Interplay of the charge density wave transition with topological and superconducting properties

Abstract

Exotic phenomena due to the interplay of different quantum orders have been observed and the study of these phenomena has emerged as a new frontier in condensed matter research, especially in the two-dimensional limit. Here, we report the coexistence of charge density waves (CDWs), superconductivity, and nontrivial topology in monolayer 1H-MSe₂ (M = Nb, Ta) triggered by momentum-dependent electron–phonon coupling through electron doping. At a critical electron doping concentration, new 2 × 2 CDW phases emerge with nontrivial topology, Dirac cones, and van Hove singularities. Interestingly, these 2 × 2 CDW phases are also superconducting. Our findings not only reveal a route towards realizing nontrivial electronic characters by CDW engineering, but also provide an exciting platform to modulate different quantum states at the confluence of CDWs, superconductivity, nontrivial topology, and electron–phonon coupling.

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New concepts

The combination of topological physics and electron–phonon coupling (EPC) in quantum materials has emerged as a new frontier of condensed matter research. Here, we report that electron-doped 1H-NbSe₂ and 1H-TaSe₂ are charge density waves (CDWs) with nontrivial topology and superconductivity. In addition, EPC is demonstrated to play a central role in both the formation of these new CDWs and the change of the superconducting critical temperatures. The EPC-driven topological phase transitions in this work provide an avenue to understand the interplay between many-body instabilities and topology, which can inspire the finding of new topologically nontrivial superconductors in the future.

1. Introduction

Many-body instabilities and topological physics are two attractive topics in condensed matter research.^1–3 It is interesting to explore the interplay between these phenomena in a single quantum material,^4–6 which leads to many exotic properties. For example, the Weyl nodes in (TaSe₄)₂I are opened by charge density wave (CDW) transition below 263 K, resulting in an axion insulator.^4,7 On the other hand, pressure-driven superconductivity in Weyl semimetal WTe₂ and MoTe₂ allows their topological edge states to be superconducting, which can be useful for quantum computation.^5,8 Furthermore, the coexistence of CDWs, superconductivity, and nontrivial topology has been reported in the recently discovered Kagome metals AV₃Sb₅ (A = K, Rb, and Cs).^9,10 The electronic bands near the Fermi energy of AV₃Sb₅ are fascinating and feature both Dirac cones and van Hove singularities (VHSs).^11–13 Such novel electronic structures are characterized by odd Z₂ indices,¹⁰ which can stimulate the anomalous Hall effect in a frustrated magnetic sublattice.¹⁴ Besides, the in-plane 2 × 2 CDW and superconducting orders occur below the respective critical temperatures.^10,15 Their formation mechanisms and mutual interplay have generated tremendous attention.^16–18 However, the coexistence of these quantum orders in AV₃Sb₅ stems from the unique Kagome lattice, which hinders the further exploration of this frontier. Therefore, it is urgently desired to realize rich electronic characters in materials with a trivial electronic band.

CDW engineering opens a new era for manipulating electronic characters. CDWs are a symmetry-breaking phenomenon below the transition temperature T_CDW.¹⁹ The band folds in its reduced Brillouin zone (BZ) provide an opportunity to form Dirac cones. Besides, the subtle lattice distortion of a CDW phase can further modify the electronic structure.²⁰ More importantly, CDW distortion is very sensitive to external stimuli, which provides great flexibility for modulating the electronic and phononic properties by CDW transition.^21,22 Because of the large specific surface area, CDWs in two-dimensional (2D) materials are particularly sensitive to external stimuli and can be easily manipulated. New exotic phenomena induced by external stimuli triggered CDW transition await to be explored.

Monolayer 1H-NbSe₂ and 1H-TaSe₂ are hot 2D transition metal dichalcogenides (TMDs) that draw a great deal of attention due to the many-body instabilities.^23–32 NbSe₂ and TaSe₂ in the following refer to the monolayer phase unless stated otherwise. NbSe₂ undergoes a CDW transition with 3 × 3 periodic distortion at T_CDW ∼ 145 K as well as a superconducting transition at T_C ∼ 3 K.²⁸ Similar 3 × 3 CDW and superconductivity also exist in TaSe₂ with T_CDW ∼ 130 K³² and T_C ∼ 2.2 K,³³ respectively. Recent experiments indicate that NbSe₂ becomes a 2 × 2 CDW under electron doping,³⁴ while such transition has not been reported for TaSe₂ yet. More importantly, the mechanism for this CDW transition and the electronic properties of the 2 × 2 CDW structure are still elusive.

In this work, based on first-principles calculations and mean-field theory, we identify 1H-MSe₂ (M = Nb, Ta) as an ideal platform to investigate CDWs, superconductivity, and topology, which are all sensitive to carrier concentration. Momentum-dependent electron–phonon coupling (q-EPC) is found to account for the phase transition from 3 × 3 to 2 × 2 CDW under electron doping. We determine two possible 2 × 2 CDW structures, which may exist when the doping concentration reaches 0.2 electrons per formula unit (e⁻/f.u.). Besides, our study explores the effects of electron doping on the CDW gaps and density of states (DOS) at the Fermi surface. In addition, electron doping produces opposite effects on superconductivity in NbSe₂ and TaSe₂. The superconducting transition temperature decreased in NbSe₂ while increased in TaSe₂. Our investigation suggests that the variation of the total electron–phonon coupling (EPC) strength under doping is responsible for this opposite trend in superconductivity. Moreover, we find in both NbSe₂ and TaSe₂, that one of their two 2 × 2 CDW phases is topologically nontrivial with Dirac cones and van Hove singularities. These results show that monolayer NbSe₂ and TaSe₂ can be an EPC-driven CDW with nontrivial topology and superconductivity.

2. Results and discussion

2.1. A CDW transition under electron doping

The structure of normal monolayer 1H-MSe₂ (M = Nb, Ta) is shown in Fig. 1a, which has a P6̄m2 space group. The metal atoms are sandwiched between two Se layers with the trigonal prismatic cells. The phonon dispersion of normal MSe₂ (M = Nb, Ta) displays imaginary frequencies around 2/3ΓM (Fig. 1d), suggesting the formation of a 3 × 3 CDW. And it has been argued that there are two possible 3 × 3 CDW structures (Fig. S2g and h, ESI†), which are coined as 3 × 3 hollow and 3 × 3 filled CDW structures with cation clusters centered on the hollow sites and Se atoms of the honeycomb lattice, respectively.^20,35,36 With the increase of electron doping concentration, the position of imaginary phonons gradually evolves from 2/3ΓM to the M point (Fig. 1d), which implies transition from 3 × 3 to 2 × 2 CDW. Similarly, we found two possible 2 × 2 CDW structures when the doping concentrations are above 0.2 e⁻/f.u., and we denote them as 2 × 2 filled (Fig. 1b) and 2 × 2 hollow (Fig. 1c) CDW structures according to the location of the centers of the cation clusters.

Fig. 1 (a) Schematic diagram of normal MSe₂ (M = Nb, Ta) monolayer in the top and side views. (b) 2 × 2 filled and (c) 2 × 2 hollow CDW structures. (d) The phonon spectra of NbSe₂ (up) and TaSe₂ (down) under different electron doping concentrations. Since the optical phonon modes hardly change, we only show acoustic phonon modes for the systems. (e) The formation energies for the 3 × 3 filled, 3 × 3 hollow, 2 × 2 filled, and 2 × 2 hollow CDW structures as a function of electron doping concentration. The high-symmetry points are indicated in Fig. S1 (ESI†).

To further verify CDW transitions under electron doping, the energetic hierarchy of these CDW structures is calculated at various doping concentrations. The CDW formation energies ΔE are calculated by E_CDW/x − E₀, where E_CDW and E₀ are the total energies of the CDW and the 1 × 1 normal phases, respectively, and x is the number of formula units in the CDW phase. Within the doping concentration range studied in this work, Fig. 1e clearly shows transitions from 3 × 3 hollow CDWs to 2 × 2 filled CDWs in both NbSe₂ and TaSe₂ at a critical doping concentration of around 0.2 e⁻/f.u., in agreement with phonon calculations (Fig. 1d).

2.2. CDW transition mechanism

Although there are some preliminary results for the understanding of the CDW formation in bulk 2H-MSe₂, a systematic and quantitative analysis for 2D counterparts is still missing. This is particularly important given the potential changes that may occur in the 2D limit, where the reduced dimensionality typically enhances the nesting condition, and the electron doping suppresses intraband screening to weaken the coupling between electrons and phonons. As such, it is crucial to investigate the formation mechanism for monolayer 2 × 2 CDWs under electron doping.

It is known that the softened phonon frequency ω_q can be obtained by:^37,38

ω_q² = Ω_q² − 2Ω_qΠ_q, (1)

where Ω_q is the bare phonon frequency, and Π_q is the real part of the phonon self-energy, the latter of which carries the information from both EPC and Fermi surface nesting (FSN). In the static limit, Π_q has the form:Here, f(ε_k) is the Fermi–Dirac function with energy ε at the k point, and g_k,k+q is the EPC matrix element, which can be calculated via:³⁹where φ_k is the electronic wavefunction with wavevector k, and V is the Kohn–Sham potential. The imaginary phonon frequency, suggesting a CDW formation, appears when Π_q is larger than (1/2)Ω_q. The peak location of Π_q corresponds to the dip position of ω_q, which can be converted to the CDW vector in the reciprocal space, and the CDW lattice in the real space.²¹ Due to the low energy physics studied in this work, the coupling is considered between the softened phonon branch and the single electronic band which crosses the Fermi level (Fig. S2a and d, ESI†). Taking |g_k,k+q|² ∼ |g_q|², which is known as q-EPC and can be extracted from Π_q to represent the EPC contribution,²⁶ the remaining part is:where χ_q is the real part of the bare electronic susceptibility, which captures the Fermi surface topology as the FSN effect.⁴⁰

Fig. 2a and e display the distribution of the softened phonon frequency ω_q in the first BZs of NbSe₂ and TaSe₂ under 0.3 e⁻/f.u. doping concentration, respectively. The distribution of imaginary phonon frequencies (see the dark blue areas in Fig. 2a and e) clearly shows a valley shape centered at the M points. Meanwhile, complementary patterns can be found for the peak of the real part of the phonon self-energy Π_q (see the yellow areas in Fig. 2b and f). Furthermore, we decompose the contribution of the real part of the phonon self-energy Π_q into the pure effect of EPC and FSN by calculating |g_q|² and χ_q separately. For doped cases, it is interesting to note that |g_q|² and Π_q have very similar patterns in the first BZs, both of which have maxima at M (Fig. 2b, c, f and g). However, their χ_q display peaks around 1/2ΓM (Fig. 2d and h), which deviates from the dips of ω_q in Fig. 2a and e and the peaks of Π_q in Fig. 2b and f. Thus, |g_q|² provides the main contribution to Π_q and determines the pattern of phonon softening, confirming q-EPC as the driving force for the doped 2 × 2 CDW formation. Furthermore, combining the pristine CDW formation analyses in SI–III (ESI†), we conclude that q-EPC is responsible for the CDW transition from 3 × 3 to 2 × 2 CDW under electron doping.

Fig. 2 (a) The softened phonon frequency ω_q, (b) the real part of the phonon self-energy Π_q, (c) q-EPC |g_q|² and (d) the real part of the bare electronic susceptibility χ_q in the first BZ of NbSe₂ under 0.3 e⁻/f.u. doping concentration. (e)–(h) Same as panels (a)–(d) but for TaSe₂ under 0.3 e⁻/f.u. doping concentration.

2.3. Electronic structures near the Fermi surface

The Fermi surface of the pristine NbSe₂ 3 × 3 hollow CDW is shown in Fig. 3a, which displays full CDW gaps in the K pockets. Such electronic gap opening also decreases the DOS at the Fermi level N_F (Fig. 3c). When NbSe₂ is under electron doping, the shrinkage of the K pocket is accompanied by a transition from a full to partial gap opening. Meanwhile, the electronic states at the Fermi level increase, as shown in Fig. 3c. The pristine 3 × 3 hollow CDW of TaSe₂ displays a similar Fermi surface to NbSe₂. However, the spectral intensity of the K pocket becomes more blurred (Fig. 3d), corresponding to the greater reduction in N_F (Fig. 3f). After the electron doping, the K pocket becomes more pronounced (Fig. 3e), which is also reflected in the enhancement of N_F (Fig. 3f).

Fig. 3 (a) The simulated Fermi surface of pristine NbSe₂ in the ground-state 3 × 3 CDW phase. (b) The simulated Fermi surface of NbSe₂ in the ground-state 2 × 2 CDW phase under 0.3 e⁻/f.u. doping concentration. (c) Electronic DOS of the normal, 3 × 3 hollow CDW for pristine NbSe₂ and 2 × 2 filled CDW under 0.3 e⁻/f.u. doping concentration. (d)–(f) Same as panels (a)–(d) but for TaSe₂. The Fermi surfaces of the normal phases of NbSe₂ and TaSe₂ are shown in Fig. S2b, c and e, f (ESI†), respectively.

2.4. Superconductivity under electron doping

The superconductivity of the NbSe₂ CDW is further explored. According to the Allen–Dynes modified McMillan theory,³⁹ the Eliashberg spectral function is obtained by:where λ_qν is the EPC strength of phonon mode ν with momentum q, and ω_qν is the phonon frequency. The total EPC strength is given by:The superconducting transition temperature T_C can be calculated by:Here, ω_log is the logarithmically averaged phonon frequency, and μ* is the effective Coulomb parameter which is estimated from the DOS at the Fermi energy: μ* ≈ 0.26N_F/(1 + N_F).^6,41

Fig. 4a and b show the calculated phonon spectra of 2 × 2 filled CDWs for NbSe₂ and TaSe₂, respectively, under 0.3 e⁻/f.u. doping concentration. The absence of an imaginary phonon frequency indicates the stability of these two structures. Fig. 4a shows that the Se vibrations concentrate on the middle-frequency region (i.e., 17–22 meV), while the Nb and Se atoms have similar contributions to the phonon DOS (PHDOS) outside this region. It is noted that the phonons between 8 meV and 15 meV have strong EPC strength λ_qν, and these phonons contribute 67% of the total λ. Combined with the high-frequency region contributions, the total λ of the doped NbSe₂ 2 × 2 filled CDW is calculated to be 0.54 (Fig. 4a). Using μ* = 0.17 estimated from N_F, we obtain T_C = 0.9 K for the doped NbSe₂ CDW. The phonon spectrum of the doped TaSe₂ CDW is less dispersive than that of the doped NbSe₂ CDW with the highest phonon frequency changed from 37 meV for NbSe₂ to 32 meV for TaSe₂ (Fig. 4a and b), due to the heavier mass of Ta atoms.⁴² In addition, Ta vibrations mainly provide the low-frequency PHDOS between 6 meV and 15 meV, further contributing to 75% of the total λ with the help of strong EPC strength λ_qν in this region (Fig. 4b). The high-frequency region is dominated by Se vibrations, which contribute 25% to the total λ. Thus, the total λ and T_C of the doped TaSe₂ 2 × 2 filled CDW is calculated to be 0.82 and 3.6 K, respectively. Such T_C is close to the intrinsic superconducting transition temperature in other VB group 2D TMDs, including 1H-NbS₂ and 1H-TaS₂, both of which have T_C around 3 K.^43,44

Fig. 4 (a) The phonon spectrum, PHDOS, Eliashberg spectral function α²F(ω) and frequency-dependent integrated EPC λ(ω) of the 2 × 2 filled CDW under 0.3 e⁻/f.u. doping concentration for NbSe₂. The phonon spectrum highlighted in red indicates the strong EPC strength λ_qν. (b) Same as panel (a) but for TaSe₂. Comparison of (c) the superconducting transition temperature T_C, (d) the effective Coulomb parameter μ* and (e) the total EPC strength λ between the pristine 3 × 3 hollow CDW and 0.3 e⁻/f.u. doped 2 × 2 filled CDW. The calculated T_C and λ for pristine CDWs are used for comparison.^20,33

Interestingly, although CDWs in both NbSe₂ and TaSe₂ have been suppressed under electron doping, the superconductivity of NbSe₂ and TaSe₂ displays an opposite trend. The calculated T_C of NbSe₂ changes from 4.5 K to 0.9 K, while that of TaSe₂ increases from 2.2 K to 3.6 K (Fig. 4c). To gain physical insight into this difference, we start with the BCS theory. According to the BCS theory, the phonon-mediated electrons form Cooper pairs to overcome Coulomb repulsion between electrons. In particular, the Coulomb repulsion between paired electrons is represented by μ*,⁴⁵ while the electron–phonon interaction is evaluated by λ.⁴⁶

Fig. 4d shows that the effective Coulomb parameter μ* for both NbSe₂ and TaSe₂ increases under electron doping. This is reasonable as electron doping reduced the plasmon energy and enhances the effective Coulomb repulsion.^33,47 However, the effective Coulomb repulsion is detrimental to the formation of superconductivity as it retards the formation of Coulomb pairs. In addition, the increment of μ* in both NbSe₂ and TaSe₂ cannot explain the opposition in superconducting temperature transitions.

Another mechanism that must be considered is electron–phonon interaction. The total λ of NbSe₂ has been reduced 36% to 0.54 through doping, while in doped TaSe₂, the total λ increases 11% to 0.82. The trend of the total λ is consistent with the trend of the transition temperature T_C. Therefore, we propose that the EPC plays an important role in the change of superconductivity under electron doping.

Fig. 5a and d show the electronic band structure of the 2 × 2 hollow CDWs of NbSe₂ and TaSe₂ without spin–orbital coupling (SOC), where Weyl nodes are protected by the mirror symmetry on the ΓM* path. Besides, two VHSs emerge at their M points. The combination of the Dirac cone and VHSs in the vicinity of the Fermi level is strongly reminiscent of the band of the Kagome metal, which drives diverse correlated phenomena.^12,48,49 The SOC opens a band gap of about 15 meV and 59 meV (Fig. 5b and e), resulting in the nontrivial topological phases in NbSe₂ and TaSe₂. The trivial bands overlapping with topological Dirac bands allow the identification of a Z₂ topological metal, which is essential for the coexistence of topological and superconducting properties.⁵⁰ A material with both surface topological Dirac cones and bulk superconductivity may intrinsically establish the proximity effect, which has promising applications.⁵¹ According to the bulk-edge correspondence, the nonzero Z₂ = 1 invariant is closely related to the nontrivial helical edge states that emerge inside the gap between two energy bands in a semi-infinite system. As presented in Fig. 5c and f, there are Dirac cone edge states emerging within the continuous nontrivial band gap. Furthermore, the 2 × 2 hollow CDW phases for NbSe₂ and TaSe₂ are also superconductors with T_C = 1.2 K and 4.4 K, respectively (see details in the SI–IV, ESI†). Therefore, the 2 × 2 hollow CDW phases are 2D superconductors with clearly visible topological edge states near the Fermi level, which are rarely reported in 2D systems. More importantly, in marked contrast to the exotic electronic structure rooted in the peculiar Kagome lattices, the one presented here is induced by CDW engineering, which can be easily achieved in experiments.^34,52–54 This could be a general strategy and applied to other systems.

Fig. 5 The band structures of the NbSe₂ 2 × 2 hollow CDW under 0.3 e⁻ f.u.⁻¹ doping concentration (a) without and (b) with SOC. The continuous band gap is shaded. (c) The edge states of the 2 × 2 hollow CDW are projected on the (100) edge. Red colors indicate higher local DOS and blue colors represent the bulk band gap. (d)–(f) Same as panels (a)–(c) but for TaSe₂. The high symmetry points are indicated in Fig. S1 (ESI†).

3. Conclusion

In conclusion, we first demonstrate that CDW engineering can be a convenient technique to modulate topology and superconductivity, to the best of our knowledge. The formation of 2 × 2 CDWs is caused by the modulation of q-EPC under electron doping, which also weakens the CDW gap opening. In addition, the superconducting transition temperature decreases in doped NbSe₂ while increasing in doped TaSe₂. This is understood by the change of the total EPC strength λ under electron doping. Thus, we propose that EPC plays a dominant role in both CDW and superconductivity under doping. Prominently, the 2 × 2 hollow CDW phases for both NbSe₂ and TaSe₂ under 0.3 e⁻/f.u. doping concentration are superconductors with nontrivial topology, suggesting that CDW engineering based on the modulation of q-EPC is a powerful tool to introduce nontrivial electronic properties. Our work not only deepens the understanding of CDW, but also presents a guide for developing topological and superconducting properties. Future works should apply the same strategy to other CDW materials, using the high-throughput method to discover topological electrons and phonons induced by CDW engineering.

4. Method

The first-principles calculations were performed using the Quantum Espresso package^55,56 with a norm-conserving pseudopotential.⁵⁷ The vacuum layer was set to be 15 Å to minimize the interaction between adjacent layers. The plane wave energy cutoff was 80 Ry. All geometries were relaxed until residual forces and the energy difference are smaller than 4 × 10⁻⁶ Ry/bohr and 10⁻¹⁰ Ry, respectively. A Gamma centered 16 × 16 × 1 (8 × 8 × 1) and 8 × 8 × 1 (4 × 4 × 1) k-grid (q-grid) was chosen for the calculations of normal phase and 2 × 2 CDW phases, respectively. The EPW code⁵⁸ was employed for the electron–phonon coupling property with fine k and q meshes of 64 × 64 × 1. The real part of the bare electronic susceptibility was calculated at CDW transition temperature 145 K with a 111 × 111 × 1 electron-momentum grid. The electron doping was simulated by adding electrons to the system with a compensating uniform positive background. Considering that the lattice of NbSe₂ is pinned by the substrate in the experiments, the structures were re-optimized with the same lattice constant under different doping concentrations. The edge states were calculated by using the Wannier functions and the iterative Green's function method.^59–62

Author contributions

Z. W. conceived the idea. Z. W., J. Y., and C. C. performed the calculations, and wrote the manuscript under the guidance of J. Z., K. P. L., and Y. P. F. All authors contributed to the discussion and the revision of the manuscript.

Conflicts of interest

The authors declare no competing financial or non-financial interests.

Acknowledgements

C. C. is thankful for the support from Shuimu Tsinghua Scholar Program. This research project was funded by the Ministry of Education, Singapore, under its MOE AcRF Tier 3 Award MOE2018-T3-1-002, and MOE AcRF Tier 2 Award MOE2019-T2-2-030. Computations were supported by the Center of Advanced 2D Materials (CA2DM) HPC infrastructure.

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Footnote

† Electronic supplementary information (ESI) available: Spectral function in a one-band mean-field model, general properties of monolayer 1H-MSe₂, CDW formation mechanism in pristine MSe₂, and superconducting properties in 2 × 2 hollow CDWs. See DOI: https://doi.org/10.1039/d3nh00207a

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