6. Ionic Crystals

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Definition of an ionic crystal composed of atoms A and B

Cohesion results from transfer of an electron from element A to element B, producing Cohesion results from transfer of an electron from element A to element B, producing closed-shell ions

closed-shell ions AA++ and and BB!!

which attract each other by Coulomb forces and repel which attract each other by Coulomb forces and repel each other at short-range due

each other at short-range due to Pauli repulsion. to Pauli repulsion. A schematic picture A schematic picture of an ionic solidof an ionic solid is shown in Fig. 1.

is shown in Fig. 1.

Fig. 1. Schematic picture of an ionic solid with the NaCl type structure Fig. 1. Schematic picture of an ionic solid with the NaCl type structure

Metal halides Metal halides

LiF, LiCl, NaF, NaCl, NaBr, KF, KBr, AgCl, AgBr, AgI etc. LiF, LiCl, NaF, NaCl, NaBr, KF, KBr, AgCl, AgBr, AgI etc. crystallize in the sodium chloride or cesium chloride structure. crystallize in the sodium chloride or cesium chloride structure.

Cesium chloride

Cesium chloride: : Simple Simple cubic cubic lattice lattice of of lattice lattice of of Cs Cs ions ions with with Cl Cl ions ions on on simplesimple cubic lattice displaced relative to the Cs lattice by

cubic lattice displaced relative to the Cs lattice by 11₂₂ 111111

vector.

vector. Alternatively, a bcc Alternatively, a bcc lattice witlattice with Cs in cube h Cs in cube corners andcorners and Cl at cube centers (Fig. 2a).

Cl at cube centers (Fig. 2a).

Sodium chloride

Sodium chloride A A simple simple cubic cubic lattice lattice of of alternate alternate positive positive and and negative negative ions.ions. Each species is forming a face-centered-cubic lattice and these Each species is forming a face-centered-cubic lattice and these are displaced with respect to each other by

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Fig.

Fig. 2a. 2a. CsCl CsCl structurestructure

Fig.

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Metal oxides Metal oxides

Sodium chloride

Sodium chloride MgO, MgO, CaO, CaO, SrO, SrO, BaO, BaO, MnO, MnO, FeO, FeO, CoO, CoO, NiO, NiO, TiO, TiO, VO, VO, CrO,CrO, ZnO, CdO

ZnO, CdO

Spinel

Spinel MgAlMgAl22OO44, FeAl, FeAl22OO44, ZnCr, ZnCr22OO44, ZnFe, ZnFe22OO44

Perovskite

Perovskite BaTiOBaTiO33, SrTiO, SrTiO33(see Fig. 3).(see Fig. 3).

Corundum

Corundum AlAl22OO33 i. e sapphirei. e sapphire (hexagonally close-packed oxygen planes(hexagonally close-packed oxygen planes

with aluminum filling two-thirds of the available octahedral with aluminum filling two-thirds of the available octahedral sites).

sites).

Fig. 3

Fig. 3 Structure Structure (cubic symmetry) (cubic symmetry) of SrTiOof SrTiO33

Electrostatic neutrality of ionic crystals Electrostatic neutrality of ionic crystals

When studying defects in ionic crystal the block has to remain electrostatically When studying defects in ionic crystal the block has to remain electrostatically neutral i. e. no surplus of a positive or negative charge must occur.

neutral i. e. no surplus of a positive or negative charge must occur. Examples:

Examples: Vacancies Vacancies occur occur in in pairs pairs of of vacancies vacancies at at positive positive and and negative negative ions.ions. Frenkel defect - vacancy +

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INTERACTIONS BETWEEN IONS

Interaction between ions in an ionic crystal can be divided into two principal parts: Interaction between ions in an ionic crystal can be divided into two principal parts:

Long-range Coulomb interaction

The energy associated with this interaction between ions can be written as The energy associated with this interaction between ions can be written as

E E_C_C == 1 1 2 2 Z Z_iiZZ_j j rr_ij_ij i,j i,j ii!! j j

"

(I1)(I1) where

where ZZ_ii is the charge at ion i and is the charge at ion i and rr_ij_ij is is the sthe separation between eparation between ions i ions i and j. and j. TheThe summation in (I1) converges because there are positive and negative ions but the summation in (I1) converges because there are positive and negative ions but the convergence is very

convergence is very slow. slow. Summation methods, such as tSummation methods, such as the Ewald he Ewald method, describedmethod, described below, have to be employed.

below, have to be employed.

The charges associated with ions may be either full charges corresponding to isolated The charges associated with ions may be either full charges corresponding to isolated ions or

ions or partial charges. partial charges. In the In the former case former case the correspondingthe corresponding ZZ_ii is an integer multiple is an integer multiple of the electron charge and in the latter case it is a non-integer multiple of the electron of the electron charge and in the latter case it is a non-integer multiple of the electron charge corresponding, for example, to the charge found in

charge corresponding, for example, to the charge found in ab initioab initio calculations for a calculations for a given ionic crystal.

given ionic crystal. In both cases there is zerIn both cases there is zero total charge within the repeat o total charge within the repeat cell.cell.

Short range interaction

These are most commonly represented by pair potentials of the Buckingham form These are most commonly represented by pair potentials of the Buckingham form

!

ijij == AA_ij_ij exp exp

""

rr

##

ijij

$

%

&&

'

(

)

) ""

D D_rr66ijij (I2)(I2)

The first term is the (Pauli) repulsion of ions at short separations described by the The first term is the (Pauli) repulsion of ions at short separations described by the Born-Mayer potential.

Born-Mayer potential. The second term The second term is the (is the (weak) Van der Waals weak) Van der Waals attraction and itattraction and it is often

is often omitted in omitted in studies of istudies of ionic solids. onic solids. The short The short range interactions range interactions can becan be regarded as significant only for separations smaller or equal to that of the first nearest regarded as significant only for separations smaller or equal to that of the first nearest neighbors.

neighbors. The The parametersparameters AA_ij_ij,, DD_ij_ij and and !!_ij_ij can be determined, similarly as in the case can be determined, similarly as in the case of any empirical pair potentials, by fitting equilibrium quantities, e.g. lattice of any empirical pair potentials, by fitting equilibrium quantities, e.g. lattice parameter, elastic moduli, cohesive energy, of the material that are determined parameter, elastic moduli, cohesive energy, of the material that are determined experimentally and/or by

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Since both long-range Coulomb interactions and short-range repulsions are described Since both long-range Coulomb interactions and short-range repulsions are described by pair potentials model the Cauchy relations between elastic moduli (

by pair potentials model the Cauchy relations between elastic moduli ( CC₁₂₁₂ ₌₌ C C₄₄₄₄_for_for

cubic symmetry) will be satisfied in this scheme. cubic symmetry) will be satisfied in this scheme.

Rigid ion model

The ions are considered as possessing either full or partial charges and the total The ions are considered as possessing either full or partial charges and the total energy of the system is composed of the Coulombic term (I1) and the term given by energy of the system is composed of the Coulombic term (I1) and the term given by the short-range pair potential

the short-range pair potentials (I2). s (I2). This determines the This determines the total energy of total energy of the solid as athe solid as a function of atomic positions and atomistic calculations (molecular statics, molecular function of atomic positions and atomistic calculations (molecular statics, molecular dynamics or Monte Carlo) can be carried out as in the case of other pair-potential dynamics or Monte Carlo) can be carried out as in the case of other pair-potential models.

models.

However, the Coulombic term cannot be evaluated by a direct summation over ions However, the Coulombic term cannot be evaluated by a direct summation over ions since the convergence would be extremely slow and Ewald type methods, discussed since the convergence would be extremely slow and Ewald type methods, discussed below, must be employed.

below, must be employed. Example: Potentials for Al

Example: Potentials for Al22OO33 discussed in the study of P. R. Kenway: J. American discussed in the study of P. R. Kenway: J. American

Ceram. Soc.

Ceram. Soc. 7777, 349, 1994., 349, 1994.

Shell model

An important difference between ionic crystals and metals is that ionic crystals are An important difference between ionic crystals and metals is that ionic crystals are insulators and internal electric fields can be present in the vicinity of defects. insulators and internal electric fields can be present in the vicinity of defects. Therefore, defect energies may depend sensitively on the electronic polarization of Therefore, defect energies may depend sensitively on the electronic polarization of the l

the lattice. attice. This iThis is particularly s particularly significant in significant in the the case of case of charged defects. charged defects. Hence, theHence, the empirical model needs to be compatible with

empirical model needs to be compatible with dielectric propertiesdielectric properties of the materials of the materials studied if these effect

studied if these effects are to be taken into account. s are to be taken into account. This can only be achieved if This can only be achieved if oneone allows for

allows for polarization of ionspolarization of ions. . In this case In this case we can evaluatwe can evaluate, and/or fie, and/or fit, the t, the low- andlow- and high-frequency dielectric constants

high-frequency dielectric constants !!

o

o and and !!"", respectively, in the framework of the, respectively, in the framework of the

model.

model. (For a (For a brief explanation of brief explanation of dielectric properties see dielectric properties see Appendix).Appendix).

The most successful empirical model through which the polarizability of ions has The most successful empirical model through which the polarizability of ions has been introduced into the atomic level analysis of ionic crystals, is the shell model, been introduced into the atomic level analysis of ionic crystals, is the shell model, first advanced by B. G. Dick and A. W. Overhauser (Phys. Rev.

first advanced by B. G. Dick and A. W. Overhauser (Phys. Rev. 112112, , 90, 190, 1958). 958). It It isis shown schematically in Fig. 4.

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Core

Spring

Shell

Fig. 4.

Fig. 4. The schematic pictThe schematic picture of the shell ure of the shell model of an ion.model of an ion.

In the framework of this model ions can be polarized and each ion i, possessing a In the framework of this model ions can be polarized and each ion i, possessing a charge Z

charge Z_ii, consists of two components:, consists of two components: (i)

(i) A A core having core having the the same msame mass ass as as the the ion ion i i and a and a chargecharge ZZ ii c c

(ii)

(ii) A A 'shell' 'shell' having having no no mass mass and and a a chargecharge ZZ ii s s .. The total charge of the ion

The total charge of the ion ZZ

ii == ZZii c c + + ZZii s s ..

The core and the shell are coupled by a harmonic spring with the force constant

The core and the shell are coupled by a harmonic spring with the force constant ! ! _ii soso

that the force acting between the core and the shell depends only on their relative that the force acting between the core and the shell depends only on their relative displacement, s,

displacement, s, and no and no directional forces ardirectional forces are involved. e involved. (In this (In this sense the sense the shell isshell is spherical).

spherical).

Interactions involved in the shell model Interactions involved in the shell model

(i) The Coulomb interactions: include shell-shell, shell-core and core-core (i) The Coulomb interactions: include shell-shell, shell-core and core-core interactions between different ions.

interactions between different ions.

(ii) Short range shell - shell repulsive interactions between different ions described by (ii) Short range shell - shell repulsive interactions between different ions described by potentials given by equation (I2). No short-range core - core interactions are usually potentials given by equation (I2). No short-range core - core interactions are usually included.

included.

(iii) Core - shell interactions inside a particular ion: Described by the harmonic spring (iii) Core - shell interactions inside a particular ion: Described by the harmonic spring linking the core and the shell, characterized by a spring constant

linking the core and the shell, characterized by a spring constant ! ! _ii ..

Fitting of shell parameters Fitting of shell parameters ZZ

ii s s and and ! ! _ii Empirical

Empirical parameters: parameters: Low- Low- and and high-frequency high-frequency dielectric dielectric constantsconstants !!

o

o and and !!""

! !

o

o the transverse optical frequency the transverse optical frequency

The dipole moment induced at an ion, i, due to the displacement, s, of the shell is The dipole moment induced at an ion, i, due to the displacement, s, of the shell is equal to

equal to p p_ii == ZZ

ii ss

ss . . Its potential Its potential energy energy in in an an external external electric electric field,field, EE_a_a_{, is}_{, is}

U

U_pot_pot ==

_!

p p

iiEEaa ==

!

ZZii ss

ss EE_aa . . At the At the same time same time the self-energy of the ion, the self-energy of the ion, arising due toarising due to the harmonic spring linking the core and the shell, is

the harmonic spring linking the core and the shell, is UU

self self == 1 1 2 2! ! iiss 2 2 . . Minimization Minimization ofof

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the total energy,

the total energy, UU_pot_{pot +}+ UU_self_self, with respect to s yields, with respect to s yields ss ==

Z Z ii s s E E a a !

! _ii so that the so that the

polarizability of the ion i, defined as polarizability of the ion i, defined as !!

ii == p pii // EEaa is is ! ! ii == (Z (Z_iiss))22 " " ii (I3) (I3)

The polarizabilities of the ions,

The polarizabilities of the ions, !!_j_j, are directly related to the two dielectric constants, are directly related to the two dielectric constants

and the transverse optical

and the transverse optical frequency. frequency. For example, in the For example, in the case of metal case of metal helides of thehelides of the type NaCl with polarizabilities of the corresponding two elements

type NaCl with polarizabilities of the corresponding two elements !!₁₁ and and !!₂₂

(Sangster, M. J. L., Schröder, U. and Arwood, A. M., J. Phys. C

(Sangster, M. J. L., Schröder, U. and Arwood, A. M., J. Phys. C 1111, 1523, 1978), 1523, 1978)

! ! 1 1 ++ !!22 == 3 3"" 4 4## $ $ % % &&11 $ $ % % ++22 + + $ $ % % ++ 22 µ µ''_o_o22(($$₀₀ ++ 2)2)((ZZ &

& ((_Z_{Z ))}22 _(I4.1)_(I4.1)

and for metal helides such as CaF

and for metal helides such as CaF22,, SrCl SrCl22

! ! 1 1 ++ 2 2!!22 == 3 3"" 4 4## $ $ % % &&11 $ $ % % ++ 2 2 + + $ $ % % ++ 2 2 2 2µµ''_o_o22(($$₀₀ ++ 2) 2) (Z

(Z& & ((_Z_{Z ))}22 _(I4.2)_(I4.2)

!

! is the volume of the repeat cell, is the volume of the repeat cell, µµ the reduced mass of the two particles the reduced mass of the two particles11,,

Z Z== ZZ

1

1 == ZZ22 the ionic charge and Z' so called Szigetti charge (B. Szigetti, Proc. Roy. the ionic charge and Z' so called Szigetti charge (B. Szigetti, Proc. Roy. Soc. London A

Soc. London A 204204, 51, 1950) given in this case by the relation, 51, 1950) given in this case by the relation

((_Z_{Z ))}!! 22 ₌₌ 99µµ""oo 2 2_#_# 4 4$$ % % o o & & %%'' ((%% ' ' ++ 2)2) 2 2 (I4.3)(I4.3)

Analogous relations can also be used for oxides of the type MgO, CaO etc. (see, for Analogous relations can also be used for oxides of the type MgO, CaO etc. (see, for example,

example, M. J. M. J. L. Sangster aL. Sangster and A. M. nd A. M. Stoneham, Philos. MagazineStoneham, Philos. Magazine 4343, 597, 1981)., 597, 1981). We now make the assumption that the free ion polarizabilities

We now make the assumption that the free ion polarizabilities !!

ii are properties are properties that depend only on the ions and not on the crystal in which they are placed.

that depend only on the ions and not on the crystal in which they are placed. ByBy

analyzing a large number of ionic compounds that combine the same elements, the analyzing a large number of ionic compounds that combine the same elements, the values of

values of !!

ii are determined are determined by the least square fitby the least square fit. . For example, for alkali For example, for alkali helides wehelides we

1 1 11 µ µ = = 1 1 m m₁₁ ++ 1 1 m m₂₂

!

"

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consider all combinations of Li, Na, K, Rb, Cs with F, Cl, Br, I, i. e. compounds: LiF, consider all combinations of Li, Na, K, Rb, Cs with F, Cl, Br, I, i. e. compounds: LiF, LiCl, LiBr, LiI; NaF, NaCl, NaBr, NaI; KF, KCl, KBr, KI; RbF, RbCl, RbBr, RbI; LiCl, LiBr, LiI; NaF, NaCl, NaBr, NaI; KF, KCl, KBr, KI; RbF, RbCl, RbBr, RbI; CsF, CsCl,

CsF, CsCl, CsBr, CsI. CsBr, CsI. Thus Thus we use we use experimental data,experimental data, !!

o

o,, !!"" and and !!oo, for twenty, for twenty

compounds to determine polarizabilities

compounds to determine polarizabilities !!

ii for for eight eight elements. elements. Using Using thesethese polarizabilities and equation (I3), the shell charges

polarizabilities and equation (I3), the shell charges ZZ ii s s

and corresponding spring and corresponding spring constants

constants ! ! _ii are then determined such as to fit with the shell model dielectric are then determined such as to fit with the shell model dielectric

constants constants !!

o

o and and !!"" and the transverse optical frequency and the transverse optical frequency !!oo; for details see Sangster,; for details see Sangster,

M. J. L., Schröder, U. and Arwood, A. M., J. Phys. C

M. J. L., Schröder, U. and Arwood, A. M., J. Phys. C 1111, 1523, 1978., 1523, 1978.

Frequently, but not always, only the negative ions (owing to the surplus of electrons) Frequently, but not always, only the negative ions (owing to the surplus of electrons) are treated as polarizable while the positive ions are regarded as point charges; the are treated as polarizable while the positive ions are regarded as point charges; the shell model is then applied only to negative ions.

shell model is then applied only to negative ions.

Fitting of parameters of short-range potentials given by equation (I2). Fitting of parameters of short-range potentials given by equation (I2).

Parameters

Parameters AA_ij_ij,, DD_ij_ij and and !!_ij_ij can be ascertained, similarly as in the case of empirical can be ascertained, similarly as in the case of empirical pair potentials for metals, by fitting the experimentally determined equilibrium pair potentials for metals, by fitting the experimentally determined equilibrium properties of

properties of the material the material and/or results of and/or results of DFT based DFT based calculations. calculations. For example,For example, potentials for oxides have been construct

potentials for oxides have been constructed by ed by M. J. L. M. J. L. Sangster and A. M. StSangster and A. M. Stonehamoneham (Phil. Magazine

(Phil. Magazine 4343, 597, , 597, 1981). 1981). During this During this fitting fitting the the parameters of parameters of the the shell,shell, determined from fitting the dielectric properties, are, of course, used.

determined from fitting the dielectric properties, are, of course, used.

Since this is still a pair potential model the Cauchy relations between elastic moduli Since this is still a pair potential model the Cauchy relations between elastic moduli ((CC₁₂₁₂ ₌₌ C C₄₄₄₄_{for cubic}_{for cubic symmetry) are satisfied.}_{symmetry) are satisfied. A modification, the}_{A modification, the so called}_{so called} _breathing_breathing

shell model,

shell model, has been developed in which non-spherical deformation of the shell is has been developed in which non-spherical deformation of the shell is included (M. J. L. Sangster: J. Phys. Chem. Solids

included (M. J. L. Sangster: J. Phys. Chem. Solids 3434, 355, 1973; also C. R. A., 355, 1973; also C. R. A. Catlow, I. D. Faux and M. J. Norgett, J. Phys. C

Catlow, I. D. Faux and M. J. Norgett, J. Phys. C 99, , 419, 1976). 419, 1976). In In this this model model thethe Cauchy relations need not be satisfied.

Cauchy relations need not be satisfied.

Atomistic calculations employing the shell model

In the framework of the shell model each ion possesses six degrees of freedom. In the framework of the shell model each ion possesses six degrees of freedom. Three are associated with the position of the core and three with the position of the Three are associated with the position of the core and three with the position of the shell.

shell. The energy of The energy of the system the system and corresponding interatomic forces and corresponding interatomic forces are composedare composed from the following contributions:

from the following contributions:

Coulomb interaction between different ions Coulomb interaction between different ions (a)

(a) Core Core – – CoreCore (b)

(b) Shell Shell – – ShellShell (c)

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In general, if there are N ions in the system and each is composed of a core and a In general, if there are N ions in the system and each is composed of a core and a shell, interactions between 2N different charges must be considered. Since Coulomb shell, interactions between 2N different charges must be considered. Since Coulomb interactions are very long range, evaluation of the corresponding interaction energies interactions are very long range, evaluation of the corresponding interaction energies requires

requires a a special special treatment treatment employing Eemploying Ewald wald summations. summations. This This approach approach isis described in more details below.

described in more details below.

Short-range interaction between shells Short-range interaction between shells

Interactions between shells associated with individual ions are described by potentials Interactions between shells associated with individual ions are described by potentials of the type (I2) and if there are M different types of ions then there are, in general, of the type (I2) and if there are M different types of ions then there are, in general,

1 2

1 2 M(M(MM++11)) such such potentials. potentials. For example For example in a in a binary compound composed of binary compound composed of ionsions

A and B there are three potentials

A and B there are three potentials !!AAAA,,!!BBBB andand!!ABAB..

No short-range interactions between the cores are usually considered since the cores No short-range interactions between the cores are usually considered since the cores are screened by shells.

are screened by shells.

Core - shell interactions inside an ion Core - shell interactions inside an ion

Described by harmonic spring with a spring constant

Described by harmonic spring with a spring constant ! ! _ii that links the core with its that links the core with its

shell. shell.

All these interactions have to be included when evaluating the energy of the system All these interactions have to be included when evaluating the energy of the system and forces

and forces on cores and on cores and shells. shells. In molecular In molecular statics calculations statics calculations the energy the energy of theof the system is then minimized with respect to the positions of both cores and shells of the system is then minimized with respect to the positions of both cores and shells of the ions.

ions. In molecular dynamics In molecular dynamics calculations both cores and calculations both cores and shells may move or shells may move or only theonly the unit core plus shell move.

unit core plus shell move.

Treatment of Long-range Coulomb interactions

(See C. Kittel: Introduction to Solid State Physics)

We consider a

We consider a periodic structureperiodic structure with s point charges, with s point charges, ZZ_j_j, in the repeat cell and the, in the repeat cell and the sum of the charges in every repeat cell is zero, i. e.

sum of the charges in every repeat cell is zero, i. e. ZZ_j j

j j==11

ss

!

== 0 0 . . Periodic Periodic boundaryboundary

conditions are

conditions are naturally assumed. naturally assumed. To To compute compute the energy,the energy, EE_C_C_{, associated with the}_{, associated with the}

Coulomb interaction of these charges we need to evaluate the potential,

Coulomb interaction of these charges we need to evaluate the potential, !!_ii, at each, at each charge site i in the repeat cell induced by all the other charges in the structure that charge site i in the repeat cell induced by all the other charges in the structure that includes all t

includes all the periodically repeating cellhe periodically repeating cells. s. The interaction energy of tThe interaction energy of this system ofhis system of charges is then charges is then E E C C == 1 1 2 2 Z Z ii!!ii ii

"

(I5)(I5)

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Ewald method for evaluation of

!!_ii

-

- Three-dimensionally

Three-dimensionally periodic

periodic

structures

The electrostatic potential experienced by charge i in the presence of all the other The electrostatic potential experienced by charge i in the presence of all the other charges can, in principal, be calculated by direct summation of the Coulomb charges can, in principal, be calculated by direct summation of the Coulomb potentials.

potentials. However, convergence However, convergence of such of such summation summation is is very slow very slow and Eand Ewaldwald proposed the following trick in which the potential is decomposed into two parts:

proposed the following trick in which the potential is decomposed into two parts: (a) The potential,

(a) The potential, !!aa, arising from Gaussian densities of charges centered at the sites, arising from Gaussian densities of charges centered at the sites

of point charges, each giving the same total charge as the corresponding point charge. of point charges, each giving the same total charge as the corresponding point charge. This is shown schematically in Fig. 5.

This is shown schematically in Fig. 5. For the charge at positiFor the charge at position n this density ison n this density is22

!!

_aann (r) (r) == Z Z n n 1 1

"

33

_##

33 /2/2 exp exp

$$

r r

$

r r _n_n 22

"

22

%%

&&

''

((

))

**

(I6)(I6)

The width of the Gaussian, determined by

The width of the Gaussian, determined by !!, is chosen such as to assure fastest, is chosen such as to assure fastest possible convergence in the summation.

possible convergence in the summation. (b) The potential,

(b) The potential, !!bb, arising from point charges and additional Gaussian, arising from point charges and additional Gaussian

distributions of charges of opposite signs centered at the positions of the charges. distributions of charges of opposite signs centered at the positions of the charges. However, no charge is placed at the position where the potential is being evaluated, i. However, no charge is placed at the position where the potential is being evaluated, i. e. position marked

e. position marked 0 in Fig. 0 in Fig. 5. This i5. This is shown schematically is shown schematically in Fig. 6. n Fig. 6. The GaussianThe Gaussian densities are given by expressions analogous to (I6).

densities are given by expressions analogous to (I6).

0 0 -1-1 -2-2 1 1 2 2 Posit

Positions ions of of point charges point charges Gaussian d

Gaussian densitiensitieses of charge

of charge

Fig. 5.

Fig. 5. Gaussian densities of Gaussian densities of charge centered at charge centered at positions of individual point positions of individual point chargescharges giving the same t

giving the same total charge as point otal charge as point charges. charges. The position at The position at which the potential iswhich the potential is calculated is marked 0.

calculated is marked 0.

2

2_{The Gaussian density is normalized such that}_{The Gaussian density is normalized such that} ₄₄_!_{! ""}

a a n n (r)r (r)r22drdr # #

$ $

== Z Z n n..

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0 0 -1-1 -2-2 1 1 2 2 Point charges Point charges Positions of Positions of point charges point charges Gaussian densities Gaussian densities of charge of charge Fig. 6.

Fig. 6. Gaussian densities of Gaussian densities of charges centered at charges centered at positions of individual point positions of individual point chargescharges giving the opposite total charge as do the point charges, together with point charges. giving the opposite total charge as do the point charges, together with point charges. The position at which the potential is calculated is marked 0 and no charge is placed The position at which the potential is calculated is marked 0 and no charge is placed at this position. at this position. Potential Potential _!_! a a

We expand this potential into a Fourier series We expand this potential into a Fourier series33

! ! aa((rr)) == CCK K exp(iexp(iK K ""rr)) K K

#

(I7.1)(I7.1) where

where KK are the reciprocal lattice vectors of the periodic structure considered are the reciprocal lattice vectors of the periodic structure considered 44_..

Similarly, the charge density invoking this potential,

!

_aa ==

!

aa k k ,, jj j j==11 ss

"

k k

"

, where the, where the summation over j extends over one repeat cell and summation over k extends over summation over j extends over one repeat cell and summation over k extends over various repeat cells, can be expanded as

various repeat cells, can be expanded as

3

3_{The potential possesses the periodicity of the structure so th}_{The potential possesses the periodicity of the structure so th at}_at_!_!

a

a ( (rr)) == !!aa ( (rr ++ r rnn)). . The same appliThe same applies to all es to all otherother quantities calculated

quantities calculated..

4

4_{A reciprocal vector}_{A reciprocal vector} _K_K_{is generally given as}_{is generally given as} KK == jj_iibb_ii

ii==11

3 3

!

where where j j_ii are integers and are integers and

b b 1 1 == 2 2!! a a 2 2 "" aa33 a a 1 1##((aa22 "" a a33)) ,,bb 2 2 == 2 2!! a a 3 3 "" a a11 a a 1 1##((aa22 "" a a33)) ,,bb 3 3 == 2 2!! a a 1 1 "" aa22 a a 1 1##((aa22 "" aa33))

are the basis are the basis vectors of the reciprocal lattice.

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!

_aa((rr))==

!

K K exp(iexp(iK K

" "

rr)) K K

#

(I7.2)(I7.2) !

!aa and and !!aa are linked by the Poisson equation are linked by the Poisson equation !!

2 2

"

"aa == # #44$%$%_a_a. . Inserting Inserting (I7.1) (I7.1) andand

(I7.2) into the Poisson equation yields (I7.2) into the Poisson equation yields

K K 22CC_K_Kexp(iexp(iK K

! !

rr)) K K

"

== 44

#

$

K K exp(iexp(iK K

!!

rr)) K K

"

(I8)(I8) and, therefore, and, therefore, C C K K == 4 4_!"_!" K K K K 22 (I9)(I9)

By definition, the Fourier coefficient of the charge density

By definition, the Fourier coefficient of the charge density !!aa is is

!"

_K_K ==

""

aa((rr)exp()exp(

##

iiK K

$ $

rr)d)drr repeat repeat cell cell

% %

(I10)(I10) where

where !! is the is the volume of the volume of the repeat cell. repeat cell. Here the Here the densitydensity !!_a_a ( (rr)) originates from the originates from the

charges within the

charges within the repeat cell as repeat cell as well as from well as from the charges in althe charges in all other celll other cells. s. Hence,Hence, this integral is the same as when integrating the density originating in the repeat cell, this integral is the same as when integrating the density originating in the repeat cell, multiplied by

multiplied by exp(exp(

_!

iiKK

_""

r r)), over the whole space, i. e., over the whole space, i. e.

!

!""

_K_K ==

""

aa j j ((rr

##

rr_j_j)) j j==11 ss

$

exp(exp(

##

iiK K

% %

rr)d)drr All space All space

& &

(I11)

Inserting (I6) into (I11) yields Inserting (I6) into (I11) yields

!"

_K_K == 1 1

#

33

$$

3/ 23/ 2 ZZ j j expexp

%%

rr

%

rr_j_j 22

#

22

%%

i iK K

& &

rr

''

((

))

**

++

,,

All space All space

- -

j j==11 ss

.

ddrr = = 1 1

#

33

$$

3/23/2 ZZ j jexp(exp(

%%

iiK K

& &

rr j j) ) eexxpp

%%

//

2 2

#

22

%%

i iK K

& &

''

((

))

**

++

,,

All space All space

- -

j j==11 ss

.

dd (I12) (I12)

where the substitution

where the substitution rr

_!

rr_j j == was made. was made. The lThe last integral ast integral can be can be evaluated withevaluated with

the help of complex variables and we obtain the help of complex variables and we obtain

(13)

!"

KK == S( S(KK)exp)exp

##

$

22 K K22 4 4

%

&

& ''

(

)

) *

*

(I13.1)(I13.1) where where S( S(K K )) == ZZ j jexp(exp(!!iiK K " "rr j j)) j j==11 ss

#

(I13.2)(I13.2)

is the structure factor in which charges

is the structure factor in which charges ZZ_j_j are the form factors. are the form factors.

After inserting (I13.1) into (I9), equation (I7.1) yields for the potential at the position After inserting (I13.1) into (I9), equation (I7.1) yields for the potential at the position II

!

_aa((rr_ii)) == 4 4

""

#

S S_ii((K K )) K K 22 expexp

$$

%

22 K K 22 4 4

&

'

' ((

)

*

* +

+

K K

,

(I14)(I14)

where the structure factor

where the structure factor SS_ii((KK)) is evaluated such that the origin is taken at the is evaluated such that the origin is taken at the position i.

position i. Owing to the Owing to the exponential dependence only a fexponential dependence only a few shortest reciprocal ew shortest reciprocal latticelattice vectors need to be included in this summation.

vectors need to be included in this summation.

Potential Potential _!_!_b_b

The potential at the position i arising from a charge at a position

The potential at the position i arising from a charge at a position rr_j j, can be written as, can be written as

Z Z_j_j 11 r r _j_j

!!

1 1 r r _j_j

""

b b (r)dr (r)dr

!

""

_b_b(r(r )) r r dr dr r r _j_j # #

$ $

0 0 r r _j_j

$ $

%%

&&

''

((

))

**

(I15) (I15)

when taking the

when taking the position i as position i as the origin. the origin. The first term The first term arises from the arises from the point charges,point charges, the second term from the part of the Gaussian distribution lying inside the sphere of the second term from the part of the Gaussian distribution lying inside the sphere of radius

radius rr_j_j and the third part from the part of the Gaussian distribution lying outside and the third part from the part of the Gaussian distribution lying outside this

this sphere. sphere. Substituting Substituting forfor !!_b_b((rr)) in (I15) the Gaussian function analogous to (I6) in (I15) the Gaussian function analogous to (I6)

and summing over all the charge positions, both in the repeat cell and in all other and summing over all the charge positions, both in the repeat cell and in all other cells, yields cells, yields

!

_b_b(r (r _ii)) == Z Z_n_n r r _ii

"

r r _n_n n n##ii

$

FF r r ii

"

r r nn

%%

&

'

((

)

*

+

(I16.1)(I16.1) where where

(14)

F(x) F(x)== 2 2

!

exp(exp(

"

yy 2 2 )dy )dy x x

$ $

(I16.2)(I16.2)

and the summation extends over all the charge positions both in the repeat cell at the and the summation extends over all the charge positions both in the repeat cell at the origin and all

origin and all other cells. other cells. Obviously, F converges very rapidly to Obviously, F converges very rapidly to zero as x increaseszero as x increases and thus only a small number of cells neighboring the repeat cell at the origin need to and thus only a small number of cells neighboring the repeat cell at the origin need to be included.

be included.

The potential at the position i is then

The potential at the position i is then !!_ii == ! !_a_a ( (rr_ii))++ ! !_b_b ( (rr_ii))" " !!₀₀((rr_ii)) where where

!

₀₀((rr_ii))== 4 4

""

rr

##

rr_ii 2 2

$$

_aa((rr

#

rr_ii)) r r

##

rr ii 0 0 % %

& &

d(d(rr

#

rr_ii)) == 2 2ZZ_ii

'

' ""

is the contribution of the Gaussian density at the position i that was incorporated into is the contribution of the Gaussian density at the position i that was incorporated into the case (a) and must be subtracted since only contributions of charges other than the the case (a) and must be subtracted since only contributions of charges other than the charge at i are to be included

charge at i are to be included..

Hence, the potential at the s

Hence, the potential at the site i isite i is

!

_{ii =}= 44

""

#

S S_ii((K K )) K K 22 expexp

$$

%

22 K K 22 4 4

&

'

' ((

)

*

* +

+

K K

,

$$

22ZZii

%

% ""

++ Z Z_n_n rr_ii

$

rr_n_n n n--ii

,

FF rrii

$

rrnn

%%

&

'

((

)

*

+

(I17)(I17) The parameter

The parameter !! is arbitrary and is arbitrary and !!_iidoes not does not depend on its depend on its choice. choice. However, the convergence ofHowever, the convergence of the sums in (I17) does and the trick is to choose

the sums in (I17) does and the trick is to choose !! such that these sums converge rapidly such that these sums converge rapidly55_..

Ewald method for two dimensional periodicity

An analogous Ewald type summation of the Coulomb energies has been developed An analogous Ewald type summation of the Coulomb energies has been developed (D. E. Parry, Surface Science

(D. E. Parry, Surface Science 4949, 433, 1975) when only two-dimensional periodicity, 433, 1975) when only two-dimensional periodicity is present as, f

is present as, for example, in tor example, in the case of ihe case of interfaces. nterfaces. The summations are The summations are then donethen done plane by plane in the crystallographic planes parallel to the periodic plane, e. g. the plane by plane in the crystallographic planes parallel to the periodic plane, e. g. the

5

5 _{In an ideal ionic crystal the Coulomb energy can be written as}_{In an ideal ionic crystal the Coulomb energy can be written as} EE_C_C == !!

" "

a

a , where a is the lattice, where a is the lattice parameter and parameter and !! == a a 2 2 Z Z_ii""ii ii==11 s s

#

is the Madelung constant; the summation over i extends over all the is the Madelung constant; the summation over i extends over all the ions in the repeat cell

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interface (see also R. E. Watson, J. W. Davenport, M. L. Perlman and T. K. Sham, interface (see also R. E. Watson, J. W. Davenport, M. L. Perlman and T. K. Sham, Phys. Rev. B

Phys. Rev. B 2424, 1791, 1981)., 1791, 1981).

In the case of an interface or free surface the bloc is divided into two regions as In the case of an interface or free surface the bloc is divided into two regions as follows:

follows: Region

Region I: I: In In this this region full region full atomic atomic relaxation relaxation is is taking taking place place and and the the repeat repeat cell cell ofof the planes parallel to the boundary is defined by the vectors

the planes parallel to the boundary is defined by the vectors bb

1 1 II and and bb 2 2 II

so that in the so that in the region I any vector in a plane parallel to the boundary can be written as their linear region I any vector in a plane parallel to the boundary can be written as their linear combination.

combination. Region

Region II: II: In this region atoms In this region atoms are not relaxed and are not relaxed and they occupy ideal latticethey occupy ideal lattice positions.

positions. This region This region may, however, may, however, be shifted be shifted as a as a rigid block. rigid block. The repeat The repeat cell ofcell of the planes parallel to the boundary is defined by the vectors

the planes parallel to the boundary is defined by the vectors bb

1 1 II II and and bb 2 2 II II

so that in the so that in the region II any vector in a plane parallel to the boundary can be written as their linear region II any vector in a plane parallel to the boundary can be written as their linear combination. combination. In general vectors In general vectors bb 1 1 II and and bb 2 2 II

are not the same as vectors are not the same as vectors bb

1 1 II II and and bb 2 2 II II

but, for example, but, for example, in the case of a grain boundary with the smallest available repeat cell, the vectors in the case of a grain boundary with the smallest available repeat cell, the vectors determining this cell will be the same in both the region I and the region II.

determining this cell will be the same in both the region I and the region II.

The Coulomb energy of a charge at the position j in the region I may be decomposed The Coulomb energy of a charge at the position j in the region I may be decomposed into contributions arising from its interaction with charges in regions I and II, into contributions arising from its interaction with charges in regions I and II, respectively, i. e respectively, i. e.. Z Z_j_jZZ_k_k r r _jk_jk k k

!

== Z Z_j_jZZ_k_k r r _jk_jk k from I k from I

!

++ Z Z_j_jZZ_k_k r r _jk_jk k from II k from II

!

(I18)(I18)

According to Parry the contribution from the region I is According to Parry the contribution from the region I is

Z Z_j_jZZ_k_k r r _jk_jk k from I k from I

!

== 4 4

""

ZZ_j_j A A_II ZZk k exp exp

%%

##

K K _II 22 44

$

22

&&''

((

))**

K K _II 22 K K _II

!

k fromI k fromI

!

coscos

_&&

%%

K K _IIrr_jk_jk

_)) #

((

#

2Z 2Z j j

2 2

$

""

+ + Z Z_j_j ZZ_k_k 1 1

##

erf erf

( (

rr_jk_jk

##

R R _!_!II_m_m

))

rr_jk_jk

##

R R _lm_lmII k from I k from I

!

! !_,m_,m

!

(I19) (I19) where

where KK_II is the reciprocal lattice vector related to planar lattice based on is the reciprocal lattice vector related to planar lattice based on bb

1 1 II and and bb 2 2 II ,, R R_! !mm II =

= !!bb₁₁II ++ mmbbII₂₂ ( (!! and m are integers), and m are integers), AA_II the area of the repeat cell in region I and the area of the repeat cell in region I and

the convergence parameter

(16)

The contribution from the region II is The contribution from the region II is

Z Z_j_jZZ_k_k r r _jk_jk k from II k from II

!

== 2 2

"

ZZ_j_j A A_II_II ZZk k exp exp

#

K K _II_II (z(z_j_j ++zz k k ) )

$$

%%

&&

''

K

K _II_II

$$%%

11

#

exp(exp(

#

K K _II_II dd))

&&''

K K _II_II ( (00

!

k from II k from II

!

coscos

$$

_%%

K K _IIIIrr_jk_jk

&&

_''

(I20)(I20) where

where ((rr_n_n,z,z_n_n ) ) are cylindrical coordinates of the position of charge n with the z axis are cylindrical coordinates of the position of charge n with the z axis

perpendicular do the interface,

perpendicular do the interface, KK_II_II is the reciprocal lattice vector related to the planar is the reciprocal lattice vector related to the planar lattice based on lattice based on bb 1 1 II II and and bb 2 2 II II

,, AA_II_II the area of the repeat cell in region II and d is the the area of the repeat cell in region II and d is the magnitude of the repeat vector in the z direction, perpendicular to the interface.

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APPENDIX – POLARIZATION IN IONIC CRYSTALS

Macroscopically the applied electric field,

Macroscopically the applied electric field, EEaa, induces polarization of the medium, induces polarization of the medium

which is described by the polarization vector

which is described by the polarization vector PP == !!EE_a_a, where, where !! is the electric is the electric

susceptibility.

susceptibility. The polaThe polarization vector rization vector is defiis defined as ned as the dithe dipole moment pole moment per unitper unit volume.

volume. The The dipole moment associated dipole moment associated with a with a chargecharge qq_j j is is pp_j j == q q_j jrr_j j, where, where rr_j j is the is the

position vector of the charge

position vector of the charge qq_j_j and the polarization vector is and the polarization vector is PP== 1 1 ! ! pp_j j j j

"

, where, where the summation extends over all the dipole moments within the repeat cell and

the summation extends over all the dipole moments within the repeat cell and !! isis the volume of this cell.

the volume of this cell.

The dielectric displacement vector

The dielectric displacement vector DD == EEaa ++ 44!!PP satisfies the same Poisson equation satisfies the same Poisson equation

in the dielectric medium as does

in the dielectric medium as does EEaa in the vacuum and we define the dielectric in the vacuum and we define the dielectric

constant,

constant, !!, via the relation, via the relation DD == !!EE_a_a ( (!! == 11++ 44"#"#).).

If

If pp_j j is the dipole moment associated with the ion j, then the is the dipole moment associated with the ion j, then the polarizabilitypolarizability of this of this

ion,

ion, !!_j_j, is defined by the relation, is defined by the relation

p p_j_j == !!

j

jEElocloc(( j)j)

where

where EE_loc_loc(( j)j) is the is the local electric local electric field at field at the position of the position of ion j. ion j. This field This field is notis not necessarily just the applied field

necessarily just the applied field EEaa but because of the effects of polarization of the but because of the effects of polarization of the

medium it is given by the so-called Lorentz relation

medium it is given by the so-called Lorentz relation EElocloc == EEaa ++

4 4!!PP 3 3 == " " ₊₊ 2 2 3 3 EEaa.. (See e.g. C. Kittel: Introduction to Solid State Physics).