Ab initio modeling of the mechanical and electronic ...cas.umkc.edu/physics/ching/1.pdf · Ab...

34
Ab initio modeling of the mechanical and electronic properties of a prismatic model of intergranualr glassy films in β-Si 3 N 4 Wai-Yim Ching University of Missouri-Kansas City, USA Lujan Seminar, Lujan Center, Los Alamos National Laboratory Los Alamos, March 30, 2010 1

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

Atom-specific N-K and O-K edges in IGF region

30

N-609bulk

N-412

N-577

N-578N-783

O-907 O-793

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

Thanks

34