First Principles Theory of Disordered Alloys and Alloy Phase Stability (original) (raw)
Carlomethods or the CVM. The difficulty with such an approacnistnatcomplex electronically mediatedinteractions aremapped ontoaneffective classical Hamiltonian. Unfortunately, thereisno apriori guarantee thatsucha procedure iseither uniqueor " rapidly convergent. In addition, since theparameters areextracted from calculations on smallunitcell systems, thereispossible thattheinteractions contain contributions (e.g. fromtheMadelnng energy) thatwill excessively favor suchstructures withrespect tothe disordered phase. Inthese lecture noteswe shall reviewtheLDA-KKR-CPA method fortreating the electronic structure and energetics ofrandom alloys and theMF-CF and GPM theories ofordering and phasestability thathavebeenbuilt on theLDA-KKR-CPA description ofthedisordered phase.Thus,we takethepoint ofviewthatmuch can be learned about metallic alloys by first studying theelectronic structure and energetics ofideal random solid solutions, which,forentropic reasons, arethenatural hightemperature solid state phasesand thento investigate their instabilities to theeither phaseseparation or to theformation ofspecific orderedphases. We shall stress thata direct connection can oftenbe made betweenspecific features intheelectronic structure associated withthe random solid solution and thedriving mechanismsbehindspecific ordering phenomena. Consequently, our understanding ofphasestability willbe underpinned by the same electronic structure thatisresponsible fordetermining theresidual resistivity and other properties ofthedisordered phaseand thatcan be experimentally verified usingoptical spectroscopies, positron annihilation and otherprobes. These lecture notesare structured as follows. In section 2 we layout the basic LDA-KKR-CPA theory oftheelectronic structure and energetics ofrandom alloys and some examplesof itsapplications to theelectronic structure and energie_ ofrandom alloys arepresented. In section 3 we reviewtheprogress thathas beenmade overthe last few yearsin understanding the mechanismsbehindspecific ordering phenomena observed inbinarysolid solutions basedon theMF-CF and GPM theories ofordering and phasestability. We will giveexamplesofa variety ofordering mechanisms:Fermi surface nesting, band filling, off diagonal randomness, charge transfer, size difference or local strain fluctuations, and magnetic effects. Ineachcasewe will trytomake thelink betweenthespecific ordering phenomenon and the underlying electronic structure of thedisordered phase.Insection 4 we will review theresults ofsome recent calculations on theelectronic structure of_-phaseNicAl1_c alloys usinga version oftheLDA-KKR-CPA codes that has been generalized to systems having complex lattices. In section 5 " we provide a few concluding remarks. 2 Theory of Random Substitutional Alloys 2.1 LDA-KKR-CPA The LDA-KKR-CPA method for calculating the energy and other properties of random solid solution alloys rests on three theoretical developments: the local density approximation to density functional theory, multiple scattering theory for solving the effective single particle SchrSdinger equation that is at the heart of the LDA-DFT self-consistent field equations, and the coherent potential approximation for treating the effects of disorder on the electronic structure i.e. for accomplishing the task of configuradonal averaging inherent in the calculation of observables. 2.1.1 Local Density Approximation and Random Alloys Density functional theory (DFT) is, in principle, an exact method for calculating the energetics of an electron system in the field of the atomic nuclei [4],[5], [21],[22],[6]. The. central result of DFT is that the total ground state energy, ELo], of a system of electrons in the presence of the external field provided by the nuclei is a unique functional ELo] = TIp] + U[p] + E,c[p] of the electron density, p(r-'),where Tip], U[p] and E,c[p] are the kinetic, potential and exchange correlation energies respectively. Furthermore, E[p] takes on its minimum value for the correct ground state p(r-').This minimum principle taken together with the constraint foo dar p(r) = N, the total number of electrons in the system leads to a set of self-consistent field equations whose solution yield the ground state charge density and hence the ground state energy. These basic equations of DFT are made into a practical computational method by making the local density approximation (LDA) in which the unknown, but exact, exchange correlation functional for the inhomogeneous interacting electron gas appropriate to the solid is approximated, at each point in space, r, by the exchange correlation functional, E_A[p], appropriate to an interacting but homogeneous electron gas having the density found at that point. Given the specification of a solid in terms of a set of atomic positions, {R/}, and corresponding nuclear charges, {Zi}, of the atoms occupying these sites, the practical applications the LDA involves the solution of a set of Hartree like, Kohn-Sham selfconsistent field equations that take the form [-V 2-I-v,s! (F;p(e; {P_}; {Zi}))] tb, Cr-') = _&,C r-') (1) J where the crystal potential ve!t(F; p(F; { R/); { Zi })) takes the form @ _2Z _ dFp(_') , [_-R'i[ + 2 IF-JI + v.=(r;'°p(r-')) (2) and where p(F; {R_}; {Zi})is given in terms of the eigen-solutions of eq. 1 as I¢,.(r-')12f(e.-p) (3)