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Green'sfunction calculation
For a given k-point ,spin component ,complex energy ,and denoting PLs by or ,let us define the following matrices:
k-space Overlap matrix linking PLs and (must hold). k-space Hamiltonian matrix linking PLs and (must hold). k-space secular matrix linking PLs and (must hold). Retarded Green's function matrix linking PL to . Retarded propagator matrix linking PL to . Advanced propagator matrix linking PL to . Self-energy matrix for PL with respect to its right/left PLs.In the AO representation, the element (,)of any of the above matrices, ,will be denoted by superscripts: .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, and eigenvectors, :
| 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 , and matrices for the coupled BL.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: with or | ||
Propagationaway from the interface PLs: or | ||
Propagationtowards the interface: or | ||
Propagationfrom interface PL to the oppositesub_BL: , or , | ||
Propagationfrom one sub_BL to the other: , or , |
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