Theory and Parameter Free Calculations of EELS and X-ray Spectra

J.J. Rehr1, J. J. Kas1, K. Jorissen2, and F. Vila1 1Department of Physics, University of Washington, 2Amazon Web Services

Seattle, WA

M&M Conference Columbus, OH 24-28 June, 2016

Theory and Parameter Free Calculations of EELS and X-ray Spectra

• GOAL: Ab initio theory & interpretation of core level EELS & XAS

IR VIS UV X-ray

• TALK: Advanced Codes & Workflow tools

Theory: FEFF/Green’s function + OCEAN/BSE

If I can’t calculate it,

I don’t understand it.

(Often attributed to R. P. Feynman)

Challenge: Full spectrum electron- and X-ray spectra

Cu loss function

Challenge: Improve on Hydrogenic model of EELS*

*L. Pauling, Proc. Roy Soc. A London 114, 181 (1927)

L. Konrad et al., MCM2015

~ 35 years later: Ab initio optical constants*

*http://feffproject.org/feff/desy

Cu loss function

Dielectric function

Energy Loss (EELS)

Absorption coefficient

Refractive index

Reflectivity

X-ray scattering factors

Hamaker constants

Optical constants

𝜖 = 𝜖1 + 𝑖𝜖2

−Im 𝜖−1

𝜇

𝑛 + 𝑖 𝑘

𝑅

𝑓 = 𝑓0 + 𝑓1 + 𝑖 𝑓2

𝜖 𝑖𝜔

fcc Al

UV Deep core

http://feffproject.org/feff/desy

Broad Spectrum Calculations: Theory vs. Expt.

Photon energy or Energy Loss (eV)

EELS Theory a lá Fermi’s golden rule

𝜕2𝜎

𝜕𝐸𝜕Ω𝐸,𝑸 = 𝜉

𝑘𝐹𝑘𝐼𝐼 𝑘𝐼 𝑉 𝑘𝐹𝐹

2 𝛿 𝐸𝐼 − 𝐸𝐹𝐼,𝐹

≈ 𝑆 𝑞, 𝜔

𝑉 = 1 𝑟 − 𝑟′ Coulomb interaction

Dynamic Structure Factor ~ Loss function

𝑆 𝑞, 𝜔 ~ 𝐼𝑚 𝜖−1 𝒒,𝜔

Computational bottleneck:

Too many states!

All state 𝑘𝐹, 𝑘𝐼 (probe) and 𝐹, 𝐼 (sample) contribute

Paradigm shift: Real-space Green’s

Function Approach

Golden rule via Green’s functions:

Golden rule via wavefunctions

Efficient:

No sums over final states!

𝜇 𝐸 ~ 𝑖 𝜖 ∙ 𝐫 𝑓 2𝑓

𝛿 𝐸 − 𝐸𝑓

𝜇 𝐸 ~ −1

𝜋 Im 𝑖 𝜖 ∙ 𝐫′𝐺 𝐫′, 𝐫, 𝐸 𝜖 ∙ 𝐫 𝑖

𝐺 = 1 𝐸 − ℎ − Σ

Ψ

J. J. Rehr & R.C. Albers, Rev. Mod. Phys. 72, 621 (2000)

XAS & EELS

Real-space Green’s function theory

http://feffproject.org/feff/desy

Core-hole SCF potentials Self-energy DW factors Efficient Core level spectra: EELS, XAS, XES

89 atom BN cluster

FEFF8

Parallel (MPI) FEFF

MPI: “Natural parallelization” Each proc. does a few energies Needed for state-of-the-art XANES simulations

𝜇 𝐸 ~ −1

𝜋 Im 𝑖 𝜖 ∙ 𝐫′𝐺 𝐫′, 𝐫, 𝐸 𝜖 ∙ 𝐫 𝑖

EELS in FEFF9: Impurity Green’s function

N K-edge ELNES of GaN*

Real space

*K. Jorissen & J. J. Rehr Phys. Rev. B 81, 245124 (2010)

EELS in periodic systems w/o supercell*

Recip. space

Expt

w/o supercell

+++

Relativistic corrections to EELS*

*P. Schattschneider et al., Phys. Rev. B 2005

and the abs. tensor is

Relativistic Coulomb interaction

𝜕2𝜎

𝜕𝐸𝜕Ω𝐸,𝑸 = 𝜉

𝑘′

𝑘

1

𝑄2 − 𝐸 ℏ𝑐 2 𝐼 𝒓 ∙ 𝑸′ 𝐹 2 𝛿 𝐸𝐼 − 𝐸𝐹 − 𝐸

𝐼,𝐹

𝑸′ = 𝑸 − 𝑄𝑧𝛽2𝒆𝑧

𝜕2𝜎

𝜕𝐸𝜕Ω𝐸,𝑸 = 𝑓 𝑄 𝑄𝑖

′𝑄𝑗′𝜎𝑖𝑗 𝐸

3

𝑖,𝑗=1

where

𝜎𝑖𝑗 𝐸 = 𝜉′𝐸 𝐹 𝑟𝑖 𝐼 𝐼 𝑟𝑗 𝐹 𝛿 𝐸𝐼 − 𝐸𝐹 − 𝐸

𝐼,𝐹

Relativistic EELS

Relativistic Magic Angle in FEFF9 agrees with expt.

Non-relat. Relat.

Incorrect Correct

2.4 mrad 0.6 mrad

FEFF9: Advanced methods

Cu K-edge

FEFF 8.4 FEFF 9

FEFF9: Advanced methods

+

Σ

x-ray

Inelastic Losses Self-energy Real-Space

Green’s Function

Screened core-hole

Ab initio: Mean free paths Self-energies Debye-Waller factors RPA core-hole screening Multi-electron excitations

J. J. Kas et al., PRB 76, 195116 (2007)

Cu Lo

ss F

un

ctio

n

Many-pole (full) much better than HL (dashed) vs better theory (dot-dashed)

Many-Pole Self-Energy (MPSE) Extension of Hedin-Lundqvist (HL) plasmon-pole (pp) GW SE model

Σ approximated as sum of PP models matched to loss function

MPSE corrections in BSE and DFT*

MgAl2O4 (ΔSCF-DFT+MPSE) LiF (BSE+MPSE)

*J. J. Kas et al., J. Phys. Conf. Series 190, 012009 (2009)

Energy (eV)

MPSE MPSE

V. Mauchamp, M. Jaouen, JJR, and J. J. Kas, U. Poiters, Preprint (2010)

Question: Can we improve

the theory?

AI2NBSE: Abinit+NIST BSE+UW MPSE Plane-wave, pseudo-potential GW/BSE code

UW + NIST collaboration

Answer: GW/BSE Ab initio optical & low-loss spectra

Si

Ab initio GW/BSE core-spectra: ELNES & XANES

OCEAN: Obtaining Core Excitations from Abinit and NBSE

LiF: F K edge SrTiO3: Ti L2,3 edge

UW + NIST Collaboration

Are we there yet? Optical constants: THz to X-ray

UV-VIS

X-ray

Challenge: Efficient phonon spectra Can we compute vibrational properties ab initio?

Ab initio Debye Waller Factors: 𝐞−𝟐𝝈𝑹𝟐𝒌𝟐

𝜎𝑅2 𝑇 =

ℏ

2𝜇𝑅 1

𝜔coth𝛽ℏ𝜔

2

∞

0

𝜌𝑅 𝜔 𝑑𝜔

𝜌𝑅 𝜔 = −2𝜔

𝜋Im 0

1

𝜔2 − D+ 𝑖휀0 ≅ 𝑤𝑣𝛿 𝜔 − 𝜔𝑣

𝑁

𝑣=1

Projected Vibrational DOS: Use efficient pole model

EXAFS MSRDs:

Helmholtz Free Energy:

𝐹 𝑇 = 𝐸 + 𝑘𝐵𝑇 ln 2 sinh𝛽ℏ𝜔

2

∞

0

𝜌𝑖 𝜔 𝑑𝜔

𝑁𝑐𝑜𝑜𝑟

𝑖

Ψ

Vila et al, Phys. Rev. B 76, 014301 (2007)

Vibrational Density of States

Cu

Typical Results

ZrW2O8: Complex unit cell

ZrW2O8: MSRDs

Phonon-contributions to final-state broadening in Cu

Satellites visible only at low T

User-friendly Java GUI: JFEFF

VESPA - Virtual EELS Software Package

Goal: Integrated EELS software for STEM using CORVUS Workflow Manager

Prototype: Virtual STEM in the Cloud State-of-the-art EELS modeling

FEFF9+JFEFF

AI2NBSE and OCEAN

TELNES

Precompiled & Optimized on the Amazon EC2

K. Jorissen et al. www.feffproject.org/feffproject-scc.html

CONCLUSIONS • Goals:

– Mostly achieved or in sight

• Complementary theoretical techniques: – Real-space

– Supercell/reciprocal space

• Next generation theory & codes including: – Many-body effects

– Relativistic effects

• Broad spectrum response: – IR VIS UV X-ray

From left to right: Ken Nagle Yoshi Takimoto Kevin Jorissen Towfiq Ahmed Hadley Lawler Aleksi Soininen Fernando Vila Adam Sorini Alex Ankudinov Micah Prange John Vinson (Shauna Story) John Rehr Josh Kas (Egor Clevac)

Acknowledgments: Thanks to DOE BES for $$ Rehr Group + L. Reining, M. Guzzo, E. Shirley,

K. Gilmore, et al.

Theory and Parameter Free Calculations of EELS and X-ray Spectra

J.J. Rehr1, J. J. Kas1, K. Jorissen2, and F. Vila1 1Department of Physics, University of Washington, 2Amazon Web Services

Seattle, WA

Thank you!

Thank you!