1. Feb 23, 2010, Tsukuba-Edinburgh Computational Science Workshop, Edinburgh Large-Scale Density-Functional calculations for nano-meter size Si materials Jun-Ichi Iwata Center for Computational Sciences University of Tsukuba

2. Outline Quantum Mechanical (First-Principles) Simulation in Solid-State Physics Density-Functional Theory W. Kohn (Nobel Prize in 1998) Density-Functional simulations for large systems Real-Space DFT program code for Parallel Computation -RSDFT- Applications of RSDFT for Si nano materials >10,000-atom system

3. First-Principles Calculation in Material Physics We describe material properties from the behavior of electrons and ions. ions → classical, electrons → quantum We solve the Schrodinger equation for electronic ground state Density-functional theory is a powerful tool for this purpose.

4. Density-Functional Theory electron density Energy Functional We get stable atomic & electronic structures. （ minimize ） P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864. W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133. minimize with respect to Potential Kohn-Sham equation → We have to solve this equation self-consistently ( Nonlinear eigenvalue problem )

5. M. T. Yin and M. L. Cohen Phys. Rev. B26, 5668 (1982). Exchange functional in Local-Density Approx. DFT calc. Expt. Lattice Constant ( Å ) 5.375.41 Bulk Modulus (Mb) 0.9770.988 Si （ in diamond structure ） Performance of DFT with simple approximation quantitatively good results Correctly describe various properties

6. Proteins （ cytochrome c oxidase ） ～ 30,000 atoms Nano structures (Si pyramid) ～ 100,000 atoms A. Ichimiya et al., Surf. Sci. 493, 555 (2001). Everybody wants to apply the DFT for Large systems Usually, we treat 10- to 1000-atom systems by DFT. However, we need to treat larger systems. to study large objects (nano structures, proteins) to make the atomic model more realistic

7. Real-Space DFT program code (RSDFT) Solve Kohn-Sham equation (eigenvalue problem) → Computational costs ～ O(N 3 ) Developed for parallel computers

8. discretize function Column vector Laplacian → Higher-Order Finite-Difference Higher-order finite difference pseudopotential method J. R. Chelikowsky et al., Phys. Rev. B, (1994) Real-Space Method continuous spacediscrete space Typical number of grid points ： 10,000 ～ 1,000,000 ( ⇔ Reciprocal-Space (Plane-Wave) Method )

9. Real-Space Finite-Difference Sparse Matrix FFT free (FFT is inevitable in the conventional plane-wave code) Kohn-Sham eq. (finite-difference) 3D grid is divided by several regions for parallel computation. Higher-order finite difference Integration MPI_ISEND, MPI_IRECV MPI_ALLREDUCE RSDFT – suitable for parallel first-principles calculation - MPI ( Message Passing Interface ) library CPU0 CPU8CPU7 CPU6 CPU5 CPU4CPU3 CPU2CPU1

10. Convergence behavior for Si 10701 H 1996 The largest system in the present study → Si 10701 H 1996 Massively Parallel Computing Computational Time (with 1024 nodes of PACS-CS) 6781 sec. × 60 iteration step = 113 hour Based on the finite-difference pseudopotential method (J. R. Chelikowsky et al., PRB1994) Highly tuned for massively parallel computers Computations are done on a massively-parallel cluster PACS-CS at University of Tsukuba. (Theoretical Peak Performance = 5.6GFLOPS/node) with our recently developed code “RSDFT” Iwata et al, J. Comp. Phys. (2010) Real-Space Density-Functional Theory code (RSDFT) Grid points = 3,402,059 Bands = 22,432

11. Conjugate-Gradient Method Gram-Schmidt orthonormalization Density, Potentials update Subspace Diagonalization Total Computational Cost ～ O(N 3 ) O(N 3 ) O(N) O(N 2 ) Flow chart Calc. Ionic Potentials Input initial configuration of Ions Hellman-Feynman Force Move ions Convergence Check Electronic structure optimization must be performed in each atomic optimization step Atomic structure optimization Electronic structure optimization Algorithm → subspace iteration method (Rayleigh-Ritz method) yes

12. Algorithm １ → Subspace Iteration Method （ Rayleigh-Ritz Method ） M-dimensional eigenvalue problem We need smallest N( ≪ M) eigen-pairs Minimize Reyleigh quotients by Conjugate-Gradient Method wave function update Initial guess Problem

13. Algorithm 2 O(MN 2 ) O(N 3 ) Subspace Diagonalization O(MN 2 ) （ Ritz vectors ） Gram-Schmidt Orthogonalization → as a basis set ← initial guess for the next iteration O(MN 2 ) Calc. Matrix Elements

14. Gram-Schmidt orthogonalization Time (sec) GFLOPS/node Old algorithm 661 (710) 0.70 (0.65) New algorithm 111 (140) 4.30 (3.50) Time & Performance for Gram-Schmidt O(N 3 ) part can be computed at 80% of the theoretical peak performance! ～ Active use of Level 3 BLAS in O(N 3 ) computation ～ → Collaboration with computer scientists much improve the performance of the RSDFT! Theoretical peak performance = 5.6 GFLOPS/node Part of the calculations can be performed as Matrix × Matrix operation! Algorithm of GS

15. PACS-CS(5.6GFLOPS/node) 256nodes → time for O(N 2 )-part and O(N 3 )-part become comparable Elapsed time for 1 step of iteration O(N 2 ) O(N 3 )

16. Application 1 Nano-meter size Si quantum dots

17. Si quantum dot is a promising material for several device applications  Memory  Single-electron transistor  Optical Device Clarifying the relation between the “Dot size” and “Band gap” is important for controlling the device properties.  System size is very large! A model of the Si quantum dot of 6.6 nm diameter （ Si 7055 H 1596 ） First-principles calculations are useful for such studies? → Yes, but …

18. (eV) Experimental fit curve From STS measurement B.Zanknoon et al., Nano letters 8, 1689 (2008). The ΔSCF gap seems to be closer to the ΔKS gap … Band Gaps 300 atoms>10,000 atoms

19. Application ２ Si nanowires

20. IEDM2005IEDM2006 Diameter of NW10 nm8 nm Gate length30 nm15 nm Vdd1.0 V I_on (n) 2.64 mA/  m1.4 mA/  m I_on (p) 1.11 mA/  m1.94 mA/  m I_off (n) 3.1 nA/  m2.0 nA/  m I_off (p) 0.0056 mA/  m1.0 nA/  m Samsung Si nanowire devices

21. 4 nm diameter （ 425 atoms) 10 nm diameter （ 2341 atoms ） 20 nm diameter （ 8941 atoms ） There may be an optimum diameter in the region of 10 nm ～ 20 nm. Several size of Si nanowires

22. d=1nm Si21H20 （ 41 atoms ） Eg=2.60eV (LDA Bulk : 0.53eV)  X Band Structure and DOS of SiNW (d=1nm)

23. d=4nm Si341H84 （ 425 atoms ） Eg=0.81eV (LDA Bulk=0.53eV)  X Band Structure and DOS of SiNW (d=4nm)

24.  X Si1361H164(1525 atoms), Eg=0.61eV Band Structure and DOS of SiNW (d=8nm)  X Bulk Si Eg=0.53eV

25. Si12822H1544 （ 14,366 atoms ） ･ 10nm diameter 、 3.3nm height 、 (100) ･ Grid spacing ： 0.45Å (~14Ry) ･ # of grid points ： 4,718,592 ･ # of bands ： 29,024 ･ Memory ： 1,022GB ～ 2,044GB Si12822H1544 Top View Side View Si nano wire with surface roughness

26. PACS-CS1024 nodes （ peak performance ： 5.6 GFLOPS/node ） Subspace diagonalization ： 4600 sec. Gram-Schmidt ： 2300 sec. Conjugate-Gradient Method ： 3700 sec. Total Energy calc. ： 1200 sec. Total(1 step) ： 12,000 sec. DOS of SiNW with roughness DOS of Bulk Si d=10nm （ with roughness ） Si12822H1544(14,366 atoms) Eg=0.57eV

27. Application3 Si divacancy

28. Structure of Si divacancy ： Small-yellow balls : vacancies (no atoms) Green balls : Si atoms with dangling bonds. There are two possibilities for the structure of Si divacancy. Resonant-Bond type Large-paring type ･ Both “Large-paring” and “Resonant-Bond” structure were found. ･ Large-Paring type is the most stable (RB type is a local minimum) More recent LDA calculation (Oguet et al., 1999) EPR experiment (Watkins & Corbett, 1965) LDA calculation (Saito & Oshiyama, 1994) Large-Paring type Resonant-Bond type is stable (Large-Paring type was not found) What is the stable structure ? Model size ～ 60 atoms Model size ～ 300 atoms →Model Size dependence ?

29. Si divacancy d ac, d ab (Å) Model size (# of atoms) Large-paring Resonant-Bond Small-Paring Structures converge at 998-atom model. LP structure appears at 510 or larger models. RB structure is most stable, but the energy difference is very small (<10 meV) J.-I. Iwata, et al., Phys. Rev. B 77 (2008) 115208 Structure of Si divacancy ： Small-yellow balls : vacancies (no atoms) Green balls : Si atoms with dangling bonds. There are two possibilities for the structure of Si divacancy. Resonant-Bond type Large-paring type

30. We have developed Real-Space DFT program code for large systems by utilizing the massively parallel computers Collaboration with computer scientist much improve the performance of RSDFT (Especially, O(N 3 )-part calculation with BLAS 3) By using a few hundred ～ 1000CPUs, we have achieved the first-principles calculation for ・ Si 1000-atom system with atomic structure optimization ・ Self-Consistent electronic structures of Si 10,000-atom systems By using large atomic models → eliminate the model-size dependence We have applied the RSDFT for nano-meter scale Si materials (SiNW, SiQD) I think the RSDFT becomes an useful tool for future device development Summary