First principles studies of materials under extreme condition Tadashi Ogitsu

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First principles studies of materials under
extreme conditionTadashi Ogitsu
FADFT2007 ISSP 7/24/06

• Quantum Simulations Group
• Lawrence Livermore National Laboratory

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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 Fahy Tyndall National
Institute, University College, Ilreland David
Fritz and David Reis University of Michigan Ann
Arbor (experiments)
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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?

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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. Koopmans theorem)

Goal!
Computational cost
Tight binding
DFT
QMC
Rigorousness
Good cost efficiency made DFT popular, but need
further improvement
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Why large scale simulations?

• Complex material elemental boron (8/1/07 at
  1700)
• Finite size effects
• Canonical ensemble
• Long time scale simulations
• A simple calculation could be expensive
• Eg. ?2(?)
• Non-equilibrium (and/or non-adiabatic) dynamics?

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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 -gt
  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

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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, Ecut45Ry, ?-point sampling

TgtTmelt
TltTmelt

• J. 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)

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Ab-initio two-phase MD at P100GPa
T2300K melt
T2200K solidify
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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 lt 30GPa

Liquid
Solid (Rocksalt)
X
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Quantum Monte Carlo Corrections to theDFT
Melting Temperature of LiH (Tmdft790K,
Tmexp965K)
Liquid LiH (TTM)
Solid LiH (TTM)
-92.2
-91.4
-92.6
-91.8
DFT equilibrium volume
Total Energy (au)
-93.0
-92.2
Internal Energy Correction
QMC equilibrium volume
-93.4
-92.6
-93.0
-93.8
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24
25
26
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23.0
23.5
24.0
24.5
Simulation Cell Volume (au)
Simulation Cell Volume (au)

• 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 (exp965K)

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

• 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

F (V,T) U(V) ZPE(V) FH(V,T) G(P,T)
F(V,T)PV P -dF/dV
LiH NaCl phase
?-point phonon
(a) This work
LO 1080 1071
TO 606 593
(a) PRB 28 3415 (1983)
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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)

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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

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Ab-initio two-phase methodComputational 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)
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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

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Melting line of hydrogen
Metallic hydrogen under pressure Wigner and
Huntington (1935)
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A hypothetical scenario towards the low-T liquid
liquid
liquid H
liquid H2
solid
solid
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Measured melting line of hydrogen Gregoryanz et
al. PRL 90 175701 (2003)
?
Exp can reach the P, T range of interest Exp
could not locate the melting point above 44GPa
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Ab initio melting curve supports low-T liquid
scenario
Experiments Gregoryanz et al. PRL (2003). Datchi
et al. PRB (2000). Diatschenko et al PRB (1985).
Theory Bonev, Schwegler, Ogitsu and Galli,
Nature, 2004.
Non-molecular fluid
Metallic super fluid at around 400GPa? Babaev,
Subda and Ashcroft, Nature 2004 Babaev and
Ashcroft, PRL 2005
Molecular fluid
Solid
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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

solid
liquid
MLWF spread distribution at Tm
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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)

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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
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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

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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)
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The Jupiter Laser Facility at LLNL
Probe
Pump
Reflected Probe
Transmitted Probe
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Schematics of experiment
Pump laser pulse
?t
T
R
50nm thick free standing gold foil
Probe laser pulse Broad band ?400800 nm

• ?t0 electrons are excited by 3.1eV photons
• ?tgt0 Transmission and Reflection (T, R) gives
  ?2(?)
• Electronic states evolve (Auger, el-el and el-ph
  scattering)
• Atomic configuration evolves (energy dissipation
  from electrons)

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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

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Parallel pair of bands (l?l?1) 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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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
Super-cell Kubo-Greenwood No-inelastic el-ph
scattering counted

• 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

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Procedure of ab-initio ?2(?) calculation
Two temperature model

• 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

Kubo-Greenwood with elevated Tel
Tion(t)
Ab-initio MD at T
?2(?)
Tel(t0) ?el-ph
Note TD-DFT-MD plus non-adiabatic correction
might provide the direct answer
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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
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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

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How fast do electrons thermalize?
E120?J/cm2

• 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 Pings
  experiments
• Residual seems to grow as a function of input
  energy

E300?J/cm2
Thermalization time scale as a function of input
energy should be re-examined
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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)