Basic Theory

A central concept in the KKR Green's function method is the use of the Dyson equation

$$\mathbf{G} = \mathbf{G}^r + \mathbf{G}^r \Delta V \mathbf{G}$$ (1)

to connect the Green's functions of the true physical system $\mathbf{G}$, with an arbitrary chosen reference system described by $\mathbf{G}^r$, which have a difference in the potential $\mathbf \Delta V$. The idea of the screened KKR is the construction of a reference system for which the Green´s function decays exponentially in real space. This would lead to short ranged interactions limited to neighbouring atoms only. For this purpose we use a lattice of strongly repulsive muffin-tin potentials. This way we can accomplish screening and thus a tight-binding, band diagonal, form for the structural Green's function matrix by virtue of the reference system. The true physical system is not required to have short range interactions. The KKR and Screened KKR methods and their properties are described in detail [1,2,3,4,5,6,7,8,9,10,11] [12], [13] and in references therein. Here we will briefly discuss the application of the SKKR to layered structures and semiinfinite systems, the case of 3D bulk systems is similar, the only difference is that the 2D Fourier transform has to be replaced by a 3D Fourier transform.

The Green's function is expressed in the usual cite-centered angular momentum expansion

$$G({\bf r} + {\bf R}_n , {\bf r}'+ {\bf R}_{n'} ;E ) = \delta_{nn'... ...m_{L,L'} R^n_L ({\bf r} ;E ) \, G^{nn'}_{LL'}(E ) \, R^{n'}_{L'}({\bf r}' ;E )}$$ (2)

where $G^n_s$ is the Green function for a single scattering potential in cell $n$ in an otherwise free space. Multiple scattering contributions are contained in the second term through the so-called structural Green function $G^{nn'}_{LL'}(E)$. The index $L$ denotes the angular momentum indices $(l,m)$ and $R^{n}_{L}({\bf r};E)$ are regular partial-wave solutions of the Schrödinger equation for the potential $V^n({\bf r})$ and energy $E$.

For layered systems we apply a 2D-Fourier transform [13]:

$$G_{LL'}^{r \; ii'} ({\bf q}_\parallel ; E ) = \sum_{\nu'} e^... ... {\bf\chi}_{\nu'} )} \; G_{LL'}^{r \; ii' \; \nu - \nu' }(E )$$ (3)

with the 2D wave vector ${\bf q}_\parallel$, and ${\bf\chi}_\nu$ are 2D lattice vectors. Then the Dyson equation takes the form:

$$G_{LL'}^{ii'} ({\bf q}_\parallel ; E ) = G_{LL'}^{r \; ii'} ... ..._{l''}^{i''} (E) \; G_{L''L'}^{i''i'} ({\bf q}_\parallel ; E )$$ (4)

where $i$ and $\nu$ denotes the layer index and the index for the atomic site in the layer respectively. $\Delta t^{n}_{l} (E)$ is the difference in the scattering t-matrix between the true and the reference system. The $t$-matrix is obtained through:

$$t^{n}_l (E) = \int^{R_{c}}_{0}\; dr \,r^2 \; j_l(\sqrt{E}r) \;V^n(r)\; R^n_l(r;E),$$

where $j_l(\sqrt{E}r)$ are spherical Bessel functions, and $R_c$ denotes the muffin-tin sphere (or ASA sphere when the Atomic Sphere Approximation is used). We consider here only spherical potentials but the extension to non-spherical, full potential, is straightforward.

The calculation of the Green's function requires a matrix inversion. To simplify the formulas we drop all angular momentum and cite indices. Equation 4 can be easily transformed to:

$$G = \Delta t^{-1} - \Delta t^{-1} \tau \Delta t^{-1}$$ (5)

where

$$\tau = (G^r - \Delta t^{-1})^{-1}$$ (6)

Due to the exponential decrease of the screened structure constants, the coupling can be limited to few neighbouring shells, so that the 2D Fourier transform of the reference Green's function has the structure of a band matrix coupling only neighbouring layers. Therefore, one can simplify the matrix $G^{r, ii'}$ by grouping several atomic layers into so-called principal layers [14,15], the coupling of which is limited to the first nearest principal layers only. Assuming a system of identical repulsive potentials in a lattice, the reference Green's function has the form:

$$G^{r \;\alpha \beta} = G^{r \; 00} \delta_{\alpha, \beta} + ... ...\delta_{\alpha ,\beta-1}+ G^{r \; 10} \delta_{\alpha ,\beta+1}$$ (7)

where $\alpha, \beta$ are principal layer indices.

For the case of a finite slab system the $G^r$ has a finite rank, is band diagonal and has the form of eq. 7, while $\Delta t^{-1}$ is diagonal in the cite index and in the principal layer representation eq.4 has the form of the Dyson equation for a linear chain with nearest neighbour coupling only. For such systems, the diagonal blocks of the real system Green´s function $G^{nn}$, which are required to obtain the charge density, $n({\bf r}; E) = -\frac{1}{\pi}ImTrG({\bf r}, {\bf r};E)$ can be calculated [16], [17] with efford that scales linear with the number of principal layers in the slab $N$. This $N$-scaling behaviour is one of the main advantages of the screened KKR approach.

The band diagonal form of the Green's function allows us to treat also semiinfinite systems. In this case we need to invert an infinite matrix. We can divide the system in three regions: an intermediate region (I), embedded into two (unperturbed) semi-infinite left (L) and right (R) halfspaces. The regions R and L are characterized by bulk potentials, so that the new selfconsistency process affects only the potentials of the region I. Using an inversion-by-partitioning tequnique it is easy to see that, embedding region I into the semiinfinite L and R host media would only affect the $G^{11}$ and $G^{NN}$ blocks of the Green's function matrix. This embedding information is included in the surface Green's function $G^{11}_{surf}$ (left halfspace), $G^{NN}_{surf}$ (right halfspace) which we obtain using an iterative procedure, the so called decimation method, as described in ref [14,15]. Usually 5-6 iterations are enough, to obtain well converged surface Green's functions. Only a few ${\bf q}_\parallel$ points (i.e. close to Van Hove singularities) require additional effort.