metales alcalinos terreos

Computational Material Science
Gonzalo Gutiérrez
Departament of Physics
Universidad de
Chile.
[email protected]
http://fisica.ciencias.uchile.cl/~gonzalo
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Outline
1. Introduction
- Simulation of Materials
- Characteristic lengths and times in materials
- Adiabatic approximation
2. Electronics Structure Calculation (T=0 K)
- Density Functional Theory (DFT), Hohenberg-Kohn Theo.
- Kohn-Sham Eqs., Local Density Approximation (LDA)
-Example: alumina
3. Moving the atoms
-Classical MD
- Ab-initio MD
- Example: germania
4. Conclusions
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Tabla Periódica de los
Elementos
Metales
alcalinos
Otros
metales
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Metales
alcalinos terreos
Gases
nobles
Metales de
transición
Metales de
tierras raras
haluros
Otros no
metales
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Computer simulation
Real
material
Theory
Experiments
Computer
simulation
Experimental
results
Test of
the theory
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Models
(reductionism)
Computer
simulation
Analitically
soluble
Test of
the model
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Space and time in materials
continium
als r
i
r
te e e
a
M gin
En
ys
>10-3m
microstructural
Ph d istry
a n em
ch
atomics
10-1010-13
10-1210-16
10-9-10-6m
electronics
10-11-10-8m
Å
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nm
µm
length
>10-6
10-610-10
10-6-10-3m
ics
time (s)
mm
m
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Splitting different characteristics time
H Ψ =E Ψ
• Equation of the system
• Solid state physics hamiltonian:
(Tion + Te + Ve−e +Ve−ion + Vion−ion )Ψ(r1,...,rN ; R1,...,RM ) = EΨ(r1,...rN ; R1,...,RM )
• Adiabatic approximation: splitting of the ionics and electronics
degree of freedom
Mion ~2000-10000 me ⇒ τion>>τe
• Writing Ψ(r;R) =Φ(r) χ (R), the problem split in two parts:
electron
(Te +Ve−e +Ve−ion+Vion−ion)Φ(r1,...rN ;{R})=ε({R})Φ(r1,...rN;{R})
ion
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(Tion +ε({R}))χ(R) = E' χ(R)
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Electrons and ions
Electrons (Te + Ve−e + Ve−ion +Vion−ion )Φ(r1,...rN ;{R}) = ε({R})Φ(r1,...rN ;{R})
• knowing ε(R) we can have:
-Ground state (T=0)
-Crystalline structure and PV diagram
-Elastic, optics, magnetics, etc, properties
Iones
-Potential energy surface
-Interatomic potential
⇒ interatomic forces
(Tion + ε ({ R}) )χ ( R ) = E ' χ ( R )
This is a quantum problem, but classical physics work pretty well!
- to look for an effective interatomic potential ε(R)
- to treat the ions as classical particles, i.e. obeying Newton Laws
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Electrons : solution of the Schrödinger Eq.
• Hamiltonian of a many electrons system:
 N h2 2
 r r
r r
∑−

∇ j + Ve −e + Vext Φ(r1,.., rN ) = EΦ( r1,.., rN )
 j =1 2 m

•To look for the GS by means of a
variational method:
~ r r
Φ(r1 ,.., rN )
~ ~
E0 [Φ0 ] = MinΦ~ Φ H Φ
3N dimensions trial function
• Approximate Methods : one electron selfconsistent equations
r r
Hartree: Φ ( r1 ,.., rN ) = φ1....φN
 h2 2 Z

n( r ' )
 −
∇ − +∫
dr ' φ i = ε iφ i , con n ( r ) = ∑ | φ i |2
r
| r − r '| 
i
 2m
Hartree-Fock:
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r r
1
Φ (r1 ,.., rN ) =
det [φ1 ....φ N i ]
N
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Density Functional Theory
The electronic density n(r) as the basic variable
Hohenberg-Kohn (1964):
1. The external potential v(r) is totally determined from the
electronic density n(r).
E [n ] = ∫ n (r )v(r )dr + T [n ] + Ve−e [n]
2. For the trial density ñ(r) we obtain E0 ≤ E[ñ]
N.B. a) We had a 3N variables trial function Φ(r1..rN) , but now only
a 3 variables trial function ñ(r)
b) DFT gives a strict formulación of the search of the GS from
the electronic density ñ(r) .
For example, Thomas-Fermi can be derived from DFT.
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Electrons II: Kohn-Sham Eqs.
• Total energy reads as (Kohn & Sham, 1965)
E{R}[ñ]=Ts[ñ]+EHartree[ñ]+Eext[ñ]+ Exc[ñ]
•Minimization of E{R}[ñ] is equivalent to solve
Kohn-Sham Eqs.
Selfconsistent
solution
r r
r
 h2 2
− 2 m ∇ + veff (r ) ϕi (r ) = ε iϕ i (r )
r
r
r
n(r ' ) r
veff = ∫ r r dr ' + Vext (r ) + vxc (r )
| r − r'|
N
r
r 2
n( r ) = ∑i =1 | ϕ i (r ) |
r
r r
• LDA approximation: Exc [n] ≈ ∫ n(r )ε xc (n(r )) dr
with
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ε xc = −
0.458
0.44
−
rs
rs + 7.8
homogeneous
electron gas
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Electrones III: implementation
all-electron full-potential
all-electron muffin-tin
pseudopotenciales
jellium
LDA
GGA
non-relativistic
relativistic
I
I
II
 h2 2
r
r  k
k k
−
∇
+
V(
r
)
+
v
(
r
)
ϕ
=
ε
ϕ
xc
i
i
i
 2m

linearized augmented
periodic
no-periodic
spin-polarized
non spin-polarized
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plane wave: LAPW
scattering functions
(ex.:. Hankel): LMTO
plane wave
LCAO
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Implementation: selfconsistent cycle
nini (r )
q (r ) = n(r ) + Z ( Rα )
2
∇ VC (r ) = 4π q(r)
v xc (r) = v xc[n(r)]
Electronics
cycle
H kl = k | H | l
S kl = k | l
(H − εS )c = 0
{ε i }, {cij }
ϕ i = ∑ j cij j
n(r) = ∑ i,oc | ϕi |2
Rα
Poisson Eq.
Matrix
elements
Dynamics
Find the de eigenvectors
and eigenvalues
KS orbitals in terms of
base functions
Etot [n( r), Rα ], F = −
∂E
∂R
Calculation of properties
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Technical issues
•Input: crystalline structure, atomic numbers
•Accuracy in energy, in forces
•Full potential calculation:
• FPLAPW (Wien2k),
• FPLMTO
•Pseudopotential calculation
•PPPW method (VASP, CPMD, ABINIT)
•Other expansion: Gaussian, SIESTA
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Example: Al2O3
Importance:
-ceramic material important both in geosciences and technology
-many technological applications, because
extreme hardness, high melting point, low electrical conductivity.
Structural phase transitions:
liquid → γ → δ, θ→ α-alumina
amorphous(a.o) → γ → θ→ α-alumina
Transition aluminas are used as adsorbents, catalysts or catalyst
support, coatings, etc, because of their fine particle size,
high surface and catalytic activities of their surfaces.
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γ-Al2O3
Ref: G. Gutiérrez et al: -PRB 65, 012101 (2002)
-to be submitted (2004)
α and γ-Al2O3 are the most common phases,
but the structure of g is still unknown.
•Aluminum surface oxidize in the atmosphere, forming oxide films.
Also, one can make a surface treatment to aluminum by
anodic oxidation.
•How is the structure of such films?
•Are they amorphous-like, or they have a γ structure, as has been
suggested because the low coordination that present Al atoms at
the surface?
•The first step to answer these question is to determine the bulk
γ-Al2O3 structure
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γ-Al2O3 : structure
• Defective spinel: cubic unit cell
32 O in a fcc array, and 211/3 Al , placed in
tetra- and octahedral positions at random.
described by [ ] 2 Al 1 O
23
213
32
•This formula is deduced from the figure.
An integer number of Al atoms is obtained
if one enlarges this unit cell three times:
160 at=[]8 Al64 O96
These Al vacancies occupy octahedral
or tetrahedral cation sites?
In what proportion?
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Al
O
Mg
Unit cell of the spinel MgAl2O4:
56 atoms: 8 Mg in tetra, 16 Al in
octahedral, and 32 O in fcc
position.
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Situation so far
• Experiments: neutron, electron and x-ray diffraction
has led to conflicting conclusions:
•vacancies situated entirely in tetrahedral positions
octahedral
both, 63:37
•Theoretical works: not conclusive results either.
•MD and MC 64% of vacancies in thetrahedral sites
•Empirical pair pot+ab-initio, MD (’99): vacancies in
octahedral positions.
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Searching for the minimum energy structure
Total energies for a large number of configurations consistent with the crystallographic
specifications need to be calculated
Two main ways:
1) In most MD simulations, authors have increased the unit cell
2) to consider as a tolerable stoichiometric approximation a unit cell with 56at=21
Al + 32 O.
•Our approach: reduction of the unit cell
•Instead of considering a cubic unit cell, one can consider
a primitive cell with less particles!
Noting that ([]8Al64O96)/4=[]2Al16O24 only 40 atoms
•In order to have it
•perfect spinel, Al3O4, has 56 at., so the primitive has 14
at. Increase by three, remove 2 Al, and you get Al2O3
•There are two Al vacancies, which can be both
tetrahedral, both octahedral, or one tetra and one octa.
•There are only 153 configurations for Al vacancies
(in comparison to 1010 for the 160 at. unit cell).
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Method
After a detailed structural analysis of the 153 configurations, one get only 14
non-equivalent config: 4 OO, 6 TO and 4 TT.
•We performed total-energy calculations for all these 14 configurations, allowing
for relaxation.
Method: ab-initio total energy calculation
DFT within LDA and exchange-correlation functional of Ceperley and Alder
Plane wave pseudopotentials code (VASP)
Kinetic energy cutoff Ecut at 400 eV.,
Two kind of k-point sampling, resulting two kind calculation
Low accuracy: only the g point, converged to 0.03 eV/(formula unit)
High accuracy, using a (3x3x9) k-point mesh, 0.004 eV/(formula unit)
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Total energy results
Total energy for the 14
minimum configurations
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γ-Al2O3 :total energy curve
Bulk modulus: 219 GPa
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Density of states
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Moving the atoms
•IDEA: atoms “are” classical particles, they follow Newton laws
Interatomic potential V(r)
F = −∇V (r1 ,..., rN )
Temperature : equipartition theorem
T = mv 2 / 3k
•Numerical solution of the coupled ordinary differential equations
•Results: r1(t),....,rN (t); v1(t),....,vN(t)
•Physical properties: average over configurations, like in statistical mechanics
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Molecular Dynamic program
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Inicial conditions:
{ri(t0),vi(t0)}N
Choice of interatomic
potential ⇒forces
Numerical solution to
the eqs. of motion
Physical properties: temporal
average over configurations:
〈….....〉t
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N~2000
V
EK=3/2NkBT
• empirical pot
• ab-initio ε(R)
Verlet
V (r ) =
q q CC
− 6 + B exp(ar)
r
r
i
j
i
j
∆t:10-15 s : time step
•Termodynamics prop.
• structural properties
• dynamical prop.
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Interatomic potentials
A general expresion for an interatomic potential should
contains one-body, two body, three body,...and N body terms
r N N rr N N N rr r
V = ∑v1(ri ) +∑∑v2 (ri ,rj ) +∑∑ ∑v3(ri , rj ,rk )+.........
N
i=1
i=1 i≠ j
i=1 i≠ j i≠ j≠k
For simplicity, people in general use only two and three body terms.
The idea is to write a general functional form and to fit it according
some experimental data: elastic constants, vacancy energy, etc.
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Simple interatomic potential
Ex: Lennard -Jones potential: good for noble gases
 σ
Φ (rij ) = 4ε 
 rij

σ
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
σ  
 −  

r  

 ij  
12
6
ε
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Interatomic potential II
A good interatomic potential should have the following properties
•
FLEXIBILITY
2. PRECISION
3. TRANSFERABILITY
4. COMPUTATIONAL EFFICIENCY
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DM algorithm
Time discretization
r
r
r
1r
2
r
t
r
t
v
t
a
t
(
+
∆)
=
(
)
+
(
)∆
+
(
)∆
i
i
i
i
1 ∂V
r
2
ai = −

r
r
r
a
t
a
t
r
r
(
)
+
(
+
∆)
m
∂
r
i
v ( t + ∆) = v ( t) + i
i i
∆
i
 i
2
ri(∆)
ri(2∆)
ri(0)
Time stepping: Velocity Verlet algorithm
Given
1.
2.
3.
4.
5.
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r
r
(ri (t),vi (t))
r
{
r
Compute
as a function of i (t)}
r
∆
r
∆r
vi (t + ) ← v i (t) + ai (t)
2
r
r
r2 ∆
ri (t + ∆) ← ri (t) + vi (t + )∆
2
r
r
r
ai (t)
Compute ai (t + ∆) as a function of
r
r
∆ ∆r
vi (t + ∆ ) ← v i (t + ) + ai (t + ∆ )
2
2
{ri (t + ∆)}
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Physical Properties
Structural properties: pair distribution function g( r)
static strucuture factor S(q)
Thermodynamics properties pV = NkT +
heat capacity
W
Dynamical properties diffusion coeficient D
Velocity autorrelation function VACF(t)
Vibrational density of states D(w)
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Amorphous Al2O3 : results I
Pair distribucion function
Coordination number
• Al-O distance: 1.76 Å (exp.=1.8 Å)
• Coord. Nr: Al-O: 4.25, O-Al:2.83 (cutoff: 2.2 Å)
Ref: G. Gutiérrez et al: PRB 65, 104202 (2002)
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Amorphous Al2O3 : angle distributions
Al
O
Summary:
-Short range order consists of AlO4 tetrahedra.
-there are 3- and 4-fold rings, with planar 2- and 3-fold ring.
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Structural trasformation of amorphous Al2O3
under pressure
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pres s u r e
• In the Al2O3-SiO2 system (important in geoscience and ceramic) present
a both a Si and Al coordination change, from tetrahedral to octahedral
Coordination under pressure bajo.
In SiO2 that change is wellknown. What does it happen in alumina?
• Phase transformation in liquid and amorphous material:
poli-amorphysm
C’
II
líquid
C
I
Observed in: H2O!!
SiO2, GeO2, P,
Al2O3-Y2O3
gas
temperature
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Amorphous Al2O3 under pressure: results
Ref: G. Gutiérrez, Submitted (2004)
4.2
3.9
3.175
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Al2O3under pressure II
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Angular distribution
Coordination of Al changes according density ⇒ OK exp.
• a-a transformation in alumina: coord 4 → coord.6
• according our simulation, it does happen at pressure 15-25 Gpa.
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Ions + electrons: ab-initio MD
The only interaction there exists is the Coulomb interactions!
The idea of ab-initio MD is to calcul everything from the solid state hamiltonian
1. Electronic structure calculation to obtain the total energy e( R) in
terms of the positions of the ions.
2. Forces for each ∆t, forces are obtained from the electronic str. calc.
∂
∂
FI = −
EKS ({ϕ i},{Ri}) ⇒
ε({R}) = −
∂RI
∂RI
∂
∂
=−
ϕ i | HKS | ϕ i = −∑ ϕ i
HKS ϕi
∑
∂RI i
∂RI
i
Teo.Hellman-Feynman
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Amorphous Al2O3 : DM ab-initio
Preliminary Results
Pair distribution functions
Ref: S. Davis & G. Gutiérrez, in progress
Electronic Density of states
Detalles: Codigo VASP, 10 ps, 600 K,80 at., (NVT),
solo punto Γ, 5min/paso.
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