Nonequilibrium absorption in bulk silicon

In this tutorial you will learn the basic concepts for computing changes in the optical properties of a semi-conductor in presence of a non-equilibrium electrons and holes distribution in conduction and valence band respectively. This tutorial is based on the results published in Phys. Rev. B

Under construction

The material: Silicon

We will study nonequilibrium absorption in bulk silicon. The same material used for this [Silicon|GW tutorial]

• FCC lattice
• Two atoms per cell (8 electrons)
• Lattice constant 10.183 [a.u.]
• Plane waves cutoff 15 Rydberg
• Direct gap 3.4 eV at Gamma
• Indirect gap 1.1 eV between Gamma= (0 0 0) and a point X', close to X=(0 1 0)

Silicon Band Structure

Tutorial files and Tutorial structure

Follow the instructions in Tutorials#Files and download/unpack the Silicon.tar.gz. Once the tutorial archive file is unzipped the following folder structure will appear

COPYING  README  Silicon/

with the Solid_Si folder containing

> ls Silicon/ 
PWSCF/  YAMBO/

In the Pwscf folder the student will find an input/output directory with input/output files for pw.x. The Silicon pseudopotential file is also provided.

> ls PWSCF/
convergence_scripts  input  output  psps

In the convergence_scripts you will find some useful shell scripts to run the ground state convergence runs for Silicon. The YAMBO folder contains the Yambo input/output files and core databases.

> ls YAMBO/
2x2x2/  4x4x4/  6x6x6/  8x8x8/  Convergence_Plots_and_Scripts/  GAMMA/

The core databases are provided for several k-points grids. In addition the folder Convergence_Plots_and_Scripts contains some scripts used for the [Silicon|GW tutorial] . Here we will just use the 8x8x8 (which is still very far from convergence) folder for computing (nonequilibrium) optical properties.

To run the tutorial you will need the standard executables

yambo
ypp 

plus the executables of the real time module of the Yambo code

yambo_rt
ypp_rt

Equilibrium optical properties

Enter the folder 8x8x8

> cd 8x8x8

For this step you can either compute static screening at equilibrium, or use the screening computed for the GW step in the Bethe-Salpeter

Energy shift in the band structure

The GW method is the standard approach to compute quasi-particle corrections. However, in presence of non-equilibrium carriers, or even at finite temperature the formulation of the GW self-energy needs to be refined. This is due to its frequency dependence, which results in extra terms when performing the analytic continuation from the Keldish contour (non-equilibrium case) or from the Matsubara axis (finite temperature case) to the real time/frequency axis. See for example this reference for the finite temperature case. On the other hand the COHSEX self-energy, being static, avoids this complication. This is why we will compute changes in the QP corrections within the COHSEX approximation.

Renormalization of the exciton binding energy

We finally perform a BSE calculation loading both the nonequilibrium carriers and the quasi-particle energy shifts.