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

We start from a density-functional theory (DFT) calculation. Then we apply MBPT to obtain the QP energies within the GW approximation and the absorption spectra from the solution of the BSE. The QP band structure will show the effect of the screened electron-electron interaction, while the absorption spectrum within the BSE accounts for the effect of the electron-hole interaction. When QP corrections are not available, they can be included using the scissor operator KfnQP_E, and the numerical values for "corrections" can be deduced from a GW calculation. For more information about solving the Equilibrium BSE equation refer to the following tutorial: Calculating optical spectra including excitonic effects: a step-by-step guide

(to delete open) In this case, do we actually compute the qp eng withing gw approximation? or we directy solve the bse eq (to delete close)

Enter the folder 8x8x8, and initialize yambo in the respective folder

$ cd 8x8x8
$ yambo

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

Static screening at equilibrium

First, to solve the BSE equation we need to calculate the statically screened electron-electron interaction (W), which is related to static diaelectric screening [math]\displaystyle{ \epsilon^{-1} }[/math].

[math]\displaystyle{ \Large \epsilon^{-1}_{\mathbf{G},\mathbf{G'}} (\mathbf{q}, \omega = 0) = \delta_{\mathbf{G},\mathbf{G'}} + v(\mathbf{v} + \mathbf{G}) \chi_{\mathbf{G},\mathbf{G'}(\mathbf{q}}, \omega = 0) }[/math]

To do this create the following input file:

$ yambo -X s -F 01_EQ_BSE_screening.in

with the following contents:

01_EQ_BSE_screening.in file (Expandable)

em1s                                 # [R][Xs] Statically Screened Interaction
% BndsRnXs
   1 |  30 |                         # [Xs] Polarization function bands
%
NGsBlkXs= 3                    Ry    # [Xs] Response block size
% LongDrXs
 1.000000 | 0.000000 | 0.000000 |    # [Xs] [cc] Electric Field
%

Where we changed the response block size NGsBlkXs = 3 Ry and number of polarization function bands BndsRnXs to be in the range of 1 to 30. Now run the file:

$ yambo -F 01_EQ_BSE_screening.in -J EQ_BSE

Solving the Bethe-Salpeter equation

To solve the Bethe-Salpeter equation, the latter is usually rewritten in the space of transitions between valence and conduction states as the (pseudo)eigenvalue problem for a two-particle Hamiltonian. For a non spin-polarized system and in the [math]\displaystyle{ q = 0 }[/math] limit, the two-particle Hamiltonian matrix elements are given by

[math]\displaystyle{ \Large H_{vc\mathbf{k},v'c'\mathbf{k'}} = (\varepsilon_{c\mathbf{k}} - \varepsilon_{v\mathbf{k}})\delta_{vv'}\delta_{cc'}\delta_{\mathbf{k},\mathbf{k'}} + (f_{c\mathbf{k}} - f_{v\mathbf{k}})\left[ 2V_{vc\mathbf{k},v'c'\mathbf{k'}} - W_{vc\mathbf{k},v'c'\mathbf{k'}}\right] }[/math]

where [math]\displaystyle{ vc\mathbf{k} }[/math] indicates the pair of quasiparticle states [math]\displaystyle{ v\mathbf{k} }[/math] and [math]\displaystyle{ c\mathbf{k} }[/math]. The first term on the RHS is the quasiparticle energy differences (diagonal only). The second term is the kernel which is the sum of the electron-hole exchange part V (which stems from the Hartree potential) and the electron-hole attraction part W (which stems from the screened exchange potential). The electron-hole attraction part W, depends on [math]\displaystyle{ \epsilon^{-1} }[/math] which was calculated in the previous section.

After calculating the equilibrium Bethe-Salpeter kernel, the resulting two-particle Hamiltonian can be diagonalized to compute the macroscopic dielectric function, which can be used to get absorption and EELS spectra. Calculation of BS kernel and diagonalization of Hamiltonian can be done in one step.

To generate the input file for the calculation of BSE kernel, run the following command:

$ yambo -o b -k sex -F 02_EQ_BSE_kernel_diag.in -y d -J EQ_BSE

with the following input

02_EQ_BSE_kernel_diag.in file (Expandable)

optics                           # [R] Linear Response optical properties
bss                              # [R] BSE solver
bse                              # [R][BSE] Bethe Salpeter Equation.
K_Threads=0                      # [OPENMP/BSK] Number of threads for response functions
BSKmod= "SEX"                    # [BSE] IP/Hartree/HF/ALDA/SEX/BSfxc
BSEmod= "resonant"               # [BSE] resonant/retarded/coupling
BSSmod= "d"                      # [BSS] (h)aydock/(d)iagonalization/(s)lepc/(i)nversion/(t)ddft`
BSENGexx= 2085             RL    # [BSK] Exchange components
BSENGBlk=-1                RL    # [BSK] Screened interaction block size [if -1 uses all the G-vectors of W(q,G,Gp)]
% BSEQptR
 1 | 1 |                             # [BSK] Transferred momenta range
%
% BSEBands
   2 |  7 |                         # [BSK] Bands range
%
% BEnRange
  2.20000 | 5.00000 |         eV    # [BSS] Energy range
%
% KfnQP_E
 0.70000 | 1.000000 | 1.000000 |      # [EXTQP BSK BSS] E parameters (c/v)
%
% BDmRange
 0.100000 | 0.100000 |         eV    # [BSS] Damping range
%
BEnSteps= 200                    # [BSS] Energy steps
% BLongDir
 1.000000 | 0.000000 | 0.000000 |        # [BSS] [cc] Electric Field versor
%
BSEprop= "abs"                   # [BSS] Can be any among abs/jdos/kerr/asymm/anHAll/magn/dich/photolum/esrt
BSEdips= "none"                  # [BSS] Can be "trace/none" or "xy/xz/yz" to define off-diagonal rotation plane

Where BSEBands was changed to include bands from 3 to 8, which defines the basis of quasiparticle pairs that are used to describe excitons. 2 valence and 4 conduction bands where included. BEnRange was increased to from 0-10 eV to 0-16 eV, and BEnSteps was increased from 100 to 200. The scissors operator KfnQP_E was used to match the positon calculated spectra with experimental values.

To run the file:

$ yambo -F 02_EQ_BSE_kernel_diag.in -J EQ_BSE

After the calculation has finished, the absorption spectra can be plotted with the following:

gnuplot commands (Expandable)

$ gnuplot
...
gnuplot> set x label "Energy (eV)"
gnuplot> set y label "Absorption" 
gnuplot> plot 'o-EQ_BSE.eps_q1_diago_bse' u 1:2 w l t '-k sex'

and the following plot will results (the "experimental" and "-k hartree" plots are added additionally for comparison)

Plot of the computed spectra (Expandable)

Generating non-equilibrium carriers

To simulate non-equilibrium absorption, we need to define the number of non-equilibrium carriers, specifically electrons in the conduction band and holes in the valence band. These carriers follow the Fermi-Dirac distribution given by:

[math]\displaystyle{ \Large f(E) = 1 / (exp((E - μ) / k_B T) + 1) }[/math]

Here:

• E is the energy level,
• [math]\displaystyle{ \mu }[/math] is the chemical potential (specific to electrons or holes),
• [math]\displaystyle{ T }[/math] is the effective temperature,
• [math]\displaystyle{ k_B }[/math] is the Boltzmann constant.

For the simulation, both electrons and holes are characterized by their own [math]\displaystyle{ \mu_i }[/math] and [math]\displaystyle{ T_i }[/math] .

The distribution of non-equilibrium carriers can be generated using Yambo’s post processing for real-time module (ypp_rt), which allows modeling of these conditions.

To generate to non-equilibrium carriers databases create input_ypp1 input_ypp2 files, with the following content:

input_ypp1 (Expandable)

RTDBs                      	# [R] Real-Time databases
Select_Fermi               	# [R] NEQ DBs according to Fermi distribution
% RTBands
   1 | 8 |               	# [RT] Bands range
%
% RTmuEh
-0.80 | 0.80 | eV  	        # [RT] Chemical potentials hole | electron
%
% RTtempEh
 0.04 	| 0.04 	| eV  	    # [RT] Effective temperature hole | electron
%
RTautotuneThr= 0.000100    	# [RT] Threshold to match no. of pumped holes and electrons.

input_ypp2 (Expandable)

RealTime                   	# [R] Real-Time Post-Processing
RToccupations              	# [R] Analize time-dependent occupations
RTenergy                   	# [R] Post-Processing kind: function of energy
TimeStep= 0.000000 	fs  	# Time step
% TimeRange
-1.000000 |-1.000000 | fs  	# Time-window where processing is done
%
#IncludeEQocc              	# [NEGF] Include also equilibrium occupations
#SkipFermiFIT              	# [NEGF] Do a Fermi Fit of occupations (and lifetimes ratio)
%QPkrange                	#  generalized Kpoint/Band indices
  1| 3|  1|8|
%

And then run the files:

$ ypp_rt 
$ ypp_rt -F input_ypp1 -J carriers -C carriers 
$ ypp_rt -F input_ypp2 -J carriers -C carriers 

The file input_ypp1 will generate a ndb.RT_carriers database which will contain the Fermi distribution of the electrons and the hole according to the specified chemical potential, RTmuEh, and for a specific temperature given by RTtempEh. The file input_ypp2 will generate a series of plots from the carriers database.

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.