||
Green'sfunction calculation
For a given k-point ,spin component
,complex energy
,and denoting PLs by
or
,let us define the following matrices:
The construction of the Green's functionfor a given BL will depend on its type. Below, we omit the ,
and
dependance for all matrices.
[method=spectral] | [method=inversion] |
First solve the diagonalizationproblem to obtain the eigevalues, ![]() ![]()
| Directly invert the secular equation: |
Both methods yield identical results, theirmain difference being the compulational resources that each requires. For[method=spectral],the main cost is the diagonallzation and the RAM required to store allthe eigenvectors but, in turn, is evaluated very fast at any energy. For [method=inversion],a matrix inversion is always required at each energy. Hence, for calculations
involving several energy points, [method=spectral]should be preferable as long as there are no RAM memory resctrictions.
We use the so-called Surface Green's FunctionMatching (SGFM) technique. If one denotes by ,
and
the Green's function matrices for the isolated sub_BLs, the Dyson equation:
allows to solve all
leading to matrix inversion-free expressions.
By projecting the Dyson equation at allPLs, and after a bit of manipulation of the formulas, the final expressionsfor all ,
,
and
matrices are the following:
PLs (![]() ![]() | [method=sgfm] | [method=sgfm_1] |
Interface PLs:
| ||
Diagonal G's: ![]() ![]() ![]() | ||
Propagationaway from the interface PLs: ![]() ![]() | ||
Propagationtowards the interface:![]() ![]() | ||
Propagationfrom interface PL to the oppositesub_BL: ![]() ![]() or ![]() ![]() | ||
Propagationfrom one sub_BL to the other: ![]() ![]() or ![]() ![]() |
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