Charge-density waves in NbS₂

This repository contains the data and source code associated with the paper:

• Timo Knispel, Jan Berges, Arne Schobert, Erik G. C. P. van Loon, Wouter Jolie, Tim Wehling, Thomas Michely, and Jeison Fischer, Unconventional charge-density-wave gap in monolayer NbS₂, arXiv:2307.13791

Experimental setup

STM and STS were carried out at a base operating temperature of T₀ = 0.35 K after in-situ transfer from the preparation chamber. STS was performed with the lock-in technique. STM images were taken in constant current mode.

Computational setup

All DFT and DFPT calculations were performed using Quantum ESPRESSO 7.1. Similarly recent versions should work equally well. (Note that the ab initio step can be skipped since we provide all relevant output files.) To interface the EPW code to our elphmod package, the file EPW/src/ephwann_shuffle.f90 has to be modified. After the determination of the Wigner-Seitz points, just before the Bloch-to-Wannier transform, add the following lines:

IF (ionode) THEN
  OPEN (13, file='wigner.dat', action='write', status='replace', &
    access='stream')
  WRITE (13) dims, dims2
  WRITE (13) nrr_k, irvec_k, ndegen_k
  WRITE (13) nrr_g, irvec_g, ndegen_g
  CLOSE (13)
ENDIF

Our elphmod and StoryLines packages and all other Python requirements can be installed in a virtual environment:

python3 -m venv venv
source venv/bin/activate
python3 -m pip install -r requirements.txt

A LaTeX installation, preferably TeX Live, is required to typeset the figure.

List of scripts

Workflow of the DFT, DFPT, and model calculations:

• run.sh

Initial structural relaxation:

• cdw_relax.py (create contents of directory xyz)

Variation of parameters:

• change_smearing.py (rerelax CDW structures for different smearings)
• change_doping.py (rerelax CDW structures for different dopings)
• change_scaling.py (rescale CDW displacements)

Characterization of selected CDW phases:

• compute_dos.py (calculate electron density of states)
• compute_a2f.py (calculate phonon density of states and Eliashberg function)
• compute_modes.py (calculate and identify amplitude and phase modes)

Plotting:

• plot.py (create Figures 5, S13, S14, and S15)
• plot_phases.py (create Figure S11)
• plot_doping.py (create Figure S12)

Contents of directory exp

Experimental data shown in the main text:

• Figure 1a.txt
• Figure 1b.txt
• Figure 2a.txt
• Figure 2a inset.txt
• Figure 2b.txt
• Figure 2b inset.txt
• Figure 2c.txt
• Figure 2c inset.txt
• Figure 2d.txt
• Figure 2d left inset.txt
• Figure 2d right inset.txt
• Figure 2e.txt
• Figure 2e inset.txt
• Figure 2f.txt
• Figure 2f inset.txt
• Figure 2g.txt
• Figure 2g inset.txt
• Figure 2h.txt
• Figure 3a.txt
• Figure 3b.txt
• Figure 3c.txt

In Figure 1a, the size was reduced from 250 nm × 250 nm to 250 nm × 166 nm. This reduction is not included in the source data.

In Figure 1b, the minimum height on the graphene surface is corrected to 0 nm.

In the insets of Figure 2a–c and Figure 2e–g, the images were clipped with a 400% zoom.

In the insets of Figure 2d, the images were clipped as a circle in the same location.

Figure 2h shows the FFT intensity of first-order 3 × 3 CDW spots (I_CDW) vs. first-order Bragg spots (I_atom) at different temperatures. The spot intensity was extracted by line profiles along the six high symmetry directions in the FFT. For each temperature, the average ratio I_CDW/I_atom of all six directions was taken. FFTs were extracted from constant current dI/dV maps as shown in Figure 2e–g.

In Figure 3a, the size was reduced from 3.5 nm × 3.5 nm to 2.7 nm × 2.7 nm. This reduction is not included in the source data.

In Figure 3b, the data was plotted with an arbitrary offset between the spectra at different positions, for clarity. This offset is not included in the source data.

In Figure 3c, the data was plotted with an offset of 10 nS between the spectra, for clarity. This offset is not included in the source data.

Contents of directory dft

Quantum ESPRESSO input files:

• scf.in (self-consistent DFT calculation with pw.x)
• nscf.in (non-self-consistent DFT calculation with pw.x)
• ph.in (DFPT calculation with ph.x)
• epw.in (Wannierization with epw.x)

Relevant Quantum ESPRESSO output files:

• NbS2_hr.dat (hopping parameters)
• NbS2_wsvec.dat (superlattice vectors to correct Wigner-Seitz cell)
• dyn0 ... dyn19 (force constants)
• NbS2.epmatwp (electron-phonon coupling)
• wigner.dat (lattice vectors of electron-phonon coupling)

Contents of directory xyz

Relaxed CDW structures:

• t1.xyz
• t2.xyz (at doping of -0.1 e/f.u.)
• t1p.xyz
• t2p.xyz
• hexagons.xyz
• stars.xyz (at doping of -0.1 e/f.u.)

Contents of directory model

Data shown in Figure 4 (T1 phase):

• smearing+0.0050_t1_hr.dat
• scaling+0.0_dos.dat (also shown in Figures S13, S14, and S15)
• scaling+0.1_t1_dos.dat
• scaling+0.2_t1_dos.dat
• scaling+0.3_t1_dos.dat
• scaling+0.4_t1_dos.dat
• scaling+0.5_t1_dos.dat
• scaling+0.6_t1_dos.dat
• scaling+0.7_t1_dos.dat
• scaling+0.8_t1_dos.dat
• scaling+0.9_t1_dos.dat
• smearing+0.0050_t1_dos.dat
• scaling+0.0_dos_zoom.dat (also shown in Figures S13, S14, and S15)
• scaling+0.1_t1_dos_zoom.dat
• scaling+0.2_t1_dos_zoom.dat
• scaling+0.3_t1_dos_zoom.dat
• scaling+0.4_t1_dos_zoom.dat
• scaling+0.5_t1_dos_zoom.dat
• scaling+0.6_t1_dos_zoom.dat
• scaling+0.7_t1_dos_zoom.dat
• scaling+0.8_t1_dos_zoom.dat
• scaling+0.9_t1_dos_zoom.dat
• smearing+0.0050_t1_dos_zoom.dat (also shown in Figure S12)
• smearing+0.0150_t1_dyn.dat (to identify amplitude and phase modes)
• smearing+0.0130_t1_ifc.dat
• modes_t1.dat
• smearing+0.0050_t1_a2f.dat (also shown in Figure S12)

Data shown in Figure S11 (CDW phases):

• doping.dat
• smearing.dat

Data shown in Figure S12 (doping dependence):

• doping-0.02_t1_dyn.dat (only atomic positions)
• doping-0.01_t1_dyn.dat (only atomic positions)
• smearing+0.0050_t1_dyn.dat (only atomic positions)
• doping+0.01_t1_dyn.dat (only atomic positions)
• doping+0.02_t1_dyn.dat (only atomic positions)
• doping-0.02_t1_dos_zoom.dat
• doping-0.01_t1_dos_zoom.dat
• smearing+0.0050_t1_dos_zoom.dat (also shown in Figure 4)
• doping+0.01_t1_dos_zoom.dat
• doping+0.02_t1_dos_zoom.dat
• doping-0.02_t1_a2f.dat
• doping-0.01_t1_a2f.dat
• smearing+0.0050_t1_a2f.dat (also shown in Figure 4)
• doping+0.01_t1_a2f.dat
• doping+0.02_t1_a2f.dat

Data shown in Figure S13 (hexagons phase):

• smearing+0.0050_hexagons_hr.dat
• scaling+0.0_dos.dat (also shown in Figures 4, S14, and S15)
• scaling+0.1_hexagons_dos.dat
• scaling+0.2_hexagons_dos.dat
• scaling+0.3_hexagons_dos.dat
• scaling+0.4_hexagons_dos.dat
• scaling+0.5_hexagons_dos.dat
• scaling+0.6_hexagons_dos.dat
• scaling+0.7_hexagons_dos.dat
• scaling+0.8_hexagons_dos.dat
• scaling+0.9_hexagons_dos.dat
• smearing+0.0050_hexagons_dos.dat
• scaling+0.0_dos_zoom.dat (also shown in Figures 4, S14, and S15)
• scaling+0.1_hexagons_dos_zoom.dat
• scaling+0.2_hexagons_dos_zoom.dat
• scaling+0.3_hexagons_dos_zoom.dat
• scaling+0.4_hexagons_dos_zoom.dat
• scaling+0.5_hexagons_dos_zoom.dat
• scaling+0.6_hexagons_dos_zoom.dat
• scaling+0.7_hexagons_dos_zoom.dat
• scaling+0.8_hexagons_dos_zoom.dat
• scaling+0.9_hexagons_dos_zoom.dat
• smearing+0.0050_hexagons_dos_zoom.dat
• smearing+0.0150_hexagons_dyn.dat (to identify amplitude and phase modes)
• smearing+0.0130_hexagons_ifc.dat
• modes_hexagons.dat
• smearing+0.0050_hexagons_a2f.dat

Data shown in Figure S14 (T1' phase):

• smearing+0.0050_t1p_hr.dat
• scaling+0.0_dos.dat (also shown in Figures 4, S13, and S15)
• scaling+0.1_t1p_dos.dat
• scaling+0.2_t1p_dos.dat
• scaling+0.3_t1p_dos.dat
• scaling+0.4_t1p_dos.dat
• scaling+0.5_t1p_dos.dat
• scaling+0.6_t1p_dos.dat
• scaling+0.7_t1p_dos.dat
• scaling+0.8_t1p_dos.dat
• scaling+0.9_t1p_dos.dat
• smearing+0.0050_t1p_dos.dat
• scaling+0.0_dos_zoom.dat (also shown in Figures 4, S13, and S15)
• scaling+0.1_t1p_dos_zoom.dat
• scaling+0.2_t1p_dos_zoom.dat
• scaling+0.3_t1p_dos_zoom.dat
• scaling+0.4_t1p_dos_zoom.dat
• scaling+0.5_t1p_dos_zoom.dat
• scaling+0.6_t1p_dos_zoom.dat
• scaling+0.7_t1p_dos_zoom.dat
• scaling+0.8_t1p_dos_zoom.dat
• scaling+0.9_t1p_dos_zoom.dat
• smearing+0.0050_t1p_dos_zoom.dat
• smearing+0.0150_t1p_dyn.dat (to identify amplitude and phase modes)
• smearing+0.0115_t1p_ifc.dat
• modes_t1p.dat
• smearing+0.0050_t1p_a2f.dat

Data shown in Figure S15 (T2' phase):

• smearing+0.0050_t2p_hr.dat
• scaling+0.0_dos.dat (also shown in Figures 4, S13, and S14)
• scaling+0.1_t2p_dos.dat
• scaling+0.2_t2p_dos.dat
• scaling+0.3_t2p_dos.dat
• scaling+0.4_t2p_dos.dat
• scaling+0.5_t2p_dos.dat
• scaling+0.6_t2p_dos.dat
• scaling+0.7_t2p_dos.dat
• scaling+0.8_t2p_dos.dat
• scaling+0.9_t2p_dos.dat
• smearing+0.0050_t2p_dos.dat
• scaling+0.0_dos_zoom.dat (also shown in Figures 4, S13, and S14)
• scaling+0.1_t2p_dos_zoom.dat
• scaling+0.2_t2p_dos_zoom.dat
• scaling+0.3_t2p_dos_zoom.dat
• scaling+0.4_t2p_dos_zoom.dat
• scaling+0.5_t2p_dos_zoom.dat
• scaling+0.6_t2p_dos_zoom.dat
• scaling+0.7_t2p_dos_zoom.dat
• scaling+0.8_t2p_dos_zoom.dat
• scaling+0.9_t2p_dos_zoom.dat
• smearing+0.0050_t2p_dos_zoom.dat
• smearing+0.0150_t2p_dyn.dat (to identify amplitude and phase modes)
• smearing+0.0070_t2p_ifc.dat
• modes_t2p.dat
• smearing+0.0050_t2p_a2f.dat

Relevant force-constants data (not needed for plotting):

• smearing+0.0050_t1_ifc.dat
• smearing+0.0050_hexagons_ifc.dat
• smearing+0.0050_t1p_ifc.dat
• smearing+0.0050_t2p_ifc.dat