6. Ionic Crystals

IONIC CRYSTALS 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 ! closed-shell ions A   and B   which attract each other by Coulomb forces and repel each other at short-range due to Pauli repulsion. A schematic picture of an ionic solid is shown in Fig. 1. +

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

 Metal halides

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

Simple cubic lattice of lattice of Cs ions with Cl ions on simple cubic lattice displaced relative to the Cs lattice by 12 111 vector. Alternatively, a bcc lattice with with Cs in cube corners and Cl at cube centers (Fig. 2a).

Sodium chloride

A simple cubic lattice of alternate positive and negative ions. Each species is forming a face-centered-cubic lattice and these are displaced with respect to each other by 12 110  (Fig. 2b).

2

Fig. 2a. CsCl structure

Fig. 2b. NaCl structure 2

3

 Metal oxides

Sodium chloride

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

Spinel

MgAl2O4, FeAl2O4, ZnCr2O4, ZnFe2O4

Perovskite 

BaTiO3 , SrTiO3 (see Fig. 3).

Corundum

Al2O3 i. e sapphire (hexagonally close-packed oxygen planes with aluminum filling two-thirds of the available octahedral sites).

Fig. 3 Structure (cubic symmetry) of SrTiO3

Electrostatic neutrality of ionic crystals 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. Examples: Vacancies occur in pairs of vacancies at positive and negative ions. Frenkel defect - vacancy + interstitial.

3

4

INTERACTIONS BETWEEN IONS 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 EC

1 =

2

" i,j i!  j

Z i Z j rij

 

(I1)

where Z i   is the charge at ion i and rij   is the separation separation between ions i and j. The summation in (I1) converges because there are positive and negative ions but the convergence is very slow. Summation methods, such as the the Ewald method, described below, have to be employed. The charges associated with ions may be either full charges corresponding to isolated ions or partial charges. In the former case the corresponding Z i  is an integer multiple 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 ab initio calculations for a given ionic crystal. In both cases there is zero zero total charge within the repeat cell.

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

! ij

=

$  r '   D ij " 6  )  # %  ij (  r

Aij exp& "

(I2)

The first term is the (Pauli) repulsion of ions at short separations described by the Born-Mayer potential. The second term is the (weak) (weak) Van der Waals attraction and it is often omitted in studies of ionic ionic solids. The short range interactions can be regarded as significant only for separations smaller or equal to that of the first nearest neighbors. The parameters Aij , D ij  and !ij  can be determined, similarly as in the case of any empirical pair potentials, by fitting equilibrium quantities, e.g. lattice parameter, elastic moduli, cohesive energy, of the material that are determined experimentally and/or by ab initio calculations.

4

5

Since both long-range Coulomb interactions and short-range repulsions are described by pair potentials model the Cauchy relations between elastic moduli ( C 12  C 44  for 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 energy of the system is composed of the Coulombic term (I1) and the term given by the short-range pair potentials potentials (I2). This determines the total energy of the solid as a 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 models. 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 below, must be employed. Example: Potentials for Al 2O3  discussed in the study of P. R. Kenway: J. American Ceram. Soc. 77, 77, 349, 1994.

Shell model 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. Therefore, defect energies may depend sensitively on the electronic polarization of the lattice. lattice. This is is particularly significant in the case of charged defects. Hence, the empirical model needs to be compatible with dielectric properties  properties   of the materials studied if these effects effects are to be taken into account. This can only be achieved if one allows for polarization of ions. evaluate, and/or fit, fit, the low- and ions. In this case we can evaluate, high-frequency dielectric constants !  and ! , respectively, in the framework of the model. (For a brief explanation of dielectric properties see Appendix). "

o

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, first advanced by B. G. Dick and A. W. Overhauser (Phys. Rev. 112, 1958). It is 112, 90, 1958). shown schematically in Fig. 4.

5

6

Spring Core

Shell

Fig. 4. The schematic picture picture of the shell model of an ion. In the framework of this model ions can be polarized and each ion i, possessing a charge Zi, consists of two components: (i)

A core having the same mass mass as the ion i and a charge

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

Zi

=

c

Zi

s

Zi

c

Zi

.

s

Zi .

+

The core and the shell are coupled by a harmonic spring with the force constant ! i so that the force acting between the core and the shell depends only on their relative displacement, s, and no directional forces are are involved. (In this sense the shell is spherical).  Interactions involved in the shell model

(i) The Coulomb interactions: include shell-shell, shell-core and core-core interactions between different ions. (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 included. (iii) Core - shell interactions inside a particular ion: Described by the harmonic spring linking the core and the shell, characterized by a spring constant ! i .

s

Fitting of shell parameters Zi  and

Empirical parameters:

! 

i

Low- and high-frequency dielectric constants !o

!o

 and

!"

 the transverse optical frequency

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

U pot

=

=

s

Zi s .

Its potential energy in an external electric field,

Ea ,

is

s

 ! p i Ea  ! Zi s Ea . At the same time the self-energy of the ion, arising due to =

the harmonic spring linking the core and the shell, is 6

Uself 

1 =

2

! 

i

s

2

. Minimization of

7 s

the total energy, U pot

+

U self  , with respect to s yields

s

Z i Ea =

! 

  so that the

i

polarizability of the ion i, defined as

!

i

!

i

 p i / E a  is

=

=

(Z si )2 "  i

 

(I3)

The polarizabilities of the ions, !  j , are directly related to the two dielectric constants and the transverse optical frequency. For example, in the case of metal helides of the type NaCl with polarizabilities of the corresponding two elements ! 1   and ! 2 (Sangster, M. J. L., Schröder, U. and Arwood, A. M., J. Phys. C 11, 11, 1523, 1978) !1 + ! 2 =

3" $% & 1 4# $% + 2

$% +

+

2 µ'o

2

($ 0 + 2)

2

( Z & Z( )  

(I4.1)

2

(I4.2)

and for metal helides such as CaF 2, SrCl2 !1 +  2 !2 =

!  Z

3" $ % & 1 4# $ % +  2

+

$ % +  2 2 µ' o

2

($ 0 +  2)

(Z & Z ( )  

is the volume of the repeat cell, µ  the reduced mass of the two particles 1,

=

Z1

=

Z2

 the ionic charge and Z' so called Szigetti charge (B. Szigetti, Proc. Roy.

Soc. London A 204, 204, 51, 1950) given in this case by the relation 2

(Z ! ) =

9µ" 2o# % o & % ' 4$

(%' + 2) 2

 

(I4.3)

Analogous relations can also be used for oxides of the type MgO, CaO etc. (see, for example, M. J. L. Sangster and and A. M. Stoneham, Philos. Magazine 43, 43, 597, 1981). We now make the assumption that the free ion polarizabilities

!

i

 are properties

that depend only on the ions and not on the crystal in which they are placed. By analyzing a large number of ionic compounds that combine the same elements, the values of ! i  are determined by the least square fit. fit. For example, for alkali helides we

1

1

= µ

!  1 # "  m

1

1

+

m2

$ , where m  and m &  %  1

2

 are the masses of the two ions.

7

8

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; CsF, CsCl, CsBr, CsI. Thus we use experimental data, ! , !   and ! , for twenty o

compounds to determine polarizabilities

!

i

for eight elements.

polarizabilities and equation (I3), the shell charges constants

! 

i

"

s

Zi

o

Using these

  and corresponding spring

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

constants !  and !  and the transverse optical frequency ! ; for details see Sangster, M. J. L., Schröder, U. and Arwood, A. M., J. Phys. C 11, 11, 1523, 1978. o

"

o

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 shell model is then applied only to negative ions. Fitting of parameters of short-range potentials given by equation (I2).

Parameters Aij , D ij   and !ij   can be ascertained, similarly as in the case of empirical pair potentials for metals, by fitting the experimentally determined equilibrium properties of the material and/or results of DFT based calculations. For example, potentials for oxides have been constructed constructed by M. J. L. Sangster and A. M. Stoneham Stoneham (Phil. Magazine 43, 43, 597, 1981). During this fitting the parameters of the shell, determined from fitting the dielectric properties, are, of course, used. Since this is still a pair potential model the Cauchy relations between elastic moduli ( C 12  C 44  for cubic symmetry) are satisfied. A modification, the so called breathing shell model,  has been developed in which non-spherical deformation of the shell is included (M. J. L. Sangster: J. Phys. Chem. Solids 34, 34, 355, 1973; also C. R. A. Catlow, I. D. Faux and M. J. Norgett, J. Phys. C 9, 419, 1976). In this model the 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. Three are associated with the position of the core and three with the position of the shell. The energy of the system and corresponding interatomic forces are composed from the following contributions: Coulomb interaction between different ions (a) (b) (c)

Core – Core Shell – Shell Core – Shell 8

9

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 interactions are very long range, evaluation of the corresponding interaction energies requires a special treatment employing Ewald Ewald summations. This approach is described in more details below. Short-range interaction between shells 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, 1 2 M(M M(M + 1)   such potentials. For example in a binary compound composed of ions A and B there are three potentials ! AA ,! BB and ! AB . No short-range interactions between the cores are usually considered since the cores are screened by shells. Core - shell interactions inside an ion Described by harmonic spring with a spring constant

! 

i

  that links the core with its

shell. All these interactions have to be included when evaluating the energy of the system and forces on cores and shells. In molecular statics calculations the energy of the system is then minimized with respect to the positions of both cores and shells of the ions. In molecular dynamics calculations both cores and shells may move or only the unit core plus shell move.

Treatment of Long-range Coulomb interactions (See C. Kittel: Introduction to Solid State Physics) We consider a periodic structure with structure with s point charges, Z j , in the repeat cell and the s

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

! Z j

=

 0 . Periodic boundary

 j 1 =

conditions are naturally assumed. To compute the energy, E C , associated with the Coulomb interaction of these charges we need to evaluate the potential, ! i , at each charge site i in the repeat cell induced by all the other charges in the structure that includes all the the periodically repeating cells. cells. The interaction energy of this this system of charges is then EC

1 =

2

"Z ! i

i

9

i

 

(I5)

0

Ewald method for evaluation of structures

- Three-dimensionally periodic

!i

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 potentials. However, convergence of such summation is very slow and Ewald Ewald proposed the following trick in which the potential is decomposed into two parts: (a) The potential, ! , arising from Gaussian densities of charges centered at the sites of point charges, each giving the same total charge as the corresponding point charge. This is shown schematically in Fig. 5. For the charge at position position n this density is 2 a

!an (r)

2 % r  $  r n (   $ *   Z n 3 3 /2  exp ' $ 2 ' "# " * & )

1

=

The width of the Gaussian, determined by possible convergence in the summation.

!,

(I6)

is chosen such as to assure fastest

(b) The potential, ! b , arising from point charges and additional Gaussian 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. e. position marked 0 in Fig. 5. This is is shown schematically in in Fig. 6. The Gaussian densities are given by expressions analogous to (I6).

Gaussian densiti d ensities es of charge

2

-1

0

1

-2

Positions Positions of  point charges

Fig. 5. Gaussian densities of charge centered at positions of individual point charges giving the same total total charge as point charges. The position at which the potential is calculated is marked 0.

#

2

The Gaussian density is normalized such that

4!

$ " (r)r dr n

a

0

10

2

=

 Z n .

1

Gaussian densities of charge

2

1

0

-2

-1

Positions of  point charges

Point charges

Fig. 6. Gaussian densities of charges centered at positions of individual 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 at this position.

 Potential ! a

We expand this potential into a Fourier series 3 ! a (r )

=

#C

K 

exp(iK " r )  

(I7.1)

K 

where K  are the reciprocal lattice vectors of the periodic structure considered 4. s

Similarly, the charge density invoking this potential, !a

=

" "!

k,j a

, where the

k   j 1 =

summation over j extends over one repeat cell and summation over k extends over various repeat cells, can be expanded as 3 The

potential possesses the periodicity of the structure so th at ! a (r )

=

! a (r

+  r n ) .

The same applies applies to all other

quantities calculated. 3

4A

reciprocal vector K  is generally given as

K

=

! j i bi  

where  j i   are integers and

i 1 =

b1

=

 2 !

a2 " a 3 a 1 # (a 2 "  a 3 )

, b2

=

 2!

a 3 "  a 1 a 1 # (a 2 "  a 3 )

vectors of the reciprocal lattice. 11

, b3

=

 2!

a1 " a 2 a 1 # (a 2 " a 3 )

 

are

the

basis

2

!a (r )

=

#!

K 

exp(iK  " " r )  

(I7.2)

K 

2

linked by the Poisson equation ! " a (I7.2) into the Poisson equation yields ! a  and !a  are

" K  C 2

K 

exp(iK  ! ! r)

=

4#

K 

"$

K 

=

 #4 $% a . Inserting (I7.1) and

exp(iK ! r )  

(I8)

K 

and, therefore, 4!"

CK 

=

K 

2

K 

 

(I9)

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

!"

K 

=

% "

a

(r)exp( #iK  $  $ r )dr  

(I10)

repeat cell

where

!  is

the volume of the repeat cell. Here the density !a (r)  originates from the

charges within the repeat cell as well as from the charges in all all other cells. cells. Hence, this integral is the same as when integrating the density originating in the repeat cell, multiplied by exp(! iK " r) , over the whole space, i. e. s

!"

K 

&  $ "

 j a

=

(r # r j ) exp( #iK  %  % r )dr  

(I11)

All space  j 1 =

Inserting (I6) into (I11) yields

!"

K 

1 =

#3 $3/ 2 1

=

#3$ 3/2

where the substitution

' r % r 2 *  j , %   & Z j exp ) %  i K  & r dr 2 ) , #  j 1 All space )( ,+ s ' /  2  & r j )  & Z j exp( %iK  & exp ) % 2 % iK  & ) #  j 1 All space ( s

r

. -  =

.

- 

=

 ! r j

=

* ,d , +

(I12)

was made. The last last integral can be evaluated with

the help of complex variables and we obtain 12

3

!"

=

K

%   $2K 2 (   S(K)exp ' # &  4 *  ) 

(I13.1)

where s

S(K )

=

# Z exp( !i  j

 " r j )   K  "

(I13.2)

 j 1 =

is the structure factor in which charges Z j are the form factors. After inserting (I13.1) into (I9), equation (I7.1) yields for the potential at the position I

!a (ri )

4" =

#

,

Si (K ) K 2

K 

&   %2 K 2 )  exp ( $   '  4 +  * 

(I14)

where the structure factor Si (K)   is evaluated such that the origin is taken at the position i. Owing to the exponential dependence only a few few shortest reciprocal lattice vectors need to be included in this summation.

 Potential ! b

The potential at the position i arising from a charge at a position r  # % ( (r ) " 1 1  b Z j ' !  (r)dr  ! dr *   "   !  b ' r  j r  j * r  0 r  '& *)

r j

, can be written as

 j

$ 

$ 

(I15)

 j

when taking the position i as the origin. The first term arises from the point charges, the second term from the part of the Gaussian distribution lying inside the sphere of radius r j   and the third part from the part of the Gaussian distribution lying outside this sphere. Substituting for !b (r)   in (I15) the Gaussian function analogous to (I6) and summing over all the charge positions, both in the repeat cell and in all other cells, yields

! b (r i )

=

Zn

$ r  " r  n #i

i

n

where 13

&  r i " r n )  F( +   % '  * 

(I16.1)

4 #

F(x)

2 =

! $ 

exp( " y )dy   2

(I16.2)

x

and the summation extends over all the charge positions both in the repeat cell at the origin and all other cells. Obviously, F converges very rapidly to zero as x increases and thus only a small number of cells neighboring the repeat cell at the origin need to be included. The potential at the position i is then ! i =  ! a (ri ) +  ! b (ri ) " ! 0 (ri )  where %

!0 (ri )

=

4"

&  0

2

r

# ri $a (r # ri ) d(r # ri ) r # r i

2 Zi

=

' "

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 charge at i are to be included . Hence, the potential at the site s ite i is

!i =

4"

#

,

Si (K ) K 2

K 

&   %2 K 2 )  2 Zi exp ( $ $ '  4 +  *  % "

+

, n -i

Zn

 $ rn i

r

&  ri $ rn )  F( +   % '  * 

(I17)

The parameter !  is arbitrary and ! i does not depend on its choice. However, the convergence of the sums in (I17) does and the trick is to choose !  such that these sums converge rapidly 5.

Ewald method for two dimensional periodicity An analogous Ewald type summation of the Coulomb energies has been developed (D. E. Parry, Surface Science 49, 49, 433, 1975) when only two-dimensional periodicity is present as, for for example, in the the case of interfaces. interfaces. The summations are then done plane by plane in the crystallographic planes parallel to the periodic plane, e. g. the "

5

In an ideal ionic crystal the Coulomb energy can be written as a

parameter and !

=

2

EC

=

!

a

, where a is the lattice

s

#Z " i

i

 is the Madelung constant; the summation over i extends over all the

i 1 =

ions in the repeat cell . 14

5

interface (see also R. E. Watson, J. W. Davenport, M. L. Perlman and T. K. Sham, Phys. Rev. B 24, 24, 1791, 1981). In the case of an interface or free surface the bloc is divided into two regions as follows: Region I: In this region full atomic relaxation is taking place and the repeat cell of I I the planes parallel to the boundary is defined by the vectors b 1  and b 2  so that in the region I any vector in a plane parallel to the boundary can be written as their linear combination. Region II: In this region atoms are not relaxed and they occupy ideal lattice positions. This region may, however, be shifted as a rigid block. The repeat cell of II II the planes parallel to the boundary is defined by the vectors b 1  and b 2  so that in the region II any vector in a plane parallel to the boundary can be written as their linear combination. I

I

II

II

In general vectors b 1  and b 2  are not the same as vectors b 1  and b 2  but, for example, 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. 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, respectively, i. e . Z j Z k  Z j Z k  Z j Z k    (I18) = + r  r  r   jk   jk   jk  k  k from I k from II

!

!

!

According to Parry the contribution from the region I is

Z jZ k 

! k from I

r  jk 

4"Z j

=

AI  Z j

+

I

K I  is I

k fromI

k from I

K I

K  I

(

1 # erf  k 

K I

'&

Z k 

!!Z ! ,m

where

! !

exp % #

r

 jk 

r

 jk 

#

2

4$

2

2

I R  !m

(  2Z j2 $ *) % cos K I r jk  ( # & ) "

)

 

(I19)

# R Ilm

the reciprocal lattice vector related to planar lattice based on I

I

m are integers), AI  the area of the repeat cell in region I and the convergence parameter !  have the same meaning as in the previous case. R !m

=

!b 1

+

mb 2  ( !  and

I

b 1  and b 2 ,

15

6

The contribution from the region II is

! k from II

Z j Z k  r  jk 

=

2"Z j A II

!Z! k 

k from II

K   ( 0 II

exp $ # K 

II

& % K II (z j + z k  )' cos $K  r &   % II  jk  ' $%1 # exp( # K II d ) '&

(I20)

where (rn ,z n )  are cylindrical coordinates of the position of charge n with the z axis perpendicular do the interface, K II  is the reciprocal lattice vector related to the planar II

II

lattice based on b 1   and b 2 , AII  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.

16

7

APPENDIX – POLARIZATION IN IONIC CRYSTALS

Macroscopically the applied electric field, E , induces polarization of the medium which is described by the polarization vector P !E , where !   is the electric susceptibility. The polarization polarization vector is defined defined as the dipole dipole moment per unit volume. The dipole moment associated with a charge q j  is p  j  q  j r j , where r j  is the a

=

a

=

position vector of the charge q j   and the polarization vector is P  1  ! " p j , where =

 j

the summation extends over all the dipole moments within the repeat cell and the volume of this cell. The dielectric displacement vector in the dielectric medium as does constant, !, via the relation If

p  j   is

ion,

! j

D

=

D

is

4 !P  satisfies

the same Poisson equation   in the vacuum and we define the dielectric

=

Ea

!E a

Ea

!

 ( !

=

+

1+ 4"# ).

the dipole moment associated with the ion j, then the polarizability of polarizability of this

, is defined by the relation

 p j

=

!

 j

E loc ( j)

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

E a  but

because of the effects of polarization of the

medium it is given by the so-called Lorentz relation (See e.g. C. Kittel: Introduction to Solid State Physics).

17

E loc

=

Ea

+

4 !P 3

=

" +  2

3

Ea.