Example: Benzenedithiol junction: Wide-Band-Limit

Download green_WBL.run

In the wide-band limit (WBL) the coupling to the leads is assumed to be independent of energy. Therefore one does not need to calculate any self-energies. This also means that the eigenspace of the Green’s function is independent of energy. It can therefore be diagonalized in advance, greatly speeding up the calculation of the DOS and the transmission.

In the example $AMSHOME/examples/adf/green_Al/green_WBL/green_WBL.run of green, the transmission of benzenedithiol junction in the wide-band limit (WBL) is calculated. In order to model the molecule-metal interface, we do need to include a few gold layers in the calculation. However, unlike before, only a single atomic layer as the principal layer is used.

Because a single atomic layer is an unnatural configuration for gold, a minor amount of smearing is necessary to make the calculation converge. The molecule is sandwiched in between the electrodes just like before (see Fig. 2 in the example for benzenedithiol). However, this time each atomic layer of gold gets its own fragment. The reason for this configuration is that if the WBL is used on the entire gold contact the result is an an unphysical coupling to the leads; even the gold atoms contacting the molecule would have a direct coupling to the environment. A much better result can be obtained by only using the WBL on the back-most atomic layer and letting the electrons propagate naturally through the rest of the contact. Because the WBL is computationally so inexpensive, we can easily calculate the DOS and transmission for 10,000 points instead of 1000.

A comparison of the resulting transmission with the calculation with self-energies is shown in the following figure:

../_images/green_WBL.png

The WBL shows good agreement with the non-WBL transmission around the Fermi energy (-0.195 Hartree or -5.306 eV). Note that the quality of the WBL depends on the choice of the coupling (ETA). For this particular contact geometry we obtain good agreement for ETA = 0.02 Hartree, but a better value may be found for other electrodes. Finally, the WBL can be incrementally improved by adding more gold layers to the extended molecule. For many layers it converges to the calculation with full self-energies.

#!/bin/sh

# In the wide-band limit (WBL) the coupling to the leads is assumed to be independent of energy.
# Therefore one does not need to calculate any self-energies.
# This also means that the eigenspace of the Green's function is independent of energy.
# It can therefore be diagonalized in advance,
# greatly speeding up the calculation of the DOS and the transmission.

# In the example $AMSHOME/examples/adf/green_Al/green_WBL.run of green,
# the transmission of  benzenedithiol junction, example green_BDT.run,
# in the wide-band limit (WBL) is calculated.
# In order to model the molecule-metal interface, we do need to include a few gold layers
# in the calculation.
# However, unlike in green_Au.run, only a single atomic layer as the principal layer is used.

# In green_BDT.run 3 layers of gold atoms from 1 fragment.
# In this example green_WBL.run each layer of gold atoms has its own fragment.
# Because a single atomic layer is an unnatural configuration for gold,
# a minor amount of smearing is necessary to make the calculation converge.
# The molecule is sandwiched in between the electrodes just like before
# (see the example for benzenedithiol, example green_BDT.run).
# However, this time each atomic layer of gold gets its own fragment.
# The reason for this configuration is that if the WBL is used on the
# entire gold contact the result is an an unphysical coupling to the leads;
# even the gold atoms contacting the molecule would have a direct coupling to the environment.
# A much better result can be obtained by only using the WBL on the back-most atomic layer
# and letting the electrons propagate naturally through the rest of the contact.
# Because the WBL is computationally so inexpensive, we can easily calculate
# the DOS and transmission for 10,000 points instead of 1000.

cp $AMSHOME/examples/adf/green_Al/Au.5p .
cp $AMSHOME/examples/adf/green_Al/Au.5p.dirac .

$AMSBIN/dirac < Au.5p.dirac
mv TAPE12 t12.rel

AMS_JOBNAME=Au $AMSBIN/ams <<eor
System
   Atoms
      Au 0.0 0.0 0.0
   End
end
Task SinglePoint
Engine ADF
  corepotentials t12.rel
  end
  create Au   file=Au.5p
  relativity
    level scalar
    formalism ZORA
  end
  xc
    lda SCF VWN
  end
EndEngine
eor

AMS_JOBNAME=layer $AMSBIN/ams <<eor
System
  atoms
     Au        0.000000   -4.996959    0.000000
     Au        0.000000   -2.498480   -1.442498
     Au        0.000000   -2.498480    1.442498
     Au        0.000000    0.000000   -2.884996
     Au        0.000000    0.000000    0.000000
     Au        0.000000    0.000000    2.884996
     Au        0.000000    2.498480   -1.442498
     Au        0.000000    2.498480    1.442498
     Au        0.000000    4.996959    0.000000
  end
end

Task SinglePoint

Engine ADF
  fragments
     Au  Au.t21
  end
  occupations Smearq=0.001
  relativity
    level scalar
    formalism ZORA
  end
  symmetry NOSYM
  title Principal layer
  xc
    lda SCF VWN
  end
EndEngine
eor

AMS_JOBNAME=molecule $AMSBIN/ams <<eor
System
  atoms
     C        -1.400000    0.000000    0.000000
     C        -0.700000    0.000000   -1.200000
     C        -0.700000    0.000000    1.200000
     C         0.700000    0.000000   -1.200000
     C         0.700000    0.000000    1.200000
     C         1.400000    0.000000    0.000000
     H        -1.200000    0.000000   -2.200000
     H        -1.200000    0.000000    2.200000
     H         1.200000    0.000000   -2.200000
     H         1.200000    0.000000    2.200000
     S        -3.200000    0.000000    0.000000
     S         3.200000    0.000000    0.000000
  end
end

Task SinglePoint

Engine ADF
  basis
     type DZP
     core Large
     createOutput None
  end
  relativity
    level scalar
    formalism ZORA
  end
  symmetry NOSYM
  title Benzenedithiol
  xc
    lda SCF VWN
  end
EndEngine
eor

AMS_JOBNAME=fock $AMSBIN/ams <<eor
System
  atoms
     Au       -9.911177   -6.662612    0.000000 adf.f=left
     Au       -9.911178   -4.164133   -1.442498 adf.f=left
     Au       -9.911178   -4.164133    1.442498 adf.f=left
     Au       -9.911178   -1.665653   -2.884996 adf.f=left
     Au       -9.911178   -1.665653    0.000000 adf.f=left
     Au       -9.911178   -1.665653    2.884996 adf.f=left
     Au       -9.911178    0.832826   -1.442498 adf.f=left
     Au       -9.911178    0.832826    1.442498 adf.f=left
     Au       -9.911178    3.331306    0.000000 adf.f=left
     Au       -7.555589   -4.996959    0.000000 adf.f=left.2
     Au       -7.555589   -2.498480   -1.442498 adf.f=left.2
     Au       -7.555589   -2.498480    1.442498 adf.f=left.2
     Au       -7.555589    0.000000   -2.884996 adf.f=left.2
     Au       -7.555589    0.000000    0.000000 adf.f=left.2
     Au       -7.555589    0.000000    2.884996 adf.f=left.2
     Au       -7.555589    2.498480   -1.442498 adf.f=left.2
     Au       -7.555589    2.498480    1.442498 adf.f=left.2
     Au       -7.555589    4.996959    0.000000 adf.f=left.2
     Au       -5.200000   -3.331306    0.000000 adf.f=left.3
     Au       -5.200000   -0.832826   -1.442498 adf.f=left.3
     Au       -5.200000   -0.832826    1.442498 adf.f=left.3
     Au       -5.200000    1.665653   -2.884996 adf.f=left.3
     Au       -5.200000    1.665653    0.000000 adf.f=left.3
     Au       -5.200000    1.665653    2.884996 adf.f=left.3
     Au       -5.200000    4.164133   -1.442498 adf.f=left.3
     Au       -5.200000    4.164133    1.442498 adf.f=left.3
     Au       -5.200001    6.662612    0.000000 adf.f=left.3
     C        -1.400000    0.000000    0.000000 adf.f=molecule
     C        -0.700000    0.000000   -1.200000 adf.f=molecule
     C        -0.700000    0.000000    1.200000 adf.f=molecule
     C         0.700000    0.000000   -1.200000 adf.f=molecule
     C         0.700000    0.000000    1.200000 adf.f=molecule
     C         1.400000    0.000000    0.000000 adf.f=molecule
     H        -1.200000    0.000000   -2.200000 adf.f=molecule
     H        -1.200000    0.000000    2.200000 adf.f=molecule
     H         1.200000    0.000000   -2.200000 adf.f=molecule
     H         1.200000    0.000000    2.200000 adf.f=molecule
     S        -3.200000    0.000000    0.000000 adf.f=molecule
     S         3.200000    0.000000    0.000000 adf.f=molecule
     Au        5.200001   -6.662612    0.000000 adf.f=right.3
     Au        5.200000   -4.164133   -1.442498 adf.f=right.3
     Au        5.200000   -4.164133    1.442498 adf.f=right.3
     Au        5.200000   -1.665653   -2.884996 adf.f=right.3
     Au        5.200000   -1.665653    0.000000 adf.f=right.3
     Au        5.200000   -1.665653    2.884996 adf.f=right.3
     Au        5.200000    0.832826   -1.442498 adf.f=right.3
     Au        5.200000    0.832826    1.442498 adf.f=right.3
     Au        5.200000    3.331306    0.000000 adf.f=right.3
     Au        7.555589   -4.996959    0.000000 adf.f=right.2
     Au        7.555589   -2.498480   -1.442498 adf.f=right.2
     Au        7.555589   -2.498480    1.442498 adf.f=right.2
     Au        7.555589    0.000000   -2.884996 adf.f=right.2
     Au        7.555589    0.000000    0.000000 adf.f=right.2
     Au        7.555589    0.000000    2.884996 adf.f=right.2
     Au        7.555589    2.498480   -1.442498 adf.f=right.2
     Au        7.555589    2.498480    1.442498 adf.f=right.2
     Au        7.555589    4.996959    0.000000 adf.f=right.2
     Au        9.911178   -3.331306    0.000000 adf.f=right
     Au        9.911178   -0.832826   -1.442498 adf.f=right
     Au        9.911178   -0.832826    1.442498 adf.f=right
     Au        9.911178    1.665653   -2.884996 adf.f=right
     Au        9.911178    1.665653    0.000000 adf.f=right
     Au        9.911178    1.665653    2.884996 adf.f=right
     Au        9.911178    4.164133   -1.442498 adf.f=right
     Au        9.911178    4.164133    1.442498 adf.f=right
     Au        9.911177    6.662612    0.000000 adf.f=right
  end
end

Task SinglePoint

Engine ADF
  fragments
     left        layer.t21
     left.2      layer.t21
     left.3      layer.t21
     molecule    molecule.t21
     right.3     layer.t21
     right.2     layer.t21
     right       layer.t21
  end
  relativity
    level scalar
    formalism ZORA
  end
  symmetry NOSYM
  title Benzenedithiol
  xc
    lda SCF VWN
  end
EndEngine
eor

$AMSBIN/green << eor
DOS fock.results/adf.rkf
TRANS fock.results/adf.rkf
EPS -0.5 0 10000
ETA 1e-6
LEFT
    FRAGMENT left
    ETA 2e-2
END
RIGHT
    FRAGMENT right
    ETA 2e-2
END
NOSAVE DOS_B, TRANS_B
eor

echo ""
echo "Contents of DOS_A:"
cat DOS_A
echo "END"
echo ""
echo "Contents of TRANS_A:"
cat TRANS_A
echo "END"