André Schleife

André Schleife Department of Materials Science and Engineering

Andre Schleife @ MatSE @ UIUC ・ Email: [email protected] ・ Web: http://schleife.matse.illinois.edu ・ @aschleife

CCMS 2014: Quantum Interactions, Excited Electrons, and Their Real-Time Dynamics

Yesterday you (should have) learned this:


1. deterministic (set of rules): Molecular Dynamics!2. random (samples from a distribution): Monte Carlo

http://upload.wikimedia.org/wikipedia/commons/e/ea/Simple_Harmonic_Motion_Orbit.gif


http://upload.wikimedia.org/wikipedia/commons/e/ea/Simple_Harmonic_Motion_Orbit.gif

(c) ICAMS: http://www.icams.de/cms/upload/01_home/01_research_at_icams/length_scales_1024x780.png


CCMS 2014: Computational Methods for Atomistic Length and Time Scales

Length Scales

http://www.icams.de/cms/upload/01_home/01_research_at_icams/length_scales_1024x780.png

• Web site: http://schleife.matse.illinois.edu/teaching.html !

• Literature recommendations:!• “Electronic Structure: Basic Theory and Practical Methods”,

Richard Martin


• Goals for today:!• Density Functional Theory • Excited electronic states • Real-time electron dynamics


http://schleife.matse.illinois.edu/teaching.html

couples to individual particles

ion KE + interactionelectron+ionelectron KE+interaction

“Atomic units”: ℏ = me = e = 1

Energy in Hartree = 27.2 eV = 316,000 K Lengths in Bohr radii = 0.529 Å = 5.29×10-9 cm

H = �N

eX

i=1

1

2

r2

i +X

i< j

1

rij�

NeX

i=1

NIX

I=1

ZI

riI�

NeX

I=1

me

2MIr2

I +X

I<J

ZIZJ

rIJ+ (external fields)

Accuracy needed to address questions at room temp.: 100 K = 0.3 mHa = 0.01 eV

Many Decimal Places! Solving this is difficult!

Forces: QM Electronic-Structure Problem



�

⇤�12

n⇧

i=1

⇤2i �

n⇧

i=1

Z

ri+

n�1⇧

i=1

n⇧

j=i+1

1|ri � rj |

⇥

⌅�(r1, . . . , rn) = E �(r1, . . . , rn)

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 1078 numbers • single precision 1 number = 4 Bytes • all data stored worldwide in 2015: 8 zettabyte = 8*1021 Bytes

Why is Many-Electron QM so hard?



ionic Hamiltonianelectron Hamiltonian

H = �NeX

i=1

12r2

i +X

i< j

1rij�

NeX

i=1

NIX

I=1

ZI

riI�

NeX

I=1

me

2MIr2

I +X

I<J

ZIZJ

rIJ

• make use of the fact that nuclei are so much heavier than electrons • factor total wave function into ionic/electronic parts (adiabatic approx.)

Does not require classical ions (though MD uses that) Eliminate electrons from dynamics and replace by an effective potential

electrons remain in instantaneous ground state in response to ion motion

Born Oppenheimer Approximation (1927)



Overview: Many-electron Problem

(quantum-mechanically) interacting electrons and ions

• ions almost always classical • (empirical) potentials • classical molecular dynamics

(r1 . . . rn,R1 . . .Rm)

Born-Oppenheimer: (r1 . . . rn,R1 . . .Rm) = (R1 . . .Rm)�(r1 . . . rn)

• Hartree: • Hartree-Fock: Slater determinant • Density Functional Theory

�(r1 . . . rn) = �(r1)⇥ · · ·⇥ �(rn)

(a) Forces on ions for an electronic potential energy surface

R1 . . .Rm

(b) (quantum-mechanically) interacting electrons for fixed ions

�(r1 . . . rn)



• Schrödinger equation: everything is a “function(al)” of wave function

Density Functional Theory

Vext[�0] =�

Vext(r) �(r)d3r• We know it for one quantity:

E = h | H | i

�(r1, r2, . . . , rn)�(r)

Function of 3N variablesFunction of 3 variables

⇢0• First Hohenberg-Kohn Theorem: everything is “function(al)” of

ground-state density

Ekin = h |� ~22m

r2 | i

• Problem: Functional is generally unknown: Ekin[⇢0] Vel�el[⇢

0] ?

• Examples:



• Second Hohenberg-Kohn theorem gives us another quantity:

E[�(r)] = F [�(r)] +�

Vext(r)�(r)d3r � E0

Second Hohenberg-Kohn Theorem

• variational principle guides us to find ground-state density and ground-state energy

• But: How to do that in practice? Kohn-Sham approach (XC functional)



Density Functional Theory: Flow Chart

(c) Wolfgang Goes, TU Vienna



Ab-initio Molecular Dynamics

(quantum-mechanically) interacting electrons and ions

Born-Oppenheimer:

(r1 . . . rn,R1 . . .Rm)

(r1 . . . rn,R1 . . .Rm) = (R1 . . .Rm)�(r1 . . . rn)

Forces on ions for an electronic potential energy surface from

DFT

R1 . . .Rm

(quantum-mechanically) interacting electrons for fixed

ions on DFT level�(r1 . . . rn) = �KS(r1)⇥ · · ·⇥ �KS(rn)



Under pressure Si displays 11 crystal phases • LDA correctly predicts the energetic order of all these phases

Silicon Crystal Phases

Compression16 GPa 36 GPa 42 GPa 79 GPa

Si(I) diamondZ=4

11 GPa 13 GPa

Si(II) −tin!Z=6

Si(V) hexagonalSi(XI) ImmaZ=6 Z=8 Z=10

Si(VI) orthorhombic Si(VII) hcpZ=12 Z=12

Si(X) fcc

Decompression

Si(XII) R8 Si(IV) hex. diamondZ=4Z=4

Si(III) BC−8Z=6!−tinSi(II)

9 GPa >480 K

Slow pressure release

Z=4

2 GPa

Fast pressure release

Si(VIII) and Si(IX) tetragonal

Under pressure silicon displays 12 crystal phases with a steady increase

of coordination and a transition from insulating to metallic.

Phys. Rev. B 24, 7210 (1981), ibid. 49, 5329 (1994), ibid. 69, 134112 (2004)

Example: Silicon



Andre Schleife @ MatSE @ UIUC � Email: [email protected] � Web: http://schleife.matse.illinois.edu � @aschleife


Optoelectronics and semiconductor technology: !•  lasers and light-emitting diodes!•  transparent electronics (displays, …) !

Energy-related applications: !•  photocatalytic water splitting!•  transparent electrodes: solar cells!•  piezoelectronics!

Plasmonics!

© Corning!

© Georgia Tech.!

•  absorption properties?!•  attosecond photonics? !•  radiation-induced defects?!

Open questions for real applications: !

Excited Electrons? !



Optoelectronics from First-Principles Theory !

Non-adiabatic electron-ion dynamics !

atomic geometry! electronic band structure! optical absorption!

Excited electronic states!

Excited Electrons!!



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 !



wurtzite ZnO: local-density approximation !

band gap too small!

d-bands too high!

band gap improved!

d-bands improved!

HSE+G0W0:!

•  reliable description of the electronic structure !predictive power (oxides, nitrides, …) !

•  good approximation that includes quasiparticle effects !

wurtzite ZnO: HSE hybrid functional!wurtzite ZnO: hybrid functional and quasiparticle effects !

•  Local-density approximation insufficient for electronic properties!

Egexp=3.4 eV!

EgLDA=0.7 eV!EgHSE=2.1 eV!EgHSE+GW=3.2 eV!




Non-adiabatic electron-ion dynamics!


Excited Electrons! !



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: !



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 SnO2, In2O3, …)!

A. Schleife et al.; Phys. Rev. B 80, 035112 (2009) !F. Fuchs, A. Schleife et al.; Phys. Rev. B 78, 085103 (2008) !

MgO:!




Non-adiabatic electron-ion dynamics!


Excited Electrons! !



Examples:!

excited-electron dynamics!

•  solar cells on satellites!•  nuclear reactors!•  magnetic confinement/inertial

confinement fusion !•  surface adsorption!

Radiation damage: !

•  slow projectiles: ionic stopping dominant !•  fast projectiles: interaction with electronic

system important stopping mechanism !

Goal:!•  parameter-free framework to model stopping !



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)!

beyond Born-Oppenheimer is necessary: !



•  energy transferred to electrons, stopping power: !

•  capture electron dynamics on atto-second time scale!



•  systematic convergence to ensure high accuracy !•  excellent agreement for slow projectiles !•  deviations at higher velocities: off channeling !!•  investigate initial-state dependence !

Comparison to database with experiments (TRIM): !

predictive!!

hydrogen: v=1.0 a.u.≈25 keV!

A. Schleife et al. (submitted)!



Promising perspective: !

•  accurate parameter-free description achievable !

•  essential: excited electronic states and their dynamics !

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 !

•  future: development based on these methods !

© www.itrc.narl.org.tw!

André Schleife

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