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SLACS-INFM/CNR Sardinian Laboratory for Computational Materials Science
www.slacs.itSLACS
Atomically informed modeling of the microstructure evolution of nanocrystalline materials
A. Mattoni
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
SLACSCNR-INFM
CRSLNLR Regional Laboratories
Atomistic investigation: large scale molecular dynamics simulations Large scale electronic structure calculationsContinuum modeling: models for growth, interface mobilities
http://www.slacs.it
•Division: Material Physics (Microstructure evolution of nanostructured materials)(6 members,
www.dsf.unica.it/colombo)
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
OUTLINE
The microstructure of interest for nanocrystalline materialsBoundaries between order/disordered phase
The theoretical frameworkMolecular dynamics atomistic simulations
Modeling the growth of nanocrstals embedded into an amorphousmatrix
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Molecular dynamics
The material of interest is The material of interest is described as an assembly of described as an assembly of molecular constituentsmolecular constituents
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Molecular dynamics
An interatomic depending on atomic positionsAn interatomic depending on atomic positions
€
rF i = mi
r ˙ ̇ r i
€
rF i = −
r ∇ iU(
r r 1,...,
r r N )
€
U(r r 1,...,
r r N )
The interatomic forces are calculated accordinglyThe interatomic forces are calculated accordingly
Newton’s equations of motion are integrated Newton’s equations of motion are integrated numerically (“Verlet velocity”) numerically (“Verlet velocity”)
Choose dt “judiciously” (~1fs) and iterate in time (“ad nauseam”)Choose dt “judiciously” (~1fs) and iterate in time (“ad nauseam”)€
r(t + dt) = r(t) + v(t)dt + 0.5a(t)(dt)2
€
v(t + dt) = v(t) + 0.5[a(t) + a(t + dt)]dt
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Interatomic potentials
€
U(r r 1,...,
r r N ) = V2(rij )
i< j
∑
€
V2(rij ) = 4εσ
r
⎛
⎝ ⎜
⎞
⎠ ⎟
12
−σ
r
⎛
⎝ ⎜
⎞
⎠ ⎟6 ⎡
⎣ ⎢
⎤
⎦ ⎥
””6-12” Lennard-Jones potential: repulsive core 6-12” Lennard-Jones potential: repulsive core 1/r1/r1212 ; VdW attraction 1/r ; VdW attraction 1/r66 r>r r>reqeq
””6-12” Lennard-Jones potential: prototypical 6-12” Lennard-Jones potential: prototypical interatomic force model for close-packed metalsinteratomic force model for close-packed metals
QuickTime™ e undecompressore TIFF (Non compresso)
sono necessari per visualizzare quest'immagine.
Professor Sir John Lennard-Jones (FRS), one of the founding fathers of molecular orbital theory
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Interatomic potentials
€
U(r r 1,...,
r r N ) = V2(rij )
i< j
∑ + V3(rij ,rik,cosϑ ijk )i< j<k
∑
Stillinger-Weber potential for anysotropic Stillinger-Weber potential for anysotropic covalent bonding (1985)covalent bonding (1985)
QuickTime™ e undecompressore TIFF (Non compresso)
sono necessari per visualizzare quest'immagine.
F. StillingerDepartment of Chemistry Princeton University Princeton, NJ 08540
T.A. Weber
€
U(r r 1,...,
r r N ) = V2(rij ,Zi)
i< j
∑ + V3(rij ,rik,cosϑ ijk,Zi)i< j<k
∑
(EDIP) Environment dependent interatomic (EDIP) Environment dependent interatomic potential (1998)potential (1998)
€
Z i= u(rli)l≠ i
∑
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
MD comes of age…K. Kadau et al. Int. Journal of Modern Physics C 17 1755 (2006)
B. J. Alder and T. E. Wainwright,
J. Chem. Phys.27,1208(1957)
Stillinger-WeberStillinger-WeberLennard-JonesLennard-JonesTersoffTersoff
EDIPEDIP
320 BILLION ATOM SIMULATION ON BlueGene/LLos Alamos National Laboratory
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
MD comes of age… more or less
Compromise between Compromise between accuracy accuracy and and computational workloadcomputational workload
The bottleneck of standard molecular dynamics: The bottleneck of standard molecular dynamics: time and length scalestime and length scales
In order to properly reproduce fracture related properties of covalent In order to properly reproduce fracture related properties of covalent materials of group IV materials (Si, Ge, C) it is necessary to take into materials of group IV materials (Si, Ge, C) it is necessary to take into account interactions as long as the second nearest neighbors distanceaccount interactions as long as the second nearest neighbors distance
A. Mattoni, M. Ippolito and L. Colombo, B 76, 224103 (2007)
ReliabilityReliability of the model potentials of the model potentials
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Computational Effort
CMPToolCMPTool: : a set of highly efficient parallel a set of highly efficient parallel numerical libraries for computational materials numerical libraries for computational materials science developed in collaboration with science developed in collaboration with CaspurCaspur, , RomeRomeGroup of materials scienceGroup of materials science (M. Rosati, S. Meloni, L. Ferraro, M. Ippolito)(M. Rosati, S. Meloni, L. Ferraro, M. Ippolito)
Typical simulation parametersTypical simulation parametersnumber of atomsnumber of atoms > > 10 1055 Runs as long asRuns as long as 6 10 6 1066 iterations (6 ns) iterations (6 ns)
1ns annealing of 100000 atoms1ns annealing of 100000 atoms requires of the order of requires of the order of 1000 CPU1000 CPU hours on state-of-the-art AMD - Opteron Dual core Linux clusterhours on state-of-the-art AMD - Opteron Dual core Linux cluster
A. Mattoni et al. Comp. Mat. Sci. 30 143 (2004)S. Meloni et al. Comp. Phys. Comm. 169 462 (2005)
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Nanocrystalline materials
Crystalline materials
0-D
Points:I,V, clusters, dots
Lines:Dislocations
1-D
Interfaces:Grain boundaries
2-D 3-D
Amorphous materials
In the amorphous phase (isotropic) the concept of dislocation is lost The microstructure evolution is controlled by: Recrystallization, normal grain growth
Plastically deformed materials
Ion implantation
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Mixed phase nanocrystalline systems
Nanocrystalline materials (nc-Si) Nanocrystalline materials (nc-Si) may be prepared through the may be prepared through the crystallization of amorphous crystallization of amorphous (disordered) (disordered) nc grains are embedded into a second nc grains are embedded into a second phase matrixphase matrix
Experimentally it is found that the smallest grain size is obtained when the amorphous samples are annealed at a crystallization temperature that is close to half the bulk melting temperatureQ. Jiang, J. Phys.: Condens. Matter 13 (2001) 5503–5506
nc
Embedding amorphous matrix
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Nc-Si for photovoltaics
Nano-crystalline silicon (nc-Si) Nano-crystalline silicon (nc-Si) consists in a distribution of grains consists in a distribution of grains embedded into an amorphous embedded into an amorphous matrixmatrix
Observation of domains separated by amorphous boundaries and (in some cases texturing)
Bright field TEM micrograph
S. Pizzini et al.Mat. Sci. Eng. B 134 p. 118 (2006)
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Modeling the a-/nc- evolution
During annealing of amorphous bulk it is difficult to deconvolve nucleation from growth (impurities, control the temperatures, grains impingement) C. Spinella et al. J. Appl. Phys. 84 5383 (1998)
Atomistic simulation as a tool to perform numerical experiment under perfectly controlled conditions of temperature and purity
What is the equation of motion of an isolated a-c boundary (planar or curved)?
Silicon as a prototype of a covalently bonded material
Mattoni and Colombo, Phys. Rev. Lett. 99, 205501 (2007)
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Why does a grain grow?
a-Si/c-Si is a metastable system
M. G. Grimaldi M. G. Grimaldi et alet al. Phys. Rev. B 44 1546 (1991). Phys. Rev. B 44 1546 (1991)
€
ga−c = ga − gc ~ 0.1 eV/atom1 kJ/mole=1.03 10-2 eV/atom
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Driving force pa-c
Driving force Driving force : specific free-energy : specific free-energy differencedifference
€
dG = −ga−cdVc + γ a−cdS < 0
€
pa−c = −dG
dVc
> 0
€
pa−c = ga−c − γ a−c
dS
dVc
€
dVc
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Equation of motion
Transition state theory
€
κ ~ Mpa−c
€
dR ~ κdt
Interface limited growth
Equation of motion of the a-c displacementEquation of motion of the a-c displacement
€
dR
dt= M(ga−c − γ a−c
dS
dV)
€
Eb
a-Sia-Si c-Sic-Si
€
pa−cVatom
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Transition State Theory
€
Eb
a-Sia-Si c-Sic-Si
€
κ =κ+ −κ−
€
κ =ωe−
Eb
kT −ωe−
Eb +Vatom pa−c
kbT
€
pa−cVatom
€
κ ~ ωVatom
e−
Eb
kbT
kbT
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟pa−c = Mpa−c
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Curved a-c boundary
€
r =R
R*
€
dR
dt= υ SPE (1−
R*
R)
The capillarity is expected to The capillarity is expected to be sizeable up to R~Rbe sizeable up to R~R** and and there give rise to anthere give rise to anAccelerated -> uniform growthAccelerated -> uniform growth
€
R* =γ a−c
ga−c
In silicon RIn silicon R**< 1 nm< 1 nm
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Planar a-c boundary
Uniform motionUniform motion: the a-c velocity is constant: the a-c velocity is constant
A. Mattoni et al. EPL 62 862 (2003)
€
vSPE (T) =M0
kbTga−ce
−Eb
kTExponential dependence on TExponential dependence on T with E with Ebb=2.6eV=2.6eVEXP G. L. Olson Mater Sci. Rep. 3, (1988) EXP G. L. Olson Mater Sci. Rep. 3, (1988)
AS N. Bernstein et al. PRB 61 6696 (2000AS N. Bernstein et al. PRB 61 6696 (2000))
QuickTime™ and aCinepak decompressor
are needed to see this picture.
€
dR
dt= vSPE (T)
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Curved a-c boundary
c-Si/a-Si: c-Si/a-Si: Isolated Crystalline fiber Isolated Crystalline fiber embedded into the amorphous phaseembedded into the amorphous phase
nc-Si/a-Sinc-Si/a-Si: Crystalline fiber embedded : Crystalline fiber embedded into an amorphous phaseinto an amorphous phase[1 0 0] case[1 0 0] case
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Characterization of the a-nc system
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Structure Factor
T/Tm
1.00.5 1.50.0
amorphous
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Crystallinity
Crystallinity of a mixed a-Si/nc-SiCrystallinity of a mixed a-Si/nc-Si: relative number of crystalline atoms: relative number of crystalline atoms
€
Θ=χCΘC + (1− χ C )Θα
€
χC =Θ − Θα
ΘC − Θα α
α
Θ−ΘΘ−Θ
≈)(
),(
T
tT
C
€
χC (T, t)∝ A = πR2
€
R(t,T) =χ C (t,T)
πL2
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Power law model
€
υ(R) = Mλ
R
⎛
⎝ ⎜
⎞
⎠ ⎟
1
q−1
ga−c (1−R*
R)
€
υ ~ R1−
1
q
Power law model Power law model the model describes both decreasing and increasing the model describes both decreasing and increasing nonuniform growthnonuniform growth
€
υ(R) = M ga−c (1−R*
R)
€
υ ~ 1
€
R ~ t
€
R ~ t q
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Fiber recrystallization
€
R ~ t q
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Fiber recrystallization
ThereThere is a is a dependence of dependence of the the growth growth exponents on exponents on temperaturetemperature and there is a and there is a clear transition clear transition close to the close to the amorphous amorphous meltingmelting
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Fiber recrystallization
€
υ =∂R
∂t(t,T) =
∂R
∂t(t(R,T),T) = υ (R,T)
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Characterization of defects
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
A simple explanation
€
dR ~ a
€
dR
dt~ υ τ
n f
2π
a
R
€
n f ~ Rα +1
€
dR
dt~ Rα
€
dR
dt~
1
R
€
n f ~ n0
€
n f ~ R
€
dR
dt~ υ 0
€
dt ~L f
υ τ
€
dR
dt~
a
L f
υ τ
SLACS-INFM/CNR
Rome, December 13, 2007
MATHEMATICAL MODELS FOR DISLOCATIONS
Conclusions
• Molecular dynamics simulation are emerging as a powerfool tool to Molecular dynamics simulation are emerging as a powerfool tool to help the characterization of the microstructure evolution of help the characterization of the microstructure evolution of nanostructured materialsnanostructured materials
• An atomically informed continuum model is found to describe An atomically informed continuum model is found to describe recrystallization in both the cases of isolated grain and distribution of recrystallization in both the cases of isolated grain and distribution of grainsgrains
Contact: [email protected]
EU-STREP “NANOPHOTO”CASPUR-ROME and CINECA-BOLOGNA computational support
A. Mattoni and L. Colombo, Phys. Rev. Lett. 99, 205501 (2007)
M. Fanfoni and M. Tomellini, Phys. Rev. B 54, 9828 (1996)
www.dsf.unica.it/colombo)
C. Spinella et al. J. Appl. Phys. 84 5383 (1998)