Observation of multiple charge density wave phases in epitaxial monolayer 1T-VSe2 film

As a special order of electronic correlation induced by spatial modulation, the charge density wave (CDW) phenomena in condensed matters attract enormous research interests. Here, using scanning–tunneling microscopy in various temperatures, we discover a hidden incommensurate stripe-like CDW order besides the ( 7×3 ) CDW phase at low-temperature of 4 K in the epitaxial monolayer 1T-VSe2 film. Combining the variable-temperature angle-resolved photoemission spectroscopic (ARPES) measurements, we discover a two-step transition of an anisotropic CDW gap structure that consists of two parts Δ 1 and Δ 2. The gap part Δ 1 that closes around ∼ 150 K is accompanied with the vanish of the ( 7×3 ) CDW phase. While another momentum-dependent gap part Δ 2 can survive up to ∼ 340 K, and is suggested to the result of the incommensurate CDW phase. This two-step transition with anisotropic gap opening and the resulted evolution in ARPES spectra are corroborated by our theoretical calculation based on a phenomenological form for the self-energy containing a two-gap structure Δ 1 + Δ 2, which suggests different forming mechanisms between the ( 7×3 ) and the incommensurate CDW phases. Our findings provide significant information and deep understandings on the CDW phases in monolayer 1T-VSe2 film as a two-dimensional (2D) material.

Combining the in-situ variable-temperature (VT) ARPES and scanning-tunneling microscopic (STM) techniques, here we investigate the multiple CDW phases and their transitions with temperatures in the monolayer 1T-VSe 2 film grown on bilayer graphene (BLG) substrate by molecular beam epitaxial (MBE) method. We found that besides the already reported ( √ 7 × √ 3) CDW reconstruction, another hidden incommensurate CDW phase with period of ∼ 2a can coexist with it at ∼ 4 K in monolayer 1T-VSe 2 film. As a consequence of the interference between these two CDW phases, the coexistence of ( √ 7 × √ 3) and (2 × √ 3) reconstruction as observed before can also be found but in short range. Using VT-STM and room-temperature (RT) STM, we found that the ( √ 7 × √ 3) phase will vanish at ∼ 165 K but the incommensurate CDW phases become more complicated and can survive up to room-temperature of 300 K. Through the analysis of the CDW gap evolutions at different momentum positions from the VT-ARPES spectra, we found that the CDW gap along the Γ -M direction exhibits a monotonic temperature dependence and vanishes at ∼ 150 K, associated with the disappearance of the ( √ 7 × √ 3) reconstruction observed by STM. Along the M-K direction, the CDW gap is also reduced with temperature, but does not vanish at 150 K, instead it extends to ∼ 343 K. Combining with the theoretical calculations using a phenomenological form for the self-energy containing a two-gap structure, we show that the CDW gap exhibits highly anisotropic momentum and temperature dependence, and shows a two-step transition along the M-K direction. We suggest the distinct two parts in the two-gap structure are related with the ( √ 7 × √ 3) CDW phase and incommensurate stripe-like CDW phases, and they may process different physical mechanism behind.

Methods
The MBE growth and experimental measurements of the 1T-VSe 2 films were performed in a combined MBE-STM-ARPES ultra-high vacuum (UHV) system with a base pressure of ∼ 2 × 10 −10 mbar (1 bar = 10 5 Pa). The STM in the MBE-STM-ARPES system is a Pan-style one and performed at room-temperature. The BLG substrate was obtained by flash-annealing the 4H-SiC (0001) wafer to ∼ 1250 • C for 60 cycles. [33] The V flux was produced from an electron-beam evaporator. The high purity Se (99.9995%) was evaporated from a standard Knudsen cell. The BLG substrate was kept at 280 • C during the growth. The surface morphology was characterized by the in-situ reflection high-energy electron diffraction RHEED and RT-STM.
The in-situ x-ray photoelectron spectroscopy (XPS) and VT-ARPES measurements were performed via a shared Scienta Omicron DA30L analyzer. The monochromatic x-ray (SIGMA) was generated from an Al electrode excitation source (Al α , 1486.7 eV), and the ultraviolet (UV) light source was generated by a helium lamp (Fermi instruments) with a SPECS monochromator (He I,21.218 eV). The samples were cooled down to ∼ 7 K by a close-cycle cryogenerator during the measurements, and the sample temperature can be controlled by an in-situ inner heater in the manipulator.
The ultra-low-temperature (ULT-) STM, VT-STM, and variable-temperature low-energy electron diffraction (VT-LEED) were performed ex-situ at Nano-X, Suzhou Institute of Nano-Tech and Nano-Bionics (SINANO), China. The ULT-STM is the UNISOKU Co, USM 1300 with lowest temperature of ∼ 4 K. The VT-STM is the Scienta Omicron Co, VT-AFM XA50/500 with variable temperature operation from 30 K to 300 K. The VT-LEED is the Scienta Omicron Co, LEED 600 MCP with variable temperature operation from 80 K to 300 K.
The first-principles calculations were performed using the QUANTUM ESPRESSO package base on density functional theory (DFT). [34] The generalized gradient approximation with the Perdew-Burke-Ernzerhof functional was used to describe the electron exchange and correlation effects. [35] A plane-wave energy cutoff of 80 Ry (1 Ry = 13.6055923 eV) and a 16 × 16 × 1 k mesh was employed. Freestanding films were modeled with a 23-Å vacuum gap between adjacent layers in the supercell. The in-plane lattice parameter a was fixed to the experimentally reported value of 3.35Å. [36] Structures were fully optimized until ionic forces and energy difference are less than 10 −3 eV/Å and 10 −5 eV.

Results and discussion
3.1. The hidden incommensurate CDW phase in monolayer VSe 2 The structure of the 1T-VSe 2 unit cell is represented as a ball and stick model shown in Fig. 1(a). The triangle formed by the top layer of Se atoms is rotated by 180 • relative to the bottom Se layer. The RHEED image of a monolayer 1T-VSe 2 film grown on BLG substrate is shown in the upper panel of Fig. 1(b). The sharp RHEED patterns prove that the film was well-crystalized. In the lower panel of Fig. 1(b), the XPS spectrum shows the characterized binding energies of Se 3d 5/2 (∼ 56 eV), Se 3d 3/2 (∼ 57 eV), V 2p 3/2 (∼ 512 eV), and V 2p 1/2 (∼ 520 eV) orbitals, indicating our film is indeed consisted of V and Se atoms.  To further determine the surface morphology of the grown film, we took a 300 nm×300 nm STM image scanned at roomtemperature [ Fig. 1(c)]. The grown 1T-VSe 2 film formed large-scale flat monolayer domains with a coverage of ∼ 80%. Few bilayer 1T-VSe 2 islands were formed on the monolayer 1T-VSe 2 surface, but they will not affect our VT-ARPES and VT-STM measurements due to their rather small sizes. From the height profile line shown in Fig. 1(d), we can see that the height of second 1T-VSe 2 layer island to the monolayer domain is about ∼ 0.67 nm, which agree with the lattice constant c of the 1T-VSe 2 monolayer. [36] However, the height of the monolayer 1T-VSe 2 to the BLG substrate is obviously larger as ∼ 0.95 nm, which is due to the large interlayer spacing between the TMDCs film and BLG substrate. [29,37,38] Figure 2(a) is the atom-resolved STM image on the monolayer 1T-VSe 2 taken at ∼ 4 K. The typical CDW reconstructions that are same to previous reports can be clearly observed. [27][28][29]31] We found that these CDW reconstructions are consisted of the period of ( √ 7 × √ 3) depicted by the red dashed lines and the period of (2 × √ 3) depicted by the blue dashed lines, but both these two CDW periods are in short range. To better understand these CDW periods in monolayer 1T-VSe 2 , we took fast Fourier transform (FFT) shown in Figs. 2(d) and 2(g), which contains complicated diffraction spots depicted by the black/blue/red circles in Fig. 2(g). In previous reports, different interpretations on these spots were suggested. [27,29,31] Here we found that the six spots in the black circles show wave vector of q 0 = (2/ √ 3) · (1/a) (where a = 3.35Å is the in-plane lattice constant of monolayer 1T-VSe 2 ), which is originated from the intrinsic (1 × 1) lattice of monolayer 1T-VSe 2 . While the spots in the red circles constitute a period grid [red dashed grid in Fig. 2 3) CDW reconstruction. However, unlike usual CDWs, the rest spots interlacing with the red grid constitute two period grids [blue dashed grids in Fig. 2(g)]. Interestingly, these two blue grids are also consisted of wave vectors of q 1 and q 2 , but with shifts of ±q 3 to the zero point. The Fig. 2 3) wave vectors of q 1 and q 2 disappear. However, the wave vector q 3 ≈ (1/2)q 0 still exists with a slightly different misangle of ∼ 15 • [same to the mis-angle in real space image of Fig. 2(b)], and more incommensurate wave vectors (e.g. q 4 ≈ (1/4.2)q 0 ) emerge. Unfortunately, we are unable to attribute these spots to a period grid strictly, but they are located along the blue dashed lines with spacing of q 3 ≈ (1/2)q 0 . Therefore, we suggest these spots and incommensurate wave vectors represent the stripe-like incommensurate (∼ 2a) CDW phase in short range. Notably, similar stripe-like incommensurate CDW phases were widely observed in the 1T-TaS 2 , [39] 1T-TaSe 2 , [40,41] LaTe 2 , [42] TbTe 3 , [43] SmTe 3 , [44] Ta 4 Pd 3 Te 16 , [45] NdO 1−x F x BiS 2 , [46] and cuprates. [47] (a) When the temperature rises to room-temperature of 300 K, the RT-STM image in Fig. 2(c) shows that the stripelike incommensurate CDW phase in short range still exists but exhibits different formations. In the FFT image shown in Figs. 2(f) and 2(i), we can still observe the wave vector q 3 ≈ (1/2)q 0 with a mis-angle around ∼ 10 • , accompanied with other different incommensurate wave vectors (e.g. q 5 ≈ (1/3.2)q 0 , q 6 ≈ (1/2.7)q 0 ). These newly emerged wave vectors may represent the short stripes depicted by the cyan dashed lines in Fig. 2(i), while the incommensurate (∼ 2a) CDW order with wave vector q 3 ≈ (1/2)q 0 can be found by the blue dashed lines but is not very obvious.

107301-3
From the above analysis on the atom-resolved STM images and their FFT images, we suggest the coexisting of the ( √ 7 × √ 3) and a hidden incommensurate (∼ 2a) CDW phases in monolayer 1T-VSe 2 at ultra-low temperature of 4 K. The ( √ 7 × √ 3) phase will disappear at 165 K, implying that a CDW transition happens blow 165 K. Considering the observation of the ( √ 7 × √ 3) phase at 76 K in previous reports, [27,28,31] here we suggest the CDW phase transition point is between 76 K to 165 K.
To confirm that the reconstructions shown in the STM images are indeed originated from the CDW phases rather than the lattice structure phase transition, we carried out the low-energy electron diffraction (LEED) taken at various temperatures. Figure 3 show the LEED images taken at 80 K, 200 K, and 300 K. The (1 × 1) diffraction patterns from the BLG substrate can be clearly observed and are depicted by the white dashed hexagons. Besides, the spots depicted by the red dashed hexagons are originated from the (1 × 1) lattice of the grown 1T-VSe 2 . Feng et al. have reported that a weak diffraction pattern of the CDW order in monolayer 1T-VSe 2 can be observed in the LEED taken at 40 K, [30] but here we did not find any features on the CDW reconstructions. This may be due to the relative poorer resolution limit of our LEED system. However, for the TMDCs materials, the crystalline structure phase transitions usually can be easily distinguished by the electron diffraction method, e.g., the 2H to 1T phase transition in monolayer WSe 2 will produce an additional pattern in the electron diffraction images. [37,48] Here, except the (1 × 1) pattern from the 1T-VSe 2 , no other diffraction pattern was found in the VT-LEED measurements, indicating that no crystalline phase transition happens in 1T-VSe 2 . Thus, the stripelike CDW structures observed in STM are not from crystalline phase transition such as 1T to 1T .

Two-step CDW gap structure transition
To investigate the CDW transition with temperature in detail, we performed VT-APRES measurements to study how the band structures evolve with the CDW transition in the monolayer 1T-VSe 2 film at various temperatures. Figure 4(a) shows the constant-energy-mapping at the binding energy of −0.1 eV below the Fermi level at the temperature of 7 K. Six elliptic pockets can be clearly observed around the six M points of the hexagonal Brillouin zone (BZ), which is consistent with the calculated Fermi surface from the previous reports. [36,49,50] Figure 4(b) shows the ARPES spectra along the Γ -M-K directions. We can see that the band disperses towards the Fermi level at the momentum positions marked by the red α point and blue β point, at which the CDW gaps can be observed and extracted from the symmetrized energy distribution curves (EDCs) of the ARPES spectra. Figures 4(c) and 4(d) show the symmetrized EDCs with subtraction of Fermi function at the momentum positions of the α and β points, respectively. These EDCs were taken at temperatures from 7 K to 34 K. Usually we can take the distance between the two symmetric peaks to be twice of the CDW gap (2 × ∆ ). [31] When the temperature rises, the peaks of the EDC will gradually flatten around the 0 eV, indicating that the CDW gap closes. The temperature dependences of the CDW gaps extracted from Figs. 4(c) and 4(d) were plotted in Figs. 4(e) and 4(f). Remarkably, we found that the gaps at different momentum positions show quite distinct behaviors. At the momentum position marked as the α point near the Γ point, the CDW gap exhibits a monotonic temperature dependence and gradually decreases from ∼ 30 ± 5 meV to zero around ∼ 150 K; while at the momentum position marked as the β point, the CDW gap decreases from ∼ 62 ± 5 meV to ∼ 30 ± 8 meV at ∼ 150 K, then it shows a stable decrease with temperature in an extended range and finally begins to drop since ∼ 310 K, showing a closing trend around ∼ 340 K. In Fig. A1 in Appendix A, we also plot the zoom-in symmetrized EDCs at the α and β points near the transition temperature of ∼ 150 K and ∼ 340 K to show the close and the closing trend of the CDW gap. The error bars of the gaps data are set as where k B is the Boltzmann constant, ε s is the resolution limit (∼ 5 meV) of the ARPES system. According to our above STM results, one may ascribe the low temperature gap as resulting from the ( √ 7 × √ 3) CDW phase, while the intermediate temperature gap as from the stripe-like incommensurate CDW phase. Therefore, we suggest a two-gap formula at the mean-field level to describe the temperature dependence of the CDW gap, [31] where A is a proportional constant and Θ is the unit step function. At the momentum position marked as the α point near the Γ point, only ∆ 1 is included. The fitting result to the experimental data is shown in Fig. 4(e) as the red line, which shows a well agreement to the original experimental data. According to the fitting result, we get ∆ 1 = 30±6 meV and T C1 = 151±6 K. At the momentum position marked as the β point, both ∆ 1 and ∆ 2 are included, and we use ∆ (T ) = ∆ 1 (T ) + ∆ 2 (T ) to fit the 107301-4 data. The fitting result is shown as the red line in Fig. 4(f), we get a very good fitting result with ∆ 1 = 32 ± 2 meV, ∆ 2 = 30 ± 1 meV, T C1 = 153 ± 2 K, and T C2 = 343 ± 2 K. The errors of the fitting results are set in 68% confidence interval. Here the two fitting results of ∆ 1 and T C1 are within the fitting errors. The combination of the experimental results and the theoretical fitting indicates that there exist two distinct CDW gaps with highly anisotropic gap distributions in the momentum space, in particular a two-step gap transition at the β point along the M-K direction in the monolayer 1T-VSe 2 , one is suggested to be associated with the ( √ 7 × √ 3) CDW with a transition temperature T C1 of ∼ 151 K (denoted by ∆ 1 ), while another may be associated with the stripe-like incommensurate CDW phases with a fitted transition temperature T C2 of ∼ 343 K (denoted by ∆ 2 ). Notably, the CDW gap ∆ 2 shows a highly anisotropic momentum dependence, which has no trace near the Γ point (at the α point) but can be clearly observed near the M point (at the β point).
where k B is the Boltzmann constant, ε s is the resolution limit (∼ 5 meV) of the ARPES system.
With the two-gap form, we can go further to make a comparison to the experimental ARPES spectra by using the phenomenological self-energy expression developed originally for high-T C cuprates, [51] where ∆ (T ) is the CDW gap, Γ 1 the single-particle scattering rate, Γ 0 the inverse particle-hole pair lifetime, and ε ( ) the single-particle dispersion. Using Eq. We note that some previous works on the CDW gap in monolayer 1T-VSe 2 also reported a similar two-step gap transition, [27,32] but they attributed one step of the gap transition to the metal-insulator transition [27] or the pseudogap. [32] Differently, with the unveiling of the hidden incommensurate CDW order, here we suggest the CDW gap consisted of two gap parts (∆ 1 + ∆ 2 ), which gives a well explanation and a comprehensive description on the two-step CDW transition. Interestingly, a similar two-stage CDW transition associated with the incommensurate CDW phase was also reported in the cuprates of La 2−x Ba x CuO 4 . [47] Besides, we note that by simply treating the scattering rates Γ 1 and Γ 0 as constants in our calculations when using Eq. (2), we get a good agreement to the experimental data. It suggests that the single-particle scattering rate Γ 1 and the inverse pair lifetime Γ 0 may show less or even no temperature dependence, and also affect less the CDW phase transitions and gap evolutions in monolayer 1T-VSe 2 . This is in contrast to the case in high-T C cuprates, [51] where both Γ 1 and Γ 0 assumes a strong temperature dependence.
3.3. Discussion on the possible physical mechanism of the CDW phases in monolayer 1T-VSe 2 Previous works have suggested that the ( √ 7 × √ 3) CDW reconstruction can be attributed to the Fermi surface nesting of the elliptic Fermi pockets around the M points in the monolayer 1T-VSe 2 , [31,50] while the proposals for the physical mechanism of the incommensurate CDW phase vary from the imperfect nesting in SmTe 3 [44] to the Coulomb interactions between domain walls, and the inter-layer interaction the in monolayer MX 2 . [41] Recent theoretical calculations reveal that the electron-electron correlations, momentum-dependent electron-phonon coupling, together with the Fermi-surface nesting would play important roles in the complicated multiple CDW phases in monolayer VSe 2 . [50,52] In order to investigate the possible physical mechanism of the multiple CDW phases in monolayer VSe 2 from experimental aspect, next we will investigate how the Fermi pockets and the electronic structures evolve during the CDW transition in the following.
Figures 6(a)-6(c) show the secondary differential ARPES spectra taken at 7 K, 200 K, 340 K, respectively. A sharp valence band denoted as the γ band at the Γ point (depicted by the cyan dashed curve) can be clearly observed, showing a movement towards the Fermi level with increasing temperatures. Figure 5(d) show the temperature dependence of the top of the γ band in energy (E V1 ). We can see that E V1 has a linear temperature dependence above ∼ 70 K, which can be attributed to the temperature-dependent charge transfer from the substrate to the grown film. The movement of the γ band in energy will affect the shape and size of the elliptic Fermi pockets around the M point profoundly. To evaluate the changing of the elliptic Fermi pockets, we plot the temperature dependence of the momentum positions of the pocket apex (k AP ) and the pocket widths (k W ) in Figs. 6(e) and 6(f), respectively. Notably, k AP also moves monotonically with the increasing of temperature, but exhibits a distinct kink around ∼ 150 K (see the red dashed lines). While k W shows nearly no temperature dependence until a little bit less than 150 K, then exhibits a step-like drop around ∼ 150 K. Since the ( √ 7 × √ 3) CDW phase was considered due to the Fermi nesting of these elliptic Fermi pockets, [31,50] and the Fermi nesting vectors are highly depending on the size and shape of the elliptic Fermi pockets, thus we suggest that the changing of the pocket shape (k AP ) and the shrinking of the pocket size (k W ) with increasing temperature will destroy the nesting wave vector of ( √ 7 × √ 3), resulting the disappearance of the ( √ 7 × √ 3) phase at the high temperatures of 165 K and 300 K in the STM images and the closing of the CDW gap ∆ 1 above ∼ 150 K. However, the formation of the incommensurate CDW phase was believed not to be related to the Fermi surface nesting. [41,44] Therefore, the changing of the elliptic Fermi pockets will not eliminate the incommensurate CDW phase and the gap ∆ 2 above ∼ 150 K, but only slightly affect the formations of the stripe-like incommensurate CDW order and the corresponding wave vectors. When the temperature is above T C2 = 343 K, the thermo fluctuation has an energy scale of k B T C2 ≈ 30 meV, which is same to the value of ∆ 2 = 30 ± 1 meV. Even though we did not directly observe the closing of the CDW gap at the β point at 340 K, but the gap closing trend around the 340 K suggested that the incommensurate CDW gap would be suppressed by the thermo fluctuation that has a same energy scale to the ∆ 2 .

Conclusion
In summary, we unveil a hidden stripe-like incommensurate (∼ 2a) CDW phase behind the ( √ 7 × √ 3) CDW phase in the epitaxial monolayer 1T-VSe 2 film. The co-existence and interference between the hidden incommensurate CDW phase and the ( √ 7 × √ 3) phase result the short-range ( √ 7 × √ 3) and (2 × √ 3) CDW reconstructions at 4 K. Using the VT/RT-STM measurements, we found that the ( √ 7 × √ 3) CDW will disappear at 165 K, but the incommensurate CDW phase can survive even at room temperature. Combining the ARPES measurements and theoretical analysis, we suggest that these two CDW phases exhibit a two-step CDW gap transition. Such similar two-step gap transition has also been reported but with different explanations of metal-insulator transition [27] or pseudogap. [32] Significantly, here we suggested that the CDW gap consists of two parts, one is corresponding to the ( √ 7 × √ 3) reconstruction results a full gap part ∆ 1 ∼ 32 meV with a transition temperature T C1 ∼ 153 K, while the other is corresponding to the incommensurate CDW order shows a highly momentum dependence and results a partial gap structure ∆ 2 ∼ 30 meV with a transition temperature T C1 ∼ 343 K. Our results illustrate an unusual CDW phenomenon with multiple reconstruction phases and a two-step CDW gap transition in the epitaxial monolayer 1T-VSe 2 , giving a deep and comprehensive understanding on the CDW mechanism in monolayer 1T-VSe 2 .
The symmetrized EDCs around the transition temperatures are shown in the following figures.