Bandgap of Si in GW

Description: calculation of the bandgap of Si using various flavours of GW.

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To do GW calculations we have to follow a 3-step procedure.

Step 1: a DFT groundstate calculation

Everything starts with a standard DFT groundstate calculation (in this case PBE).

• INCAR

ISMEAR =  0
SIGMA  =  0.05
GGA    = PE

• KPOINTS

6x6x6
 0
G
 6 6 6
 0 0 0

• POSCAR

system Si
5.430
0.5 0.5 0.0
0.0 0.5 0.5
0.5 0.0 0.5
2
cart
0.00 0.00 0.00
0.25 0.25 0.25

Step 2: obtain DFT virtual orbitals

To obtain a WAVECAR file with a reasonable number of virtual orbitals (50-100 per atom) we need to restart from the previous groundstate calculation with ALGO=Exact, and manually set the number of bands by means of the NBANDS-tag. To obtain the corresponding WAVEDER file we additionally specify LOPTICS=.TRUE.

• INCAR

ALGO = Exact
NBANDS  = 64
LOPTICS = .TRUE.
NEDOS = 2000
## you might try
#LPEAD = .TRUE.

ISMEAR =  0
SIGMA  =  0.05
GGA    = PE

Mind: make a copy of your WAVECAR and WAVEDER files, as we will repeatedly need them in the following. For instance

cp WAVECAR WAVECAR.LOPTICS
cp WAVEDER WAVEDER.LOPTICS

Step 3: the actual GW calculation

Restart from the WAVECAR and WAVEDER files of the previous calculation, with

• INCAR

## Frequency dependent dielectric tensor including
## local field effects within the RPA (default) or
## including changes in the DFT xc-potential (LRPA=.FALSE.).
## N.B.: beware one first has to have done a
## calculation with ALGO=Exact and LOPTICS=.TRUE.
## and a reasonable number of virtual states (see above)
ALGO = GW0 ; LSPECTRAL = .TRUE. ; NOMEGA = 50
#LRPA = .FALSE. 
## be sure to take the same number of bands as for
## the LOPTICS=.TRUE. calculation, otherwise the
## WAVEDER file is not read correctly
NBANDS = 64

 QP shifts <psi_nk| G(iteration)W_0 |psi_nk>: iteration 1
 for sc-GW calculations column KS-energies equals QP-energies in previous step
 and V_xc(KS)=  KS-energies - (<T + V_ion + V_H > + <T+V_H+V_ion>^1  + <V_x>^1)

 k-point   1 :       0.0000    0.0000    0.0000
  band No.  KS-energies  QP-energies   sigma(KS)   V_xc(KS)     V^pw_x(r,r')   Z            occupation

      1      -6.4888      -6.8243     -10.9766     -10.4570     -17.5189       0.6458       2.0000
      2       5.4800       5.1162     -11.8859     -11.4060     -12.7290       0.7580       2.0000
      3       5.4800       5.1162     -11.8859     -11.4060     -12.7290       0.7580       2.0000
      4       5.4800       5.1162     -11.8859     -11.4060     -12.7290       0.7580       2.0000
      5       8.0443       8.3296      -9.7235     -10.1038      -5.7364       0.7502       0.0000
      6       8.0443       8.3296      -9.7235     -10.1038      -5.7364       0.7502       0.0000
      7       8.0443       8.3296      -9.7235     -10.1038      -5.7364       0.7502       0.0000
      8       8.8407       9.2475     -10.5130     -11.0594      -6.0662       0.7445       0.0000

 k-point   2 :       0.1667    0.0000    0.0000
  band No.  KS-energies  QP-energies   sigma(KS)   V_xc(KS)     V^pw_x(r,r')   Z            occupation

      1      -6.1276      -6.4734     -11.0208     -10.4905     -17.3978       0.6521       2.0000
      2       3.0946       2.6991     -11.4452     -10.9063     -13.1354       0.7340       2.0000
      3       5.0279       4.6595     -11.7159     -11.2282     -12.6625       0.7552       2.0000
      4       5.0279       4.6595     -11.7159     -11.2282     -12.6625       0.7552       2.0000
      5       7.8309       8.1065      -9.8441     -10.2097      -5.8680       0.7539       0.0000
      6       8.6943       8.9816      -9.8669     -10.2533      -5.6768       0.7436       0.0000
      7       8.6943       8.9816      -9.8669     -10.2533      -5.6768       0.7436       0.0000
      8      10.9341      11.2678     -10.4716     -10.9278      -5.5632       0.7316       0.0000

     ..         ..           ..           ..           ..           ..           ..           .. 
     ..         ..           ..           ..           ..           ..           ..           .. 
     ..         ..           ..           ..           ..           ..           ..           .. 
     ..         ..           ..           ..           ..           ..           ..           .. 


 k-point  16 :      -0.3333    0.5000    0.1667
  band No.  KS-energies  QP-energies   sigma(KS)   V_xc(KS)     V^pw_x(r,r')   Z            occupation

      1      -2.2240      -2.5911     -11.4857     -10.9550     -16.0663       0.6917       2.0000
      2      -2.2240      -2.5911     -11.4857     -10.9550     -16.0663       0.6917       2.0000
      3       1.8279       1.3698     -10.7735     -10.1380     -12.9637       0.7209       2.0000
      4       1.8279       1.3698     -10.7735     -10.1380     -12.9637       0.7209       2.0000
      5       8.2346       8.4128      -9.3111      -9.5472      -5.1975       0.7546       0.0000
      6       8.2346       8.4128      -9.3111      -9.5472      -5.1975       0.7546       0.0000
      7      12.2605      12.5170      -9.7969     -10.1486      -4.3607       0.7294       0.0000
      8      12.2605      12.5170      -9.7969     -10.1486      -4.3607       0.7294       0.0000

To quickly find the QP-energy of the highest lying occupied state, try

grep "     4     " OUTCAR | sort -n -k 3 | tail -1 | awk '{print $3}'

and for the lowest lying unoccupied state,

grep "     5     " OUTCAR | sort -n -k 3 | head -1 | awk '{print $3}'

Beyond the random-phase-approximation

To include local field effects beyond the random-phase-approximation in the description of the frequency dependent dielectric response function (local field effects in DFT) add the following line to your INCAR file:

LRPA = .FALSE.

and again restart from the WAVECAR and WAVEDER files from step 2.

Beyond G₀W₀: GW₀

The most usual step beyond single-shot GW (G₀W₀) is to iterate the quasi-particle energies in the Greens functions. This is the socalled GW₀ approximation. To have VASP do, for instance, 4 iterations of the QP-energies in G, add the following line to the INCAR file:

NELM = 4

and again restart from the WAVECAR and WAVEDER files from step 2.

To quickly find the QP-energy of the highest lying occupied state after 4 iterations of the QP energies in G, try

grep "     4     " OUTCAR | tail -16 | sort -n -k 3 | tail -1 | awk '{print $3}'

and for the corresponding lowest lying unoccupied state,

grep "     5     " OUTCAR | tail -16 | sort -n -k 3 | head -1 | awk '{print $3}'

Download

Si_bandgap_GW.tgz

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