The Atomic-Sphere Approximation ASA

The original material for the chapter may be found in the unpublished notes by Andersen [6.1] and in his paper [6.2]. 

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We have used the multisublattice generalization of the coherent potential approximation (CPA) in conjunction with the Linear-MufRn-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed for the local spin density approximation (LSDA) the Vosko-Wilk-Nusair parameterization". 

Second, using the fully relativistic version of the TB-LMTO-CPA method within the atomic sphere approximation (ASA) we have calculated the total energies for random alloys AiBi i at five concentrations, x — 0,0.25,0.5,0.75 and 1, and using the CW method modified for disordered alloys we have determined five interaction parameters Eq, D,V,T, and Q as before (superscript RA). Finally, the electronic structure of random alloys calculated by the TB-LMTO-CPA method served as an input of the GPM from which the pair interactions v(c) (superscript GPM) were determined. In order to eliminate the charge transfer effects in these calculations, the atomic radii were adjusted in such a way that atoms were charge neutral while preserving the total volume of the alloy. The quantity (c) used for comparisons is a sum of properly... 

The muffin-tin potential around each atom in the unit cell has been calculated in the framework of the Local-Spin-Density-Approximation using the ASW method. The ASW method uses the atomic sphere approximation (ASA), i.e. for each atom a sphere radius is chosen such that the sum of the volumes of all the overlapping spheres equals the unit cell volume. The calculation yields the expected ferromagnetic coupling between Cr and Ni. From the self-consistent spin polarized DOS, partial and total magnetic moment per formula unit can be computed. The calculated total magnetic moment is 5.2 pg in agreement with the experimental value (5.3 0.1 e calculations presented here have been performed... 

Electronic structure determinations have been performed using the self-consistent LMTO method in the Atomic Sphere Approximation (ASA). 

TABLES 1 and 2 show the calculated and measured results of splitting energies in WZ and ZB structures, respectively. Suzuki et al derived the values of A and Ar for WZ and ZN GaN and AIN from a full-potential linearised augmented plane wave (FLAPW) and band calculation [3,4], Another result with LAPW calculation was given by Wei and Zunger [5], Kim et al [6] determined them by the linear muffm-tin orbital (LMTO) method within the atomic sphere approximation (ASA). Majewski... 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

The investigations of Asada et al. and Christensen - were carried out with linear-muffin-tin orbitals within the atomic sphere approximation (LMTO- AS A) Within the muffin-tin model suitable s, p and d basis functions (muffin-tin orbitals, MTO) are chosen. In contrast to the APW procedure the radial wave functions chosen in the linear MTO approach are not exact solutions of the radial Schrodinger (or Dirac) equation. Furthermore, in the atomic sphere approximation (ASA) the radii of the atomic spheres are of the Wigner-Seitz type (for metals the spheres have the volume of the Wigner-Seitz cell) and therefore the atomic spheres overlap. The ASA procedure is less accurate than the APW method. However, the advantage of the ASA-LMTO method is the drastic reduction of computer time compared to the APW procedure. 

With the later introduction of the Linear Muffin-Tin Orbital (LMTO) method [46, 47, 48, 49, 50], a formulation of the multiple scattering problem in terms of Hamiltonians was introduced. This provided another way to gain more knowledge about the KKR method, which, although elegant, was not so easily understood. In the LMTO method one had to use energy linearizations of the MTOs to be able to put it into a Hamiltonian formalism. The two methods (KKR and LMTO) were shown [51] to be very closely related within the Atomic Sphere Approximation (ASA) [46, 52], which was used in conjunction with the LMTO method to provide an accurate and computationally efficient technique. 

Fig.1.4. Self-consistent energy-band structure for bcc tungsten obtained by the LMTO method within the atomic-sphere approximation (ASA) using local-density theory for exchange and correlation. Relativistic effects are included except spin-orbit coupling which is neglected...

As a final example of the role of the self-interaction component in some band gaps obtained with the exact exchange are listed in Table 2.8. All-electron OPM results [58] based on the Korringa-Kohn-Rostoker (KKR) method and the atomic sphere approximation (ASA) are compared with full potential plane-wave pseudopotential (PWPP) data [59] for C, Si and Ge. In addition to the x-only data, also the values resulting from the combination of the exact with either LDA or GGA correlation are listed. The singleparticle contribution A, i.e. the direct band gap, is separated from the xc-contribution A c to the band gap Eg,... 

The electronic structure is determined using the ab initio all-electron scalar-relativistic tight-binding linecir muffin-tin orbital (TB-LMTO) method in the atomic-sphere approximation (ASA). The nnderlying lattice, zincblende structure, refers to an fee Bravais lattice with a basis which contains a cation site (at a(0,0,0)), an anion site (at o(j,, )), and two interstitial sites occupied by empty spheres (at a(, 5, h) and a(, , )) which in turn are necessary for a correct description of open lattices . ... 

Linearized band structure methods were developed in the 1970s the linearized augmented plane wave (LAPW) method (36), the linear combination of muffin-tin orbitals (LMTO) method (37), the augmented spherical wave (ASW) method (38), and some others. In the LAPW method a warped muffin tin potential is frequently used, in which the real shape of the crystal potential in the interstitial region between the atomic spheres is taken into account. In the LMTO and ASW approaches the atomic sphere approximation (ASA) is frequently applied, in which— contrary to the muffin-tin approximation—overlapping atomic spheres are used. The crystal potential in the spheres is again assumed to be spherically symmetric. The sum of the atomic sphere volumes must be equal to the total volume of the unit cell. No interstitial space remains. 

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