Example: DOS and transmission: Aluminium¶
As an example of a non-self-consistent Green’s function calculation, we will look at the density of states (DOS) and transmission of an infinite 1D chain of Aluminum atoms.
First we need to perform a single-point calculation with ADF on a principal layer, consisting, in this case, of four atoms. Since bulk Aluminum has an FCC structure with a lattice constant of 4.05 Angstrom, the nearest neighbor distance is approximately 2.83 Angstrom. green requires SYMMETRY NOSYM, so we have the following input file for the principal layer:
$ADFBIN/adf << eor TITLE Principal layer ATOMS Al -4.290000 0.000000 0.000000 Al -1.430000 0.000000 0.000000 Al 1.430000 0.000000 0.000000 Al 4.290000 0.000000 0.000000 END SYMMETRY NOSYM BASIS Type DZP Core Large CreateOutput None END eor mv TAPE21 layer.t21
The bulk contact geometry consists of three principal layers:
$ADFBIN/adf << eor TITLE Bulk ATOMS Al -15.730000 0.000000 0.000000 f=left Al -12.870000 0.000000 0.000000 f=left Al -10.010000 0.000000 0.000000 f=left Al -7.150000 0.000000 0.000000 f=left Al -4.290000 0.000000 0.000000 f=center Al -1.430000 0.000000 0.000000 f=center Al 1.430000 0.000000 0.000000 f=center Al 4.290000 0.000000 0.000000 f=center Al 7.150000 0.000000 0.000000 f=right Al 10.010000 0.000000 0.000000 f=right Al 12.870000 0.000000 0.000000 f=right Al 15.730000 0.000000 0.000000 f=right END SYMMETRY NOSYM FRAGMENTS left layer.t21 center layer.t21 right layer.t21 END SCF Iterations 100 END eor mv TAPE21 bulk.t21
Notice that we have increased the number of SCF iterations. The combination of SYMMETRY NOSYM with a 1D chain of metal atoms generally leads to convergence problems. This is the main reason why the principal layer consists of only four atoms. Fortunately, for larger 3D contacts, the convergence is generally better.
From the bulk TAPE21 file green can calculate the self-energies of the left and right contacts. As discussed in the introduction, the self-energy of the left contact needs the center and right fragments of the bulk calculation, and the self-energy of the right contact needs the center and left fragments. Since we need a self-energy matrix for every energy for which we want to calculate the DOS and transmission, already here we have to specify the energy range. We take 1000 points between -0.4 and 0 Hartree.
$ADFBIN/green << eor SURFACE bulk.t21 FRAGMENTS center right END EPS -0.4 0 1000 ETA 1e-6 eor mv SURFACE left.kf $ADFBIN/green << eor SURFACE bulk.t21 FRAGMENTS center left END EPS -0.4 0 1000 ETA 1e-6 eor mv SURFACE right.kf
Since we want to calculate the DOS and transmission of bare aluminum, we can reuse the bulk.t21 file for the extended molecule. We couple the left self-energy to the “left” fragment and the right self-energy to the “right” fragment in bulk.t21. Since we performed restricted ADF calculations, there is no difference between spin-A and spin-B and we can omit spin-B from the calculation.
$ADFBIN/green << eor DOS bulk.t21 TRANS bulk.t21 EPS -0.4 0 1000 ETA 1e-6 LEFT left.kf FRAGMENT left END RIGHT right.kf FRAGMENT right END NOSAVE DOS_B, TRANS_B eor
The resulting DOS and transmission are shown in the following figure:
As would be expected for a 1D system, the DOS shows Van Hove singularities at the band edges. Apart from oscillations due to the finite size of the system in ADF, the transmission only reaches integer values. Between approximately -0.35 and -0.15 Hartree, only the sigma channel contributes to the transmission. Above -0.15 Hartree also the two pi channels start to contribute.