[QE-users] Linear Dispersion Behavior of Graphene Supercells: Seeking Explanations and Solutions

[David Yao ANSI]

Please here is an image of the band structure plot I had for 3x3x1pristine graphene supercell. 
Is it normal to have this? And why do I see the linearity around the K high symmetry point in papers? 

https://tinyurl.com/3x3image 

And this is the content of input files: 
SCF 

&CONTROL 
calculation = "scf" 
outdir = "../temp/" 
prefix = "project" 
pseudo_dir = "../pseudo" 
restart_mode = "from_scratch" 
/ 

&SYSTEM 
a = 7.4017956849 
c = 18 
degauss = 1.00000e-02 
ecutrho = 3.60000e+02 
ecutwfc = 9.00000e+01 
ibrav = 4 
nat = 18 
ntyp = 1 
occupations = "smearing" 
smearing = "marzari-vanderbilt" 
/ 

&ELECTRONS 
conv_thr = 1.00000e-08 
mixing_beta = 7.00000e-01 
diagonalization = "david" 
electron_maxstep = 200 
/ 
K_POINTS {automatic} 
15 15 1 0 0 0 

ATOMIC_SPECIES 
C 12.01070 C.pbe-n-kjpaw_psl.1.0.0.UPF 


ATOMIC_POSITIONS {crystal} 
C 0.0000000000 0.0000000000 0.2499999997 
C 0.1111111269 0.2222222537 0.2499999994 
C -0.0000009773 0.3333325787 0.2499999998 
C 0.1111119864 0.5555564932 0.2499999999 
C -0.0000009773 0.6666664440 0.2499999998 
C 0.1111111269 0.8888888731 0.2499999994 
C 0.3333335560 0.0000009773 0.2499999998 
C 0.4444443680 0.2222221840 0.2500000005 
C 0.3333335560 0.3333325787 0.2499999998 
C 0.4444435068 0.5555564932 0.2499999999 
C 0.3333330000 0.6666670000 0.2499999993 
C 0.4444435068 0.8888880136 0.2499999999 
C 0.6666674213 0.0000009773 0.2499999998 
C 0.7777778160 0.2222221840 0.2500000005 
C 0.6666670000 0.3333330000 0.2500000025 
C 0.7777778160 0.5555556320 0.2500000005 
C 0.6666674213 0.6666664440 0.2499999998 
C 0.7777777463 0.8888888731 0.2499999994 



Bands 

&CONTROL 
calculation = "bands" 
outdir = "../temp/" 
prefix = "project" 
pseudo_dir = "../pseudo" 
/ 

&SYSTEM 
a = 7.4017956849 
c = 18 
degauss = 1.00000e-02 
ecutrho = 3.60000e+02 
ecutwfc = 9.00000e+01 
ibrav = 4 
nat = 18 
ntyp = 1 
nbnd = 144 
occupations = "smearing" 
smearing = "marzari-vanderbilt" 
/ 

&ELECTRONS 
conv_thr = 1.00000e-08 
mixing_beta = 7.00000e-01 
diagonalization = "david" 
electron_maxstep = 200 
/ 


K_POINTS {crystal_b} 
4 
0.0000000000 0.0000000000 0.0000000000 50 !G 
0.3333333333 0.3333333333 0.0000000000 30 !K 
0.5000000000 0.0000000000 0.0000000000 40 !M 
0.0000000000 0.0000000000 0.0000000000 0 !G 

ATOMIC_SPECIES 
C 12.01070 C.pbe-n-kjpaw_psl.1.0.0.UPF 


ATOMIC_POSITIONS {crystal} 
C 0.0000000000 0.0000000000 0.2499999997 
C 0.1111111269 0.2222222537 0.2499999994 
C -0.0000009773 0.3333325787 0.2499999998 
C 0.1111119864 0.5555564932 0.2499999999 
C -0.0000009773 0.6666664440 0.2499999998 
C 0.1111111269 0.8888888731 0.2499999994 
C 0.3333335560 0.0000009773 0.2499999998 
C 0.4444443680 0.2222221840 0.2500000005 
C 0.3333335560 0.3333325787 0.2499999998 
C 0.4444435068 0.5555564932 0.2499999999 
C 0.3333330000 0.6666670000 0.2499999993 
C 0.4444435068 0.8888880136 0.2499999999 
C 0.6666674213 0.0000009773 0.2499999998 
C 0.7777778160 0.2222221840 0.2500000005 
C 0.6666670000 0.3333330000 0.2500000025 
C 0.7777778160 0.5555556320 0.2500000005 
C 0.6666674213 0.6666664440 0.2499999998 
C 0.7777777463 0.8888888731 0.2499999994 


From: "David Yao ANSI" <dyansi1 at st.knust.edu.gh> 
To: users at lists.quantum-espresso.org 
Sent: Wednesday, May 31, 2023 5:33:43 PM 
Subject: Linear Dispersion Behavior of Graphene Supercells: Seeking Explanations and Solutions 

Dear All, 

In studying quantum capacitance of doped graphene, I recently made band structure calculations on pristine graphene using supercells of varying sizes. However, I encountered an unexpected result when increasing the supercell size from 2x2x1 to 4x4x1: the linear dispersion behavior around the K - high symmetry point gradually disappeared. This came as a surprise to me since previous articles I read reported a linear dispersion around the K point for similar supercell sizes. I did some also with a gamma Brillouin zone sampling during scf but had same results. 

What are possible reasons behind this and potential approaches to achieve the desired linear behavior in supercell calculations? 
Are there specific techniques or considerations that need to be taken into account when working with larger supercells of graphene? 

Thank you in advance for your assistance and expertise. 

Regards, 
David Ansi 
Master's student 
University of Sci. and Tech.(KNUST), Ghana 

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