First principles studies of materials under extreme condition

Tadashi Ogitsu

Quantum Simulations Group

Lawrence Livermore National Laboratory

FADFT2007 ISSP 7/24/06

This work was performed under the auspices of the U.S. Dept. of Energy at the University of California/ LLNL under contract no. W-7405-Eng-48.

Collaborators

Andrea Trave, Alfredo Correa, Jonathan DuBois, Kyle Caspersen,

Eric Schwegler, and Andrew Williamson (Physic Ventures)

Lawrence Livermore National Laboratory (theory)

Gillbert Collins, Andrew Ng, Yuan Ping

Lawrence Livermore National Laboratory (experiment)

David Prendergast

Molecular Foundry, Lawrence Berkeley Laboratory

François Gygi and Giulia Galli

University of California, Davis

Stanimir Bonev

Dalhousie University, Canada

Eamonn Murray and Steven FahyTyndall National Institute, University College, Ilreland

David Fritz and David ReisUniversity of Michigan Ann Arbor (experiments)

Outline of my talk:

• DFT: my viewpoint

• Why we need large scale simulations?

• Phase diagram of materials under pressure

– Temperature and pressure are extremely high

– Equilibrium property

• Dynamical response of materials upon ultra fast laser pulse

– Ultra fast (sub ps) time resolved measurement

– Non-equilibrium dynamics (electrons and ions)

– Non-adiabatic?

DFT: my viewpoint

• Rigorous theory for the ground state, but…

– We need approximations (LDA/GGA, pseudopot) to apply it to a real system

– The KS eigenvalues are not supposed to represent electron excitation in theory, while 104,000 papers on DFT band structure (as of 7/11/07) are found by google

– So confusing… (as of April 1989)

• Why justified for excited state?

– Huge amount of literature seem to suggest qualitatively ok (sort of defacto standard)

– For a certain limit, some theoretical requirement is satisfied (eg. Koopman’s theorem)

Rigorousness

DFT

QMC

Tight binding

Com

puta

tiona

l cos

t Goal!

Good cost efficiency made DFT popular, but need further

improvement

Why large scale simulations?

• Complex material: elemental boron (8/1/07 at 17:00)

• Finite size effects

• Canonical ensemble– Long time scale simulations

• A simple calculation could be expensive– Eg. 2()

• Non-equilibrium (and/or non-adiabatic) dynamics?

Phase diagram of materials under pressure: Significance of ab-initio approach

• Phase boundaries are rich in physics

– Crossing line of Gibbs free energies of different phases

• Change in structure (static total energy)

• Potential energy surface (ion dynamics -> entropy)

• Electronic structure (direct and indirect)

• Important applications in various sub-field of physics

– Modeling of interior of planets

– Fundamental questions in condensed matter physics

– Designing a novel material

Method: melting line calculation

• Two-phase simulation method (nucleation is already introduced)

• Ab initio method (GP and Qbox codes by Gygi at UC Davis)

– Density Functional Theory with PBEGGA

– Planewave expansion, nonlocal pseudo potential for ions

– 432 atom cell, Ecut=45Ry, -point sampling

T>Tmelt

T<TmeltJ. Mei and J. W. Davenport, Phys. Rev. B 46, 21 (1992)A. Belonoshko, Geochim. Cosmochim. Acta 58, 4039 (1994)J. R. Morris, C.Z. Wang, K.M. Ho, and C.T. Chan, Phys. Rev. B 49, 3109 (1994)

Ab-initio two-phase MD at P=100GPa

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T=2300K: melt

T=2200K: solidify

LiH melting lineOgitsu et al. PRL 91, 175502 (2003)

• LiH: simple yet its phase diagram is not well understood

– Ionic crystal with rocksalt structure (B1)

• What is left?

Tmexp = 965 K

? TmGGA = 795 K (18% lower than exp)

B1 phase stable up to 100GPa (exp)

? No B2 (CsCl) phase found

- All the other alkali hydride exhibit B1-B2 transition < 30GPa

Liquid

Solid (Rocksalt)X

Quantum Monte Carlo Corrections to theDFT Melting Temperature of LiH (Tm

dft=790K, Tmexp=965K)

Liquid LiH (T=TM)

Simulation Cell Volume (au)

23 24 25 26 27 28-93.0

-92.6

-92.2

-91.8

-91.4

Solid LiH (T=TM)

23.0 23.5 24.0 24.5

Tot

al E

nerg

y (a

u)

-93.8

-93.4

-93.0

-92.6

-92.2

DFTQMC

Simulation Cell Volume (au)

DFTQMC

• QMC predicts corrections to the internal energy and equilibrium volume

• These equation of state corrections are larger in the solid than the liquid

• Preliminary results predict an increase in TM from 790K to 880K (exp=965K)

• QMC predicts corrections to the internal energy and equilibrium volume

• These equation of state corrections are larger in the solid than the liquid

• Preliminary results predict an increase in TM from 790K to 880K (exp=965K)

QMC equilibriumvolume

DFT equilibriumvolume

Internal EnergyCorrection

F (V,T) = U(V) + ZPE(V) + FH(V,T)

G(P,T) =F(V,T)+PV; P = -dF/dV

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ZPE(V ) =12

hϖs,q

(V )s,q

∑

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FH(V ,T ) = kT ln[1− exp{−hϖ

s,q(V ) /kT}]

s,q

∑

•Free energy surface of phases match at the phase boundary

•Free energy surface, G(P,T), of solid can be well described by harmonic phonon model

Solid/solid phase boundary: Quasi Harmonic Approximation (QHA) Karki, Wentcovitch, Gironcoli, and Baroni, PRB 62, 14750 (2000)

-point phonon

(a) PRB 28 3415 (1983)

(a)This

work

LO 1080 1071

TO 606 593

LiH: NaCl phase

LiH phase diagram

Theory:

• B1-B2 boundary determined by ab initio QHA method

• B2-liquid boundary determined by ab initio two-phase method

Experiments:

• Low-T B1-B2 boundary is being explored by DAC experiments (Spring-8)

• High-T B2-liquid boundary by isentrope experiments (LLNL)

Property of LiH fluid under pressure

• Strong correlation between Li and H dynamics

– Velocity distributions reflect the mass difference

– Diffusion constants of Li and H are almost the same

• Dynamical H2 (Hn) formation observed

at high temperature

– Charge state of H2 in LiH fluid is

nearly neutral

– Ionicity of LiH fluid is weakened upon dynamical H2 formation

Ab-initio two-phase method:Computational cost

• Two approaches successfully mapped liquid/solid phase boundaries of materials in ab-initio level

– Two-phase: Ogitsu et al. PRL 2003

– Potential switching: Sugino and Car PRL 1995

• Which is more cost efficient?

– Two-phase method is computationally intensive, while potential switching method demands intensive human labor (many many MD runs on P, T and the switching parameter space)

Example with LiH: Each two-phase simulation was roughly 2-10 ps MD run with 432 atoms cell

In total, to map out the melting line for 0-200 GPa, the CPU cost equivalent to a half year with a linux cluster (128 cpu) was used (2002-2003)

Note: low density costs more (nature of planewave + faster dynamics at higher pressure)

Summary on LiH phase diagram

• It has been demonstrated first time that ab-initio two-phase method is feasible

– LDA/GGA seems to underestimate the melting temperature

– QMC corrections look promissing

• B1/B2/liquid phase boundaries of LiH have been calculated for a wide range of P, T space

• Property of compressed LiH fluid has been studied from first-principles

– Correlated Li and H dynamics

– Dynamical Hn clustering yielding weakening of ionicity

Melting line of hydrogen

Pressure

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Atomic hydrogen solid

bcc structure (?) - metal

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bcc

bcc

bcc

bcc

bcc

bcc

GroupI

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Molecular hydrogen solid

hcp structure - insulator

GroupVII

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r

Metallic hydrogen under pressure [Wigner and Huntington (1935)]

Large zero-point motion => possible low-T liquid state?[Brovman, Kagan, Kholas, JETP (1972)]

Two possible scenarios [N. Ashcroft, J Phys. Cond. Matt. (2000)]

Insulator

Liquid

Metal

Liquid

Insulator Metal

liquid

solid

A hypothetical scenario towards the low-T liquid

H2

€

K ~ EH2H H

€

dH2~ dH2 −H2

H Hsolid

liquid: H2

liquid: H

Measured melting line of hydrogen: Gregoryanz et al. PRL 90 175701 (2003)

?

Exp can reach the P, T range of interestExp could not locate the melting point above 44GPa

Ab initio melting curve supports low-T liquid scenarioExperiments:Gregoryanz et al. PRL (2003).Datchi et al. PRB (2000).Diatschenko et al PRB (1985).

Theory:Bonev, Schwegler, Ogitsu and Galli, Nature, 2004.

Molecular fluid

Non-molecular fluid

Solid

Metallic super fluid at around 400GPa?Babaev, Subda and Ashcroft, Nature 2004Babaev and Ashcroft, PRL 2005

Reminder: experiment can reach to the P, T rangeCould we suggest how to detect melting?

• Change in distribution comes from the tail of MLWF spread

– Net overlap is changing at high P

• Stronger asymmetry observed in liquid MLWFs at high P

– Suggest enhanced IR activity in liquid

MLWF spread distribution at Tm

liquidsolid

Summary on the melting line of hydrogen

• Maximum in melting line of hydrogen is found by ab-initio two-phase method

• The negative slope is explained by weakening of effective inter molecular potential. Dissociation of molecule is not necessary

• IR activity measurement might be able to detect the high pressure melting curve (given that the condition is experimentally accessible)

Why higher pressure phase has not been well understood? Limit in computational approach

• Does LDA/GGA work?

– ~200GPa might be OK (Pickard and Needs, Nature Physics Jul, 2007)

– No well established reference system to compare with

• Quantum effect of proton

– DFT/path-integral (maybe DMC/path-integral) is feasible, however, within adiabatic approx.

– Full (elec & ion) path-integral: lowest temperature record is about 5000K

• Non-adiabatic electron-phonon coupling

– Crucial if metallic

Breakthrough in computational approach needed

What is limiting high pressure experiments?

• To reach high P, T itself is challenging (diamonds break)

• Small sample

• Probe signal needs to go through diamond/gasket

– S/N ratio problem

• Direct structural measurement (X-ray, neutron) cannot reach too high pressure

– X-ray cannot determine the orientation of H2 (X-ray scatter off electrons)

• Most reliable experimental techniques, Raman/IR, provide only indirect information to the structure

– Hidden challenge for theory: How do we know the structure? [Pickard and Needs, Nature Phys 2007]

By Russel Hemley at Carnegie Institution

Dynamical response of materials upon ultra fast laser pulse

• Advance in the pump and probe experiments made sub pico second time resolution possible with

– Ultra-fast Electron Diffraction

– Dielectric function measurement

– Raman/IR

– X-ray

• Time evolution of phase transition, chemical reaction (breaking/making a bond) can be directly measured!

• Big challenge for theory since

– Non-equilibrium

– Adiabatic approximation might be breaking down

Time evolution of electron diffraction of Al

At t = 0, the laser pulse (70 mJ/cm2) is induced

Siwik et al. Science 302, 1382 (2003)

The Jupiter Laser Facility at LLNL

Reflected ProbeTransmitted Probe

Pump

Probe

Schematics of experiment

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Epulse = 2.9 ×106 J / kg,150 fs FWHM ,λ = 400nm (3.1eV )

50nm thick free standing gold foil

Pump laser pulse

Probe laser pulseBroad band =400~800 nm

t

1. t=0: electrons are excited by 3.1eV photons2. t>0: Transmission and Reflection (T*, R*) gives 2()

1. Electronic states evolve (Auger, el-el and el-ph scattering)2. Atomic configuration evolves (energy dissipation from electrons)

R*

T*

Time evolution of 2() of 50nm Au film triggered by a laser pulse [Ping et al, PRL 96, 25503 (2006)]

• Fine time resolution, simple and reliable technology

• Interpretation of results is challenging due to missing information

– Electronic states

– Atomic configurations

€

Epulse = 2.9 ×106 J / kg,150 fs FWHM ,λ = 400nm (3.1eV )

1. For 1.2-4 ps, 2(), does not change (quasi steady state)2. The inter-band transition peak at 2.5eV is present in the quasi steady state3. The peak is enhanced from ambient condition

Parallel pair of bands (ll1) contribute on a peak in 2()

• Inter-band transition no-momentum change

– Kubo-Greenwood formalism

• Intra-band transition require change in momentum

– Electron-phonon coupling

– The transportation function (2F()) to DC conductivity and the Drude form

Ef

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2(ω ) ~< ki |

v p | kj >

Ei − Ejk, i, j

∑ ( fi − fj )δ (Ei − Ej −ω )

0

1

2

3

4

5

6

7

1 1.5 2 2.5 3 3.5 4

Photon energy (eV)

Current formalisms for 2() does not describe low and high energy regimes seamlessly

• Inter-band transition no-momentum change

– Kubo-Greenwood formalism

• Intra-band transition require change in momentum

– Electron-phonon coupling

– The transportation function (2F()) to DC conductivity and the Drude form

Ef

•In a disordered system, elastic scattering becomes dominant, therefore, Kubo-Greenwood formula is good enough

•The quasi-steady state of warm dense gold: ordered or disordered?

-The inter-band transition peak suggests presence of long range order

Super-cell + Kubo-GreenwoodNo-inelastic el-ph scattering countedSuper-cell + Kubo-GreenwoodNo-inelastic el-ph scattering counted

Procedure of ab-initio 2() calculation

Underlying assumptions: •Electrons are in thermal equilibrium•Heating of ions is slow (el-ph coupling of gold is small)•Ions are also in thermal equilibrium

Two temperature model:

Tion(t) Ab-initio MD at T 2()Tel(t=0) + el-ph

Note: TD-DFT-MD plus non-adiabatic correction might provide the direct answer

Kubo-Greenwood with elevated Tel

Comparison of exp and ab-initio 2()

No enhancement of inter-band peak observed in ab-initio 2()

• Missing el-ph coupling (eg. intra-band transition)?• Thermalized electrons (Fermi distribution) incorrect?• Inter-band peak implies long range order of lattice?

Note: single 2 calculation generate 1TB data

Summary

• Ab-initio 2() does not agree with experimental measurement

– No inter-band peak above 2eV

• There are many assumptions to be re-examined

– Electron distribution function

– Application of Kubo-Greenwood formalism to small (Drude) regime

– Electron-phonon coupling constant upon excited electrons

How fast do electrons thermalize?

• There seems to be a general consensus on electronic thermalization time scale of several hundred femto second

• Only one quantitative experimental measurement on gold found [PRB 46, 13592 (1992)]

– Residual in high energy is not explained

– Energy density is very small compared to Ping’s experiments

– Residual seems to grow as a function of input energy

E=120J/cm2

E=300J/cm2

Thermalization time scale as a function of input energy should be re-examined

Concluding remark

• Physics under extreme condition provide exciting and challenging problems to computational physics community

– Significance of computational approach in high-pressure physics has been and will be growing

– Ultra-fast pump and probe experimental technique provide exciting new physics that challenges theory. Novel computational approaches will be needed

Ab-initio MD beyond BO approximation

Seamless transport calculation formalism (elastic and in-elastic el-ph scattering)