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Density Functional Theory for Electrons in Materials Richard M. Martin. Prediction of Phase Diagram of Carbon at High P,T. Bands in GaAs. Outline. Pseudopotentials Ab Initio -- Empirical Bloch theorem and bands in crystals
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Density Functional Theoryfor Electrons in MaterialsRichard M. Martin Prediction of Phase Diagram of Carbon at High P,T Bands in GaAs Comp. Mat. Science School 2001 Lecture 2
Outline • Pseudopotentials • Ab Initio -- Empirical • Bloch theorem and bands in crystals • Definition of the crystal structure and Brillouin zone in programs used in the lab (Friday) • Plane wave calculations • Iterative methods: • Krylov subspaces • Solution by energy minimization: Conjugate gradient methods • Solution by residual minimization (connnection toVASP code that will be used by Tuttle) • Car-Parrinello ``ab initio’’ simulations • Examples Comp. Mat. Science School 2001 Lecture 2
Bloch Theorem and Bands • Crystal Structure = Bravais Lattice + Basis Atoms Crystal Points or translation vectors Space group = translation group + point group Translation symmetry - leads to Reciprocal Lattice; Brillouin Zone; Bloch Theorem; ….. Comp. Mat. Science School 2001 Lecture 2
a2 b2 b2 a2 a1 b1 a1 Wigner-Seitz Cell b1 Brillouin Zone Real and Reciprocal Lattices in Two Dimensions Comp. Mat. Science School 2001 Lecture 2
a3 a2 a1 Simple Cubic Lattice Cube is also Wigner-Seitz Cell Comp. Mat. Science School 2001 Lecture 2
z a3 y a1 a2 X Wigner-Seitz Cell Body Centered Cubic Lattice Comp. Mat. Science School 2001 Lecture 2
z y a2 a3 X a1 Wigner-Seitz Cell One Primitive Cell Face Centered Cubic Lattice Comp. Mat. Science School 2001 Lecture 2
z y X ZnS Structure with Face Centered Cubic Bravais Lattice NaCl Structure with Face Centered Cubic Bravais Lattice Comp. Mat. Science School 2001 Lecture 2
z z X X R L y L L K G S y G X D X W S W M U D X K U X X X z H A z D L H L K T P G D y G H S S H N M K X y x Brillouin Zones for Several Lattices Comp. Mat. Science School 2001 Lecture 2
Example of Bands - GaAs • GaAs - Occupied Bands - Photoemission Experiment - Empirical pseudopotential • “Ab initio” LDA or GGA bands almost as good for occupied bands -- BUT gap to empty bands much too small T.-C. Chiang, et al PRB 1980 Comp. Mat. Science School 2001 Lecture 2
Transition metal series • Calculated using spherical atomic-like potentials around each atom • Filling of the d bands very well described in early days - and now - magnetism, etc. • Failures occur in the transition metal oxides where correlation becomes very important L. Mattheisss, PRB 1964 Comp. Mat. Science School 2001 Lecture 2
Standard method - Diagonalization • Kohn- Sham self Consistent Loop • Innner loop: solving equation for wavefunctions with a given Veff • Outer loop: iterating density to self-consistency • Non-linear equations • Can be linearized near solution • Numerical methods - DIIS, Broyden, etc. (D. Johnson)See later - iterative methods Comp. Mat. Science School 2001 Lecture 2
Empirical pseudopotentials • Illustrate the computational intensive part of the problem • Innner loop: solving equation for wavefunctions with a given Veff • Greatly simplified program by avoiding the self-consistency • Useful for many problems • Description in technical notes and lab notes Comp. Mat. Science School 2001 Lecture 2
Iterative methods • Have made possible an entire new generation of simulations • Innner loop: This is where the main computation occurs • Many ideas - all with both numerical and a physical basis • Energy minimization - Conjugate gradients • Residual minimization - Davidson, DIIS, ... • See lectures of E. de Sturler Used in Lab Comp. Mat. Science School 2001 Lecture 2
Car-Parrinello Simulations • Elegant solution where the optimization of the electron wavefunctions and the ion motion are all combined in one unified algorithm Comp. Mat. Science School 2001 Lecture 2
Example • Prediction of Phase Diagram of Carbon M. Grumbach, et al, PRB 1996 • Above ~ 5 Mbar C prdicted to behave like Si - Tmelt decreases with P Comp. Mat. Science School 2001 Lecture 2
Conclusions • The ground state properties are predicted with remarkable success by the simple LDA and GGAs. • Accuracy for simple cases gives assuarnce in complex cases • Iterative methods make possible simulations far beyond anything done before • Car-Parrinello “ab initio” simulations • Greatest problem at present: Excitations • The “Band Gap Problem” Comp. Mat. Science School 2001 Lecture 2