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Transcript of Ab initio modeling of the mechanical and electronic ...cas.umkc.edu/physics/ching/1.pdf · Ab...
Ab initio modeling of the mechanical and electronic properties of a prismatic model of
intergranualr glassy films in β-Si3N4
Wai-Yim Ching
University of Missouri-Kansas City, USA
Lujan Seminar, Lujan Center, Los Alamos National Laboratory
Los Alamos, March 30, 2010
1
Outline:
1. Motivation
2. Methods
2
3. Physical properties of IGFa) IGF model constructionb) Mechanical propertiesc) Electronic and optical propertiesd) Theoretical tensile experimente) Post failure analysis f) Atom-specific XANES/ELNES spectral calculation
4. Some conclusions5. What more can be done?
What is an IGF?
More difficult to sinter
than oxide ceramics
Si3N4/SiO2 Sintering aids
(Al2O3/MgO/Y2O3/Ln2O3)
Solutes Residual Glass
Bulk Si3N4 grain IGF Bulk Si3N4 grain
3
Motivation
• Microstructures such as IGF control most physical properties in polycrystalline ceramics.
• Difficult to perform experiments on IGF. Simulation studies may be the only way to obtain atomistic information.
• Influence of IGF on the mechanical properties β-Si3N4.
• Tensile “experiment” on IGF models to reveal failure behavior.
• Understand electronic structure, bonding, optical dielectric functions, potential distribution etc of IGF models.
• Analyze the post failure behavior and properties of the IGF model under increasing strain.
• To analyze IGF structure using core-level spectroscopy.
4
Methods
• Two density functional theory based ab initio methods used:
• VASP (Vienna Ab initio Simulation Package):
Plane-wave basis using pseudopotentials.
Used for structural relaxation, mechanical properties and tensile experiment using supercomputers.
5
• OLCAO (Orthogonalized linear combination of atomic orbitals) • Local orbital with Gaussians for basis expansion.• Used for electronic structure, bonding and optical properties
calculations.• Particularly efficient for very complex systems.
An IGF models in β-Si3N4 with basal plane was studied previously
♠ The model has 798 atoms sandwiched between basal planes of β-Si3N4.
1). Paul Rulis, Jun Chen, Lizhi Ouyang and W.Y.
Ching, Xiaotao Su, S.H. Garofalini, Phys. Rev. B71,
235317-1-10 (2005).
2). J. Chen, P. Rulis, L. Ouyang, A. Misra, and W.Y.
Ching, Phys. Rev. Lett. 95, 256103 (2005).
3). W. Y. Ching, Jun Chen, Paul Rulis, Lizhi Ouyang,
and Anil Misra, J. Materials Science, 41 (16) 5061-
5067, (2006).
4) A. Misra, L. Ouyang, J. Chen and W.Y. Ching,
Philosophical Magazine A. 87(25), 3839-3852
(2007).
6
♠ The basal model is not very realistic and quite defective at the interface.
A new 907-atom model with prismatic planes (prismatic model).
This prismatic model is the focus of this talk!
a) Construction of the prismatic IGF model
The prismatic model was created in several steps:
1) Preliminary design of the model.
2) Molecular dynamics.
3) Simulated annealing.
4) VASP relaxation.
5) Each steps of the construction were done very carefully to ensure the model is stable.
(a) And (b) shows the model in two orientations.
(c) Shows the radial pair distribution function of the atoms in the IGF part of the model.
7
Structure and composition of the prismatic IGF model
8
Bonding statistics:
Bulk crystal part: Si 4-fold to N, N 3-fold to Si.
IGF part: Major configurations: Si-O4, Si-NO3;
Minor configurations: Si-N4,Si-N2O2,Si-N3O.
Defective structures: Si-NO2; Si-N2O;Si-O5.
Model dimension: a = 14.533 Å, b= 15.225 Å, c= 47.420 Å.
IGF thickness: ~17 Å; Volume fraction: Bulk crystal = 65.4% ; IGF = 34.6%.
Crystal orientation: The prismatic plane (10-10) parallel to the IGF layer.
Z direction of the model perpendicular to the IGF layer.
Number of atoms: Total 907
Bulk crystal: 679 (299 Si, 380 N);
IGF:228 (72 Si, 32 N, 124 O).
IGF composition: Si/(N+O)=0.46,
N(N+O) =0.21,
O/(N+O) =0.79.
• The elastic constants Cij are obtained from elastic stress σi by applying ± 1% of elastic strain εj to the model: σi = ∑Cijεj using VASP and fully converged.
• The Cij values are used to obtain bulk structural parameters based on the Voigt and Reuss approximation (Voigt-Reuss-Hill scheme) to obtain:
Bulk modulus K,
Shear modulus G,
Young’s modulus E
Poisson ratio η
Sound velocity v
9
b) Mechanical and Elastic properties
Mechanical properties of the IGF model and bulk β-Si3N4 crystal
IGF model: (in GPa) z-axis perpendicular to IGF
C11 C22 C33 C12 C13 C23 C44 C55 C66
411.8 322.3 305.7 76.7 75.0 119.8 96.3 86.7 88.9
10
Unit (GPa) IGF Model β-Si3N4 Glass Portion
Bulk modulus K: 175.4 244.2 51.0
Shear modulus G: 103.0 130.8 51.4
Young’s modulus E: 258.4 332.9 115.3
Poisson ratio η: 0.255 0.273 0.123
IGF model is weaker than β-Si3N4 crystals due to the glassy portion.
β-Si3N4: (in GPa) z-axis (c-axis) perpendicular to prismatic direction.
C11 C22 C33 C12 C13 C23 C44 C55 C66
431.3 431.3 551.7 178.1 108.0 - 104.5 - 126.6
Exp. 433±3 433±3 574±3 195±8 127±5 - 108±2 - 119±4
R. Vogelgesang, M. Grimsdich and J.S. Wallace, Appl. Phys. Lett., 76, 982 (2000)
Glass-portion: (in GPa) z-axis perpendicular to IGF
C11 C22 C33 C12 C13 C23 C44 C55 C66
161.5 127.3 81.0 20.7 18.2 15.5 54.9 61.0 42.1
Longitudinal elastic sound velocities in IGF model
Calculation based on Christoffel wave equation using elastic tensor
1) Longitudinal component of the sound velocity (green) is isotropic in
Ky-Kz plane but anisotropic in Kx-Ky & Kz-Kx planes.
2) This implies the x-direction is unique, not the z-direction.
3) The sound velocity in polycrystalline Si3N4 is not affected by IGFs. It
depends more on the crystalline orientations.
mkjijklmi ullCuv 2
Vs =10,300-10,600 m/s Vs =10,300-12,000 m/s Vs =10,100-12,000 m/s
11
c) Electronic structures and bonding
• Electronic structure calculations using the OLCAO method based on the VASP relaxed IGF model.
12
• Properties calculated:– DOS, PDOS
– Localization Index
– Mulliken effective charge
– Bond order and its distribution
– Electrostatic potential distribution
– Optical dielectric function (anisotropy)
Total DOS & partial DOS (PDOS)
Energy (eV)
13
Energy (eV)
β-Si3N4
IGF
PDOS of Si and N in the
crystal part of the IGF
model
Comparison of TDOS of IGF
model and bulk β-Si3N4 crystal.
Total and PDOS of O, Si
and N in the glassy part
of the IGF model
Localization index (LI) of electron states in IGF model
14
LI ranges from 1 to 1/N. States near the band gap are highly localized
(LI large). CB states are completely delocalized (LI ~ 1/N). There are 7
highly localized states near the top of the VB related to defective
structures. 6 occupied, 1 unoccupied.
2
,
, ,
n n
n i j i j
i j
LI C C S
Mulliken effective charges
←bulk crystal →← IGF →← bulk crystal →
Average values of Q* Bulk crystal IGF region
Si 2.41 2.02
N 6.19 6.18
O -- 7.03
The IGF region is more ionic than the bulk region
* *
,
, ,
.n n
i j i j
i n occ j
Q C C S
Information on charge transfer!
15
0 5 10 15 20 25 30 35 40 45 50
0
1
2
6
7
8
N O
N10-c20
Eff
ecti
ve C
harg
e
Atomic c Axis Position (Å)
Si
Bond Order in IGF model
Siα,jβ is the overlap matrix between ith orbital of the αth atom and the jth orbital of
the βth atom. The C’s are the eigenvector coefficients of the nth eigenstate.
*
,
, ,
.n n
i j i j
i n occ j
C C S
16
Strength of bonds indicated by color.
♠ Within IGF, very strong and weak
bonds are possible.
♠ Strength of the bond mainly
depends on the bond length.
♠ Si-O bonds are weaker than Si-N
bonds.
♠ The bond strength for atoms at the
crystalline/glass interface is
diffusive.
Bond Order:
Electrostatic Potential distribution in IGF
17
♠ For the prismatic IGF model: ΔV ~ 7.41 eV.
♠ For the basal IGF model (PRL, 2005): ΔV ~ 2.58 eV.
♠ ΔV depends on the nature of the glassy composition and
crystal orientations. These are approximate values.
The space charge model is a well established concept in ceramics.
Electrostatic potential difference validates the space charge model.
Optical dielectric function of the IGF model
Energy (eV) Energy (eV)
18
ε2-xxε2
ε1
ELF
ε2-yy
ε2-zz
Energy (eV)
ε2-xx
ε2-yy
ε2-zz
The strong anisotropy in ε2 of the IGF model is not due to the
presence of IGF, It is related to crystal orientation of bulk Si3N4.
β-Si3N4IGF model IGF model
d) Theoretical tensile experiments
• Uniaxial extension of IGF in the direction perpendicular to IGF plane
• Step-wise extension with full relaxation at each strain until the film is fully separated
• Extract stress data and plot stress vs strain
• Analyze post failure behavior of the IGF
19
d) Theoretical tensile experiments
• Uniaxial extension of IGF in the direction perpendicular to IGF plane
• Step-wise extension with full relaxation at each strain until the film is fully separated
• Extract stress data and plot stress vs strain
• Analyze post failure behavior of the IGF
20
Stress vs. strain for the prismatic IGF modelStress vs strain under uniaxial
extension in z direction: zz, xx,
yy components
The same stress vs strain data
as on the left before breaking
point.
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
0
2
4
6
8
10
12
14
Str
ess (
GP
a)
Strain
Conclusions: (1) Initial relation between strain/stress linear, change slope at s=0.05.
(2) Maximum stress = 13.9 GPa at strain =0.094.
(3) Post-failure stress decreases not very rapidly.
(4) After breaking, residue stress exist. Fractured sample resembles coated
surficial film on bulk Si3N4 crystal.
0.00 0.05 0.10 0.15 0.20
0
2
4
6
8
10
12
14
Str
ess (
GP
a)
Strain
21
Evolution of physical properties under strain
• Evolution of DOS
• Change in the bonder order
• Change in the electrostatic potential difference.
Structural models at strain of:
0.00, 0.039, 0.094, 0.148, 0.239
22
Evolution of IGF properties under strain
0
150
300(a)
S=0.000
0
150
(b)
S=0.039
0
150
(c)
S=0.094
0
150
(d)
S=0.148
-25 -20 -15 -10 -5 0 5 10 15
0
150
PD
OS
Sta
tes
/[e
V C
ell]
Energy (eV)
(e)
S=0.239
05
10152025
(a)
S=0.000
05
101520 (b)
S=0.039
05
101520 (c)
S=0.094
05
101520 (d)
S=0.148
0 10 20 30 40 50 60
05
101520
c Axis (Å)
(e)
S=0.239
(a) DOS
Conclusions: 1). Overall feature of DOS change little with strain.
2). Most changes related to defect states, surfaces states.
3). Decease in bond order density in the IGF region with
increase strain, effect of bond elongation and weakening.
23
(b) Bond order density
Evolution of electrostatic potential under strain
24
(a) Change in the electrostatic
potential across the IGF as a function
of strain up to s= 0.094.
(b) Plot of ∆V vs. strain for five data
points in (a).
Bond order evolution under extension
25
The figure shows the evolution
of BO values in color scheme fro
strain levels of (a) 0,%; (b) 1.7%,
(c) 3.9 %, (d) 9.4%. Even at zero
strain, strong and weak Si-O
and S-N bonds exist in the IGF
region due to disorder and
variations in bond lengths.
At small strain, the bond simply
elongate a little.
Fracture occur in IGF where
there are high concentration of
weak bonds.
Evolution of strain field distribution in three directions under strain (with reference to unstrained model)
26
S=1.35%
S=2.25%
S=3.90%
S=9.42%
e) Post failure analysis for IGF model
Conclusions: Comparison with the original
strain=0 model shows similar DOS, LI, and
optical properties, ε1(0) value is slightly
smaller due to reduced density.
Energy (eV)
Energy (eV)
DOS
LI
27
ε2
ε1
f) Atom-specific XANES/ELNES spectral calculation
28
XANES: X-ray absorption Near Edge spectroscopy.
ELNES: Electron Energy-Loss Near Edge spectroscopy.
ELNES/XANES reflects electron dipole
transition from a core-orbital to the unoccupied
conduction band.
Conventional interpretation using
orbital resolved local density of states
(LDOS) of unoccupied band.
Many theoretical methods exist to
calculate/ELNES/XANES spectra of solids.
A more robust and accurate method:
Supercell OLCAO method.
Supercell-OLCAO method for XANES/ELNES calculation
29
♠ The IGF model itself is a form of supercell.
♠ Merits of the supercell-OLCAO method:
Superior to other methods or the use the Z+1 approximation.
Fermi Golden Rule for electron transition in the dipole approximation.
Inclusion of the dipole transition matrix element imposes selection rules.
Theoretical transition energy obtained from the difference in total energy in
the initial and final state calculations.
Compute all multi-center
integrals
SCF with one core e- at the bottom of the CB
SCF with all e- in the ground state
Obtain CB Wave Functions
Obtain Core/VB Wave Functions
Compute Optical transitions between initial & final states
Some conclusions
• A new prismatic IGF model is constructed with properties very different from the basal model. The prismatic model shows much stronger mechanical properties.
• Tensile experiment reveal complicated deformation and fracture behavior of the IGF model.
• Sound velocity and optical anisotropy in polycrystal ceramics are not affected by IGF but depends on crystal orientations.
• Electrostatic potential difference exists between IGF and bulk crystal, validating the space charge model in ceramics.
• Post fracture of the IGF model results in formation of surfacial films with residue stress.
• Fundamental understanding of mechanical and electronic properties of complex microstructures requires atomistic level simulation based on quantum mechanics.
31
Future direction
• Study larger IGF models to be more realistic
• Study IGF models with different N-content
• Study IGF-models with different Rare earth dopants
• Study the dependence on IGF thickness and volume fraction
• Study the temperature dependence of the mechanical properties of the IGF models
• Detailed analysis of the atom-specific XANES/ELNES calculations and use spectral image technique to make connections with experiments.
32
Acknowledgements
Electronic Structure Group (ESG) at UMKC:
Professor Wai-Yim Ching
Dr. Paul Rulis
Mr. Hongzhi Yao
Mr. Sitaram Aryal
Ms. Lei Liang
Mr. Liaoyuan Wang
Mr. Yuxiang Mo
Other Collaborators:Professor Anil Misra, University of Kansas
Professor Lizhi Ouyang, Tennessee State University
Many other international and US collaborators:
Work supported by: DOE-BES, DOE-NERSC
33