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Effect of spin-orbit interaction
on the electronic and optical properties of GaSb
by M. Palummo, M. Marsili & D. Sangalli

The scope of this tutorial is to calculate the G0W0 corrections to the DFT-LDA band-structure and the BSE optical spectrum for a bulk semiconducting material, the Gallium Antimonide. We will compare the results with and without the spin-orbit interaction.

Material : GaSb

GaSb structure
GaSb bands
GaSb absorption

Crystal Structure

Zincblende crystalline structure. Two atoms per cell, Ga and Sb (8 electrons). Lattice constant 11.38 [a.u.].

Band Structure

Ground state: plane waves cutoff 18 Ry Large spin-orbit splitting of highest occupied bands: ~ 0.7eV

Optical properties

Large spin-orbit splitting of the E1 optical peak (E1, E1+D1) visible in the experimental optical spectrum

First steps

For this tutorial we provide two state (GS) databases (DBs), one contains the GS-DBs with the effect of SOC included and one without. The idea is to repeat the same calculation for both database and compare the results and the performances of the code. As in other tutorials you can find the folder containing the input files for the pwscf code (Pwscf, in case you prefer to generate the GS-DBs by yourself) and the folder containing the GS-DBs, the reference input files and the reference output files for the yambo code (YAMBO). Instead of continuosly jump back and forth from the Without_SOC to the With_SOC folder we advice to open two terminals and keep one for the without SOC case and another for the with SOC case.

Let's have a look to the content of the folders. For example we can enter in the Solid_GaSb/With_SOC/ folder >cd Solid_GaSb/With_SOC/ >ls Pwscf YAMBO >cd YAMBO >ls 4x4x4_shifted+GAMMA 6x6x6_shifted 10x10x10_shifted 14x14x14_shifted Inputs Reference_Outputs

Inside the YAMBO folder there are indeed four folders corresponding to four differet k-points meshes. In this tutorial we will use the 4x4x4_shifted+GAMMA folder for the computation of the QP corrections within the GoWo approximation. It is a kpts grid obtained as the sum of a 4x4x4 k-points grid centered in Gamma and a 4x4x4 grid shifted. Instead for the computation of the optical properties we will use the 6x6x6_shifted folder. It is a 6x6x6 k-points grid shifted from Gamma. The other two folders, 10x10x10_shifted and 14x14x14_shifted can be used to do a minimum convergence of the optical properties against the number of k-points (pay attention that the BSE calculations with bigger grids will be very memory and time demanding!).

The first part of the tutorial is on the computation of the QP corrections. So let's enter the 4x4x4_shifted+GAMMA grid and, as seen in other tutorials run the Initialization:

>cd 4x4x4_shifted+GAMMA/ >yambo -J 01_init \ \ / / / \ | \/ | | _ \ / __ \ \ \/ / / ^ \ | \ / | | |_) | | | | | \_ _/ / /_\ \ | |\/| | | _ < | | | | | | / _____ \ | | | | | |_) | | `--" | |__| /__/ \__\ |__| |__| |______/ \______/ <---> [01] CPU structure, Files & I/O Directories <---> [02] CORE Variables Setup <---> [02.01] Unit cells <---> [02.02] Symmetries <---> [02.03] RL shells <---> Shells finder |########################################| [100%] --(E) --(X) <---> [02.04] K-grid lattice <---> [02.05] Energies [ev] & Occupations <---> [03] Transferred momenta grid <---> X indexes |########################################| [100%] --(E) --(X) <---> SE indexes |########################################| [100%] --(E) --(X) <---> [04] External corrections <---> [05] Game Over & Game summary

If now you edit the r-01_init_setup file you will find as always information on the GS properties of the system. You can no also enter the Solid_GaSb/With_SOC/YAMBO/4x4x4_shifted+GAMMA and run the setup there as well. >cd Solid_GaSb/With_SOC/YAMBO/4x4x4_shifted+GAMMA/ >yambo -J 01_init

Now you can finally compare the result of the two setup runs.

With_SOC case: [02.05] Energies [ev] & Occupations =================================== Fermi Level [ev]: 5.046557 VBM / CBm [ev]: 0.00000 0.06025 Electronic Temp. [ev K]: 0.00 0.00 Bosonic Temp. [ev K]: 0.00 0.00 El. density [cm-3]: 0.146E+24 States summary : Full Metallic Empty 0001-0008 0009-0200 Indirect Gaps [ev]: 0.06025 3.57546 Direct Gaps [ev]: 0.06025 6.87127
Without_SOC case: [02.05] Energies [ev] & Occupations =================================== Fermi Level [ev]: 4.777401 VBM / CBm [ev]: 0.000000 0.332282 Electronic Temp. [ev K]: 0.00 0.00 Bosonic Temp. [ev K]: 0.00 0.00 El. density [cm-3]: 0.146E+24 States summary : Full Metallic Empty 0001-0004 0005-0100 Indirect Gaps [ev]: 0.332282 3.620628 Direct Gaps [ev]: 0.332282 7.137342
A clear shrink of the DFT-LDA gap due to spin-orbit occurs. Note that, having 8 electrons in the fundamental cell, due to the spinorial nature of the wavefunction, with SOC we have 8 occupied bands, while only 4 without. Here a picure of the DFT band structure for the two cases.

GaSb bands DFT

To go on with the tutorial please visit the next sections:

  1. Computing the QP band structure
  2. Computing the optical properties

References

[1] GaSb on wikipedia http://en.wikipedia.org/wiki/Gallium_antimonide
[2] Y.S. Kim et al. PHYSICAL REVIEW B 82, 205212 2010
[3] I. N. Remediakis and Efthimios Kaxiras Phys. Rev. B 59 5336 (1999)