مدرسه پاییزه نظریه تابعی چگالی 12-14 آذر 1387 گروه فیزیک، دانشگاه اصفهان

1. مدرسه پاییزه نظریه تابعی چگالی 12-14 آذر 1387 گروه فیزیک، دانشگاه اصفهان

2. Introduction to Maximally-Localized Wannier Functions Nafise Rezaei, Zahra Torbatian, Hoda Jebeli S. Javad Hashemifar Physics Departments, Isfahan University of Technology

3. Bloch Theory Crystal  Translational symmetry  Bloch orbital: Extended States

4. Bloch Representation N unit cell Bloch Representation 1 unit cell

5. Bloch Quantum Number Lattice Momentum k Advantages ?

6. Bloch Representation:Band Structure Band Structure Electronic behavior in reciprocal space Optical properties Si

7. Bloch Representation:Fermi Surface de Haas-van Alphen effect Hole and Electron like states Fermi surface nesting

8. Bloch Representation:Electron dynamic Dynamics of Bloch electrons Group velocity Effective mass

9. Bloch Representation:DiffusiveTransport Boltzman non equilibrium distribution function Dynamic  Semi classical description

10. Bloch Representation: DiffusiveTransport Lattice Imperfections, Phonons Scattering

11. Bloch Quantum Number Lattice Momentum k Disadvantages ?

12. Delocalized orbitals Not appropriate for visualization of the chemical bonds A Kohn-Sham orbital for an oligopeptide molecule

13. Bloch Representation: scaling behavior Assuming a plane wave basis set Number of basis functions ~ (size)

14. Bloch Representation: scaling behavior Size x 2 Other Factors Since Bloch Functions are delocalized many Hij are nonzero All basis functions should be orthogonal Computational Time ~ (size)3

15. Bloch Representation: scaling behavior Computational Time ~ (size)3 k points are independent on each other Computational Time ~ Number of k points

16. Quantum Transport Conservation of momentum  Ballistic transport  Quantum conductance:

17. Quantum Transport www.wannier-transport.org The probability of Transmission between two ends Localized basis functions are more appropriate for calculation of Green function

18. Wannier Functions:Basic definition Fourier transformation of the Bloch functions 1 unit cell N unit cell Real space representation of solid

19. Localized Wannier Functions Real space representation of solid

20. Wannier Functions: Nature of the Chemical bonds d-eg Wannier functions of Copper I. Souza, N. Marzari, and D. Vanderbilt, Physical Review B 65, 035109 (2002)

21. Localized Wannier Functions: scaling behavior Size x 2 Computational Time ~ (size) Appropriate for Green function and quantum conductance calculations.

22. Localized Wannier Functions: scaling behavior Bloch Functions Time ~ (size)3 Time ~ (Number of k points) Bloch Functions Wannier Functions Time ~ (size) Localized Wannier Functions are appropriate basis for Linear scaling DFT codes Goedecker, Review of Modern Physics 71, 1085 (1999)

23. Linear scaling DFT Schwegler, Galli, Gygi, Hood, Physical Review Letters 87, 265501 (2001)

24. Wannier Functions:Band interpolation تعداد نقاط تبدیل فوریه درون یابی

25. Wannier Functions Fourier transformation of the Bloch functions How to find the best Wannier functions ?

26. Wannier Functions:Gauge Freedom Not determined by Schrödinger equation Wannier Functions are not unique !!!

27. Wannier Functions:Generalized A simple Unitary transformation group of isolated bands A General Unitary transformation

28. Wannier Functions:Generalized Unitary Rotation in a group of isolated bands This general gauge freedom is used to obtain the Maximally Localized Wannier Functions[1] [1] Marzari and Vanderbilt, Physical Review B 56 , 12847 (1997)

29. Localization criteria Spread functional (Variance) Some examples: Delta function Constant function

30. Reciprocal-space representation Position operator Gradient • Determining the Bloch orbitals on a regular k mesh • Using finite difference to evaluate the above derivatives

31. Finite difference method b are a set of vectors connecting a k-point to its near neighbors

32. Reciprocal-space representation Central Parameter

33. Localization procedure An infinitesimal rotation of the Bloch orbitals: infinitesimal antiunitary matrix Provides an equation of motion (e.g. conjugate gradient or steepest descent ) for the evolution of towards the minimum of Ω.

34. فرایند کمینه کردن No yes End

35. رهایی یافتن ازکمینه های موضعی اربیتال های بلوخ را روی یک مجموعه اربیتال های آزمایشی تصویر می کنیم اورتونرمال می شوند Lowdinاز طریق تبدیل از اربیتال های بلوخ حاصل به عنوان نقطه شروع فرایند کمینه کردن استفاده می کنیم.

36. Disentanglement For a group of isolated bands: U is NxN matrix Some times number of required Wannier functions is less than number of bands (e.g. valence band of metals) Hence U is NxM matrix Mixing N states to make M rotated states

37. Disentanglement procedure Two step localization procedure Find the best U for transforming N states into M states Find the best U for maximal localization of M Wannier functions

38. Energy windows Inner Energy window: The rotated Bloch states should have the same band structure as original Bloch states Outer Energy window: The rotated Bloch states may have different band structure as original Bloch states

39. Disentanglement in Silicon

40. با تشکر فراوان

41. Disentanglement procedure Step 1 • ”disentangle the N bands of interest ” from the rest • Defining an energy window(outer window), states lie within this energy window • Obtaining a set of N Bloch states by a unitary transformation amongst Bloch states Outer window Obtaining by minimizing the gauge invariant spread Nafise Rezaei 11 اردیبهشت 1387

42. Step 2 • obtain maximally-localized WFs • Within the subspace S(k) determined in Step 1 (which have a fixed ΩI) • Minimize ,using the algorithm of Marzari & Vanderbilt Nafise Rezaei 11 اردیبهشت 1387

43. Exact Constraints on the Inner Energy Window • Energy bands of this optimal subspace may not correspond to any of the original energy band • Introduce a second energy window (inner window) • States lying within inner energy window are included unchanged in the optimal subspace Outer window Inner window