Strain engineering a $4a\times \surd 3a$ charge-density-wave phase in transition-metal dichalcogenide $1\mathrm{T}\text{−}\mathrm{VS}{\mathrm{e}}_2$

We report a rectangular charge density wave (CDW) phase in strained $1\mathrm{T}\text{−}\mathrm{VS}{\mathrm{e}}_2$ thin films synthesized by molecular beam epitaxy on $c$-sapphire substrates. The observed CDW structure exhibits an unconventional rectangular $4a\times \surd 3a$ periodicity, as opposed to the previously reported hexagonal $4a\times 4a$ structure in bulk crystals and exfoliated thin-layered samples. Tunneling spectroscopy shows a strong modulation of the local density of states of the same $4a\times \surd 3a$ CDW periodicity and an energy gap of $2{\mathrm{\Delta }}_{\mathrm{CDW}}=\left(9.1\pm 0.1\right)\mathrm{meV}$. The CDW energy gap evolves into a full gap at temperatures below 500 mK, indicating a transition to an insulating phase at ultra-low temperatures. First-principles calculations confirm the stability of both $4a\times 4a$ and $4a\times \surd 3a$ structures arising from soft modes in the phonon dispersion. The unconventional structure becomes preferred in the presence of strain, in agreement with experimental findings.

Figure 1

Molecular beam epitaxy growth of strained $\mathrm{VS}{\mathrm{e}}_2$ thin films with rectangular CDW structure. (a) Structure of $1\mathrm{T}\text{−}\mathrm{VS}{\mathrm{e}}_2$ consisting of trilayer elements of Se (red) and V (green). The upper layer Se are larger in the top view. (b) HRTEM image of the cross section of the MBE grown $\mathrm{VS}{\mathrm{e}}_2$ thin film. (c) Selective area diffraction pattern from the film shown in (b), which defines the following epitaxial relationship between the $\mathrm{VS}{\mathrm{e}}_2$ film and sapphire substrate: (0001) sapphire $\left|\right|\left(0001\right)\mathrm{VS}{\mathrm{e}}_2$ and $\left(1\overline{1}00\right)$ sapphire $\left|\right|\left(1\overline{2}10\right)\mathrm{VS}{\mathrm{e}}_2$. (d) RHEED intensity oscillations observed during growth. (e) X-ray diffraction pattern of a ten-layer film of $1\mathrm{T}\text{−}\mathrm{VS}{\mathrm{e}}_2$ on $c$-sapphire. $\mathrm{VS}{\mathrm{e}}_2$ crystallizes in hexagonal structure [space group $P\overline{3}m1$ (164)] with a calculated $c$-lattice parameter of 0.590 ± 0.001 nm [53]. (f) An epitaxial model of $\mathrm{VS}{\mathrm{e}}_2$ on $c$-sapphire derived from TEM showing 30 ° rotation for the $\mathrm{VS}{\mathrm{e}}_2$ lattice relative to the $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$ lattice. Only O (light green) and Se (dark green) atoms are shown. (g) STM image of $\mathrm{VS}{\mathrm{e}}_2$ islands grown on sapphire by MBE. (h) STM image of a grain boundary imaged in one of the $\mathrm{VS}{\mathrm{e}}_2$ islands. The STM images in (g, h) are shown in a color scale from dark to light, covering a height range of 24 and 0.1 nm, respectively.

Figure 2

Rectangular CDW in MBE grown strained $\mathrm{VS}{\mathrm{e}}_2$ thin films. (a–c) STM topographic images, 14 × 14 nm, with decreasing tunneling impedance (decreasing sample-tip distance). Tunneling impedance: (a) 25 MΩ, (b) 12.5 MΩ, and (c) 6.25 MΩ. Height range for images: (a) 114 pm, (b) 117 pm, and (c) 107 pm. Tunneling current 4 nA. $T=150\mathrm{mK}$. The white box in (c) outlines the $4a\times \surd 3a$ unit cell. (d) 2D FFT of the image in (c) showing the $4a\times \surd 3a$ unit cell. (e, f) STM height profiles from (c) along the indicated white lines showing the $4a$ periodicity along the $\langle 11\overline{2}0\rangle$ direction in (e) and the $\surd 3a$ periodicity along the $\langle 1\overline{1}00\rangle$ in (f).

Figure 3

Voltage-dependent STM topographic imaging of the $\mathrm{VS}{\mathrm{e}}_2$ CDW. (a–c) Negative sample bias, −25 mV, −60 mV, and −200 mV, respectively. (d–f) Positive sample bias, 5 mV, 60 mV, and 150 mV, respectively. Image size 14 nm × 14 nm. The purple rectangle outlines the same position in all images. Tunneling impedances: (a) 6.25 MΩ, (b) 100 MΩ, (c) 333.3 MΩ, (d) 8.3 MΩ, (e) 100 MΩ, (f) 250 MΩ. Height range for images: (a) 107 pm, (b) 148 pm, (c) 113 pm, (d) 259 pm, (e) 146 pm, and (f) 121 pm. $T=150\mathrm{mK}$.

Figure 4

Tunneling spectroscopy of the rectangular CDW state in $\mathrm{VS}{\mathrm{e}}_2$ thin films. (a, b) dI/dV spectroscopy measurements (red lines) obtained on the lattice row with $4a$ periodicity, at the locations indicated by the dots in (c). Two energy gaps, ${\mathrm{\Delta }}_{\mathrm{CDW}}$ and ${\mathrm{\Delta }}_{\mathrm{I}}$, discussed in the text are defined from the inflection points in the dI/dV measurements, corresponding to the maxima and minima of the derivative of the differential conductance (orange lines). (c) STM image, 6 × 3 nm, at the locations of dI/dV spectroscopy measurements showing strong modulations in the local density of states of (d) $4a$ periodicity along the $\langle 11\overline{2}0\rangle$ direction and (e) $\surd 3a$ periodicity along the $\langle 1\overline{1}00\rangle$ direction. A CDW gap of $2{\mathrm{\Delta }}_{\mathrm{CDW}}=\left(9.1\pm 0.1\right)\mathrm{meV}$ is determined from the spectra as indicated in (a) and (b) [53].

Figure 5

Tunneling spectroscopy of the low temperature $\mathrm{VS}{\mathrm{e}}_2$ insulating state. (a) dI/dV spectra (symbols) as a function of temperature. The solid lines are smooth splines through the data points. (b) The 2Δ energy gap measured from the spectra in (a) as a function of 1/$T$. At temperatures above 500 mK the energy gap remains constant with $2\mathrm{\Delta }=\left(9.1\pm 0.1\right)\mathrm{meV}$ [53]. Uncertainties in the energy values were obtained from nonlinear peak fitting of the derivative spectra of (a).

Figure 6

Theory and experiment of a new CDW phase in strained $\mathrm{VS}{\mathrm{e}}_2$. (a) Fermi surface and 3D Brillouin zone for $\mathrm{VS}{\mathrm{e}}_2$. Phonon dispersions of the normal state of the (b) ideal and (c) strained bulk crystal of $\mathrm{VS}{\mathrm{e}}_2$. Imaginary frequencies corresponding to soft modes are shown as negative values in the plots. The red and black arrows indicate soft modes corresponding to the $4a\times 4a$ and $4a\times \surd 3a$ structures, respectively. (d) Top view of the atomic structure of the $4a\times \surd 3a$ CDW phase in monolayer $\mathrm{VS}{\mathrm{e}}_2$, which matches closely with the STM results [see Table 1]. The red spheres represent Se atoms (top layer larger spheres), and the green spheres the V atoms. The V-Se pairs with a contracted bond length are connected in the figure. The dashed lines outline the $4a\times \surd 3a$ unit cell. (e) Simulated STM image for this structure at  25 mV bias at the isosurface density of $5\times {10}^{-5}e/a_0^3$. An experimental STM image of (f) filled states at a bias of −25 mV, and (g) empty states at a bias of 5 mV. The color scale covers a range of 66 pm and 80 pm, from dark to bright, in (f) and (g). The white rectangles in (e–g) outline the same $4a\times \surd 3a$ unit cell areas. $T=150\mathrm{mK}$.