Unveiling Charge-Density Wave, Superconductivity, and Their

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Unveiling charge-density wave, superconductivity, and their competitive nature in two-dimensional NbSe2 Chao-Sheng Lian, Chen Si, and Wenhui Duan Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b00237 • Publication Date (Web): 13 Apr 2018 Downloaded from http://pubs.acs.org on April 13, 2018

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Nano Letters

Unveiling charge-density wave, superconductivity, and their competitive nature in two-dimensional NbSe2 Chao-Sheng Liana , Chen Sib∗ and Wenhui Duanacd∗ a Department

of Physics and State Key Laboratory of Low-Dimensional Quantum Physics,

Tsinghua University, Beijing 100084, People’s Republic of China b School

of Materials Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China

c Collaborative d Institute

Innovation Center of Quantum Matter, Beijing 100084, People’s Republic of China

for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China E-mail:

[email protected],+86-010-8231-3923;[email protected],+86-010-6278-5577

∗ To

whom correspondence should be addressed

1

ACS Paragon Plus Environment

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Abstract Recently, charge-density wave (CDW) and superconductivity are observed to coexist in atomically thin metallic NbSe2 . Lacking of knowledge on the structural details of CDW, however, prevents us to explore its interplay with superconductivity. Using first-principles calculations, we identify the ground state 3 × 3 CDW atomic structure of monolayer NbSe2 which is characterized by the formation of triangular Nb clusters and shows a scanning tunnelling microscopy (STM) image and Raman CDW modes in good agreement with experiments. We further demonstrate that from bulk to monolayer NbSe2 , as the layer thickness decreases, the CDW order is gradually enhanced with rising energy gain and strengthened Fermi surface gapping, while superconductivity is weakened due to the increasingly reduced Fermi level density of states in the CDW state. These results well explain the observed opposite thickness dependencies of CDW and superconducting transition temperatures and uncover the nature of competitive interaction between the two collective orders in two-dimensional NbSe2 . KEYWORDS: Two-dimensional materials, NbSe2 , charge-density wave, superconductivity, first-principles calculations

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Nano Letters

Charge-density wave (CDW), a periodic modulation of electronic densities and the atomic lattice, is an intriguing collective phenomenon in condensed-matter physics. 1,2 The layered transition metal dichalcogenide NbSe2 is known as a prototypical CDW material. 3–9 It undergoes a second order phase transition towards a nearly commensurate CDW phase at TCDW = 33 K and then enters into a superconducting state coexisting with this CDW order at Tc = 7 K. Recent experimental studies 10–13 found that both CDW and superconductivity persist in the monolayer limit of NbSe2 , thanks to the advances in fabrication of atomically thin two-dimensional (2D) materials. This sparks intense interests in exploration of the dimensionality effects on CDW, superconductivity and their mutual relation. 14–17 At present, there are still some puzzles surrounding the CDW order in monolayer NbSe2 . First, the modulation of electronic properties by CDW remains unclear. Scanning tunnelling spectroscopy measurements 11 on monolayer NbSe2 revealed a small CDW gap of 4 meV at the Fermi level, while subsequent theoretical calculations 18 based on a bulk CDW structural model proposed by Malliakas and Kanatzidis 19 did not reproduce such a gap feature. Second, it is debated whether the CDW in monolayer NbSe2 is enhanced compared to the bulk case. Xi et al. 10 observed the CDW transition temperature TCDW increases from 33 K in the bulk to 145 K in the exfoliated monolayer based on the Raman signal of CDW. On the other hand, Ugeda et al. 11 reported weakened CDW ordering with TCDW ∼ 25 K in the monolayer sample grown on graphene through STM studies. The third unresolved question concerns the interplay between the CDW and superconductivity in the 2D limit. Competitive interaction between these two collective orders is generally believed for the bulk, 20–24 although evidence to the contrary has also been discussed. 25 With the reduction of layer thickness, it is found that the superconducting transition temperature Tc monotonously decreases, 26,27 raising the issue of how this competition evolves by moving from bulk to monolayer. It should be stressed that for resolving the above puzzles, the first important step is to elucidate the CDW structural details (including the precise atomic distortions) which are still not clearly known despite a 3 × 3 CDW periodicity that has been established by STM. 11,28,29 We notice that a recent theoretical study 30 proposed several distorted CDW models of monolayer

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NbSe2 and found that the formation of CDW suppresses the magnetic instability in the undistorted lattice. 31 So far, however, the ground state CDW structure and its related physical properties, including superconductivity, are still less understood. In this Letter, based on first-principles calculations, we investigate the distorted atomic structure and lattice dynamics of the CDW state and the effects of CDW-induced electronic modulations on superconductivity in 2D NbSe2 . We determine the precise 3 × 3 CDW structure of monolayer NbSe2 with triangular clustering of Nb atoms for which the calculated STM image, Raman CDW modes and superconducting transition temperature Tc are in good agreement with experiments. This CDW phase is characterized by a wave-vector dependent gapping of the Fermi surface which results in a quasi-energy-gap at the Fermi level. By further studying the multilayer and bulk CDW phases, we find that as the layer thickness decreases the CDW energy gain increases and superconductivity is weakened because of the strengthened Fermi surface gapping in the enhanced CDW state. Our results thus constitute theoretical evidence for the competitive interaction between CDW and superconducting orders in 2D NbSe2 . Methods. Density functional theory (DFT) calculations were performed using the Vienna ab initio simulation package 32 with the projector augmented wave method. 33 We adopted the generalized gradient approximation of Perdew-Burke-Ernzerhof parametrization 34 for the exchangecorrelation functional. The plane-wave energy cutoff was set as 350 eV. For the slab model of 2D NbSe2 , a 15 Å thick vacuum layer was introduced and the vdW interaction was treated using the DFT-D3 method proposed by Grimme. 35 The geometries were optimized until the remanent atomic forces are less than 0.001 eV/Å. Lattice dynamics and electron-phonon coupling (EPC) were calculated within the density functional perturbation theory 36 as implemented in the Quantum ESPRESSO code, 37 where norm-conserving pseudopotentials 38 with a plane-wave energy cutoff of 100 Ry and a Methfessel-Paxton smearing 39 of 0.01 Ry were used. For the normal phase of monolayer NbSe2 , the phonon spectrum was obtained by Fourier interpolation of the dynamical matrices computed using a 24 × 24 × 1 k-point mesh and a 12 × 12 × 1 q-point mesh. For the monolayer 3 × 3 CDW phase, the electronic and vibrational Brillouin zones (BZ) were sampled

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(a)

(c) 1.5

Top view

Nb 4dz2 Se 4p Nb 4dx2 -y2 , dxy

a

Se

b

Nb

(b)

Energy (eV)

1.0 0.5 0.0 -0.5

Side view

-1.0

(e) (d)

Γ

M

K

Γ

300 200

-1

ω (cm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

M

Γ

100 0

K

-100

Nb-xy Se

Γ

M

K

Γ

Figure 1: (Color online) (a) Top and (b) side views of the crystal structure of monolayer NbSe2 . (c,d) Electronic band structure and Fermi surface. The blue, red and green dots represent Nb 4dz2 , Nb 4dx2 −y2 and 4dxy , and Se 4p orbitals, respectively. (e) Phonon spectrum weighted by the Nb in-plane modes and Se vibration modes. using 12 × 12 × 1 and 3 × 3 × 1 meshes in the calculation of EPC matrix elements. Low-energy candidate CDW structures. Monolayer NbSe2 , formed by a layer of Nb atoms sandwiched between two layers of Se atoms, exhibits a hexagonal honeycomb lattice with Nb and Se atoms occupying the two sublattices (see Figures 1a and 1b). We first analyze the electron and phonon spectrum of the 1 × 1 normal phase. As shown in Figures 1c and 1d, the Fermi surface of monolayer NbSe2 consists of a cylindrical hole pocket at Γ which has predominantly Nb dz2 character and triangular hole pockets at K which are mainly formed by Nb dx2 −y2 and dxy states. The calculated phonon spectrum is displayed in Figure 1e, clearly indicating the dynamical instability of the normal phase. The most unstable phonon mode involving mainly Nb in-plane vibrations appears around the wave vector qCDW = 32 ΓM (see Supporting Information Figure S1 for the phonons along ΓM calculated using a point-by-point method), consistent with the experimentally observed

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3 × 3 CDW order in monolayer NbSe2 . 11 Next we determine the equilibrium atomic structure of the CDW state. We constructed different 3 × 3 superstructures with atomic distortions suggested by the soft mode shown in Figure 1e or other randomized distortions. After total energy minimization, we found all these initial structures converge to one of the two distorted structures (the left or the right one) depicted in the insets of Figure 2a. The resulting distortions are associated with the change of the Nb-Nb distances and generate triangular Nb trimers and triangular six-Nb-atom clusters. Since all the 3- and 6-atom Nb clusters centre on the Se atoms in the left structure in Figure 2a while the Nb clusters in the right structure centre on the hollow site of the honeycomb lattice, hereafter these two distorted structures are called 3+6-filled CDW and 3+6-hollow CDW, respectively. We have calculated CDW formation energy ∆E defined as ECDW − Enor , where ECDW and Enor are the total energies of the CDW phase and the normal phase. The energy gain with respect to the normal phase is found to be 28.5 meV for 3+6-filled CDW structure and 33.0 meV for 3+6-hollow CDW structure. The large energy gain suggests the ease of the formation of the two distorted structures. In addition, we also consider other CDW periodicities using 3×1, 4×1 and 4×4 supercells and find that the corresponding distorted structures are all less stable than the 3+6-hollow CDW structure (see Supporting Information Table S1), which further verifies the 3×3 CDW order found experimentally. To reveal the stabilization mechanism of 3+6-filled and 3+6-hollow CDW structures, in Figure 2a we plot the corresponding formation energy ∆E as a function of atomic displacement δ Nb. Here, starting from an undistorted 3 × 3 structure, δ Nb is introduced by shifting the Nb trimer toward a Se atom (corresponding to negative δ Nb) or toward a hollow site (corresponding to positive δ Nb). At a given δ Nb, the total energy of the distorted structure is calculated by fixing the Nb trimer motif but fully relaxing the remaining Nb and Se atoms. Obviously, two energy minima are found at δ Nb = −0.05 Å and δ Nb = 0.08 Å, corresponding to the obtained 3+6-filled and 3+6-hollow CDW structures, respectively. The large magnitudes of the Nb atomic distortions suggest that electron-phonon coupling plays a dominant role in causing the structural and electronic instabilities. 40,41

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3+6-filled CDW

3+6-hollow CDW

60

∆E (meV)

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Nano Letters

3

40

1

2

2

1

3

20 0 CDW in Ref. 19

-20 -40 -0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

δNb (A)

(b)

(c)

Exp. STM

3+6-hollow STM

Figure 2: (Color online) (a) Calculated formation energy ∆E as a function of the Nb trimer displacement δ Nb for monolayer NbSe2 in a 3 × 3 supercell. The insets show the crystal structures of 3+6-filled CDW (left) and 3+6-hollow CDW (right) phases, with the numbers 1-3 indicating three inequivalent Nb atoms. The equilibrium Nb1-Nb1, Nb2-Nb2 and Nb2-Nb3 distances are 3.433, 3.391 and 3.415 Å for 3+6-filled CDW phase and 3.346, 3.389 and 3.424 Å for 3+6-hollow CDW phase. For comparison, the energy gain for the monolayer CDW phase based on a bulk NbSe2 CDW model in Ref. 19 relative to the normal phase is also shown (green solid line). (b) Experimental STM image of monolayer NbSe2 CDW phase. 11 (c) Simulated STM image of 3+6-hollow CDW phase at Vs = −4 mV and an iso-density value of 5.1 × 10−4 e/Å2 .

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Table 1: Calculated frequencies of Raman modes related to the CDW distortion and superconducting properties (λ , ωlog and Tc ) for the 3+6-filled and 3+6-hollow CDW structures, compared to available experimental data. 10,16,17 3+6-filled

3+6-hollow

ω (cm−1 )

Disp.

Sym.

59.5

Nb-xy

A1

A1

181.5

Nb-xy

λ

ωlog (K) 126.6

Sym. ′

A1 ′

1.09

Exp.

ω (cm−1 )

Disp.

ω (cm−1 )

71.9

Nb-xy

74.7 10

A1

189.7

Nb-xy

190.0 10

Tc (K)

λ

ωlog (K)

Tc (K)

Tc (K)

7.6

0.84

134.3

4.5

1.0-3.5 16,17

′

′

For completeness, we have also considered a distorted monolayer structure taken from a bulk NbSe2 CDW model proposed in a previous study. 19 Though this structure is more stable than the undistorted normal structure, it is energetically less favorable than both 3+6-filled and 3+6-hollow CDW structures (see Figure 2a). In addition, phonon calculations show that this structure is dynamically unstable with large imaginary frequencies near the Γ point (see Supporting Information Figure S2). Despite the possibility that it is stabilized by anharmonicity, 42 this structure is not a promising candidate CDW structure for monolayer NbSe2 due to its significant energetic instability with respect to the CDW structures studied in the present work. We have simulated STM images for the obtained 3+6-filled and 3+6-hollow CDW structures. The results for the energetically more favored 3+6-hollow structure (Figure 2c) agree well with the experimental STM image in the CDW state (Figure 2b) reported by Ugeda et al. 11 (The 3+6-filled structure fails to match with experiment, see Supporting Information Figure S3). Here, the highest three Se atoms located over the 6-atom Nb cluster give rise to three largest bright spots in the STM images, while the remaining six Se atoms having lower heights are connected with smaller spots with low intensities. Combining with above STM results, further analysis of the Raman active phonon frequencies in each distorted phase allows us to confirm the 3+6-hollow CDW structure to be the ground state CDW phase of monolayer NbSe2 observed experimentally. Two CDW-induced Raman modes were reported in a recent study by Xi et al.: 10 an amplitude mode around 74.7 cm−1 , correspond8


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(a)

0.0

300

0.2

0.4

0.0

0.5

Nb

1.0

l(w)

Se

)

200

1-

mc(

w 100

3+6-filled 0

G (b)

M

K

G 0.0

300

a

PHDOS 0.2

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F(w)

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Se

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w

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Nano Letters

100

3+6-hollow 0

G

M

(c)

G

PHDOS

a

2

F(w)

(d)

Raman A' (71.9 cm

K

Raman A'

1

-1

)

1

-1

(189.7 cm

)

Figure 3: (Color online) (a,b) Phonon spectra, Nb and Se atom-projected phonon density of states (PHDOS), Eliashberg spectral function α 2 F(ω ) and the frequency-dependent integrated EPC λ (ω ) for the 3+6-filled and 3+6-hollow CDW structures. (c,d) Atomic vibration patterns (blue ar′ rows) for the two A1 Raman modes of 3+6-hollow CDW phase at 71.9 and 189.7 cm−1 .

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ing to the amplitude variation of the collective lattice distortion, and a high-frequency weak mode located around 190 cm−1 . Figures 3a and 3b show the calculated phonon spectra for 3+6-filled and 3+6-hollow CDW structures, where the absence of negative frequency modes indicates the dynamical stabilities of the two distorted structures. For the 3+6-hollow CDW structure, a Raman ′

amplitude mode at 71.9 cm−1 with A1 symmetry is identified (see Table 1). Its vibration pattern is displayed in Figure 3c, reflecting mainly Nb in-plane displacements and the contraction or expansion of the characteristic 3- and 6-atom Nb clusters, both of which are responsible for the CDW ′

formation. Moreover, we also find another Raman A1 mode involving in-plane vibrations of Nb atoms (see Figure 3d) to emerge at 189.7 cm−1 , in good agreement with the experimentally observed high-frequency CDW mode around 190 cm−1 . However, the corresponding Raman CDW modes for the 3+6-filled CDW structure are located at 59.5 and 181.5 cm−1 , respectively (see Table 1), which (particularly the former) significantly deviate from the experimental results. 10 Superconductivity in CDW state. The lack of knowledge on the CDW structure of monolayer NbSe2 prevented a calculation of the superconducting transition temperature Tc . With the newly obtained low-energy distorted structures at hand, we can now address its superconducting properties in the framework of a phonon-mediated paring mechanism. The total electron-phonon coupling λ is calculated to be 0.84 (1.09) for the 3+6-hollow (3+6-filled) CDW structure, yielding Tc = 4.5 (7.6) K based on the logarithmically averaged phonon frequency ωlog = 134.3 (126.6) K and an effective screened Coulomb parameter µ ∗ = 0.16 43 (see Supporting Information Table S2 for the convergence of superconducting properties) using the Allen-Dynes modified McMillan equation 44,45

Tc

[ ] ωlog −1.04(1 + λ ) = exp . 1.2 λ − µ ∗ (1 + 0.62λ )

(1)

Clearly, the result for 3+6-hollow CDW structure is in better agreement with the experimental data ranging from 1.0 K to 3.5 K for samples prepared using different methods. 10–13,16,17 The calculated

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Eliashberg function α 2 F(ω ) and the frequency-dependent EPC

λ (ω ) = 2

∫ ω 2 α F(ω )

ω

0

dω

(2)

are shown in right panels of Figures 3a and 3b. In comparison with the atom-projected phonon density of states (DOS) in the middle panels of Figures 3a and 3b, one can see that for both distorted structures, α 2 F(ω ) is similar in shape to the Nb phonon DOS below the frequency of 150 cm−1 , and in the higher-frequency region it is similar to the phonon DOS of Se. This indicates that both Nb and Se vibrations significantly contribute to superconductivity, upturning the conventional wisdom that in superconducting transition metal dichalcogenides the transition metal plays a critical role while chalcogen is accessory with negligible contributions. 3 Electronic features under CDW modulation. Figure 4a shows the comparison of electronic DOS of the normal phase and the 3+6-hollow (3+6-filled) CDW phase. A striking feature for the DOS of 3+6-hollow phase is that the Fermi level (EF ) is located well within a quasi-energygap, resulting in an obvious reduction of DOS at EF with respect to the normal phase. For the 3+6-filled phase, however, its EF is found very close to a peak, giving rise to a relatively small decrease of DOS at EF . Here, the stronger modulation of DOS in the 3+6-hollow structure is closely associated with its larger atomic distortions, leading to a smaller Fermi level DOS available for the superconductivity and thus a lower predicted Tc than that in the 3+6-filled structure (see Table 1). From Figure 4b we see that for the 3+6-hollow phase the prominent change in the band structure due to CDW is the splitting of bands and the downshift of certain splitted bands around EF . Accompanied with such an electronic rearrangement, the DOS peak at −0.12 eV in the normal phase is shifted to −0.17 eV in the CDW phase (see Figure 4a). Because of the large weight of this peak in the DOS spectrum, the energy reduction of corresponding filled states strongly stabilizes the system. To better reveal the electronic features in the CDW state, we have unfolded the supercell band

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Nano Letters

(b) 1.0 Normal 3+6-filled

6

3+6-hollow 0.5

3 0

Normal 3+6-hollow

6

Energy (eV)

Total DOS (states/eV/Nb)

(a)

0.0 -0.5

3 0

-0.3

0.0

0.3

-1.0

0.6

Energy (eV)

Γ

M 3

(c)

Γ

K 3 BZ

(d) 1.0 2

Normal 3+6-hollow

Nb 4dz2

3+6-hollow 0.5

1 0 2

Nb 4dx2 -y2 , dxy

Energy (eV)

Projected DOS (states/eV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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-0.5

1 0

0.0

-0.3

0.0

0.3

-1.0

0.6

Energy (eV)

Γ

M 1

K

Γ

1 BZ

Figure 4: (Color online) (a) Electronic DOS of 3+6-filled (top) and 3+6-hollow (bottom) CDW phases. The dashed lines indicate the Fermi level. (b) Band structure of 3+6-hollow CDW phase compared to that of the normal phase in the 3 × 3 BZ. (c) The projected DOS of 3+6-hollow CDW phase for the Nb 4dz2 (top) and Nb 4dx2 −y2 and 4dxy (bottom) states. (d) The unfolded band structure in the 1 × 1 BZ for the 3+6-hollow CDW phase. The size of red dots is proportional to the spectral weight of the energy bands. structure of the 3+6-hollow CDW phase into the 1 × 1 BZ using the method proposed by Tomi´c et al. 46 The obtained results are shown in Figure 4d. Compared with the normal phase (Figure 1c), after the CDW transition the original band around EF remains essentially unaffected along the ΓM high-symmetry line but changes seriously with gap opening in the region around the K point. Such anisotropic Fermi surface gapping is closely related to the momentum-dependent electronic orbital features. As seen in Figure 1c, the electronic states along ΓM mainly consist of Nb dz2 orbitals, while those around K are dominated by Nb dx2 −y2 and dxy orbitals. The in-plane dx2 −y2 and dxy orbitals are well directed to form interactions between neighboring Nb atoms. By contrast, the dz2 orbitals pointing to the out-of-plane direction hardly generate effective Nb-Nb interaction.

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(a)

(b)

∆E (meV/layer)

CDW formation energy -24

-28

-32

1

2

3

Bulk CDW

Bulk

Layer number Normal CDW

6 Bilayer 3 0 6 Trilayer 3 0 6

(d) 3.0

NF (states/eV/Nb)

(c) Total DOS (states/eV/Nb)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Bulk

2.5 2.0

Normal CDW

1.5

3 0

-0.3

0.0

0.3

1.0

0.6

1

2

3

Bulk

Layer number

Energy (eV)

Figure 5: (Color online) (a) CDW formation energy ∆E as a function of layer number. (b) Bulk CDW crystal structure composed of 3+6-hollow (upper) and 3+6-filled (lower) monolayer CDW phases. (c) Electronic DOS of bilayer, trilayer and bulk CDW phases in comparison with the normal phase. The dashed lines indicate the Fermi level. (d) NF as a function of layer number for both the normal and CDW phases. Consequently, the former are easily affected by the in-plane distortion induced by CDW but not the latter. This is further reflected by the orbital-resolved DOS of 3+6-hollow phase (Figure 4c) where at EF the dx2 −y2 and dxy states display stronger depletion than dz2 across the CDW transition. Enhanced CDW and weakened superconductivity in 2D NbSe2 . To understand the evolution of the CDW order with layer thickness, we obtain the CDW structures for NbSe2 films with different thickness by performing structural relaxations starting from random distortions in 3×3 supercells and then calculate the CDW formation energy as a function of layer number (see Figure 5a). It is found that the monolayer 3+6-filled and 3+6-hollow CDW structures are two basic building blocks for the obtained CDW structures of multilayer NbSe2 and the bulk CDW structure is formed by alternating stacking of the two monolayer structures (see Figure 5b). With decreas-

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ing layer number, the absolute value of the formation energy |∆E| increases notably, indicating stronger CDW instabilities, in good agreement with Xi et al.’s Raman studies showing an enhancement of TCDW as the layer thickness of NbSe2 decreases. 10 In principle, |∆E| can be obtained as

δ Eel − δ Elattice , 41 where δ Eel and δ Elattice are the electronic energy gain and the lattice elastic energy cost due to the CDW distortion, respectively. As seen in Figure 5c, after the CDW formation, the electronic DOS of the bilayer NbSe2 shows a redshift of the peak around EF from −0.12 eV to −0.17 eV, similar to the monolayer CDW case (Figure 4a), suggesting nearly unchanged energy gain from monolayer to bilayer. However, in the bilayer (or multilayer) with additional interlayer Se-Se coupling, the distortion in each layer has to cost more elastic energy because of the restriction from the other layers via interlayer coupling. Evidently, from the monolayer to bilayer, the rise of δ Elattice will result in the decrease of |∆E|. Similar reason is expected for the gradually decreasing |∆E| and weakened CDW in multilayer systems. Given the observed opposite thickness dependencies of TCDW and Tc , 10 CDW and superconductivity are usually considered as two competing phenomena. However, the nature of their competitive interaction remains elusive. Figure 5d shows the change of Fermi-level DOS NF across the CDW transition as a function of layer number of NbSe2 , which provides key clues to the understanding of the interplay between the two collective orders. In the bulk NbSe2 , CDW only introduces a minor decrease of NF , implying its good coexistence and weak competition with superconductivity. However, as the layer thickness decreases, accompanied with the strongly enhanced CDW order, the reduction of NF induced by CDW becomes more and more notable (see Figure 5d), consequently, the number of charge carriers available for superconductivity gradually declines, which could explain the decrease of the experimental Tc with decreasing layer thickness. 13,27 It is clearly seen that NF serves as a vehicle for the competitive interaction between CDW and superconductivity in 2D NbSe2 : stronger CDW leads to lower NF , which further weakens superconductivity. We note that the experimental Tc shows a much more marked decrease from bilayer to monolayer, which is expected to arise from not only the change of NF but also other factors such as the enhancement of fluctuation effects and the reduced screening of the Coulomb

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interaction in the limit of monolayer thickness. Conclusions. Using first-principles calculations, we have studied the microscopic formation mechanism of CDW and its interplay with superconductivity in 2D NbSe2 . Two dynamically stable monolayer 3 × 3 CDW structures of NbSe2 (i.e., 3+6-filled and 3+6-hollow) are identified, featuring 3- and 6-atom Nb clusters centering on the Se atoms and the lattice hollow sites, respectively. The energetically more favored 3+6-hollow phase has a larger atomic distortion, which results in stronger electronic modulation (smaller NF ) and thus a lower EPC and Tc . Moreover, its STM pattern and Raman CDW modes agree well with experiments, suggesting it being the CDW ground state of monolayer NbSe2 . We also determine the multilayer and bulk CDW phases and confirm that, with decreasing layer thickness, the CDW order is enhanced with rising energy gain and competes with the superconducting order as the strength of CDW anticorrelates with the size of NF available for superconductivity. Such competitive interaction of CDW with superconductivity should be responsible for the consecutive suppression of Tc in atomically thin NbSe2 observed experimentally, which provides useful implications for clarifying the debate over mutual relation between the two collective orders in NbSe2 .

AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected]. Phone: +86-010-8231-3923. *E-mail: [email protected]. Phone: +86-010-6278-5577. Notes The authors declare no competing financial interest.

Acknowledgement This work was supported by the Basic Science Center Project of NSFC (Grant No. 51788104), the Ministry of Science and Technology of China (Grant No. 2016YFA0301001) and the National

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Natural Science Foundation of China (Grant Nos. 11674188, 11334006 and 11504015).

Supporting Information Available Phonons along ΓM of monolayer NbSe2 in the normal phase calculated using a point-by-point method, total energies of the distorted monolayer NbSe2 with other CDW periodicities, the phonon spectrum of the monolayer CDW structure based on a bulk NbSe2 CDW model in Ref. 19, the STM image of 3+6-filled CDW structure and the convergence of superconducting properties. This material is available free of charge via the Internet at http://pubs.acs.org/.

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