3.1. Spin-orbit interaction (SOI)

Spin-orbit interaction is a relativistic effect that couples the experienced magnetic field by an electron with its spin magnetic moment [A12]. This coupling lifts the degeneracy between the different spin states, causing splitting in its energies. This interaction could spin-split the valence and conduction bands of semiconductors and has been explored to design spin-based devices. In this field, the study of spin-orbit coupling is of paramount importance to control and manipulate the electron states.

In the NanoDCAL code, the linear combination of atomic orbitals (LCAO) calculations takes into account the SOI effect using relativistic norm-conserving pseudopotentials and making an on-site approximation to the spin-orbit matrix elements [FOS07]. Within the framework of non-relativistic theory, the spin should be an intrinsic physical property of the system rather than an input parameter. Thus, when relativistic effects have to be considered to perform the calculations, in NanoDCAL the spin-orbit interaction requires:

  • Relativistic pseudopotentials;

  • General spin method (the spin arrangement has to be noncollinear).

Important

Before each calculation, some physical quantities have to be well converged. Please check the set values at:

  • calculation.realspacegrids.E_cutoff

  • calculation.k_spacegrids.number

  • calculation.SCF.convergenceCriteria (\(10^{-6}\) for density matrix)

In this tutorial, we’ll show how to compute the SOI effect in general spin calculations for silicon (Si) and gallium arsenide (GaAs) crystals. For these materials the valence bands are largely p-like solutions of the Schrödinger equation. The p states possess the orbital angular momentum \(l = 1\), and the state is 3-fold degenerate (\(m_{l}\) = -1, 0, +1). The \(m_{l}\) is the projection of \(l\) on an arbitrary axis are described by different orthogonal wave functions. The following steps will be performed in this tutorial:

  1. Restricted spin SCF calculations;

  2. General spin (with SOI) SCF calculations;

  3. Band structure calculations and spin-orbit splitting analyses of the valence band maximums.

3.1.1. Importing the crystals

Begin by starting the Device Studio software and creating a new project.

  1. Open Device Studio software ➟ Create a new Project ➟ File name semiconductor ➟ press Save;

  2. In the menu bar click File ➟ Import ➟ Import Local to pop up the search box:

    • Go to DeviceStudio folder ➟ material ➟ 3Dmaterials ➟ Semiconductor ➟ select Si.hzw ➟ click open.

    • Back to DeviceStudio folder ➟ material ➟ 3Dmaterials ➟ Semiconductor ➟ select GaAs.hzw ➟ click open.

3.1.2. SCF calculations

The required input files for for both Si and GaAs crystals could be generated using Device Studio with NanoDCAL module simulator, as follows:

  1. In the menu bar of Device Studio, click Simulator ➟ NanoDCAL ➟ SCF Calculation to pop up the interface to set the parameters;

    • On the Exchange correlation choose PBE_GGA96;

    • In the K-point Sampling set n1 = 18, n2 = 18, and n3 = 18;

    • Go to the Iteration control ➟ set DensityMatrix = \(10^{-6}\), HamiltonianMatrix = \(10^{-6}\), and totalEnergy = \(10^{-6}\).

  2. For perform the SCF calculations without spin-orbit interaction ➟ click on Generate files Otherwise:

    • Go to the Spin Type ➟ select GeneralSpin;

    • In the Element Spin polarization window ➟ double click on change Rho[0 1] change to 0.001 ➟ click on spin-orbital interaction to include SOI effect ➟ click on Generate files.

The generated input files for Si and GaAs crystals could be downloaded here:

Most of the keywords are common for both no spin and general spin calculations and were already specified here. However, there are relevant keywords to include the SOI effect on the general spin calculation, that will be discussed below

system.spinType = GeneralSpin
calculation.spinOrbitInteraction.isIncluded = true
AtomType OrbitalType X Y Z SpinPolarization_r SpinPolarization_theta SpinPolarization_phi
Si  PBE-DZP 0.67883750      0.67883750      0.67883750      0.001   0       0
Si  PBE-DZP 2.03651250      2.03651250      2.03651250      0.001   0       0

The system.spinType determines the spin description in the calculation, while calculation.spinOrbitInteraction.isIncluded taken into account the spin-orbit interaction. Note that last keyword could only used for systems having the system.spinType of GeneralSpin. In addition, the initial guess spin orientation is defined in spherical coordinate system with SpinPolarization_r, SpinPolarization__theta and SpinPolarization_phi.

Tip

  • The NoSpin and GeneralSpin calculations were performed to explore the effects over the electronic structure of Si and GaAs materials.

Run the scf calculations, the NanodcalObject.mat files are required for the band structure calculations.

3.1.3. Band structure calculations

Note that both Si and GaAs crystals have the same Fd-3m space group. Therefore, the same band structure input file (the same special k points) could be used to compute the band structure for all structures. The user should be able to set and perform band structure calculations following the procedure previously described here.

After the calculations, the .xml or .mat data files could be loaded in Device Studio or MATLAB platform to plot the band structures.

Band structure analyses of Si crystal

The calculated result shows that Si crystal has an indirect bandgap (\(E_{g}\)) of about 0.54 eV. The highest occupied state (valence band maximum) is localized at the \(\Gamma\)-point while the lowest unoccupied state (conduction band minimum) is outer of higher symmetry points along the \(\Gamma\)-X direction. As one can see in Fig. 3.1.1, this indirect band gap remains the same for both NoSpin and General Spin method [TB09].

../../_images/Si-gga-bands.svg

Fig. 3.1.1 Band structures of Si with NoSpin (black) and SOI (red).

The most relevant difference between the NoSpin and General Spin method occurs on the top of the valence band maximum. To see this let’s amplify the energy values around the \(\Gamma\)-point, as shown in Fig. 3.1.2.

../../_images/Si-gga-bands-zoom.svg

Fig. 3.1.2 Amplification of the Si-NoSpin and Si-SOI band structures at \(\Gamma\)-point.

At the \(\Gamma\)-point, in the absence of SOI, there are three-fold degenerated valence bands of about -0.0277 eV. These three valence bands are commonly known as the heavy-hole (HH), light-hole (LH), and split-off (SOI) bands. The inclusion of SOI partially removes this degeneracy. In this calculation, the result shows the split a single band (split-off) moves down in energy, while the other two bands remains degenerated (HH and LH) at the \(\Gamma\)-point.

The energetic difference between the singly and doubly degenerate bands was found to be \(\Delta\) = 0.03, which is close to the experimental data (\(\Delta\) = 0.05eV) [YHS89].

Band structure analyses of GaAs crystal

Now we’ll analyze the GaAs semiconductor material. The direct band-gap is localized at the \(\Gamma\)-point, as shown in Fig. 3.1.3. As one can see, the SOI coupling affects significantly the band structure. This is somewhat expected since heavier elements are characterized by stronger SOI coupling.

Considering the results with and without SOI effects, the top of the valence band shifts upwards, while the conduction bands shift downwards leading to a reduction of the bandgap of about 0.1 eV. At the \(\Gamma\)-point, the SOI splits the triply-degenerate valence bands into a doubly-degenerated bands and a single band. We found an energy split of \(\Delta\) = 0.30 eV, in good agreement with experimental results (\(\Delta\) = 0.34eV).

../../_images/GaAs-gga-bands.svg

Fig. 3.1.3 Band structures of GaAs with NoSpin (black) and SOI (red).

[A12]

  1. Autschbach. Perspective: Relativistic effects J. Chem. Phys. 136 (2012) p. 150902.

[FOS07]

F. Fernández-Seivane, M. A. Oliveira, S. Sanvito and J. Ferrer. On-site approximation for spin-orbit coupling in linear combination of atomic orbitals density functional methods J. Phys.: Condens. Matter 19 (2007) p. 489001.