How to treat low dimensional systems

In this tutorial you will learn for a low-dimensional (2D) material how to:

• use the Random Integration Method (RIM) to avoid numerical divergence of the self-energy when low-dimensional dense k-sampling is used
• generate a truncated coulomb potential with a box-like cutoff
• use this truncated coulomb potential in the GW calculation
• use this truncated coulomb potential in the BSE calculation
• analyze the difference with corresponding calculations without the use of a truncated potetnial

Prerequisites

• Complete the Generating the Yambo databases tutorial
• SAVE folder for 2D hBN.
• yambo executable
• ypp executable
• Run Initialization

Use the RIM to avoid the numerical divergence of the self-energy when low-dimensional dense k-sampling is used

The calculation of the self-energy requires a BZ integration performed numerically on a suitable k-grid. The use of a low-dimensional k-sampling (i.e. in a 2D material a 2D k-grid like n x n x 1) implies a numerical divergence due to the q=0 term of the Coulomb potential (see i.e. the exchange self-energy espression) when a dense BZ sampling is used. To eliminate the numerical divergence YAMBO use the so-called Random Integration Method performing a Monte Carlo integration near q=0 using a set of random Q-points whose number RandQpts is given in the input

Create the input to generate the ndb.RIM database

$ yambo -F yambo_RIM.in -r 

 RandQpts= 1000000                   # [RIM] Number of random q-points in the BZ
 RandGvec= 1            RL    # [RIM] Coulomb interaction RS components 

Close input and Run yambo

$ yambo -F yambo_RIM.in -J 2D

At the end in the 2D directory you will have a new database ndb.RIM

Generate a truncated coulomb potential/ndb.cutoff database (yambo -r)

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

Creation of the input file:

$ yambo -F yambo_cut2D.in  -r

Open the input file yambo_cut2D.in

Change the variables inside as:

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

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

Run yambo:

$ yambo -F  yambo_cut2D.in  -J 2D

in the directory 2D you will find the two new databases

ndb.RIM		ndb.cutoff