Theory in Materials Science Theory in Materials Science 4 4 rd rd lecture lecture Applications of the first principles calculation Applications of the first principles calculation for the high pressure physics for the high pressure physics Professor Naoshi Suzuki & Associate Professor Koichi Kusakabe Graduate School of Engineering Science, Osaka University
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Theory in Materials ScienceTheory in Materials Science
44rdrd lecturelectureApplications of the first principles calculationApplications of the first principles calculation
for the high pressure physicsfor the high pressure physics
Professor Naoshi Suzuki & Associate Professor Koichi Kusakabe
Graduate School of Engineering Science, Osaka University
Simulation of unknown structures and novel materials!
Techniques in the molecular dynamics
• Constant-Temperature Scheme (Nose & Hoover)– Average of the kinetic energy is made constant by introduction of
interaction between the system and an imaginative heat bath.– An equation of motion with dissipative term written by an artificial
variable is solved.
• Constant-Pressure Scheme (Parrinello & Rahman)– The volume (cell parameters) of the system is made a variable.– MD and structural optimization is performed using the enthalpy,
H=E+PV.
Realization of realistic situation
The density functional theory– Hohenberg-Kohn's theorem & Kohn-Sham theory– Levy's DFT (Density Functional theory)
The energy of the ground state is given by the minimum of anEnergy functional of the single-particle density.
EGS = d 3r vext(r) nGS(r) + F[nGS]
F[n] = min
Φ → n (r) Φ|T + Vee|Φ
The Kohn-Sham equation: – h2
2m∇2 + vext(r) + e2 d 3r' n(r)|r – r'| + δExc[n]
δn(r) φi = εiφi
This equation is solvable by computation with a high speed bythe Car-Parrinello method or CG.
Characteristics of basis sets for DFT calculation
• Plane-wave expansion method with pseudo-potentials• Since plane waves are independent of position of atoms, the result is
accurate.• Accuracy of the calculation is determined by the maximum energy of plane
waves.• The kinetic energy is diagonal in the Fourier space, while the potential
energy is diagonal in the real space. Thus FFT is used.• The Hellmann-Feynman force and the quantum stress are easily obtained.
• FLAPW (Full-potential linearlized augmented plane wave)• The wave functions are expanded in spherical waves in an atomic sphere
and in plane waves out of the sphere.• Accuracy is determined by number of partial waves and the maximum
energy of plane waves.• Less ambiguity compared to the pseudo-potential method.• Pulay force has to be evaluated.
The local density approximation I.
The local density approximation II.
The local density approximation III.
The plane-wave expansion I.
The plane-wave expansion II.
• Dimension of the matrix can be O(103) or O(104).• The diagonalization is often performed using the Housholder method of the conjugate gradient method (CG).
However, • the process to find the ground state is regarded as an optimization process forthe energy functional in the function space.
The Car-Parrinello method (Conceptually different idea)
The pseudopotential I.
Ca 4s orbital
Ca 4p orbitalAll electron calc.Pseudization calc.
All electron calc.Pseudization calc.
The pseudopotential II.
The pseudopotential III.
The Hellmann-Feynman force
The calculation scheme of FPMD at a constant pressure
Input:AtomsUnit Cell
External Pressure
{RI}{a0}
{Ψi}
{F}{σ}
{ε}
a = (1 +ε ) a0
Pext
Wave Functions
HF ForceQuantum Stress
Molecular Dynamics
{RI}
Convergence check
update
using {F} {σ}
{ε}{RI} update
a = (1 + ε)a0
by CG method
Optimized {Ψ i} {ε}{RI}
Kohn-Sham eq.{
Structuraloptimization
Moleculardynamics
What is expected for FPMD?
• Accuracy– If the electronic state is well described, accurate within
the Born-Oppenheimer approximation.– See examples below.
• Power– Structural transformation is reproducible for any kind of
materials in principle.– Limited by computational resources and by accuracy of
DFT-LDA (DFT-GGA).
We can show some examples onthe prediction of new materials.
Graphite under high pressure
• Purpose– We show accuracy of FPMD by comparing real
experiments and our calculations• Examples1.Graphite-diamond transformation2.Graphite ball under high pressure
Synthesis methods of Diamond using high pressure
• High-pressure synthesis of Diamond– Static compression
• The metal solvent method Single crystals• The direct transformation Powder, Single crystals• The sintering Poly-crystals
– Shock compression (Dynamic compression)• The direct transformation
We consider static compression at low temperatures:• Less diffusion of atoms => Prediction is not so hard.• Materials design using a meta-stable state in non-equilibrium.
Graphite-Diamond transformation at high-pressure
Graphite(AB stacking)
Hexagonal Diamond(AB stacking)
Cubic Diamond(ABC stacking)
a
b
c
Compression Compression
Monoclinic
[001]
[111]
Kish graphite
Pc = 14 ~ 18 GPa ( Static )Room temperatureT. Yagi et al. ('92)
Hexagonal DiamondPolycrystalline graphite
Pc = 14 ~ 20 GPa ( Static )Room temperatureS. Endo et al. ('94)
Force inversion technique to search for the saddle point
P1
L1
L2
P2
by Tsuneyuki et al.
1) Determine a direction between L1 and L2 in multi-dimensional phase space.2) Obtain a projection of the inter-atomic force (the blue arrow) on the direction from L1 to L2.3) Make inversion of the projection in 2) and obtain an artificial force (the black arrow).4) Optimization with the force in 3) gives the saddle point.
Relative stability among several structures of carbon
At ambient pressure At high pressure
.Tota
l ene
rgy
E [e
V/a
tom
]
HD GR CD
P = 0 [GPa]
+ 0.043[eV]
+ 0.072[eV]
+ 0.414[eV]
+ 0.347[eV]
-154.820 [eV] H =
E +
PV
[eV
/ato
m]
HD GR CD
P = 20 [GPa]
- 0.208[eV]
+ 0.222[eV] + 0.148
[eV]
- 0.234[eV]
-153.852 [eV]
By Tateyama et al. (1996).
Graphite-diamond transformation in a FPMD simulation
At ambient pressure At high pressure
Formation of new bondings at the transformation
• White objects are carbonatoms and yellow iso-surfaces represent chargedensity of electrons.
• We see new bondingrepresented by bondingcharge between graphitelayers.
• Sliding of layersoccurs due to formationof sp3 bond connections.
Simulation
Electronic band structure of graphite under high pressure