Density Functional Theory and the LAPW method with Applications Andrew Nicholson Solid State Physics II Spring 2009 (Elbio Dagotto) Brought to you by:

The Road Map Introduction What is a functional? Density Functional Theory The Local Density Aproximation Why Planewaves as a basis? Problems with Planewaves The Linearized Augmented Planewave Method Applications Conclusions

Introduction

Foundations of DFT What is a functional? A functional maps a vector space to a scalar field Energy Electron Density

Energy and Electronic Density Consider a uniform electron gas, under an arbitrary external potential:

Energy and Electronic Density Combining these energies with the definition of electronic density we find: Unknown functional G[n].. what is the form of E xc ? Here we have taken a many body problem and reduced it to many single noninteracting problems!

The Local Density Approximation Assume that the electronic density varies slowly here ε xc is the exchange correlation energy per electron of a uniform electron gas

The Local Density Approximation Now we can just solve the single particle Schrodinger Equation: Where:

The Local Density Approximation From all of this the energy is just: Now we can solve all of these equations self consistantly

The Local Density Approximation

Even in these regimes problem is not completly solved: There are still a large number of single particle Schrodinger equations to solve LDA is exact in two limiting cases: 1) Slowly varying densities 2) Very high densities

Why Planewaves? Bloch's Theorem: Bloch's theorem gives a power tool to solve a large system Also.. Planewaves are easy to use! But To approximate a wave function exactly an infinite number of planewaves are needed

Problems with Planewaves Electron wave functions near the nucleus are strongly varying ( lots of wiggles) Payne et al

The Linearized Augmented Planewave Basis Problem: Need to reduce the number of planewaves Solution: Divide space into two regions From D. J. Singh

LAPW – Spherical Region

LAPW – Interstital Region Wave function and its energy derivative must be continious at the sphere boundary!

LAPW - Method

Application – Bulk Cu One of the earliest applications of this method – Koelling and Arbman 1975 Koelling and Arbman

Application - BaNi 2 As 2 Using the LAPW to calculate band structure, DOS, Fermi surface: Singh et al

Application – Ni 3 Al, Ni 3 Ga Study of two similar materials: - Similar Structure -Cu 3 Au cP4 structure - Similar lattice constants - Similar electronic structure Singh Ni 3 Al - solid Ni 3 Ga - dashed

Application - Ni 3 Al, Ni 3 Ga LDA Results for magnetism: Ni 3 Ga stronger tendancy for magnetism than Ni 3 Al

Application - Ni 3 Al, Ni 3 Ga Theory vs Experiment: LDA has failed becuase it is not sensitive to spin flucuations in the material

In Conclusion DFT takes a many body problem to many noninteracting single body problems LDA approximates the functional by assuming that the electronic density varies slowly There is still a problem approximating the electron wave functions LAPW uses spherical regions where the eigenfunctions of the radial equation are used to reduce the number of planewaves LDA calculations using the LAPW method have been sucessful in many cases

Sources D. Singh, Planewaves, Pseudopotenails and the LAPW method, 2 nd Ed. Springer M. C. Payne et al, Rev. Mod. Phys., (1992) W. Kohn, P. Hohenberg, Phys. Rev., 136, B 864 (1964) von Barth, Many Body Phenomena at Surfces, edited by D. Langretha nd H. Sughl, Academic D. Koeling, G, Arbman, J. Phys. F, 5, 2041 (1975) W. Kohn, L. J. Sham, Phys. Rev., 140 A 1133 (1965) N. W. Ashcroft, N. D. Mermin, Solid State Physics, 1976, Holt Saunders A. Aguayo, I. Mazin, D Singh. Phys. Rev. Lett., 92, (2004) A. Subedi, D. Singh,. Phys. Rev. B, 78, (2008)