Chimica Fisica dei Materiali Defects in crystals e …3 B. Civalleri - Chimica Fisica dei Materiali...

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B. Civalleri - Chimica Fisica dei Materiali a.a. 2013-2014 1

The use of slides from Prof. Piero Ugliengo’s lectures is acknowledged

Slides adapted from the lecture given at MSSC2007 by Cesare Pisani

Chimica Fisica dei Materiali e laboratorio

Modellizzazione di difetti locali in

materiali cristallini


Defects in crystals

Crystals are never perfect !

Extended defects in 3D crystals :

2D : limiting surfaces, interfaces;

stacking faults, … …ABCABCABABC…

1D : dislocation lines (edge, screw, …), …

Local (0D) defects in 3D crystals : all kind of possibilities!

B. Henderson, Defects in crystalline solids, Arnold, London (1972).

W. Hayes and A. M. Stoneham, Defects and defect processes in non metallic solids, Wiley, NY (1984)

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Typical local defects in 3D ionic crystals

cation vacancy (V)

anion vacancy (F) interstitial

Substitutional

(anion)

Substitutional

(cation)


Surface defects

0D: vertex

1D: step

0D/1D: kink edge

0D: Adsorbed

molecules

0D: surface vacancy

Mg

O

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QM simulation of local defects: fundamentals

Sf

+ +

+ R S'(q) + P {E(q)}

defect zone

• The perfect host crystal assumption:

Sf (the host system) is a perfect (and known ) crystalline structure

• The locality assumption:

Sf and S' structures coincide outside the defect zone (finite and small)

• The QM problem:

(i) determine the equilibrium geometry of S'(q) (ii) describe the change of the electronic structure in the defect zone (iii) evaluate E(q) (a finite difference between infinite energies!!)

• The self-embedding condition:

without defects, all features of S'(q) should coincide with those of Sf


Local defects in crystals: a classification of QM strategies

Modelling of the defect and solution technique are inter-related

aspects which characterize the approach to the QM problem.

A possible classification of strategies is as follows:

B. “Periodic strategies”:

Sf (and possibly S') is modelized as a perfect crystal

b1) Supercell approach

b2) Multi-level (P-ONIOM) technique

b3) Embedding techniques (perturbative, Green-functions)

b4) -Partition (group-function) techniques

A. “Non-Periodic (cluster) strategies”:

Both Sf and S' are simulated as a finite cluster

a1) Bare (bond-saturated) clusters

a2) Dressed clusters (extra terms to simulate environmental effects)

a3) Multi-level (ONIOM) technique

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Examples of use of the cluster approach

A. Sf and S' are simulated as a finite cluster

and extra terms may be added to simulate environmental effects.

Ionic solids (MgO, NaCl,…)

Covalent solids (SiO2, graphite, silicon…)

Molecular crystals (ice, benzene, urea,…)

Metals (Cu, Ni, Pd, …)


The bare-cluster approach for ionic solids: MgO(001)

CRYSTAL

1

CRYSTAL

5

21 13

Electrostatic potential along blue line

(MgO5)8-

The self-embedding condition is not-satisfied !!!

Potential is usually very distorted. Adsorption energies of

molecules may be overestimated. Corrections needed

R

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The dressed-cluster approach for ionic solids:

MgO bulk and surface

F.Illas and G. Pacchioni: Bulk and surface F centers in MgO, JCP 108 (1998) 7835

(* Ricci, Di Valentin, Pacchioni, Sushko, Shluger, Giamello, JACS 125 (2003) 738 )

O-vacancy [3s 2p 1d]

(Mg2+)6 [2s 1p]

(O2-)12 [2s 2p]

(Mg2+)8 [1s]

(O2-)24 [pseudo]

… 292 point charges*

(* or polarizable “shell-model

ions)


The cluster approach for metals

Cu

• Spin state is critically dependent on cluster size and shape

• Geometrical Relaxations are large

• “Relaxed-Drop” models are better than clusters cut out from the crystal

• Serious problems are met in converging adsorption energies

Triguero, Wahlgren, Boussard, Siegbahn, CPL 237 (1995) 550

surface states

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The cluster approach for molecular crystals

• Geometry resulting from the cut

will change dramatically upon relaxation

• Molecules at the border will move to maximize the

intermolecular interactions (H-bond)

• Optimized structures are no longer representative of the

crystal to which they initially belong

• Arbitrarly shaped clusters may develop huge dipole

moment absent in the periodic system

Bussolin, Casassa, Pisani, Ugliengo, J. Chem. Phys., 108, 9516 (1998).

unsatisfied

H-bond

unsatisfied

H-bond B. Civalleri - Chimica Fisica dei Materiali a.a. 2013-2014 12

The saturated-cluster approach for covalent solids

• N° atoms grows fast with cluster size • Large number of terminal H atoms

• Hard to enforce crystal memory into the cluster structure

Consider quartz as a prototype of covalent material. Silicon and other

semiconductors all belongs to this category

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Multi-level cluster technique: the ONIOM method Dapprich, Komaromi, Byun, Morokuma, Frisch, J. Mol. Struct.-Teochem, 461, 1 (1999).

Link atoms (H, F,…)

Model cluster (Low; High)

Real system (Low)

ETOT(High:Low) = E(High, Model) + E;

E = E(Low, Real) - E(Low, Model)

If Model Real, E 0,

If Low High , ETOT(High: High) = E(High,

Real)

All other QM quantities are defined following the same scheme. The link between (High,Model) and (Low,Real) systems is via forces on the common link atoms.


Pros and Cons of the cluster approach

The approach is flexible and user-friendly: high quality standard molecular

packages can be used (GAUSSIAN09, Turbomole, MOLPRO, NWCHEM…)

The QM level can go beyond that allowed by periodic codes (DFT,HF)

Size and shape of the cluster are critically important

Additional terms may be needed in the Hamiltonian (*)

Link atoms (H, F,…) to saturate frontier bonds are often required (*)

Boundary effects are always important (*)

All long range interactions are not included by definition

The self-embedding condition is not satisfied (*)

(*) all these problems are alleviated by the use of the ONIOM approach

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Local defects in crystals: “Periodic” approaches

B. The defect as an impurity of the host perfect crystal:

Sf is treated as a perfect crystalline structure

and S' is treated (in principle) at the same level of accuracy.

B1. The supercell (SC) approach

B2. The Periodic Multilevel (P-ONIOM) approach

B3. The embedded cluster Green-function approach

B4. The -partition approach


Local defects in crystals: the super-cell (SC) approach

Sf perfect host

n1=2, n2=2 n1=1, n2=1

S'SC

defective host

E = lim(n1, n2, n3) [ E(S'SC) – n1 n2 n3 E(Sf) ] + E(P) – E(R)

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Not only local… 1-d defects at surfaces with SC approach

l

AM Ferrari et al. "Polar and non-polar domain borders in MgO ultrathin films on Ag(001)”, Surf. Sci. 588, 160-166 (2005).


SC vs cluster approaches: substitutional C in silicon R. Orlando et al. "Cluster and supercell calculations for carbon-doped silicon" J. Phys.: Condens. Matter 8 (1996) 1123

SC-Si64 SC-CSi63

C Si

+ +

Si86CH76 Si34CH36

Si4CH12

Si-Si 2.36 Å Si-C 1.90 Å C-C 1.56 Å

C

Si H

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C/Si substitutional energy and band gap

ES = [Ecluster(Si) - Eatom(Si)] - [Ecluster(C) – Eatom(C)]

Cluster Relaxation ES (eV) Mulliken charge E(LUMO)-E(HOMO)

00 II IIII

CSi4H12 Unrelaxed 11..9900 --11..1188 00..8833 // 1133..4400 First 0.87 Second /

CSi34H36 Unrelaxed 11..9977 --11..1166 00..2233 00..1155 99..6699 First 0.81 Second 0.72

CSi86H76 Unrelaxed 11..9999 --11..1188 00..3377 --00..0088 88..6655 First 0.52 Second 0.25

Supercell Unrelaxed 11..9933 --11..2200 00..3388 --00..0033 66..2200 First 0.43 Second 0.08


The approach is "universal" (bulk & surface; ionic, covalent, metallic)

Any “periodic” code can be used

It is "simple" (the parameters are n1, n2, n3)

It allows a proper definition of E

The self-embedding condition is satisfied

No advantage is taken of the solution for the host crystal (except for E(Sf) )

The SC shape may not be the most suitable to describe the defect zone

The QM level of the treatment is that allowed by the periodic code (DFT,HF)

Lateral interaction between defects may be important

(in particular for charged defects!!)

Pros and cons of the SC approach

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O Be Li O

Axial defect

Defect-defect interactions?

Low density defective system High density defective system

[Li]0 center in beryllium oxide

Supercell Etot(BeO) EBeO

S32 (2 2 2) -1435.23633 -89.70227

S48 (2 2 3) -2152.85450 -89.70227

S64 (2 2 4) -2870.47266 -89.70227

S72 (3 3 2) -3229.28175 -89.70227

S108 (3 3 3) -4843.92262 -89.70227

S144 (3 3 4) -6458.56349 -89.70227

S180 (3 3 5) -8073.20437 -89.70227

S216 (3 3 6) -9687.84524 -89.70227

S252 (3 3 7) -11302.48612 -89.70227

S128 (4 4 2) -5740.94533 -89.70227

S192 (4 4 3) -8611.41799 -89.70227

S256 (4 4 4) -11481.89066 -89.70227

S300 (5 5 3) -13455.34061 -89.70227

Etot(BeOLi) E

-1427.83826 0.25818

-2145.45400 0.26061

-2863.07115 0.26162

-3221.89975 0.24211

-4836.53959 0.24314

-6451.18019 0.24341

-8065.82080 0.24368

-9680.46139 0.24396

-11295.10206 0.24417

-5733.56656 0.23888

-8604.03731 0.24079

-11474.50982 0.24095

-13447.96104 0.23968

Axial defect E = (Edef + EBe) – (Eper + ELi) UHF

EBeo= -89.70227 ; EBe= -14.56948 ; ELi= -7.42959 Energies in Hartree

BeO:[Li]0: effect of the supercell size

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0.235

0.24

0.245

0.25

0.255

0.26

0.265

0 50 100 150 200 250 300 350

Number of atoms

Fo

rma

tio

n e

ne

rgy

(H

art

ree

) 2 2 X=2, 3, 4

3 3 X=2, 3, 4, 5, 6, 7

4 4 X=2, 3, 4

5 5 3

BeO:[Li]0: effect of the supercell size

Axial defect UHF


P-ONIOM (periodic generalization of Multi-level approach)

P.Ugliengo, A.Damin,

CO/MgO(001),

CPL 366 (2002) 683

C. Tuma, J. Sauer,

Protonation of isobutene in zeolite ferrierite,

PCCP 8 (2006) 3955

Real, low

Model, low

Model, high

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P-ONIOM applications to surface science

NH3 adsorption on hydroxylated

silica surfaces [CPL 341 (2001) 625]

C

O

Mg

CO adsorption on MgO(001) surface

(CPL 366 (2003) 683)

C

O

Al H

O

Si

CO adsorption on HY faujasite


It permits to incorporate naturally important features of the host crystal

It allows the use of high quality QM techniques in the defect zone

Its optimal application is restricted to covalent crystals

An SC calculation for each defect geometry is needed

Cancellation of errors in the various steps isn’t warranted

and it is difficult to verify

Pros and cons of the P-ONIOM approach

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CONCLUSIONS

The study of defects in crystals (especially surface defects)

is an important field of application of periodic codes

The super-cell approach is currently the preferred choice

The use of “hybrid” (Multi-Level) models is often recommended,

to allow a high-quality treatment of the defect zone

Prospective developments include the use of -Partition techniques,

based on a Wannier-function description of the ground-state

wavefunction of the host crystal

Chimica Fisica dei Materiali Defects in crystals e …3 B. Civalleri - Chimica Fisica dei Materiali...

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