Der Einsatz der Dichtefunktionaltheorie in der Materialphysik: Ein atomistischer Blick auf moderne Werkstoffe

Transcription

1 Der Einsatz der Dichtefunktionaltheorie in der Materialphysik: Ein atomistischer Blick auf moderne Werkstoffe Slide 1

2 Inhalt I. Atomistische Modellierung von Materialien II. Dichtefunktionaltheorie (DFT) III. Anwendungen Slide 2

3 0.1 nm 1 nm 10 nm 100 nm 1 m 10 m 100 m 1 mm 10 mm 100 mm Length Scales in Materials Modelling Juan J. de Pablo and William A. Curtin, Guest Editors, MRS Bulletin 32 (Nov. 2007) Slide 3

4 Length Scales in Materials Modelling Continuum Mechanics 0.1 nm 1 nm 10 nm 100 nm 1 m 10 m 100 m 1 mm 10 mm 100 mm Finite Element Methods Juan J. de Pablo and William A. Curtin, Guest Editors, MRS Bulletin 32 (Nov. 2007) Slide 4

5 Length Scales in Materials Modelling Mesoscale 0.1 nm 1 nm 10 nm 100 nm 1 m 10 m 100 m 1 mm 10 mm 100 mm dislocation models Juan J. de Pablo and William A. Curtin, Guest Editors, MRS Bulletin 32 (Nov. 2007) Slide 5

6 Length Scales in Materials Modelling Nanoscale 0.1 nm 1 nm 10 nm 100 nm 1 m 10 m 100 m 1 mm 10 mm 100 mm dislocation plasticity and atomistic models coexist Juan J. de Pablo and William A. Curtin, Guest Editors, MRS Bulletin 32 (Nov. 2007) Slide 6

7 Length Scales in Materials Modelling Quantum Scale 0.1 nm 1 nm 10 nm 100 nm 1 m 10 m 100 m 1 mm 10 mm 100 mm Ab-initio electronic structure methods Juan J. de Pablo and William A. Curtin, Guest Editors, MRS Bulletin 32 (Nov. 2007) Slide 7

8 From First Principles Coulomb Force = Slide 8

9 From First Principles Coulomb Force = type of bonding Quantum Mechanics + crystal structure elastic constants lattice vibrations band structure Slide 9

10 Born Oppenheimer Approximation He atom (schematically) Born Oppenheimer approximation electrons QM de Broglie wavelength nuclei classical (Newton) Slide 10

11 Born Oppenheimer Approximation He atom (schematically) The Born Oppenheimer (=adiabatic) Approximation The electrons can follow the much heavier nuclei instantaneously Electronic and nuclear motion can be separated Slide 11

12 A Simple Example: H2 Molecule Total Electronic Hamiltonian H2 molecule e p+ p+ e 2 electron Schrödinger equation Slide 12

13 Many Electron Problem Total Electronic Hamiltonian e Zp+ Many electron Schrödinger equation Slide 13

14 Van Vleck Catastrophe Small molecules Wave function methods (HF, configuration interaction,...) give excellent results Number of parameters in many electron wave function: M = p3n Large molecules and solids Many electron wave function is not a legitimate concept when N > 100 Number of parameters in wave function: M = p3*100 = 10150!! Accuracy of the wave function becomes a problem! Storage of the results: B = q3*100 = bits required!! Slide 14

15 Electron Density as a Loophole Electron Density in a (10,0) single walled Carbon Nano Tube Electron density n(r) is the basic variable Density Functional Theory (DFT) provides rigorous framework All microscopic and macroscopic properties depend on n(r) Slide 15

16 Inhalt I. Atomistische Modellierung von Materialien II. Dichtefunktionaltheorie (DFT) III. Anwendungen Slide 16

17 Hohenberg Kohn Theorem universal functional of the electron density external potential due to atomic nuclei The total energy of a system of interacting electrons is a functional of the density. The energy takes its minimum at the ground state density. Slide 17

18 Hohenberg Kohn Theorem universal functional of the electron density external potential due to atomic nuclei The total energy of a system of interacting electrons is a functional of the density. The energy takes its minimum at the ground state density. Suggestion of Kohn and Sham: exchange correlation energy Slide 18

19 Kohn Sham Equations Replace the system of interacting electrons by a fictitious system of non interacting electrons with the same density Single electron Schrődinger equations with an effective potential Slide 19

20 Kohn Sham Equations Atomic nuclei Slide 20

21 Kohn Sham Equations Atomic nuclei Hartree potential Slide 21

22 Kohn Sham Equations Atomic nuclei Hartree potential classical electro static interactions Exchange correlation potential Quantum mechanical effects Slide 22

23 Kohn Sham Equations Self consistency Slide 23

24 Kohn Sham Equations Self consistency Approximations Slide 24

25 Exchange Correlation Functionals First Generation (1980) Local Density Approximation (LDA) Slide 25

26 Exchange Correlation Functionals First Generation (1980) Second Generation (1996) Local Density Approximation (LDA) Generalized Gradient Approximation (GGA) Slide 26

27 Exchange Correlation Functionals First Generation (1980) Second Generation (1996) Third Generation (1999) Local Density Approximation (LDA) Generalized Gradient Approximation (GGA) Orbital dependent exchange e.g.: PBE0 = *Hartee Fock exchange + (1 )GGA Slide 27

28 DFT in a Nutshell Walter Kohn Rev. Mod. Phys. 71, 1253 (1999) atomic nuclei electrons DFT Nobelprize 1998 Walter Kohn Slide 28

29 Numerical Solution Slide 29

30 Numerical Solution Expansion in a basis Matrix Eigenvalue Problem Property typically around 100 basis functions per atom Slide 30

31 High Performance Computing IBM Power 5+ System 74 Computing Nodes 300 Computer Cores about 1000 GByte RAM 9 TByte Storage 1.5 TeraFLOP Euro Slide 31

32 DFT Codes PW PP PAW FP LAPW plane wave pseudo potentials projector augmented wave full potential LAPW PWscf CP PAW WIEN2k clausthal.de/atp/ ABINIT VASP code.org/ Slide 32

33 Inhalt I. Atomistische Modellierung von Materialien II. Dichtefunktionaltheorie (DFT) III. Anwendungen Slide 33

34 What can be calculated? Structural Properties Lattice Parameters Elastic Constants Atomic Forces, Equilibrium Geometry Surface Relaxations Defect Structures Lattice Dynamics Vibrational Frequencies Phonon DOS, Vibrational Entropy Electron Density Charge Rearrangements Electric Field Gradients Electronic Structure Band Structure Density of States Spectroscopy Photoemission Electron Energy Loss Optical Absorption Dielectric Function Core Level Spectroscopies Raman Scattering Compton Scattering Positron Annihilation Slide 34

35 Applications A) Fe Si Alloys: Lattice Constants / Bulk Modulus B) Fe Mn Austenite Steel: Stacking Fault Energy C) Organic Semiconductors: Cohesive and Adsorption Energies Slide 35

36 Bulk Cu: Total Energy vs. Volume body centred cubic face centred cubic Slide 36

37 Bulk Modulus of Copper bcc aexp = 3.60 Å adft = 3.63 Å fcc Bexp = 142 GPa BDFT = 139 GPa Slide 37

38 FeSi Alloys Zhang, Puschnig, Ambrosch Draxl (submitted) Slide 38

39 Stacking Faults in Austenite Steel A C B A B A C B A Motivation: Development of TWIP steels which combine high strength with high formability Slide 39

40 Stacking Faults in Austenite Steel 2 cm Picture from: Ausgekochter Stahl fűr das Auto von morgen, G. Frommeyer, MaxPlanckForschung 3/2004 A C B A B A C B A Motivation: Development of TWIP steels which combine high strength with high formability Slide 40

41 Stacking Faults in Austenite Steel A C B A B A C B A Axial Interaction Model [Denteneer et al., J. Phys. C Sol. St. Phys. 20, L883 (1987)] ~ hcp + 2Fdhcp 3Ffcc SFE = F Temperature Dependence F = E T S Magnetic entropy from DFT + Monte Carlo simulations Total energy from DFT Slide 41

42 Stacking Faults in Austenite Steel Reyes Huamantinco, Ruban, Puschnig, Ambrosch Draxl, to be published Slide 42

43 Organic Semiconductors Sample of a 10x10 cm2 white OLED (from HC Starck CleviosTM PH510 PEDOT layer) Samsung ultra thin 0.05mm 4 inch OLED display ( resolution, 100,000:1 contrast, 200cd/m2) Organic Solar Cell (Linz Institute for Solar Cells) The work is part of the National Research Network Interface controlled and functionalized organic films Slide 43

44 Organic pi Conjugated Molecules Pentacene (C22H14) OFET Organic Field Effect Transistor Para Sexiphenyl (C36H26) 2.6 nm OLED Organic Light Emitting Diode Slide 44

45 Cohesive Energy of Molecular Crystals Pentacene Crystal Structure Ecrystal Slide 45

46 Cohesive Energy of Molecular Crystals Pentacene Crystal Structure Ecohesive = [ (1/2) Ecrystal Emolecule ] Slide 46

47 Cohesive Energy of Molecular Crystals Slide 47

48 Van der Waals Density Functional Exchange Correlation Energy Nonlocal Correlation Energy leading to van der Waals interaction Dion et al, Phys. Rev. Lett. 92, (2004). Slide 48

49 Cohesive Energy of Molecular Crystals Nabok, Puschnig, Ambrosch Draxl, Phys. Rev. B 77, (2008). Slide 49

50 Surface Energy of Molecular Crystals d Nabok, Puschnig, Ambrosch Draxl, Phys. Rev. B 77, (2008). Slide 50

51 Thiophene / Cu(110) d Slide 51

52 Thiophene / Cu(110) d Sony, Puschnig, Nabok, Ambrosch Draxl, Phys. Rev. Lett. 99, (2007). Slide 52

53 Summary Electronic structure calculations are the first step in a multi scale approach towards computational materials design Density Functional Theory is the standard framework for the calculation of ground state properties from first principles Various materials properties can be obtained from DFT total energy and force calculations Strong predictive power of DFT since it involves no empirical parameters Further developments in Exc are needed to improve accuracy Slide 53

54 NiTi Shape Memory Alloys Slide 54

55 Elastic Constants stress strain Hooke's Law stress = F/A ut tensio sic vis (1678) F strain = l/l Slide 55

56 Elastic Constants stress strain Hooke's Law ut tensio sic vis (1678) Cubic Crystal 3 different components Slide 56

57 Elastic Constants stress strain Hooke's Law ut tensio sic vis (1678) Monoclinic Crystal 13 different components Slide 57

58 Elastic Constants Special Strains z y x Total Energy Monoclinic Crystal 13 different components Slide 58

59 NiTi Shape Memory Alloys Golesorkhtabar, Spitaler, Puschnig, Ambrosch Draxl, to be published Slide 59