatic EPC calculations63,64,65, and find that it leads to a considerable reduction compared to the adiabatic EPC constants and Tc. Moreover, we find that Tc can be further lowered by taking spin–orbit coupling (SOC) into account, which is known to determine the possible pairing symmetries66 and to enhance the upper critical magnetic field67,68,69. With this study, we successfully fill in the missing gap in comprehending the complicated phase diagram of MoS2. We show that the Tc increases for the original H phase and reaches a maximum value in a region where a normal H phase and 2 × 2 CDW coexist. For even larger doping concentrations, the 1T′ phase and 2 × 2 CDW ordering are stabilized with Tc being lowered, which in total forms a characteristic dome structure as observed in the experiments3,36. Our model calculations based on the tight-binding approximation (TBA)70 confirm the dome structure of Tc, and additionally reveal localized polaronic distortions for carrier concentrations in the phase diagram where neither the original 1 × 1 H structure nor the 2 × 2 CDW phase are stable. We rely on the jellium model for doping, which leads to the overestimation of the Tc values62. Anharmonicity is neglected in this work but is important in proximity to a structural phase transition, to determine the exact critical doping at which a CDW transition occurs. However, the qualitative explanation of the dome structure origin in MoS2 related to structural transitions provided in this work remains valid even if anharmonicity shifts the dome to a different doping region, and if taking a gating geometry into account.

Results

MoS2 can be doped by gating or intercalation, reaching large charge densities. In Fig. 1, we show a doping-dependent phase diagram of dynamically stable structures that MoS2 assumes, as obtained from density functional theory (DFT), including the 1 × 1 H phase, 2 × 2 CDW structure with distorted atoms, and 1T′ phase. Between the regions where the 1T′ and high-doping 2 × 2 CDW phases are stable, our TBA simulations reveal polaronic distortions and partial CDW coverage. Our aim is to calculate the EPC strengths and Tc across these phase transitions.

Fig. 1: Phase diagram of electron-doped MoS2 with various dynamically stable phases.
figure 1

In the upper row, we show the three distinct phases appearing in the phase diagram together with their primitive cells. For visualization, the atomic structures are shown with exaggerated atomic displacements (ten times) and on top of a hexagonal mesh. Each of the phases from the first row is stable in some doping range, as color-coded in the phase diagram. For the gray shaded area, our TBA model predicts the stabilization of polaronic distortions and partial CDWs. The corresponding Brillouin zones are displayed in the second row with the same color code from the phase diagram.

Full size image

The low-doping 1 × 1 H phase

Our calculated phonon dispersions indicate a dynamically stable 1 × 1 H structure up to the excess charge concentration of 2.38 × 1014 cm−2, when the acoustic phonon of A1 symmetry at the M point of the Brillouin zone (BZ), shown in Fig. 2a, induces the instability. This critical doping nc is sensitive to the choice of the electronic temperature in the DFT electronic structure calculations, and varies from nc = 1.5 × 1014 cm−2 to nc = 2.4 × 1014 cm−2 when Tscf is between 100 and 800 K, as one can see in Supplementary Figs. 1, 2. Here we use Tscf = 800 K since the choice of a lower electronic temperature would require denser momentum grids. This strongly coupled acoustical phonon describes the motion of Mo atoms towards the S atom, and by increasing doping, its frequency softens. Its strong coupling to electrons manifests itself as a large broadening in the phonon spectral function Bν(q, ω) [see Fig. 2b] and as a large peak in the Eliashberg spectral function α2F(ω) [see Fig. 2c]. Here we also show how the α2F(ω) and the EPC constant λ in a nonadiabatic (NA) formalism (see Methods) differ from their adiabatic (A) counterparts63. With the largest contribution coming from the A1 soft mode, the NA frequency renormalization and broadening effects strongly renormalize the EPC properties. In fact, the NA effects reduce the EPC strength λ by half, which mostly comes from the blueshifted Kohn anomalies at the M and K points of the BZ. In the second row of Fig. 2, we show doping-dependent quantities calculated from the NA phonon spectrum.

Fig. 2: Electron-phonon properties of MoS2 in the 1 × 1 H phase.
figure 2

a Schematics of the linear combination of the strongly coupled lowest-energy acoustic phonon eigendisplacements of A1 symmetry at all q = M points, see Supplementary Fig. 3. b Phonon spectral function of MoS2 at 800 K, with the adiabatic phonon band structure denoted with dashed black line. c Eliashberg spectral function in the adiabatic (A, red) and nonadiabatic (NA, blue) approximations together with the corresponding EPC cumulative constant λ(ω). d EPC strength λ and logarithmic average frequency \({\omega }_{\log }\) computed with NA phonons. e Superconducting Tc calculated using McMillan’s (MM) formula (hollow red circles), the Allen–Dynes (AD) formula (full red dots), and the isotropic Migdal–Eliashberg (ME) formalism (golden stars). Cross symbols denote the calculations which include spin–orbit coupling.

Full size image

The doping-dependent EPC constant λ and the corresponding \({\omega }_{\log }\) for the 1 × 1 H stable part of the phase diagram in Fig. 2d form a dome, but we rule out the possibility that the SC dome arises due to \({\omega }_{\log }\) in the Tc formula26. In fact, we find that the Tc is monotonically increasing according to McMillan’s formula, the Allen–Dynes formula, as well as the isotropic Migdal–Eliashberg formalism in agreement with ref. 62 [see Fig. 2e]. For the two largest dopings, we show how the inclusion of the SOC lowers the Tc by ~30%. An increase of the Coulomb pseudopotential μ* can also reduce the value of Tc by a few percent (see Supplementary Fig. 3). Note that our computed Tc values overestimate the experiments, and that more quantitative results would be obtained with the additional inclusion of anharmonicity and proper gating geometry62.

Coexistence of 2 × 2 CDW and 1× 1 H phases

Slightly before the instability, we find that a 1 × 1 H structure coexists with a 2 × 2 CDW phase. In analogy to the 1T-TiSe271, our calculations show that the CDW in MoS2 is driven by three nonequivalent M phonon instabilities and results in a 2 × 2 unit cell with triangular distortions of Mo atoms [see Figs. 1, 2a as well as Supplementary Fig. 4], in agreement with the CDW pattern found in the recent STM measurements14. Due to the splitting of the bands [see Fig. 3a], the DOS at the Fermi level in a 2 × 2 CDW phase drops, making it energetically favorable. However, due to the existence of the two minima in the potential energy surface [see Fig. 3c], the atom relaxation in a 2 × 2 supercell results in a hysteresis loop as doping is varied [see Fig. 3d].

Fig. 3: Coexistence of the 1 × 1 H and 2 × 2 CDW phases and corresponding electron and phonon properties.
figure 3

a Electron band structures comparison of the two phases for n = 2.15 × 1014 cm−2. b Phonon dispersion comparison, with the soft mode from (e) denoted with an orange circle. c Potential energy surface between the two phases for a range of different dopings. d Hysteresis plot of the doping-dependent displacement of the Mo atoms. e Doping-dependent frequency of the Γ* mode, backfolded from the M point of the 1 × 1 unit cell. The orange vertical line corresponds to n = 2.15 × 1014 cm−2. See Fig. 1 for the definition of the starred (*) high-symmetry points.

Full size image

For the same critical doping value, we find that the 1 × 1 H structure becomes unstable due to the aforementioned M point phonon instability, while a 2 × 2 CDW structure becomes unstable due to the M* point (see Fig. 1) phonon instability. The latter is clearly visible in Fig. 3e. In the 1 × 1 H phase, the M point instability is a result of electronic transitions between the filled K and 1/2 K electron pockets on the Fermi surface. In the 2 × 2 CDW phase, band splitting leads to the occurrence of the flat band close to the Fermi level at the M point. Electron transitions between these flat-band parts cause phonon softening in the new backfolded M* point (see Supplementary Fig. 5).

The 1T′ phase

Next, we find that one possible stable structure for larger doping values is a 1T′ phase, which is separated from the 1 × 1 H phase by a large energy barrier of around 10 eV. This could explain why the transition to the 1T′ phase in experiments is not observed when MoS2 is doped by gating32,72,73. The 1T′ phase is a metastable phase, and our calculations confirm that the barrier reduces by doping74 (see Supplementary Fig. 6). We stress that the unit cell of the 1T′ phase for this doping region is not related to the ωM instability in the 1 × 1 H structure nor the \({\omega }_{{\rm M}^{* }}\) instability in the 2 × 2 CDW structure, but we find it to be dynamically stable for this doping regime. In Fig. 4a we show the electronic bands and find that the 1T′ phase is stable for a narrow doping region which could be a consequence of its Fermi surface, which is prone to nesting (see Supplementary Fig. 7). The phonon dispersion in Fig. 4b reveals many Kohn anomalies in the lowest acoustic branch, appearing for the wave vectors which nest the Fermi surface. Again, we observe large NA effects in the EPC properties [see Fig. 4c].

Fig. 4: Properties of MoS2 in the 1T′ and high-doping 2 × 2 CDW phases.
figure 4

a Electron band structure obtained for the excess charge of n = 2.62 × 1014 cm−2. b Adiabatic phonon dispersion. c Eliashberg spectral function calculated using the adiabatic (A) and nonadiabatic (NA) formula, with the corresponding cumulative EPC constant λ(ω). d Electron band structures for the largest doping values considered in this work; n1 = 3.0 × 1014 cm−2, n2 = 4.3 × 1014 cm−2, n3 = 4.6 × 1014 cm−2. e Corresponding phonon dispersions and f NA Eliashberg spectral functions with cumulative EPC constants λ(ω). g Softening and stabilization of the M* point acoustic phonon. The stabilization of this mode defines the low- and high-doping 2 × 2 CDW phase, while the region where it is unstable corresponds to the polaronic and partial CDW phases.

Full size image

Polaronic distortions and partial CDW

Remarkably, after the 1 × 1 H structure reaches instability, it does not immediately stabilize into the 2 × 2 CDW phase. As the 1 × 1 M point backfolds in the 2 × 2 Γ point, a 2 × 2 CDW phase seems to stabilize this particular instability [see Fig. 3e] but, as demonstrated above, only for the narrow doping region from 2.11 to 2.38 × 1014 cm−2. By adding more electrons, this 2 × 2 CDW phase again shows an instability in the new M* point (see Supplementary Fig. 5), suggesting the existence of structural distortions on a larger length scale.

Calculations with supercells larger than 2 × 2 are expensive within DFT and, therefore, we use a TBA model to investigate the region of our phase diagram where neither H nor 2 × 2 CDW phases are stable (see gray area in Fig. 1). Our model calculations show that stable structures indeed require much larger supercells, once the doping is increased past the first instability point. Considering an \(18\sqrt{3}\times 18\sqrt{3}\) supercell, we find a single localized distortion, a polaron, where the triangular displacement of the three Mo atoms resembles that of a single unit of the 2 × 2 CDW in Fig. 2a. With increasing doping, the number of localized distortions increases, forming first a partial and then a full \(3\sqrt{3}\times 3\sqrt{3}\) CDW.

The high-doping 2 × 2 CDW phase

At high doping, our DFT calculations show that a 2 × 2 CDW phase once again becomes stable. The soft acoustic phonon at the M* point of the 2 × 2 BZ visible in Fig. 3b starts to blueshift as a function of doping, and becomes stable for around n = 3.5 × 1014 cm−2, as depicted in Fig. 4g. The flattening of the lowest conduction band, which hosts the excess charge [see Fig. 4d], significantly lowers the DOS at the Fermi level. As doping is increased, the M* point Kohn anomaly hardens [see Fig. 4e, g], while the EPC constant reduces.

For this doping range, we also report a \(2\sqrt{3}\times 2\sqrt{3}\) CDW phase coexistence (see Supplementary Fig. 8), in agreement with the experiment14. Here again, the triangular pattern of Mo displacements is present, but in a modulated configuration where one in three triangles is more pronounced. Even though we report the stability of a \(2\sqrt{3}\times 2\sqrt{3}\) CDW phase for much larger doping values than the doping values for which the 1 × 1 H phase is dynamically stable, we note that for the 1 × 1 H phase, there is an additional adiabatic Kohn anomaly close to the K point and strong nonadiabatic effects exactly at the K point [see Fig. 2b]. Since the transition from the 1 × 1 H phase to the high-doping 2 × 2 and \(n\sqrt{3}\times n\sqrt{3}\) CDW phases is not direct, we do not observe clear signatures for both phases in the low-doping phonon band structure, but the mentioned anomalies indicate strong EPC effects for the K point mode. We also show in Supplementary Fig. 5b the newly formed Kohn anomaly along Γ*−K* path that appears for 2 × 2 CDW structure in the unstable region, also indicating a potential stability of \(n\sqrt{3}\times n\sqrt{3}\) pattern. Using the TBA model, both a perfect 2 × 2 CDW and massively distorted \(3\sqrt{3}\times 3\sqrt{3}\) CDWs are stable, indicating that there should be other competing CDW orders in the system.

Discussion

To further validate the obtained structural phase diagram as a function of doping, in Fig. 5a we compare the calculated phonon frequency of Raman-active A1g optical phonon mode across various phases with the experimental measurements done on Li-doped (i.e., electron-doped) MoS275. This A1g optical phonon is an out-of-plane mode at the center of the BZ that is strongly coupled to the excess charge carriers in the conduction valleys of MoS2, and in the 1 × 1 H phase is redshifted as a function of conduction band population65,76. However, in Raman measurements, the frequency of the A1g mode is blueshifted with the maximum value around x = 0.4 (where x is the ratio of Li atoms with respect to the MoS2 unit), and then redshifted for x > 0.5. Our theoretical results presented in Fig. 5a confirm that the phase transitions between 1 × 1 H, 1T′, and CDW phases need to be taken into account in order to qualitatively capture the observed doping-induced renormalization of A1g frequency. Additionally, the large NA frequency renormalization of the A1g mode for the 1 × 1 H phase is present65,77, and it is crucial for understanding the modifications of the phonon frequency across the whole phase diagram. We observe larger nonadiabatic renormalizations in the 1 × 1 H phase than in the 1T′ phase or the high-doping 2 × 2 CDW phase. In contrast, their strength does not change much with doping within the same phase. The change of phase impacts the band structure, modifies the electron scattering phase space, and it also leads to different EPC matrix elements. We find that weak nonadiabatic effects in the 1T′ and the high-doping 2 × 2 CDW phases are the result of smaller EPC matrix elements and a change in electronic scattering phase space.

Fig. 5: Recreating the superconductivity dome structure of Tc in doped MoS2.
figure 5

a Evolution of the A1 phonon mode frequency across the phase transition from the 1 × 1 H to the 1T′ phase in the adiabatic and nonadiabatic approximation (lower x-axis). First-principles data is compared with the experiment from ref. 75, and the upper x-axis that corresponds to the experimental data is the atomic ratio of Li(x) in LixMoS2. b Doping-dependent SC phase diagram of MoS2, with different phases color-coded in the background. The experimental results for Tc are presented with open black symbols3,36.

Full size image

In Figs. 5, 6, we show a complete phase diagram of MoS2 obtained from first principles and model calculations, respectively. Firstly, a monotonically increasing Tc with doping persists until the point of instability. In the region of 1 × 1 H and 2 × 2 CDW phase coexistence, we report the Tc to be almost the same in the two phases, and it reaches its peak value of 34.6 K. The bands’ reconstruction in a 2 × 2 CDW phase compared to the 1 × 1 H phase for the same doping leads to the softening of the M* acoustic phonon mode and a significant EPC contribution coming from the lower frequency mode. This is accompanied by changes in the electronic DOS, reaching slightly larger values in the 1 × 1 H phase (see Supplementary Fig. 9). The doping-induced increase in the DOS and the softening of the M* point acoustic phonon lead to an increase in the Tc. As the system assumes the 1T′ phase, the SC Tc is lower, in agreement with experiments78.

Fig. 6: First-principles-based tight-binding model for MoS2 monolayer.
figure 6

All calculations have been performed on an \(18\sqrt{3}\times 18\sqrt{3}\) supercell, using the approximation of a linearized, nearest-neighbor EPC. Doping is modeled via a rigid shift of the chemical potential. a–c Transition from polaronic deformation via partial to full \(3\sqrt{3}\times 3\sqrt{3}\) CDW for intermediate dopings. d Critical temperature according to the Allen–Dynes formula as a function of doping for different structural phases. The underlying effective parameters λ and \({\omega }_{\log }\) as well as the energy gain and displacement magnitude are shown in Supplementary Fig. 7. The experimental reference points are the same as in Fig. 5 and taken from refs. 3,36. e, f 2 × 2 CDW and strongly distorted phase at high doping. Arrows indicate atomic displacements, scaled by a factor of 20 for better visibility. Hexagonal insets show the structure factor \(S=| \mathop{\sum }\nolimits_{i = 1}^{{N}_{{\rm{at}}}}\exp ({\rm{i}}{{\bf{qR}}}_{i})/{N}_{{\rm{at}}}{| }^{2}\) for all supercell Γ points q inside the first Brillouin zone of the primitive 1 × 1 cell, which indicates periodicities present in the shown relaxed displacement patterns.

Full size image

When the 2 × 2 CDW phase becomes stable again for larger doping concentrations, due to hardening of the strongly coupled mode with doping and decrease of the DOS at the Fermi level, SC Tc is reduced further. Note that the doping concentrations corresponding to the experimentally observed regions of a 2 × 2 and \(2\sqrt{3}\times 2\sqrt{3}\) CDW phases14 overlap with the region where the SC Tc is decreasing36. We believe that the indirect transition from the 1 × 1 H phase to a 2 × 2 CDW (possibly via polaronic phases) is the leading explanation of the SC dome. Good qualitative agreement with the experimental dome structure3,36 further corroborates this claim. We note also that the dome structure of Tc is correlated to the softening of the M* phonon, as one can see from Fig. 4g. See also Supplementary Fig. 9 for the interplay of DOS at the Fermi level and softening of the relevant phonon mode.

The above results are confirmed by the TBA model calculations. We emphasize that the purpose of our model is a qualitative understanding, and not to provide precise quantitative values of SC Tc. Figure 6d shows Tc as a function of doping for different stable structures on the \(18\sqrt{3}\times 18\sqrt{3}\) supercell (see Methods). In agreement with the DFT results in Fig. 5, we find that the 1 × 1 H and 2 × 2 CDW structures are favored for smaller and larger dopings, respectively. For intermediate dopings, however, both are unstable toward the formation of polaronic distortions and partial CDWs as shown in Fig. 6a–c. The periodicities present in these different types of structural order can be seen in the structure factor S shown in the insets. A possible explanation is that the doping electrons tend to localize near triangular distortions involving three Mo atoms, which can only cover the full sample as in the 2 × 2 CDW if enough excess charge is available. Here the TBA model calculations confirm that the saturation (or even slight decrease) of the DOS at the Fermi level for the polaronic and CDW phases, as well as the hardening of the relevant phonon modes, lead to the reduction of SC Tc (see Supplementary Fig. 10). Our previous studies of such distortions in the MoS2 monolayer using the same model approach have revealed associated dispersionless in-gap states24,25. Recently, similar distortions and bands have also been predicted for photodoped MoS2 and other TMDs79. Also, a recent study of bulk MoTe2 argued that the laser-induced transition from 2H to 1T′ phase goes through polaron formation80. Our combined DFT and TBA simulations suggest that such polaron formation might be universal and present in equilibrium conditions, playing an important role in structural phase transitions and formation of the SC dome.

Note that even though we reproduce the dome structure of Tc in accordance with the experiments, our values of Tc are always larger than the measured ones. For instance, the maximum values obtained in refs. 3,36 are Tc = 10.9 K and Tc = 11.6 K, respectively. Here we obtain Tc = 29.9 K for n = 2.04 × 1014 cm−2, while this value is additionally reduced to Tc = 21.6 K when SOC is included. In recent theoretical work, for the same doping concentration of n = 2.02 × 1014 cm−2, a Tc = 19 K was reported62, and the importance of using anharmonic corrections and a realistic gating geometry compared to a homogeneous jellium background for reaching the more quantitative Tc was discussed. In ref. 47, the renormalization of electronic states due to electron correlations was discussed as an important ingredient to obtain the quantitative agreement with the experimental Tc. It should also be noted that the onset and the maximum value of the Tc in our calculations is shifted toward larger doping concentrations compared to the experiments. For instance, the critical doping at which the Tc is maximum and where the original 1 × 1 H structure becomes unstable is nc = 2.1 × 1014 cm−2, while in the experiments this value is around nc = 1.5 × 1014 cm−2 3. Nevertheless, this could be corrected by using the lower value for electronic temperature in DFT calculations, see also Supplementary Fig. 1. Another important factor in determining the critical doping could also be the Mo-S distance, see Supplementary Fig. 2b, which dictates the energy ratio between the K and Q valleys in the band structure65,76, and, therefore, the onset of SC and CDW states. The energy ratio between the relevant K and Q valleys is also highly sensitive to the choice of DFT functionals. For example, the PBE functional provides the value of ~250 meV, while the many-body G0W0 correction would give ~50 meV, which is closer to experimental estimations65. Interestingly, ref. 81 argues that the inclusion of quantum lattice fluctuations in pristine MoS2 reduces the energy difference between K and Q valleys compared to the adiabatic frozen calculations as obtained in the DFT. Therefore, it is possible that in the proximity to the polaronic and CDW instabilities in doped samples, these fluctuations are even stronger and also reduce the value of nc. Further, we find that the presence of a substrate would alter the critical doping value, but would still enable a CDW transition. In the Supplementary Table 1, we report a DFPT analysis of the soft mode and we observe more pronounced softening of the M point instability when a single layer of MoS2 is placed on a graphite and hBN substrates (see also Supplementary Fig. 11). Our analysis indicates that the presence of a substrate even enhances a CDW transition. Interfacial interactions between monolayer MoS2 and a substrate therefore support our conclusions on the superconducting dome origin and shift it to lower doping values, closer to its experimentally observed doping region.

There are many studies discussing EPC strengths and the corresponding Tc for doped MoS220,23,26,47,59,62,82,83, however, the dome structure and the complete phase diagram up to higher doping concentration is rarely discussed, even though intriguing SC dome appears in many different materials, including different TMDs84.

In refs. 62,83 only the saturation of Tc as a function of doping was theoretically obtained, while it was argued that the SC dome structure observed in the experiments is due to extrinsic effects, such as charge localization for higher dopings, which leads to strong disorder and therefore to effective increase of the Coulomb electron-electron repulsion, which competes with the phonon-mediated pairing85. This was modeled phenomenologically by increasing the Coulomb repulsion parameter μ* that enters the Allen–Dynes formula and Eliashberg equations from 0.1 to 0.15. In ref. 47 it is claimed that an increase of electron correlations with doping is an intrinsic effect, which suppresses the EPC and, therefore, the Tc, producing a dome structure. However, the latter studies neglect the structural phase transitions, and the corresponding modifications to the electronic structure that are clearly present in MoS2 for larger doping concentrations. In another work, the SC dome structure appears due to the parametrization of McMillan’s formula26, where a certain combination of EPC λ and \({\omega }_{\log }\) could result in the saturation and decrease of Tc. On the other hand, here we show that the Tc for the 1 × 1 H phase is always increasing as a function of doping (see also results in ref. 62), both for the Allen–Dynes and Migdal–Eliashberg approaches. In the same study, the CDW phase was predicted to exist in doped MoS2, but the authors only reported the 2 × 1 CDW pattern coming from one nonequivalent M phonon instability, while here we report on the full 2 × 2 CDW phase induced by the three nonequivalent soft phonon modes. In addition, we show that this 2 × 2 CDW phase in MoS2 is not stabilized immediately after the 1 × 1 H phase becomes unstable, but is preceded by the polaronic states and partial CDWs, as well as possibly by the 1T′ phase. Also, the EPC strength and Tc was not evaluated beyond the 1 × 1 H phase in ref. 26.

Here we show that the SC dome structure in doped MoS2 is related to the strongly coupled soft phonon modes responsible for the structural phase transitions, as Figs. 3b, e, 4g, as well as the model data in Supplementary Figs. 9, 10, suggest. Namely, when approaching the point of structural instability with doping, the relevant phonon mode becomes softer, and therefore, the EPC λ and Tc are increased. With a further increase of doping, the new structure is stabilized, the relevant phonon mode hardens, and λ and Tc decrease, which in total creates the dome structure. The importance of the soft phonon modes in elevating the Tc was discussed a long time ago for A15 compounds86,87, and it was proposed as a source of the unconventional SC dome structure in SrTiO343,44 as well as predicted to play a role in BaTiO346 and WTe245. For SrTiO3, the critical doping concentration for which the Tc reaches maximum value correlates with the point when the optical polar phonon at the center of the BZ becomes unstable, inducing ferroelectric fluctuations44. Here, instead of the ferroelectric phase, the maximal Tc is related to the CDW and polaronic phase transitions and the corresponding soft acoustic phonon modes away from the BZ center.

In conclusion, we have explored the phase diagram of electron-doped MoS2 and explained the SC dome structure observed in various transport and tunneling experiments. Across a large doping range, we have found a number of dynamically stable phases, such as the 2 × 2 CDW, the 1T′ phase, polaronic states, and partial CDWs. In agreement with existing literature, we have found a doping-induced increase in Tc. Since the same strongly coupled acoustic phonon mode is responsible for the SC transition and a CDW occurrence, the doping-dependent EPC entirely coordinates their interplay. In this work, we observe that the initial rise of EPC strength with doping is followed by a corresponding increase in the superconducting Tc, favoring a superconducting state over charge ordering. However, the strong EPC eventually leads to a soft phonon mode, inducing a lattice reconstruction. Following a steep increase, a combination of structural phase transitions and CDW formations leads to its reduction. The latter correlates with the softening and hardening of the relevant acoustic phonon modes that are both responsible for the structural phase transitions and dominate the SC pairing mechanism. Comparing the EPC doping dependence in a 1 × 1 H phase and a high-doping 2 × 2 CDW phase, it can be noted that the two regimes are the exact opposites of one another. In the first regime, doping induces an increase in EPC and softening of the acoustic mode, while it stabilizes the CDW in the second regime. Therefore, we can conclude that a CDW state suppresses an SC state in MoS2. Our work reconciles various experimental findings of CDWs in MoS2, with its doping-dependent SC dome structure, and could be helpful in comprehending the SC mechanism of other materials hosting a Tc dome, such as other correlated TMDs and SrTiO3.

Methods

Migdal–Eliashberg theory

When it comes to calculating the Tc, there exist several levels of complexity, building up on the pioneering work of Bardeen, Cooper, and Schrieffer (BCS)88. Specific details of a material’s electronic structure, phonon spectrum, and interaction strengths can be included within the scope of Migdal–Eliashberg theory56. The smoothness of the Fermi surface determines whether an isotropic set of Eliashberg equations can be used89,90:

$${\widetilde{\omega }}_{n}={\omega }_{n}+T\mathop{\int}\nolimits_{-\infty }^{\infty }{\rm{d}}\varepsilon \frac{N(\varepsilon )}{N({\varepsilon }_{F}^{0})}\sum _{m}\frac{{\widetilde{\omega }}_{m}}{{\Theta }_{m}(\varepsilon )}{\lambda }_{n-m},$$
(1a)
$${\phi }_{n}=T\mathop{\int}\nolimits_{-\infty }^{\infty }{\rm{d}}\varepsilon \frac{N(\varepsilon )}{N({\varepsilon }_{F}^{0})}\sum _{m}\frac{{\phi }_{m}}{{\Theta }_{m}(\varepsilon )}[{\lambda }_{n-m}-\mu ],$$
(1b)
$${\chi }_{n}=-T\mathop{\int}\nolimits_{-\infty }^{\infty }{\rm{d}}\varepsilon \frac{N(\varepsilon )}{N({\varepsilon }_{F}^{0})}\sum _{m}\frac{\varepsilon -{\varepsilon }_{F}+{\chi }_{m}}{{\Theta }_{m}(\varepsilon )}{\lambda }_{n-m},$$
(1c)
$${n}_{e}=\mathop{\int}\nolimits_{-\infty }^{\infty }{\rm{d}}\varepsilon N(\varepsilon )\left[1-2T\sum _{m}\frac{\varepsilon -{\varepsilon }_{F}+{\chi }_{m}}{{\Theta }_{m}(\varepsilon )}\right],$$
(1d)

with the common denominator

$${\Theta }_{n}(\varepsilon )={\widetilde{\omega }}_{n}^{2}+{\phi }_{n}^{2}+{[\varepsilon -{\varepsilon }_{F}+{\chi }_{n}]}^{2}.$$
(1e)

Here, N(ε) is the DOS per spin, ωn = (2n + 1)πT are fermionic Matsubara frequencies, \({\widetilde{\omega }}_{n}\) are renormalized Matsubara frequencies, ϕn is the SC order parameter, and χn an energy shift. The number of electrons ne is conserved, which implies that the chemical potential εF may deviate from its noninteracting value \({\varepsilon }_{F}^{0}\). The Eliashberg equations are solved iteratively for \({\widetilde{\omega }}_{n}\), ϕn, χn, and εF, using our own implementation91. The presence of an SC state (ϕn > 0) for a given temperature T is determined by the effective Coulomb interaction μ and the effective EPC

$${\lambda }_{n}=\mathop{\int}\nolimits_{0}^{\infty }{\rm{d}}\omega \frac{2\omega }{{\omega }^{2}+{\nu }_{n}^{2}}{\alpha }^{2}F(\omega ),$$
(2)

where νn = 2nπT are bosonic Matsubara frequencies. Since we are only interested in the value of Tc, it is sufficient to only consider the linearized version (ϕn → 0) of the equations.

From the linearized form of the (real-axis) Eliashberg equations, McMillan’s formula can be qualitatively obtained57. The majority of theoretical estimates of Tc rely on McMillan’s semiempirical result, which was later refined by Allen and Dynes58:

$${T}_{c}=\frac{{f}_{1}{f}_{2}{\omega }_{\log }}{1.2}\exp \left[-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{* }-0.62\lambda {\mu }^{* }}\right].$$
(3)

The strong-coupling correction \({f}_{1}={[1+{(\lambda /{\Lambda }_{1})}^{3/2}]}^{1/3}\) and the shape correction \({f}_{2}=1+(r-1){\lambda }^{2}/({\lambda }^{2}+{\Lambda }_{2}^{2})\) with Λ1 = 2.46(1 + 3.8 μ*), Λ2 = 1.82(1 + 6.3 μ*)r, and \(r={\overline{\omega }}_{2}/{\omega }_{\log }\) are needed if λ > 1.5. We refer to Eq. (3) as the Allen–Dynes formula, while the same result without the strong-coupling corrections is referred to as McMillan’s formula. EPC effects are encapsulated in the EPC constant λ and the logarithmic and second-moment average phonon frequencies \({\omega }_{\log }\) and \({\overline{\omega }}_{2}\), which are obtained as integrals over the Eliashberg spectral function:

$$\lambda =2\mathop{\int}\nolimits_{0}^{\infty }\frac{{\alpha }^{2}F(\omega )}{\omega }{\rm{d}}\omega ={\lambda }_{0},$$
(4)
$${\omega }_{\log }=\exp \left[\frac{2}{\lambda }\mathop{\int}\nolimits_{0}^{\infty }\log (\omega )\frac{{\alpha }^{2}F(\omega )}{\omega }{\rm{d}}\omega \right],$$
(5)
$${\overline{\omega }}_{2}=\sqrt{\frac{2}{\lambda }\mathop{\int}\nolimits_{0}^{\infty }\omega {\alpha }^{2}F(\omega ){\rm{d}}\omega }.$$
(6)

A third material parameter is the Coulomb pseudopotential. The effective Coulomb electron-electron interaction μ in Migdal–Eliashberg theory differs from the Morel–Anderson pseudopotential μ* of McMillan’s formula92. The latter is scaled to a typical phonon frequency, which we assume to be \({\overline{\omega }}_{2}\) following ref. 58. For a given μ*, we can then estimate

$$\mu ={[1/{\mu }^{* }-R({\overline{\omega }}_{2})]}^{-1},$$
(7)

with the cutoff-dependent rescaling term90

$$R(\omega )=\frac{1}{\pi }\mathop{\int}\nolimits_{-\infty }^{\infty }{\rm{d}}\varepsilon \frac{N(\varepsilon )}{N({\varepsilon }_{F}^{0})}\frac{1}{\varepsilon -{\varepsilon }_{F}^{0}}\arctan \left[\frac{\varepsilon -{\varepsilon }_{F}^{0}}{\omega }\right].$$
(8)

Also the fact that the summations over Matsubara frequencies are in practice truncated at a cutoff frequency ωC has to be considered. We limit the summations in Eqs. (1) to ∣ωm∣ < ωC and at the same time replace μ in Eq. (1b) by \({\mu }_{{\rm{C}}}^{* }={[1/\mu +R({\omega }_{C})]}^{-1}\) scaled to the Matsubara cutoff. Using Eq. (7), we can also write

$${\mu }_{{\rm{C}}}^{* }={[1/{\mu }^{* }-R({\overline{\omega }}_{2})+R({\omega }_{C})]}^{-1},$$
(9)

where μ* depends on doping and in multi-valley materials it also changes when calculating the inter/intra-valley repulsion. For larger doping values it has been calculated to have the mean value of 0.1393. In this work we therefore set μ* = 0.13 and use it as a constant.

The Eliashberg spectral function in Eqs. (4–6) can be written as94,95

$${\alpha }^{2}F(\omega )=\frac{1}{\pi N({\varepsilon }_{F}^{0})}\sum _{{\bf{q}}\nu }\frac{{\gamma }_{{\bf{q}}\nu }}{{\omega }_{{\bf{q}}\nu }}{B}_{\nu }({\bf{q}},\omega ),$$
(10)

where q and ν are phonon wavevector and band indices. ωqν are the statically screened phonon frequencies, as obtained in density functional perturbation theory (DFPT)96,97. In Eq. (10), α2F(ω) is written in its NA form. It can be seen as a phonon DOS, weighted, renormalized, and broadened by EPC in the form of phonon linewidths γqν obtained in the double-delta approximation98 and the NA phonon spectral function

$${B}_{\nu }({\bf{q}},\omega )=-\frac{1}{\pi }\,{\rm{Im}} \left[\frac{2{\omega }_{{\bf{q}}\nu }}{{\omega }^{2}-{\omega }_{{\bf{q}}\nu }^{2}-2{\omega }_{{\bf{q}}\nu }{\pi }_{\nu }({\bf{q}},\omega )}\right].$$
(11)

With the term “adiabatic”, we refer to the statically screened and infinitely long-lived (DFPT) phonons99. Nonadiabaticity refers to the fact that the statically screened DFPT phonon frequencies are corrected by the real part of the NA phonon self-energy

$$\begin{array}{rcl}{\pi }_{\nu }({\bf{q}},\omega )&=&\sum\limits _{{\bf{k}}nm}{\left\vert {g}_{mn\nu }({\bf{k}},{\bf{q}})\right\vert }^{2}\dfrac{{f}_{n{\bf{k}}}-{f}_{m{\bf{k}}+{\bf{q}}}}{\omega +{\varepsilon }_{n{\bf{k}}}-{\varepsilon }_{m{\bf{k}}+{\bf{q}}}+i\eta }\\ &&-\sum\limits _{{\bf{k}}nm}{\left\vert {g}_{mn\nu }({\bf{k}},{\bf{q}})\right\vert }^{2}\dfrac{{f}_{n{\bf{k}}}-{f}_{m{\bf{k}}+{\bf{q}}}}{{\varepsilon }_{n{\bf{k}}}-{\varepsilon }_{m{\bf{k}}+{\bf{q}}}}\end{array}$$
(12)

evaluated at the unrenormalized frequencies, including the broadening effects as well. In Eq. (12), q ≠ 0, k, and n, m are electronic momentum and band indices, fnk denote the Fermi-Dirac distribution functions, εnk are the electron energies, and gmnν(k, q) are the statically screened EPC matrix elements. We use the “screened-screened” approximation to the phonon self-energy for quadratic errors64.

Model calculations

To estimate the critical temperature for lattice distortions on very large supercells (here 2916 atoms), for which full DFPT calculations are unfeasible, we use the approximation of a linearized EPC as described in ref. 70, wherein it is referred to as “model III”. Additionally, we reduce the EPC to a nearest-neighbor model including only the \({d}_{{z}^{2}}\), \({d}_{{x}^{2}-{y}^{2}}\), and dxy orbitals of the transition-metal atoms, as defined in Eq. (B4) of ref. 100 for TaS2, with parameters fitted to DFPT data for MoS2. The first-principles data are taken from Appendix B of ref. 64. Doping is modeled as a rigid shift of the chemical potential. The electronic temperature is set to 5 mRy ≈ 800 K. For each carrier concentration the atomic positions are optimized, starting from both a randomly distorted 2 × 2 CDW and a single triangular displacement involving three neighboring Mo atoms, until all forces are below 10−5 Ry/bohr. Phonons and the adiabatic effective interaction parameters are then calculated for q = 0 only, with the help of sparse matrices and repeated transforms between the orbital and band basis to save memory101.

Computational details

The written equations and their ingredients are calculated by means of DFPT96 and Wannier interpolation102 of EPC matrix elements103. We use Quantum ESPRESSO97 for the DFT calculations, and the EPW code103,104,105,106 for EPC. We use the fully-relativistic norm-conserving Perdew–Burke–Ernzerhof pseudopotentials with a kinetic energy cutoff of 100 Ry from PseudoDojo107. The NA dynamic and static terms in the phonon self-energy, Eq. (12), are calculated separately, because in the dynamic term, dense k- and q-grids are used, while the grids for the static-term calculation are already determined by the DFT and DFPT calculations23,63,108. For the calculations in a 1 × 1 H phase, the relaxed lattice constant is set to 3.186 Å, with a vacuum of 16 Å. The self-consistent electron density calculation is done on a 24 × 24 × 1 k-point grid, and the phonon calculation on a 12 × 12 × 1 q-point grid. In EPW, we use 11 maximally localized Wannier functions109 with the initial projections of d orbitals on the Mo sites and p orbitals on the S atom sites. The fine sampling of the Brillouin zone for the electron-phonon interpolation is done on a 360 × 360 × 1 k-grid and 120 × 120 × 1 q-grid. Smearing in the EPW calculation is set to 20 meV. For the calculations in other phases, all the abovementioned parameters are the same, except that the unit cell is larger and the grids are accordingly smaller. In all the DFT and DFPT calculations, we use the Coulomb truncation110, while in the EPW calculation, we use the long-range terms which include 2D dipoles following ref. 111.