1. Comp. Mat. Science School 2001 Lecture 21 Density Functional Theory for Electrons in Materials Richard M. Martin Bands in GaAs Prediction of Phase Diagram of Carbon at High P,T

2. Comp. Mat. Science School 2001 Lecture 22 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 Outline

3. Comp. Mat. Science School 2001 Lecture 23 Bloch Theorem and Bands Crystal Structure = Bravais Lattice + Basis Points or translation vectors Atoms Crystal Space group = translation group + point group Translation symmetry - leads to Reciprocal Lattice; Brillouin Zone; Bloch Theorem; …..

4. Comp. Mat. Science School 2001 Lecture 24 a1a1 a2a2 b2b2 b1b1 b2b2 b1b1 Real and Reciprocal Lattices in Two Dimensions Wigner-Seitz Cell Brillouin Zone a1a1 a2a2

5. Comp. Mat. Science School 2001 Lecture 25 a1a1 Simple Cubic Lattice Cube is also Wigner-Seitz Cell a2a2 a3a3

6. Comp. Mat. Science School 2001 Lecture 26 Body Centered Cubic Lattice X y z a3a3 a1a1 a2a2 Wigner-Seitz Cell

7. Comp. Mat. Science School 2001 Lecture 27 Face Centered Cubic Lattice Wigner-Seitz Cell X y z a1a1 a3a3 a2a2 One Primitive Cell

8. Comp. Mat. Science School 2001 Lecture 28 NaCl Structure with Face Centered Cubic Bravais Lattice ZnS Structure with Face Centered Cubic Bravais Lattice X y z

9. Comp. Mat. Science School 2001 Lecture 29 Brillouin Zones for Several Lattices     X y z R M X X  X K L X X K W    U X z X y H X z y H P     N W U  M   A H K L z H y x K T

10. Comp. Mat. Science School 2001 Lecture 210 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

11. Comp. Mat. Science School 2001 Lecture 211 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

12. Comp. Mat. Science School 2001 Lecture 212 Standard method - Diagonalization Kohn- Sham self Consistent Loop Innner loop: solving equation for wavefunctions with a given V eff 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

13. Comp. Mat. Science School 2001 Lecture 213 Empirical pseudopotentials Illustrate the computational intensive part of the problem – Innner loop: solving equation for wavefunctions with a given V eff –Greatly simplified program by avoiding the self-consistency Useful for many problems Description in technical notes and lab notes

14. Comp. Mat. Science School 2001 Lecture 214 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

15. Comp. Mat. Science School 2001 Lecture 215 Car-Parrinello Simulations Elegant solution where the optimization of the electron wavefunctions and the ion motion are all combined in one unified algorithm

16. Comp. Mat. Science School 2001 Lecture 216 Example Prediction of Phase Diagram of Carbon Above ~ 5 Mbar C prdicted to behave like Si - T melt decreases with P M. Grumbach, et al, PRB 1996

17. Comp. Mat. Science School 2001 Lecture 217 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”