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ARPES spectra calculator

Synopsis

This is a code to calculate ARPES spectra for slabs with an accumulation or a depletion surface layer. The calculation proceeds in two steps:

  1. The Poisson and the Schrödinger equations are solved self-consistently to obtain eigen-pairs {En, ψn(z)}.
  2. ARPES spectra are then calculated using a Fourier transform of ψn(z), followed by a convolution with normal distributions to account for instrumental broadening.

1D finite elements (esf library) are used to discretize the Poisson and the Schrödinger equations. The Intel MKL library is used to solve linear systems, generalized eigenvalue system, compute Fourier transforms and convolution. The mathematical details can be found in this PDF file.

The results are exported into Matlab/Octave MAT-files and Gnuplot binary matrix files.

Results

All images below are presented for exposition only, no attempt has been made here to fit any experimental data.

Depletion layer (no bound states):

Depletion layer

(Large image)

Accumulation layer with one bound state:

Accumulation layer with one bound state

(Large image)

Accumulation layer with two bound states:

Accumulation layer with two bound states

(Large image)

How to build

Set MKLROOT environment variable to point to the MKL installation directory, and be sure that your CMake version is >= 3.13. Then:

git clone --recursive https://github.com/eugnsp/surface_arpes.git
cd surface_arpes
mkdir build && cd build
cmake -DCMAKE_BUILD_TYPE=RELEASE .. && make

to build the tool.

C++17 compiler is required. Tested with GCC 8.3.0 and Clang 7.0.

How to run

./arpes [file]

./arpes ../example/accum1.txt
./arpes < ../example/accum1.txt

where file is the text file with simulation parameters (see examples in the example directory). The standard input is used to read parameters if no file is specified.

You can run arpes via a shell script

echo '#!./arpes' > accum1.sh
cat ../example/accum1.txt >> accum1.sh
chmod 755 accum1.sh
./accum1.sh

Output files

The following output files are produced:

  • poisson_cl.mat and poisson_q.mat

    The Matlab files that contain the following variables:

    • vertices – mesh points zi [nm],
    • ec – Ec(zi) at mesh points [eV],
    • f – the Fermi level [eV],
    • n – the total charge density n(zi+1/2) at edge mid-points [eV].

    poisson_cl.mat corresponds to the solution obtained using the quasi-classical Thomas–Fermi approximation, poisson_q.mat corresponds to the solution obtained using the Schrödinger equation. The former one is used as an initial guess for the Poisson–Schrödinger iterative solver.

  • schrod.mat

    The Matlab file that contains the solution of the Scrödinger equation obtained at the last iteration:

    • en – the eigen-energies En,
    • psi – the eigen-functions ψn(zi).

  • arpes.mat

    The Matlab file that contains the calculation parameters and ARPES spectra. The variables are:

    • arpes_e_kz – the (E, kz) spectrum,
    • arpes_kx_e – the (kx, E) spectrum,
    • arpes_kx_kz – the (kx, kz) spectrum,
    • e_min, e_max – the energy range Emin ≤ E ≤ Emax [eV],
    • kx_max – the momentum range -kx max ≤ kx ≤ kx max [Å-1],
    • kz_max – the momentum range -kz max ≤ kz ≤ kz max [Å-1],
    • mfp – electron mean-free path λ [nm],
    • sigma_e_inst – the instrumental broadening δAE [eV],
    • sigma_kx_inst – the instrumental broadening δAk [Å-1],
    • gamma_disorder – the disorder broadening δDE [eV].

  • arpes_e_kz.dat, arpes_kx_e.dat, arpes_kx_kz.dat

    The binary files that contain ARPES spectra for visualization using Gnuplot, see shell scripts in the plot directory. The figures above were generated using the plot/plot_all.sh shell script.

External dependencies

References

  1. V.N.Strocov. Photoemission response of 2D states.
    J. Electron. Spectrosc. 229, 100 (2018), arXiv preprint (2018).
  2. S.Moser et al. How to extract the surface potential profile from the ARPES signature of a 2DEG.
    J. Electron. Spectrosc. 225, 16 (2018).
  3. A.Trellakis et al. Iteration scheme for the solution of the two-dimensional Schrödinger–Poisson equations in quantum structures.
    J. Appl. Phys. 81, 7880 (1997).

License

This code is distributed under GNU General Public License v3.0.