André Schleife Department of Materials Science and Engineering
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1 André Schleife Department of Materials Science and Engineering
2 Yesterday you (should have) learned this: Simple_Harmonic_Motion_Orbit.gif 1. deterministic (set of rules): Molecular Dynamics! 2. random (samples from a distribution): Monte Carlo
3 CCMS 2014: Computational Methods for Atomistic Length and Time Scales Length Scales (c) ICAMS:
4 Goals for today:! Density Functional Theory Excited electronic states Real-time electron dynamics Web site: Literature recommendations:! Electronic Structure: Basic Theory and Practical Methods, Richard Martin
5 Forces: QM Electronic-Structure Problem H = XN e i=1 1 X 2 r2 i + i<j 1 r ij XN e i=1 XN I I=1 Z I r ii XN e I=1 m X e r 2 I 2M + I I<J Z I Z J r IJ +(external fields) electron KE+interaction electron+ion ion KE + interaction couples to individual particles Atomic units : = me = e = 1 Energy in Hartree = 27.2 ev = 316,000 K Lengths in Bohr radii = Å = cm Accuracy needed to address questions at room temp.: 100 K = 0.3 mha = 0.01 ev Many Decimal Places! Solving this is difficult!
6 Why is Many-Electron QM so hard? 1 2 n i=1 2 i n i=1 Z r i + n 1 i=1 n j=i+1 1 r i r j (r 1,..., r n ) = E (r 1,..., r n ) Example: Fe atom Fe has 26 electrons wave function has 3 26 = 78 variables store wave function on a grid coarse grid of only 10 points along each direction to store wave function would require storage of numbers single precision 1 number = 4 Bytes all data stored worldwide in 2015: 8 zettabyte = 8*10 21 Bytes
7 Born Oppenheimer Approximation (1927) make use of the fact that nuclei are so much heavier than electrons factor total wave function into ionic/electronic parts (adiabatic approx.) H = XN e i=1 1 2 r2 i + X i<j 1 r ij XN e i=1 XN I I=1 Z I r ii XN e I=1 m X e r 2 I 2M + I I<J Z I Z J r IJ electron Hamiltonian ionic Hamiltonian electrons remain in instantaneous ground state in response to ion motion Does not require classical ions (though MD uses that) Eliminate electrons from dynamics and replace by an effective potential
8 Overview: Many-electron Problem (quantum-mechanically) interacting electrons and ions (r 1...r n, R 1...R m ) Born-Oppenheimer: (r 1...r n, R 1...R m )= (R 1...R m ) (r 1...r n ) (a) Forces on ions for an electronic potential energy surface R 1...R m (b) (quantum-mechanically) interacting electrons for fixed ions (r 1...r n ) ions almost always classical (empirical) potentials classical molecular dynamics (r 1...r n )= (r 1 ) (r n ) Hartree: Hartree-Fock: Slater determinant Density Functional Theory
9 Density Functional Theory Schrödinger equation: everything is a function(al) of wave function E = h Ĥ i ~ 2 Examples: E kin = h 2m r2 i First Hohenberg-Kohn Theorem: everything is function(al) of ground-state density 0 (r 1, r 2,...,r n ) (r) Function of 3N variables Function of 3 variables Problem: Functional is generally unknown: E kin [ 0 ] V el el [ 0 ]? We know it for one quantity: V ext [ 0 ] = V ext (r) (r)d 3 r
10 Second Hohenberg-Kohn Theorem Second Hohenberg-Kohn theorem gives us another quantity: E[ (r)] = F [ (r)] + V ext (r) (r)d 3 r E 0 variational principle guides us to find ground-state density and groundstate energy But: How to do that in practice? Kohn-Sham approach (XC functional)
11 Density Functional Theory: Flow Chart (c) Wolfgang Goes, TU Vienna
12 Ab-initio Molecular Dynamics (quantum-mechanically) interacting electrons and ions (r 1...r n, R 1...R m ) Born-Oppenheimer: (r 1...r n, R 1...R m )= (R 1...R m ) (r 1...r n ) Forces on ions for an electronic potential energy surface from DFT R 1...R m (quantum-mechanically) interacting electrons for fixed ions on DFT level (r 1...r n )= KS (r 1 ) KS (r n )
13 Example: Silicon Under pressure Si displays 11 crystal phases LDA correctly predicts the energetic order of all these phases Compression 11 GPa 13 GPa 16 GPa 36 GPa 42 GPa 79 GPa Si(I) diamond Z=4 Si(II)! tin Z=6 Si(XI) Imma Si(V) hexagonal Si(VI) orthorhombic Z=6 Z=8 Z=10 Si(VII) hcp Si(X) fcc Z=12 Z=12 Decompression Slow pressure release Fast pressure release 9 GPa 2 GPa >480 K Si(VIII) and Si(IX) tetragonal Si(II)! tin Z=6 Si(XII) R8 Z=4 Si(III) BC 8 Z=4 Si(IV) hex. diamond Z=4 Phys. Rev. B 24, 7210 (1981), ibid. 49, 5329 (1994), ibid. 69, (2004)
14 Optoelectronics and semiconductor technology:! lasers and light-emitting diodes! transparent electronics (displays, )! Plasmonics! Excited Electrons?! Energy-related applications:! photocatalytic water splitting! transparent electrodes: solar cells! piezoelectronics! Corning! Open questions for real applications:! absorption properties?! attosecond photonics?! radiation-induced defects?! Georgia Tech.!
15 Excited Electrons!! Optoelectronics from First-Principles Theory! Excited electronic states! atomic geometry! electronic band structure! optical absorption! Non-adiabatic electron-ion dynamics!
16 Photoemission spectroscopy (PES)! removal (PES) or addition (inverse PES) of an electron! important: reaction of the electrons of the system! quasiparticle equation:! approximation of the electronic self energy:! quasiparticle energies from one step of perturbation theory! HSE hybrid functional: non-local treatment of exchange and correlation!
17 wurtzite ZnO: HSE hybrid local-density hybrid functional functional! approximation! and quasiparticle effects! Local-density approximation insufficient for electronic properties! band gap improved! too small! E LDA HSE+GW g =0.7 =2.1 =3.2 ev! ev! E g exp =3.4 ev! d-bands improved! too high! HSE+G 0 W 0 :! good approximation that includes quasiparticle effects! reliable description of the electronic structure! predictive power (oxides, nitrides, )!
18 Excited Electrons!! Optoelectronics from First-Principles Theory! atomic geometry! electronic band structure! optical absorption! Non-adiabatic electron-ion dynamics!
19 Optical absorption:! electron from valence band excited into conduction band! electron-hole attraction (Coulomb potential)! macroscopic dielectric function: local-field effects! Bethe-Salpeter equation for optical polarization function! electron-hole interaction:!
20 MgO:! Excitonic effects: solution of the Bethe-Salpeter equation! leads to eigenvalue problem (excitonic Hamiltonian)! Huge matrix: rank typically > 50,000! time-propagation approach to calculate the dielectric function! excellent description of the optical properties of the oxides! predictive power (e.g. for SnO 2, In 2 O 3, )! F. Fuchs, A. Schleife et al.; Phys. Rev. B 78, (2008)! A. Schleife et al.; Phys. Rev. B 80, (2009)!
21 Excited Electrons!! Optoelectronics from First-Principles Theory! atomic geometry! electronic band structure! optical absorption! Non-adiabatic electron-ion dynamics!
22 Examples:! solar cells on satellites! nuclear reactors! magnetic confinement/inertial confinement fusion! surface adsorption! excited-electron dynamics! Radiation damage:! slow projectiles: ionic stopping dominant! fast projectiles: interaction with electronic system important stopping mechanism! Goal:! parameter-free framework to model stopping!
23 beyond Born-Oppenheimer is necessary:! time-dependent Kohn-Sham equations:! popular for excitation spectra: linear-response TDDFT! here: integrating the equations of motion: Real-time TDDFT! challenging: highly parallel implementation! Runge-Kutta explicit integration scheme! compute forces at each time step and update positions of the atoms! Ehrenfest molecular dynamics! A. Schleife et al.; J. Chem. Phys. 137, 22A546 (2012)!
24 capture electron dynamics on atto-second time scale! energy transferred to electrons, stopping power:!
25 Comparison to database with experiments (TRIM):! hydrogen: v=1.0 a.u. 25 kev! systematic convergence to ensure high accuracy! excellent agreement for slow projectiles! deviations at higher velocities: off channeling!! investigate initial-state dependence! predictive! A. Schleife et al. (submitted)!
26 Promising perspective:! essential: excited electronic states and their dynamics! accurate parameter-free description achievable! future: development based on these methods! Towards predictive materials design:! oxides and nitrides highly promisings! verifiable predictions for other material systems! experiment and theory work hand in hand through state-of-the-art calculations! fully exploit modern trends in high-performance computing!
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