Local Defects in Crystalline Materials
Claudio M. Zicovich-Wilson
Fac Ciencias, Autonomous University of the Morelos State, Mexico
Slides adapted from the lecture given at MSSC2007 by
Cesare Pisani
Dipartimento di Chimica IFM - Via P. Giuria 5 - University of Torino
The use of slides.from P. Ugliengo and B. Civalleri’s lectures is gratefully
acknowledged
1
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)
2
Typical local defects in 3D ionic crystals
Substitutional
cation vacancy (V)
(anion)
Substitutional
(cation)
interstitial
anion vacancy (F)
3
Surface defects
0D: vertex
0D: surface vacancy
O
0D: Adsorbed
molecules
0D/1D: kink edge
Mg
1D: step
4
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 self-embedding condition:
without defects, all features of S'(q) should coincide with those of Sf
• 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!!)
5
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:
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
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) Ψ-Partition (group-function) techniques
6
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,…)
Metals (Cu, Ni, Pd, …)
Covalent solids (SiO2, graphite, silicon…)
Molecular crystals (ice, benzene, urea,…)
7
The bare-cluster approach for ionic solids: MgO(001)
Electrostatic potential along blue line
CRYSTAL
CRYSTAL
21
13
5
1
(MgO5)8-
R
The self-embedding condition is not-satisfied !!!
Potential is usually very distorted. Adsorption energies of
molecules may be overestimated. Corrections needed
8
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)
9
The cluster approach for metals
Cu
surface states
•
•
•
•
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
10
The cluster approach for molecular crystals
unsatisfied
H-bond
• 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
11
11
The saturated-cluster approach for covalent solids
Consider quartz as a prototype of covalent material. Silicon and other
semiconductors all belongs to this category
• N° atoms grows fast with cluster size
• Large number of terminal H atoms
• Hard to enforce crystal memory into the cluster structure
12
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.
13
Pros and Cons of the cluster approach
 The approach is flexible and user-friendly: high quality standard molecular
packages can be used (GAUSSIAN06, Turbomole, MOLPRO, NWCHEM…)
 The QM level can go beyond that allowed by periodic codes (DFT,HF,
LMP2)
 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
14
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 Ψ-partition approach
15
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)
16
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).
17
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
+
+
C
Si
SC-Si64
SC-CSi63
Si
C
H
Si4CH12
Si34CH36
Si-Si 2.36 Å
Si-C 1.90 Å
Si86CH76
C-C 1.56 Å
18
C/Si substitutional energy and band gap
ES = [Ecluster(Si) - Eatom(Si)] - [Ecluster(C) – Eatom(C)]
Cluster
CSi4H12
CSi34H36
CSi86H76
Supercell
Relaxation
Unrelaxed
First
Second
Unrelaxed
First
Second
Unrelaxed
First
Second
Unrelaxed
First
Second
ES (eV)
1.90
0.87
/
1.97
0.81
0.72
1.99
0.52
0.25
1.93
0.43
0.08
Mulliken charge
E(LUMO)E(HOMO)
0
-1.18
I
0.83
II
/
13.40
-1.16
0.23
0.15
9.69
-1.18
0.37
-0.08
8.65
-1.20
0.38
-0.03
6.20
19
Pros and cons of the SC approach
 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,
LMP2)
 Lateral interaction between defects may be important
(in particular for charged defects!!)
20
P-ONIOM (periodic generalization of Multi-level approach)
Real, low
Model, low
P.Ugliengo, A.Damin,
CO/MgO(001),
CPL 366 (2002) 683
C. Tuma, J. Sauer,
Model, high
Protonation of isobutene in zeolite ferrierite,
PCCP 8 (2006) 3955
21
P-ONIOM applications to surface science
O
C
O
H
Al
Si
CO adsorption on HY faujasite
O
C
Mg
NH3 adsorption on hydroxylated
silica surfaces [CPL 341 (2001) 625]
CO adsorption on MgO(001) surface
(CPL 366 (2003) 683)
22
Pros and cons of the P-ONIOM approach
 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
23
Local defects: The Green-Function approach
An exact perturbative theory
D
C
D
C
H hermitian operator , z any complex number . Define :
Green operator :
(Cauchy theorem)
G(z)
⇔
.
.
PDOS [n(ε)]
G (z) = [ z I – H ] -1
.
.
⇒ Density matrix, Energy
Use a representative basis set of localized AO’s (non orthonormal)
For the Host perfect crystal : Qf(z) = z S – Hf

Qf(z) Gf(z) = I
For the defective crystal :

Q(z) G (z) = I
Q(z) = z S – Hf - V
PARTITION TECHNIQUE:
QCC
QCD
GCC
GCD
0CD
0DC
IDD
=
How to use the Host Crystal solution
via some form of the locality assumption?
ICC
QDC
QDD
GDC
GDD
24
EMBED : a Green-function code for the study of defects
[ CRYSTAL96 ] → EMBED98
[ CRYSTAL98 ] → EMBED01
Pisani, Corà, Casassa, Birkenheuer, Comp. Phys. Comm 82 (1994) 139; 96 (1996) 152
The “fundamental assumption”
of the Perturbed Cluster method:
GDD (z) = GDD(z)
D
C
That is: the density of states projected in the outer zone
is unaffected by the defect
The problem is restricted to one in the defective cluster !
PCD = fCD [ ρDD(ε) ; HCC ; HCD]
Energy,
PCC = fCC [ ρDD(ε) ; HCC ; HCD]
electronic structure, etc.
PDD = PDD
25
Pros and cons of the Green-function embedding approach
 It fully exploits the solution for the host crystal
 It allows a natural definition of the defect zone
 The QM calculation is restricted to the defect zone
 It is intrinsically self-embedding consistent
 It uses non standard techniques (algorithms are non-competitive)
 Only one-electron Hamiltonians (HF, [DFT]) are allowed
 Convergence of the SCF procedure is difficult, specially with good basis sets
 Corrective terms are needed to account for long-range effects
(charge balance, polarization of the surroundings….)
26
Ψ-Partition Technique (a)
Ψ-(group-function) PARTITION approach and the LOCALITY ANSATZ
R McWeeny, Rev. Mod. Phys. 32 (1960) 335;
Z Barandiaran, L. Seijo. JCP 89 (1988) 5739;
HA Duarte, DR Salahub, JCP 108 (1998) 743;
LN Kantarovich, Surf. Sci. 76 (2000) 511
Ψ(x1,….xN) ≈ A [Φclus(x1,….xNclus) Φfext(xNclus+1,….xNtot)]
Φfext ≈ Π(ext) Φfion
[H clus + H env ] Φclus(x1,….xNclus) = E Φclus(x1,….xNclus)
H env is assumed to depend only on the Host-Crystal solution !
27
Ψ-Partition Technique (b)
Wannier-Function Strongly-Orthogonal Partition of Host Crystal solution
Ψf = A [Φfclus Φfext]
Φfclus = Π ωi (clust) ; Φfext = Π ωi (ext)
Augment the set of cluster WFs : Ξclus ≡ {ωi(clust) }⊕ {ϕj(clust)}
The added functions ϕj(clust) (
) should be :
local, distant from the defect border, apt to describe the proper defect region.
Solve the “embedded cluster” constrained-variation-equation:
[H clus + H env ] Φclus(x1,….xNclus) = E Φclus(x1,….xNclus), using Ξclus as basis set.
The resulting wave-function:
Ψ = A [Φclus Φfext]
provides a good description of the system, satisfying
locality assumption, self-embedding condition, charge conservation
Ψ can be improved using ONIOM-like criteria
28
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 Ψ-Partition techniques,
based on a Wannier-function description of the ground-state
wavefunction of the host crystal
29