How to treat low dimensional systems

In this tutorial you will learn how to treat low-dimensional material. A small k-sampling is used to reduce the computational cost/time. Later on you can repeat this tutorial but using denser k-grids to check the convergence.

Prerequisites

• yambo executable
• ypp executable
• gnuplot, for plotting spectra
• Have Completed the First steps: a walk through from DFT to optical properties tutorial
• Have Completed the How to obtain an optical spectrum - a guide through the workflow of calculations tutorial

Avoid numerical divergences using the Random Integration Method (RIM)

In DFT runs of low-dimensional materials low dimensional k-grids are generally used. (i.e. NxNx1 for a 2D sheet perpendicular to the z direction) This can create numerical problems in the convergence of the many-body (MB) results due to the divergence at small q of the coulomb potential, which appears in all the main equations. To eliminate this problem YAMBO uses the so-called Random Integration Method which means to use a Monte Carlo Integration with Random Q-points whose number RandQpts will be given in input. For example the exchange self-energy matrix element which is:

assuming that the integrand is a smooth function of momenta it can be approximated as

where the small Brillouin Zones (sBZ) relative to a given q-point are the Brillouin Zones of the momenta vectors lattice. They are chosen in such a way to cover the whole BZ. In the RIM run-level yambo calculates the integrals of the symmetrized Coulomb potential

So we are ready to start the tutorial First go in the proper directory and have a look if all the required databases are there

$ cd .YAMBO_TUTORIALS/hBN-2D/YAMBO
$ ls ./SAVE
$ ndb.gops ndb.kindx ns.kb_pp_pwscf_fragment_1...

Create the input to generate the ndb.RIM database

$ yambo -F yambo_RIM.in -r 

and change the following variables

RandQpts = 1000000   # [RIM] Number of random q-points in the BZ
RandGvec = 100  RL   # [RIM] Coulomb interaction RS components 

N.B RandGvec=100 means to use the RIM for the first 100 G-components of the coulomb potential (Suggestion : later you can check convergence of the HF gap changing these two values) Close input and Run yambo

$ yambo -F yambo_RIM.in -J 2D

At the end you find a new database ndb.RIM and a new report file r-2D_rim_cut. Open it and look inside

[04.02] RIM integrals
 =====================

 Gamma point sphere radius         [au]:  0.08028
 Points outside the sphere             :  800231
 [Int_sBZ(q=0) 1/q^2]*(Vol_sBZ)^(-1/3) = 7.667102
                                should be < 7.795600
 [WR./2D//ndb.RIM]-------------------------------------------
  Brillouin Zone Q/K grids (IBZ/BZ):   7   36    7   36
  Coulombian RL components        : 209
  Coulombian diagonal components  :yes
  RIM random points               : 1000000
  RIM  RL volume             [a.u.]: 0.390129
  Real RL volume             [a.u.]: 0.390112
  Eps^-1 reference component       :0
  Eps^-1 components                : 0.00      0.00      0.00
  RIM anysotropy factor            : 0.000000
 - S/N 005962 -------------------------- v.04.01.02 r.00120 -

 Summary of Coulomb integrals for non-metallic bands |Q|[au] RIM/Bare:

 Q [1]:0.1000E-40.9835 * Q [2]: 0.256404 1.093779
 Q [5]: 0.444104 1.031700 * Q [3]: 0.512807 1.023425
 Q [6]: 0.678380 1.013439 * Q [4]: 0.769211 1.010447
 Q [7]: 0.888208 1.007869

The RIM and Real RL Volumes are quite similar so one million random q-points seems a reasonable number, but again you are invited, later on, to check the convergence of one of the main observables (i.e. HF gap, GW gap, absorption..) changing this number.

Close the report file and perform a calculation of the exchange self-energy to estimate the HF gap (very fast calculation) using the RIM. Generate the input

$ yambo -F yambo_HF.in -r -x  -J 2D 

In order to calculate the HF gap only for the last k-point, change the last line as

%[Variables#QPkrange|QPkrange]]                    # [GW] QP generalized Kpoint/Band indices
7|  7|  4|  5|

Close the input and run yambo

$ yambo -F yambo_HF.in -J 2D

At the end you will find the report file r-2D_HF_and_locXC_rim_cut and the output file o-2D.hf. Open them and have a look

In the output file o-2D.hf you will see

#  K-point    Band       Eo         Ehf        DFT        HF
  7.00000    4.00000    0.00000   -3.21640  -16.21949  -19.43589
  7.00000    5.00000    4.40109    9.68783  -11.10752   -5.82078

The HF gap is 12.91 eV obtained subtracting the 2 values in the fourth column Doing a similar HF calculation without generating and reading the RIM database the HF gap results to be 12.69 eV. Indeed the presence of the numerical instability is evident only using denser k-grids with respect to the one used in this Tutorial (6x6x1) Suggestion: later you can generate other SAVE directories with denser k-grids and check the HF gap. The results of the HF gap calculated with different k-grids without (noRIM) and with Random Inetgration Method (RIM) to show up the problem are reported:

            noRIM       RIM
 6x6x1      12.69eV    12.91eV
 12x12x1    12.80eV    12.89eV
 15x15x1    12.96eV    12.90eV
 45x45x1    15.52eV    12.96eV

So the home message is : use always the RIM in MB simulations of low-dimensional materials. Before start the next section please enter in the 2D directory and delete ndb.HF_and_locXC.

Generate a truncated coulomb potential

To simulate an isolated nano-material a convergence with cell vacuum size is in principle required, like in the DFT runs. The use of a truncated Coulomb potential allows to achieve faster convergence eliminating the interaction between the repeated images along the non-periodic direction (see i.e. D. Varsano et al Phys. Rev. B and .. ) In this tutorial we learn how to generate a box-like cutoff for a 2D system with the non-periodic direction along z.

In YAMBO you can use :

• spherical cutoff (for 0D systems)
• cylindrical cutoff (for 1D systems)
• box-like cutoff (for 0D, 1D and 2D systems)

The Coulomb potential with a box-like cutoff is defined as

Then the FT component is

where

For a 2D-system with non period direction along z-axis we have

Important remarks:

• the Random Integration Method (RIM) is required to perform the Q-space integration
• for sufficiently large supercells a choose L_i slightly smaller than the cell size in the i-direction ensures to avoid interaction between replicas

Create the input file:

$ yambo -F yambo_cut2D.in  -r  -J 2D 

Change the following variables as:

CUTGeo = "box z" # [CUT] Coulomb Cutoff geometry: box/cylinder/sphere X/Y/Z/XY..
% CUTBox
 0.00     | 0.00     | 32.0    |        # [CUT] [au] Box sides

Close the input file and run yambo:

$ yambo -F  yambo_cut2D.in  -J 2D

in the directory 2D you will find a new database ndb.cutoff

Use the truncated coulomb potential in a G₀W₀-PPA calculation

Generate the input reading the databases in the 2D directory

$ yambo -J 2D -F yambo_G0W0.in -p p -g n -r -k hartree -V qp
In the input  change the following variables
EXXRLvcs = 40 Ry    # [XX] Exchange RL components
NGsBlkXp = 4  Ry    # [Xp] Response block size 
BndsRnXp
  1 |  40 |                 # [Xp] Polarization function bands
QPkrange                    # [GW] QP generalized Kpoint/Band indices
 7|  7|  4| 5|

Close the input and run yambo

$ yambo -F yambo_G0W0.in -J 2D

At the end you will find a new report file r-2D_em1d_ppa_HF_and_locXC_gw0_rim_cut, open it and have a look. You will find also a new output file o-2D.qp. If you open yoi will find that now the G₀W₀ gap is 4.40 eV (LDA) + 3.72 eV (G0W0 correction) = 8.12 eV Indeed as you have learned in the previous tutorials the correlation part of the self-energy strongly reduces the HF gap. Moreover you should note that the QP correction is larger that one found in the h-BN bulk. Why?

Use the truncated coulomb potential in a BSE calculation

Generate the input for the BSE calculation:

$ yambo -J 2D -F yambo_BSE.in -r -o b -p p -y d -k sex -V all

Some remarks: the largest verbosity is used -V all and a long input file is generated; with the option -J 2D -p p the static part of the screening matrix is read from the ndb.pp databases Put the number of G-vectors in the exchange and correlation part of the kernel like in the h-BN bulk

BSENGexx= 40 Ry    # [BSK] Exchange components
BSENGBlk= 4 Ry    # [BSK] Screened interaction block size

use the simple rigid scissor to open the correct the KS energies

% KfnQP_E
3.72000000 | 1.000000 | 1.000000 |        # [EXTQP BSK BSS] E parameters  (c/v) eV|adim|adim

Set the number of bands involved in the BSE matrix to:

% BSEBands
 2 |  6 |                 # [BSK] Bands range

Increase the number of energy steps

BEnSteps= 500                # [BSS] Energy steps

To do the next Tutorial we need to write the excitonic WFs, so uncomment the following line

#WRbsWF                      # [BSS] Write to disk excitonic the WFs

Close the input file and run yambo

$ yambo -J 2D -F yambo_BSE.in 

Look at the report file r-2D_optics_bse_bsk_bss_em1d_ppa_rim_cut and use gnuplot to plot o-2D.eps_q1_diago_bse

Analyze the differences with corresponding GW and BSE calculations without the use of a truncated coulomb potential

Generate the ndb.RIM in a new directory 2D_NC

 $ yambo -J 2D_NC -F yambo_RIM.in 

Open the input file yambo_G0W0.in and set back CUTGeo= "none"

Close the input and run yambo

$ yambo -J 2D_NC -F yambo_G0W0.in 

You will find a new report r-2D_NC_optics_bse_bsk_bss_em1d_ppa_rim_cut. You are invited to see the difference with the previous one r-2D_optics_bse_bsk_bss_em1d_ppa_rim_cut

and also a new output file o-2D_NC.qp is present. Open and check the QP correction this time A value of 2.84 eV instead of 3.72 eV is obtained. Are you able to explain this result?

Now redoo the BSE calculation but without reading the cutoff database and applying the QP correction just obtained without using the cutoff.

To do that, open the yambo_BSE.in and put CUTGeo= "none" and set the QP correction of 2.84 eV, just obtained without uisng the cutoff.

% KfnQP_E
2.830000 | 1.000000 | 1.000000 |        # [EXTQP BSK BSS] E parameters  (c/v) eV|adim|adim

Close the input file and run yambo

$ yambo -J 2D_NC -F yambo_BSE.in

Plot the dielectric functions

$ gnuplot 
gnuplot> plot 'o-2D_NC.eps_q1_diago_bse' u 1:4 w l title 'GW without cutoff' ,'o-2D.eps_q1_diago_bse' u 1:4 w l title ' GW with cutoff' , 'o-2D_NC.eps_q1_diago_bse' u 1:2 w l title 'BSE without cutoff' ,'o-2D.eps_q1_diago_bse' u 1:2 w l title 'BSE with cutoff'

You can note that the energy of the peaks in the two GW spectra (with and without cutoff) are quite different, while are much more similar in the BSE spectra. Are you able to explain why? Remark: To obtain a converged optical spectrum denser mesh of k-points are required. The convergence in k-space can be done separately for GW and BSE calculations, because generally more k-points are needed to converge the exciton with respect to the quasi-particle properties.