Surface Green’s function

B. Wenzien J. Kudrnovsky, V. Drchal and M. Sob, On the calculation of the surface Green s function by the tight-binding linear-muffin tin orbital method, J. Phys. Condens. Matter 1, 9893 (1989). 

As a result, the self-energies are given by the coupling matrices Hcl, Hen, and the surface Green s functions of the semi-infinite leads, Hzl(E) and HRR(E). Explicitly, these surface Green s functions are ... 

A more elegant and convenient way than Bq.(2.174) to compute the surface Green s function from the bulk Green s functions, is to consider Hamiltonian matrix elements Hfj to belong to the Hamiltonian matrix of the closed chain and to consider... 

The transfer equation for the surface Green s function matrix... 

Note that for semi-infinite media in case of conservative scattering, a = 1, we find from Eqs. (33a), (33) and (30) that the energy flux integral for the surface Green s function matrix will be equal to zero. 

On taking into account Eq. (33), and also Eq. (28), we rewrite the transfer equation (20) for the surface Green s function matrix in the form... 

As a result, the original homogeneous transfer equation (20) for the surface Green s function matrix has been transformed into the transfer equation (37) with a modified phase matrix (36) and fictitious internal primary sources. When the transfer equation (37) is solved separately for the two internal primary source function vectors qi(r,M) and q2(r,w)givenby Eq. (39), the vector parameters g (0,//o) needed for compiling the complete solution can... 

In the following, it will be shown that the original surface Green s function matrix can be expressed immediately in terms of the surface Green s function matrix for the transformed transfer equation. 

Thus, Eq. (43) is the a-transformation formula for the surface Green s function matrix of a semi-infinite medium, which enables one to retrieve the original surface Green s function matrix, if the surface Green s function matrix of the transformed transfer equation (41) is known. With an appropriate choice of the free parameters in Eq. (36), it may be much easier to solve the transformed transfer equation (41) rather than to solve the original transport equation Eq. (20). 

Let us suppose now, that the effective single-scattering albedo of the modified phase matrix (m, v) has been reduced substantially by means of an appropriate choice of the free parameter vectors in Eq. (36). Then, with increasing optical depth, the surface Green s function matrix G (t, m 0, /Iq ) rapidly fades away, and Eq. (45) asymptotically yields... 

It is remarkable that the free function vectors Ci(m) and Cafw) determining the (p-transformation, do not appear explicitly in the a-transformation formulae for the albedo matrix or for the surface Green s function matrix, Eqs. (49) and (45), respectively. 

The surface Green s function matrix for a finite medium... 

For a medium of finite optical thickness b, the -transformation formulae become a bit more complex. On starting with the transfer equation Eq. (37) for the surface Green s function, we find a partial solution... 

Then, obviously, the transformation formulae (59) and (61) can be formulated solely in terms of the surface Green s function matrix Gc (r, w 0, //q ) ... 

While the electrodes provide the surface Green s function (g ) in the self-energy term, the linkers determine the interaction term. It is also used when calculating SDOS, i.e. SDOS = -7r-ilm[7r(g"5)]. 

T. Rother, M. Kahnert, A. Doicu, J. Wauer, Surface Green s function of the Helmholtz equation in spherical coordinates, PIER 38, 47 (2002)... 

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