Coexistence of Superconductivity with Enhanced Charge Density

Jul 5, 2019 - Coexistence of Superconductivity with Enhanced Charge Density Wave Order .... overcomes both the CDW effect and naturally reinforced Cou...
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Cite This: J. Phys. Chem. Lett. 2019, 10, 4076−4081

Coexistence of Superconductivity with Enhanced Charge Density Wave Order in the Two-Dimensional Limit of TaSe2 Chao-Sheng Lian,†,‡ Christoph Heil,§ Xiaoyu Liu,∥ Chen Si,*,⊥ Feliciano Giustino,# and Wenhui Duan*,‡,∥,∇

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†

International Laboratory for Quantum Functional Materials of Henan and School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China ‡ Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China § Institute of Theoretical and Computational Physics, Graz University of Technology, NAWI Graz, 8010 Graz, Austria ∥ Institute for Advanced Study, Tsinghua University, Beijing 100084, China ⊥ School of Materials Science and Engineering, Beihang University, Beijing 100191, China # Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, United Kingdom ∇ Collaborative Innovation Center of Quantum Matter, Beijing 100084, China S Supporting Information *

ABSTRACT: Bulk 2H-TaSe2 is a model charge density wave (CDW) metal with superconductivity emerging at extremely low temperature (Tc = 0.1 K). Here, by first-principles calculations including the explicit calculation of the screened Coulomb interaction, we demonstrate enhanced superconductivity in the CDW state of monolayer 1H-TaSe2 observed in recent experiments. Its ground-state 3 × 3 CDW phase features triangular clustering of Ta atoms and possesses a large electron−phonon coupling of λ = 0.74, yielding an order of magnitude higher superconducting Tc compared to the bulk. Upon lowering the thickness from bulk to monolayer TaSe2, the CDW intensifies with slightly decreased Fermi-level density of states, while superconductivity gets boosted via a largely increased intrinsic electron−phonon coupling strength, which overcomes both the CDW effect and naturally reinforced Coulomb repulsion. These results uncover the simultaneously enhanced CDW and superconducting orders in the two-dimensional limit for the first time and have key implications for other CDW metals like 2H-TaS2.

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simplified Coulomb pseudopotential formula8 and later to be a weakening of competing CDW order deduced from limited transport data.10 Compared to 2H-NbSe2 and 2H-TaS2, 2H-TaSe2 has a unique complex phase diagram:15−23 it exhibits an incommensurate CDW transition at 120 K, followed by a lock-in transition into a 3 × 3 commensurate CDW phase at 90 K, in which superconductivity emerges at extremely low Tc = 0.14 K. Currently, several open questions remain concerning the evolutions of its CDW and superconducting orders under reduced dimensions. (i) Recent experiments24,25 found a robust CDW in monolayer 1H-TaSe2 with increased transition temperature to 130 K and unchanged 3 × 3 symmetry of the commensurate phase. On the basis of a CDW structural model

n the phase diagrams of layered 2H-NbSe2, 2H-TaS2, and 2H-TaSe2, a charge density wave (CDW) coexists with superconductivity at low temperatures, making these systems ideal models for exploring the interplay of collective orders in low-dimensional physics.1,2 With rapid advances in exfoliation and growth techniques, many metallic transition-metal dichalcogenides (TMDs) have become available in atomically thin forms.3−12 For 2H-NbSe2, with decreasing layer thickness, the superconducting transition temperature Tc decreases gradually, accompanied by an increasing CDW transition temperature.3−7 This suggests a competition between the CDW and superconductivity, which has been well revealed from first-principles via the strengthened Fermi surface (FS) gapping in the enhanced CDW state.13,14 In sharp contrast, as 2H-TaS2 is thinned to the monolayer limit, its Tc consecutively rises,8−10 with very different CDW behaviors reported for samples of different origin.10−12 The physical mechanism of the enhanced superconductivity is still debated, proposed initially to be a reduced Coulomb repulsion by assuming a © 2019 American Chemical Society

Received: May 23, 2019 Accepted: July 5, 2019 Published: July 5, 2019 4076

DOI: 10.1021/acs.jpclett.9b01480 J. Phys. Chem. Lett. 2019, 10, 4076−4081

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The Journal of Physical Chemistry Letters proposed for bulk 2H-TaSe2 in the 1970s,26,27 the CDW order was previously predicted to be somewhat weakened in the monolayer.28 This, however, is contrary to the recent observations,24,25 casting doubt on the rationality of this model. (ii) Scanning tunneling and electrical transport measurements revealed a superconducting Tc of ∼1 K in multilayer samples,29,30 but whether intrinsic superconductivity persists in the monolayer along with a strengthened CDW is still unknown. (iii) In the case of enhanced superconductivity down to the monolayer limit, it is highly desirable to uncover the corresponding physical mechanism by clarifying the dimensionality effects on the CDW electronic structure, electron−phonon interaction, and Coulomb repulsion. In this Letter, we address these issues by studying the equilibrium atomic structure, electron−lattice coupling, and electron−electron interaction in the CDW state of monolayer 1H-TaSe2 entirely from first-principles. A 3 × 3 phase characterized by triangular Ta clusters is identified as the most stable CDW, in agreement with scanning tunneling microscopy (STM) data. Our results establish this phase as a moderate-coupling superconductor with Tc = 2.2 K, indicating the persistence and obvious enhancement of superconductivity in monolayer TaSe2. We find that, from bulk to the monolayer, the CDW order intensifies, yielding slightly decreased Fermilevel density of states (DOS) while Coulomb repulsion becomes strengthened due to reduced screening, thus ruling out these two effects as causes for the substantial Tc enhancement. A largely increased electron−phonon coupling (EPC) strength in the reduced dimensionality is further revealed to be responsible for boosting the superconductivity. Our calculations are performed within the generalized gradient approximation31 to density functional theory (DFT). We employ the Vienna ab initio simulation package32 for structural optimization and electronic structure calculations, where the effect of spin−orbit coupling (SOC) is included. The projector augmented wave method33 is used with a plane-wave energy cutoff of 350 eV. At least a 12 Å thick vacuum layer is introduced in a slab model with the van der Waals interaction treated by the DFT-D3 method.34 Electron− phonon calculations are performed by employing the Quantum ESPRESSO package35−37 using optimized norm-conserving Vanderbilt pseudopotentials38,39 with a plane-wave cutoff of 50 Ry. We obtain the phonon spectrum by Fourier interpolation of the dynamical matrices computed on a uniform 12 × 12 × 1 q grid for the normal phase and 3 × 3 × 1 q grid for the CDW phase, with their EPC matrix elements calculated using dense 48 × 48 × 1 and 12 × 12 × 1 k grids, respectively. To determine the electron−electron interaction strength μ = NF⟨⟨Vk,k′⟩⟩FS as a double FS average of Vk,k′ = ⟨k, −k|W|k′, −k′⟩, the screened Coulomb interaction W is calculated within the random phase approximation using the SternheimerGW code40−42 with an energy cutoff for the dielectric matrix of 5 Ry. The Sternheimer equations are solved on 12 × 12 × 4 and 12 × 12 × 1 (4 × 4 × 1) grids for the bulk normal phase and the monolayer normal (CDW) phase. The normal phase of monolayer 1H-TaSe2 has point group D3h and consists of a Se−Ta−Se sandwich in trigonal prismatic coordination, forming a honeycomb lattice (Figure 1a,b). At low temperature, it becomes unstable, accompanied by the appearance of a soft phonon mode involving Ta in-plane 2 vibrations around q CDW = 3 ΓM (Figure 1c), pointing to a 3 × 3 CDW instability.28 Figure 1d,e shows Brillouin zone (BZ)

Figure 1. (a,b) Crystal structure and BZ of monolayer 1H-TaSe2. (c) Phonon spectrum of 1H-TaSe2 in the normal state. (d,e) BZ maps of the phonon line width of the soft acoustic branch (d) and the FS nesting function (e), where blue (red) represents low (high) values. (f) CDW formation energy ΔE as a function of the Ta trimer displacement δTa toward a Se atom (negative δTa) or a honeycomb lattice hollow site (positive δTa) for monolayer TaSe2 in a 3 × 3 supercell. (Insets) Simulated STM images of 3+6-filled (left) and 3+6-hollow (right) CDW phases at Vb = 50 mV. (g−i) Crystal structures of 3+6-filled (g), 3+6-hollow (h), and 7+2 (i) CDW phases.

maps of the phonon line width of the soft acoustic branch and the FS nesting function. It is clearly seen that the line width has a pronounced maximum at qCDW but not the nesting function. This suggests that momentum-dependent EPC, rather than FS nesting,43−47 is the origin of the CDW order in monolayer TaSe2. The low-energy CDW structures are determined by performing full optimization of 3 × 3 superstructures with initial atomic positions being slightly randomized in different manners. Surprisingly, except for the early proposed 7+2 CDW structure26,27 characterized with seven-atom clusters (Figure 1i), we find two other new distorted structures, named 3+6filled (Figure 1g) and 3+6-hollow (Figure 1h), featuring triangular three- and six-atom Ta clusters centering on the Se atoms and the honeycomb lattice hollow sites, respectively. Their formation process is modeled in Figure 1f, where we plot the formation energy ΔE of CDW distortion as a function of atomic displacement δTa, which is introduced by moving a Ta trimer motif in the 3 × 3 supercell toward either a Se atom (negative δTa) or a hollow site (positive δTa). At each δTa, we relax all atoms except the Ta trimer motif. The 3+6-filled and 3+6-hollow structures are found to appear at the minima of the ΔE(δTa) curve at δTa = −0.08 and 0.11 Å, with large formation energies of −51.7 and −74.0 meV (calculated without SOC), and both of these structures are energetically more favorable than the 7+2 structure [inclusion of SOC does not change the energy order between the three structures (see 4077

DOI: 10.1021/acs.jpclett.9b01480 J. Phys. Chem. Lett. 2019, 10, 4076−4081

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The Journal of Physical Chemistry Letters Table 1)]. Further phonon calculations show that the 7+2 structure is dynamically unstable, while the two newly Table 1. Calculated CDW Formation Energy ΔE and Ta Atomic Distortion |δTa| for the 3+6-Filled, 3+6-Hollow, and 7+2 CDW Phases without (with) SOCa system

3+6-filled

3+6-hollow

7+2 CDW

ΔE (meV) |δTa| (Å) stable?a

−51.7 (−50.8) 0.08 (0.08) yes

−74.0 (−56.3) 0.11 (0.10) yes

−41.7 (−41.9) 0.075 (0.07) no

a

Dynamical stability.

identified structures are stable (see Supporting Information Figure S1). In addition, STM simulations (Figure 1f) reveal that, different from the 3+6-filled phase (the left inset), the 3+6-hollow phase (the right inset) displays an array of three large bright spots contributed by the three highest Se atoms on top of the six-atom Ta cluster, matching well with the experimental STM image reported by Ryu et al.24 These results establish the energetically most stable 3+6-hollow structure as the experimentally observed CDW ground state. We next assess the superconductivity of monolayer 1HTaSe2 by studying the electronic structure and EPC of the CDW state. Figure 2a shows the band structure of the 3+6-

Figure 3. Calculated electron−phonon properties of the 3+6-hollow CDW phase. (a) Phonon spectrum weighted by the mode EPC strength λqν (red dots). (b) Eliashberg spectral function α2F(ω) along with the λ(ω) = 2∫ ω0 (α2F(ω)/ω) dω. (c) Ta and Se atom-projected PHDOS. (d) Electronic DOS projected on the Ta 5dz2, Ta 5dx2−y2, Ta 5dxy, and Se 4p orbitals. (e,f) Atomic vibration patterns (blue arrows) for the A′1 and E′ modes marked in (a) with large EPCs at the Γ point. Figure 2. (a) Unfolded electronic band structure of the 3+6-hollow CDW phase in the 1 × 1 BZ. The size of the green dots represents the spectral weight of the energy bands. (b) Band structure of monolayer 1H-TaSe2 in the normal state. The dashed lines indicate the Fermi level.

vibrations barely contribute to the EPC due to the presence of reflection symmetry50,51 in 1H-TaSe2. Figure 3e,f illustrates the A1′ and E′ modes with large EPCs at Γ: they involve mainly Ta in-plane vibrations and thus can couple effectively to the Ta 5dz2, 5dx2−y2, and 5dxy electronic states dominating the FS (Figure 3d). To calculate the superconducting Tc, the Allen−Dynes McMillan equation52 is employed with Coulomb effects included via the Morel−Anderson pseudopotential53 μ* = μ/ [1 + μ log(ωel/ωph)], where μ is the electron−electron interaction strength and ωel and ωph are characteristic electron and phonon energies, chosen to be the lowest plasmon energy54,55 and the highest phonon energy of the system, respectively. Using λ = 0.74 and ω log = exp[(2/ 2 λ)∫ ∞ 0 (α F(ω) log ω/ω) dω] = 117.6 K, we obtain Tc = 2.2 K for the monolayer 3+6-hollow CDW phase based on μ* = 0.18 (μ = 0.47) calculated from first-principles. This Tc value is an order of magnitude higher than Tc = 0.14 K in bulk 2HTaSe2,21−23 indicating that superconducting order is significantly enhanced in monolayer 1H-TaSe2, in contrast to the suppression of Tc found in atomically thin 2H-NbSe2.3−7 In addition, we find that both λ and Tc of this system can be improved by applying biaxial compressive strain due to the softening of phonons contributing to superconductivity, and at a low level of −3%, the increase of Tc is as large as more than 100% (see Supporting Information Figure S2). Note that SOC is not included in the electron−phonon calculations presented

hollow CDW phase unfolded onto the 1 × 1 BZ,48,49 in comparison to that of the undistorted structure (Figure 2b). Obviously, the CDW distortion leads to energy gaps forming on the hole FS pocket around K, which is dominated by the Ta 5dx2−y2 and 5dxy orbitals. The partial gap opening, rather than the full gap opening around the FS, implies that in monolayer TaSe2 superconductivity may coexist with the CDW state. Figure 3a,b shows the phonon spectrum of the 3+6-hollow CDW phase along with the mode-resolved EPC strength λqν (represented by the size of the red dots) and the corresponding Eliashberg spectral function α2F(ω). The total 2 EPC, λ = 2∫ ∞ 0 (α F(ω)/ω) dω is calculated to be 0.74, suggesting that monolayer 1H-TaSe2 is a moderate-coupling superconductor. A comparative analysis of the Eliashberg function α2F(ω) (Figure 3b) and the atom-projected phonon DOS (PHDOS, Figure 3c) shows that the low-frequency phonons below 125 cm−1 associated with the Ta in-plane vibrations contribute to 81% of the total λ and the remaining 19% of λ is sustained by the coupled Ta−Se vibration modes in higher-frequency regions between 135 and 210 cm−1 and around 235 cm−1. It is noted that the out-of-plane Ta 4078

DOI: 10.1021/acs.jpclett.9b01480 J. Phys. Chem. Lett. 2019, 10, 4076−4081

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monolayer λ value of 0.74 calculated here from first-principles and the bulk λ value of 0.4−0.5 evaluated using the inverted McMillan equation with μ* = 0.15−0.2 as reported in previous works22,23 (it is necessary to determine λ for the bulk in the future on an equal footing with the monolayer). According to

here. We have found that SOC only slightly modifies the DOS at the Fermi level in monolayer TaSe2, and its effects on λ and Tc are very small (see Supporting Information Figure S3). To uncover the physical origin for the Tc enhancement observed in the TaSe2 monolayer limit, we first probe the change of potentially competing CDW order in the monolayer with respect to the bulk. By determining the ground-state CDW structures for systems of different thickness (see Supporting Information Tables S1−S3) we have calculated the formation energy ΔE and the Fermi-level DOS NF of the CDW state as a function of layer thickness (see Figure 4a).

λ≈

NF⟨g 2⟩ 58 , M⟨ω2⟩

both phonon softening and the enhanced ⟨g2⟩ (a

FS average over the electron−phonon matrix elements) can increase λ under reduced dimensionality. For the former, it is interesting to see whether frequency lowering occurs in future low-temperature Raman studies of thin TaSe2 films. For the latter, direct evidence could be given through phonon line width (∝|gνq|2) calculations of the soft acoustic phonon mode ν at qCDW in the normal phase, which is found to exhibit the largest EPC matrix elements between states on the FS. Our calculated results reveal that its phonon line width increases from 0.45 meV in the bulk to 0.62 meV in the monolayer. Generally, across the CDW transition, this soft acoustic mode transforms into optical modes at the reduced BZ center that largely dominate the EPC in the CDW state. It is worth distinguishing the electron−phonon enhancement mechanism in 1H-TaSe2 from that proposed for monolayer FeSe, which also shows an enhanced Tc. In the case of FeSe,59 coupling between FeSe electrons and the substrate phonons is found to be at work, indicating an extrinsic interfacial coupling mechanism, while in our TaSe2 case, it is the intrinsic EPC that gets strengthened because of solely the dimensionality reduction. In light of this, a similar intrinsic EPC mechanism might also hold for other layered 2H TMDs. Further theoretical analysis combined with experimental measurements of electron−phonon signals is called for to better understand their enhanced EPC in the twodimensional limit. In conclusion, using state-of-the-art DFT calculations and the random-phase approximation to determine the screened Coulomb interactions, we elucidated the CDW formation and phonon-mediated superconductivity in monolayer 1H-TaSe2. In contrast to a model containing seven-atom clusters,26,27 we determined the 3 × 3 phase featuring triangular three- and sixatom Ta clusters around the lattice hollow sites as the CDW ground state, matching well with STM experiments. This monolayer CDW phase is found to possess a large total EPC λ of 0.74, leading to superconductivity at an order of magnitude higher transition temperature Tc compared to that for the bulk. It is confirmed that from bulk to monolayer TaSe2 both the intensified CDW order and Coulomb repulsion impair the superconductivity and the strong Tc enhancement is rooted in a general physical scenario where the rising EPC strength plays a decisive role. These findings establish an enhanced superconducting order in the strengthened CDW state of monolayer 1H-TaSe2 and will also provide insights into the debate over enhanced superconductivity in other TMDs like 2H-TaS2.

Figure 4. (a) Formation energy ΔE and Fermi-level DOS NF of the CDW state as a function of layer number. (b) Comparison of monolayer TaSe2’s EPC λ (left) and Coulomb repulsion μ (right) with those of bulk TaSe2. The bulk λ data (blue dashed line) are extracted from refs 22 and 23 based on the inverted McMillan equation.

From bulk to monolayer TaSe2, the energy gain |ΔE| due to the formation of CDW is found to increase slowly, with the NF in the CDW state decreasing slightly. This demonstrates that a strengthened CDW order persists in monolayer 1H-TaSe2, in line with recent experimental studies showing a CDW transition temperature in the monolayer a bit higher than that in the bulk.24,25 However, the reduction in NF and hence in the density of charge carriers available for superconductivity in the monolayer rules out the CDW as a cause for the enhanced Tc. Another mechanism that needs to be considered is the role of electron−electron Coulomb interaction. Superconductivity in the TMDs is believed to be of BCS type, where electrons form superconducting Cooper pairs when phonon-mediated attraction overcomes the Coulomb repulsion. Upon lowering the layer thickness of a TMD material, electronic screening is expected to become poorer and Coulomb repulsion intensifies.56,57 Indeed, taking the normal phase of TaSe2 as an example, our calculated Coulomb repulsion strength μ increases from 0.41 for the bulk to 0.56 for the monolayer (Figure 4b). Given that the CDW order in 2H-TaSe2 weakly depends on dimensionality and would yield similar modifications of μ under different layer thicknesses, a similar increasing trend of μ with the thickness reduction is expected for the CDW phase. Evidently, the enhancement of Coulomb repulsion in the monolayer is also not possible to contribute to boosting the superconducting Tc. On the basis of above discussions, a feasible mechanism responsible for the Tc increase in the monolayer limit should be the enhancement of electron−phonon interaction. Though our phonon and EPC calculations of the CDW phase are limited to the monolayer due to the drastically increased computational cost in the bulk, a rising EPC strength (see Figure 4b) would be evidenced via the contrast between the

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b01480. Phonon spectra of monolayer TaSe2 CDW phases, biaxial strain effects on superconductivity, electronic DOS, α2F(ω) and λ(ω) of the 3+6-hollow CDW phase with SOC, ground-state CDW structures for systems of 4079

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The Journal of Physical Chemistry Letters

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different thickness, energetic stability of the distorted monolayer TaSe2 in different supercells, and convergence of superconducting results for the 3+6-hollow CDW phase (PDF)

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. Phone: +86-010-8231-3923. *E-mail: [email protected]. Phone: +86-010-62785577. ORCID

Chen Si: 0000-0002-4292-4151 Feliciano Giustino: 0000-0001-9293-1176 Wenhui Duan: 0000-0001-9685-2547 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Ministry of Science and Technology of China (Grant No. 2016YFA0301001), the National Natural Science Foundation of China (Grant Nos. 51788104, 11674188, 11874079, and 11847116), and the Beijing Advanced Innovation Center for Future Chip (ICFC). C.H. and F.G. acknowledge support by the Austrian Science Fund (FWF) Project No. J3806-N36, the Graphene Flagship (Horizon 2020 Grant No. 785219 - GrapheneCore2), the Vienna Science Cluster, and the Cambridge Service for Data Driven Discovery (CSD3) funded by EPSRC (Grant No. EP/ P020259/1).

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DOI: 10.1021/acs.jpclett.9b01480 J. Phys. Chem. Lett. 2019, 10, 4076−4081

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DOI: 10.1021/acs.jpclett.9b01480 J. Phys. Chem. Lett. 2019, 10, 4076−4081