SCREENED IONS IN SOLIDS

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SCREENED IONS IN SOLIDS

G. Morgan

To cite this version:

G. Morgan. SCREENED IONS IN SOLIDS. Journal de Physique Colloques, 1972, 33 (C3), pp.C3-

49-C3-55. �10.1051/jphyscol:1972308�. �jpa-00215042�

JOURNAL DE PHYSIQUE

Colloque C3, supplkment au no 5-6, Tome 33, Mai-Juin 1972, page C3-49

SCREENED IONS IN SOLIDS

G. J. MORGAN

The Physics Department, The University of Leeds, UK

Rkumk. -

On donne une definition exacte d'un ion

cr

kcrant6

>>

d'un solide parfait dans l'appro- ximation de Hartree. Ceci permet d'examiner les rkgles pour construire un ion

<<

kcrantk

^>) a

l'aide des propriktks diklectriques du solide car l'ion

<<

kcrant6

>>

peut Stre d6fini en fonction d'une cons- tante diklectrique effective. Si on utilise le modkle de Penn pour les Btats d'un isolant, on obtient un ion

<<

kcrant6

^D

neutre.

On discute aussi la dkfinition d'un ion

<<

kcrant6

>>

dans un solide diatomique et on examine diverses fa~ons de tenir compte de l'kchange.

Abstract. - An exact definition is given of a screened ion in a perfect solid within the Hartree approximation. This enables one to examine prescriptions for constructing a screened ion in terms of the dielectric properties of the solid since the exact screened ion may be defined in terms of an effective dielectric function. Using the Penn model of the states in an insulating solid the exact screened ion is found to be neutral as it should be.

The definition of a screened ion in a diatomic solid is also discussed and different ways of includ- ing exchange are examined.

1. Introduction. - In the Hartree approximation the simplest prescription, for the self consistent potential, in a nearly free electron metal, is to superim- pose screened ions. Each ion is screened individually in terms of the dielectric function of a free electron gas, so that if ui(q) denotes the Fourier transform of the bare ion potential, the screened potential is just vi(q)/s(q) where ~ ( q ) is the dielectric function for a free electron gas. If we now consider a solid with a band gap, one might suppose that the screened ions could be constructed in much the same way replacing ~ ( q ) by the << dielectric function

⁾⁾

of the insulating solid. Such a procedure has however no apparent logical basis, and this is reflected in the fact that this procedure leaves the ions incompletely screened. Phillips [8]

has sought to remedy this by the hypothesis that the dielectric screening charge should be supplemented by

<< bonding charge

^)>

and indeed obtains reasonable

agreement between calculated and empirical values for the pseudopotential in Si using a simple model for the << bonding charge

)>.

In this paper we construct an exact definition of the screening charge in the Hartree approximation. The inclusion of exchange is relatively straight forward but we do not consider this complication here at first, because we are primarily interested in questions con- cerning charge neutrality. In

§

2 the exact definition of screening charge is discussed and it is shown that in the Hartree approximation, the highest occupied energy levels in a solid must be coincident with the free efectron Fermi energy. I n 9 3 the screening charge is defined in terms of a n effective dielectric function and this is a

convenient way of examining Phillips' proposition concerning the screening of ions in semiconductors.

In 5 4 the question of screening and charge transfer in ionic solids is considered in terms of the states of the covalent mean crystal

^D,

and the real solid. The

<< mean crystal

^)>

being the solid in which each different

constituent ion potential is replaced by a suitable average ion potential. Finally in

§

5 two possible methods of including exchange are discussed.

2. The charge distribution. - We will begin by considering the Hartree approximation and monato- mic solids. The basic information about the charge distribution is contained in the Green function G(rrl, E) which is the solution to

where V(r) is the potential energy and

y

is a real positive infinitesimal. The contribution to V(r) from the electrons is just

where

3 i

denotes the real part, the integration ouer E is conjined to energies below the Fermi level and e is the electronic charge.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972308

C3-50 G . 3. MORGAN

We may rewrite (1) as the integral equation

where G: denotes a free space Green function which is purely real for E < - C and E

=

0 at the vacuum level. The reasons for writing (1) in this form are that we wish to define an exact screening charge which can be compared with that obtained from linear screening theory, and to express the charge distribution in terms of an effective dielectric function. The value of C is entirely arbitrary and we can choose it to suit our objectives.

A generalized definition of screening charge has been given by Benneman [3] corresponding to choos- ing - C to be the average potential in the solid, but here we will make another choice which at first sight is different. The energy levels for a free electron gas lie above the vacuum level because the smeared ion and electron charge densities cancel to give a potential which is zero. Let us now choose C so that the upper- most occupied levels in the real solid are coincident in energy with the uppermost free electron energy corresponding to G:. This is depicted schematically in figure 1.

The charge density is given by

so that substituting from eq. (3) we have

-

Free etectron Fermi l e d AE= C

Fermi in the solid

k

FIG. 1. - Curve A corresponds to a free electron band. Curve B corresponds to an energy band in the real solid, with the upper- most occupied level lying an amount BE below the free electron Fermi energy. Curve C corresponds to the Green function G $

when c = AE.

The first of the terms in (5) gives the constant charge density of the smeared out electrons, because we have adjusted the uppermost free electron level to be coincident with that of the real crystal, and this term is independent of any absolute scale for the crystal energy levels. The second term is also independent of an absolute energy scale but the final term does depend on the shift in the energy levels through the constant C.

The charge density can only depend on the nature of the wave functions and the relative separation of the energy bands, so that we must have

C%

- - 'he f ^dE f d r l G ~ ( r r l , E ) G ( r l r , E )

(6) The integral in (6) is certainly not zero so we are left with the conclusion that

A E = C = O .

(7)

This conclusion is essentially the point made by Ball [2] concerning screened ions in simple metals where it is shown that the Friedel sum for the poten- dr, G;(rr,, E) CG(rl r, E) . (5) tial be

Let us now ignore the constant charge density for the moment in (9, then we may write

where vi(r - rj) denotes an individual ion at r j tion. Neither of these replacements would be exactly and Go now denotes G ; = ~ . We have an equation which, right, but we can formally express the solution of (8) if we replaced Go by G, could be solved in terms of the in terms of an effective generalized dielectric function, generalized dielectric function for the solid, and if we namely

replaced G by Go then the solution could be expressed

in terms of the Lindhard free electron dielectric func- V(q)

=

C ^{K'(q9 q +}

^g)

^{~ ' ( q +}

^g)

⁽⁹⁾

g

SCREENED IONS I N SOLIDS C3-51

In (9) V(q) is the Fourier transform of the total Fourier transform of the total ion potential and K' potential,

g

is a reciprocal lattice vector, Vi(q) is the is the effective dielectric function which satisfies the

equation

< ^ko I

^e-iq.r

I kf nf > <

^{k f}

^nf I e+i(q+g').r

I

^k0

>

x

Kf(q + ^{g', q} +

^g)

n

'

jy;;; h ^k2 - E + ^{iq) (E..(~o}

^-

^E + ^iq)

In (10) En.&') denotes the energy corresponding to a

state k' in the n'th, band ] k' n' > denotes the corres- plane wave state and g' is a reciprocal lattice vector.

ponding wave function, while 1 k, > denotes a pure If we can assume that K'(q, q) is dominant then we may write

and

The fact that Vi can be written as a sum of individual For our present purpose we will assume that only bare ions means that we may define an individual Kf(q, q) is important. If we now consider a simple screened ion as

(*)

metal and approximate G by a free space Green func-

tion then we obtain

K'-l(q,

q) =

~ ~ ( 9 ) but we have so far ignored the constant charge density where

gL

is the Lindhard dielectric function and due to the first term in (5). This is easily dealt with

because we can simply

cc

tack

^)>

this constant charge AV(0)

=

+ 3 EF (16) density onto the ion charge density to give an extra where EF is the free electron Fermi energy. The average contribution to the potential potential due to the screened ions is just - 3 EF giving

K'hY q + g) V(0)

=

0

AT/'(q)

=

4 nepo C

^{g -}

^{(13) (17)}

g

I q + g 1 2 which will always be the case, providing Kt(q, q) is where p, is the constant charge density. The final ~ r o ~ o r t i o n a l to q2 for small

q-

Hence Benneman's potential is therefore choice (1964) that - C should be the average potential in the solid is correct, providing we recognise that the V(q)

=

x ^{Kf(q, q + vi(q + g) +} average potential is zero and that A V(q) must be added

g

to the screened ion potential.

K'(q9

q

+ g) We are now in a position to investigate the nature

f

4 nepo C

g

l q + g I 2

^-

(I4) screened ions in general situations and we next consi- der a monatomic solid with a band gap.

(') This definition of a screened ion is somewhat arbitrary in a

perfect lattice since the total potential is only defined when q is 3.

Screened

ions

in monatomic with

a

band gap*

equal to a reciprocal lattice vector.

-Returning to eq. (8) we define our individual

screened ion by

C3-52 G. J. MORGAN

If there is more than one atom in the unit cell, then because of the different behaviour of G near each ion.

each ion in the cell will be screened slightly differently We may rewrite (19) in different ways. First we could write

so that the term involving

(

G o - G ) vG would repre-

sent the correction to screening vi by the dielectric conjecture is correct, would contain the bonding function for the insulating solid and, if Phillips' charge contribution. An alternative and symmetrical

way of writing (18) is 2

ie2

v =

vi + ¹ , ( 5 ^{dB dr,} I ^dr2 I r - r 1 I ^GuG 2 ^{( G o}

^-

^{G )} 2

^{U ( G O}

^- "'1 ^{) . (20)}

The term ( G o - G ) v(GO - G ) would now give the

corrections to a procedure in which an ion is screened function and the actual dielectric function of the solid.

by the arithmetic mean of the Lindhard dielectric The real dielectric function for a solid is given by (Srinivasan, [I

01)

where

(

kn > denotes the state with wave vector k in the n'th band. Nn(k) is the occupation number for the state

(*).

For comparison purposes we write Kt-' = ~ ' ( q ) and put (10) in a similar form to (21), namely

8

ne2

] < ko I e-jq.'

[

k' n' > l2 ( ~ , . ( k ' ) - No(k)) d ( q )

=

1 + ⁷ C C

^---

q

^{k k '}

n'

h

where I k o > denotes a plane wave state and No(k) the occupation numbers for a free electron band. The expression for

E'

splits naturally into two parts. First there is the contribution when No(k)

⁼

1 and Nn.(k')

⁼

0 . The second contribution is when Nn,(k')

=

1 and

No(k)

⁼

0 .

In the first case the energy denominator does not go to zero. In the second case the denominator can go to zero at isolated points corresponding to the highest occupied levels in the zone, but these will not in general correspond to points where k'

=

k .

E = O

We will consider the limiting behaviour of

^E'

as

q k

tends to zero, using the simple model used by Penn [6]

and Srinivasan [ l o ] for the wave functions in a solid

with a gap. This model, which is based on a

⁽⁽

one

_FIG. 2. - Energy bands for the Penn model. Curve A represents

dimensional gap

1,

produced by diffraction, appears to

a free electron band, while curve B represents the E(k) relation-

give

a

reasonable form for the true dielectric function,

ship for a direction corresponding to a highest occupied level.

so is hope

that

we may use it in (22)

to

consider

^{Curve C} represents the Penn energy bands where Ez is the average energy gap, so that the uppermost energy levels are no longer

the limiting form for

8'.

This model is shown in figure 2

coincident with the free electron Fermi energy.

in the extended zone scheme. It would not be sensible

(*) Here Nn(k) and N n , ( k r ) are zero if the state is unoccupied

to make the uppermost level for the model states

and equal to unity otherwise.

coincident with the free electron Fermi energy since

SCREENED IONS IN SOLIDS C3-53

in the Penn model the band gap E6 represents the (E; - Eg)/2 below EF where EE is the minimum energy average energy gap. Thus, for a symmetrical gap, gap.

the highest occupied state for the Penn model lies In the extended zone scheme

E'

now takes the form

where E f and E - denote the energy bands for k' > k , and k' < k , respectively, and k , is the Fermi wave vector. Following the notation of Srinivasan [lo]

the matrix elements for this model are given by

where K t

=

kt - 2 k , k'lk' and the wave functions are given by

$ ;

=

(1

+ , & 2 ) - 5

(eik'.r

+ ^e+iK'.r

1 . (25) Thus each contribution in (23) is further divided into two parts, a normal process part form the first delta function in (24), and an umklapp process part from the second delta function. In the limit as

q

tends to zero, the volume of integration shrinks t o zero for N-pro- cesses. The volume of integration for U-processes is limited by the fact that only the band edge states containing a reflected plane wave component have non-vanishing matrix elements. To estimate

^E'

we suppose elm [ 6 ] ) that only states within k , EG/4EF of k, have any reflected component, and set

a: = +_

1 in this region. We then obtain

This is a very crude result but it shows that the screen- ed ion is neutral because .st

aq-'

for small

q.

The fact that the energy denominator can vanish in the exact expression (22) need cause no alarm. The points where the denominator can vanish, corresponding to the highest occupied states, are isolated. In three dimen- sional solids they are points where the density of states vanishes, and in general

k #

k' at these points.

The definition of a screen ion given here is for a perfect lattice and in general the modification of the lattice Green function due to ion displacements must be taken into account. In the case of a simple metal the concept of a screening charge which moves around with the ion arises from supposing that the Green function is approximately independent of the potential.

The main point is that (22) allows us t o ask what kind of screened ion is consistent with known gross features of the energy bands. The attractive feature of

the true dielectric function is that very reasonable forms can be calculated by means of crude models provided basic features are built into the model, such as the orthogonality of band edge states in a semicon- ductor. It remains to be seen how true this is for zt.

Is it interesting to note that in a monatomic solid containing more than one atom in the unit cell the diagonal components of K' do not contain any infor- mation about the different environments of non- equivalent atoms. For example when there are two atoms in the unit cell we may write

vl(q)

=

C ^K'(q,

^q

^{+ g) v:(q + g) (27)}

g

where v,(q) is the first type of atom and K t is to be evaluated taking the first type of atomic site as the origin. The second type of potential may be written as v2(q)

⁼

C Kt(% q +

g)

eig'd vk(q + g) (28)

g

where d is the vector joining type one and type two atoms. This suggests that in a detailed investigation of the charge distribution in covalent solids it is essential to take into account the off diagonal components of K'.

Until now we have restricted ourselves to monato- mic solids, though the general case could have been included. We prefer, however, to consider ionic solids in a slightly different way.

4. Ionic solids and charge transfer.

-

Instead of constructing screened ions in terms of deviations from a uniform charge density, it is advantageous, for solids containing more than one type of atom, to begin with the mean crystal. For simplicity let us consider a crystal containing two types of atom, and let the bare ion potentials be denoted by v i and vg. We can now construct a fictitious crystal where each different ion is replaced by the mean ion potential

In practice we may wish to replace vb and vg by pseudo-ions, but that refinement will not be considered here. We can now construct the states and energy levels for the mean crystal and the Green function will be denoted by GM. This mean crystal may be metallic or covalent in general, depending on the constituent potentials and the structure. If we now add to v&,

AUA

⁼

^{(vA -} v t ) and

=

(vk - v h )

C3 -54 G . J. MORGAN

then the Green function for the real crystal will satisfy where A V is the sum of the screened potentials Av,

the equation and Av,. We now make use of the fact that the upper-

G(rrt, E )

=

most occupied levels in the real crystal and the mean

crystal will be coincident to define an individual

=

cM(n', E ) - I dr, GM(rr,, E ) AVG(r, r', E ) (30) screened ion by 2

iez

=

AV:,, +

Z

/ 5 ^dE 5 ^dr, 5 ^dr, I r - r l ¹ I

GM A v A o r B

0 ) .

There is no problem with constant charge densities here terms of a generalized dielectric function as in (9), because we are adding a neutral combination of but here we will content ourselves with an investigation

and A&. We now may proceed to solve (30) in of the simple effective dielectric function analogous to (221,

where k,,, n, and k' n' denote states in the mean and real crystals respectively. The screened potential is

now given by

E

The interesting question is how E" will behave for small q. If it is proportional to q-2 as in a metal, then

VA

will be neutral, but if E" were constant for small q then we might identify

as the charge transfer from one atom to another. I

I A Z I is the difference in the number of bare charges

on the mean ion and type A or B ion.

^{FIG. 3. -} Average energy bands for the mean << covalent ^D crystal (M) and real crystal (C) based on a one dimensional

This is a tempting ~ r o ~ o s i t i o n but examination

model. The energy gaps for the mean and real crystals are denoted

--

-

- of (30) would appear to rule out this definition of the

by E g and EG respectively. Note that C and M should really be

charge transfer. If the covalent mean crystal has a band

shifted with respect to each other since they are only average

gap there is no problem with vanishing energy denom

bands. The states near the gap at k ~ / 2 are those which describe the charge transfer.

minators in (30) and for .

^, E"

to have a finite value at q

=

0 we require the exact orthogonality of the mean crystal states and the real crystal states. Although we can construct simple models analogous to the Penn model is depicted in figure 3 in which the states are approximately orthogonal in different bands any slight departure from non-orthogonality is sufficient to make E" aq-'

c

for small values of q.

It is clear that the addition of the potentials AvA and AnB will

cc

mix

D

the states of the mean crystal and the new states cannot be exactly orthogonal to the mean crystal states in different bands. It is also clear that the diagonal component of the effective dielectric func- tion

E"

(32) does not contain information about the environment of the different types of atom since it does

not depend on the co-ordinate origin which is also the case for the true dielectric function and

E'

defined in eq. (22).

How then may we obtain information about the charge transfer

?

Let us consider the case of NaCl.

The Fourier components of the total potential or

charge density are only defined at reciprocal lattice

vectors of the body centred lattice and the information

about the charge transfer is contained in those Fourier

components corresponding to reciprocal lattice vectors

which do not coincide with the reciprocal lattice of the

simple cubic lattice of the mean crystal. Just as with

the case of a monatomic solid with more than one type

SCREENED IONS I N SOLIDS C3-55

of atom in the unit cell it would also seem essential to include the off diagonal components of the effective dielectric function to obtain a detailed picture of the charge distribution.

Our purpose in this paper is to put forward a scheme whereby the charge distribution can be investigated using simple representative models for the energy states in solids, where the nature of these states in known in a qualitative manner, in the spirit of calcula- tions of the dielectric function (Penn [6]). However we have so far confined the discussion to the Hartree approximation and if we are to consider states in a real crystal than some method of describing exchange must be included in the definition of a screened ion.

5. The exchange potential.

-

The inclusion of exchange in the one body potential has been considered by Benneman [3] within the screened exchange approximation (e. g Phillips

[7]).

For the sake of simplicity we will confine ourselves to the pure Hartree- Fock potential and consider possible ways of including exchange in the definition of a screened ion.

The exchange operator in the H-F scheme is given in terms of the Green function by

Exactly the same argument as given in 5 2 may be pursued to show that the position of the Fermi energy with respect to the vacuum level should be the same as for an electron gas in the H-F approximation. If we now wish to set up an integral equation analogous to (3) there seem to be two main possibilities. The most obvious one is to replace Go on the right hand side of (3) by the Green function of an electron gas in the Hartree-Fock approximation (Benneman [3]) and then follow through the arguments of 5 2, 4 3 and 9: 4, including exchange. A second possibility is to leave Go as the Hartree Green function and instead of (3) we then have

G(rrl, E)

=

~ ' ( r r ' , E) - J dr' ~ ' ( r r , , E) V,, G(r, r ' , ' ~ ) (36)

Refer

[1]

BAILYN

(M.), Phys.

Rev.,

1960,117,974-984.

[2]

BALL (M.

A.), J. Phys. C., 1969, 2, 1248-1257.

[3]

BENNEMAN

(K. H.), Phys.

Rev.,

1964, 133, 1045-1061.

[4]

GILBERT

(R.)

and MORGAN

(G.

J.),

Phys. Stat. Sol., 1970, 42,267-273.

[5]

OVERHAUSER

(A. W.), Phys.

Rev.,

1970, 2, 874-876.

[6]

PENN

(D. R.), Phys.

Rev.,

1962,128,2093-2097.

[7]

PHILLIPS (J.

C.), Phys.

Rev.,

1961, 123,420-424.

where Yo, is the total potential including exchange.

This equation would be a useful starting point for defining a screened ion when the Hartree-Fock Fermi level in the solid lies below the occupied free electron levels of the Hartree approximation. In these circum- stances (Li, Na and K are cases where this would be true) the free space Green function on the right hand side of (36) makes no contribution to the Coulomb or exchange potentials and V , , may be simply defined as a sum of potentials associated with each ion.

However this will not generally be the case particularly when the exchange operator is screened and then these would then appear to be no advantage in adopting this second possible way of defining a screened ion.

The inclusion of exchange inevitably leads to the difficulties of solving an integral equation for the effective dielectric function (e. g. Sham and Ziman [l 11) and is clearly important to know the accuracy of simple approximations to the solution of this type of integral equation (e. g. Bailyn [I]). Once it is known how to construct the non-local exchange potential in a reaso- nable fashion then the construction of a local exchange potential will be straight forward, for example, by fol- lowing the suggestion of Overhauser [5] that the local form should be chosen to reproduce the same charge density as the non-local form.

6. Summary.

-

The definition of a screened ion given here for a perfect lattice has two useful features.

First it enables one to compare the exact screened ion defined in terms of an effective dielectric function with the form suggested by Phillips [8] for covalent semi-conductors. Secondly one can in general construct models for the states in a solid and enquire about the nature of the screened ion which is consistent with the model.

In the case of a monatomic covalent solid it is important to make realistic calculations of the effective dielectric function to find out if Phillips' construction is accurate and to find out how well the function may be described by a simple model like that of Penn [6].

The energy bands in solids can be intimately connected with the existence of bound states and resonances (Gilbert and Morgan [4]) and it is possible that the Penn model may not be adequate for the effective dielectric function.

[8]

PHILLIPS

(J. C.), Phys.

Rev.,

1968, 166, 832-838.

[9]

PHILLIPS (J.

C . )

and

VAN VECHTEN

(J.

A.), Phys.

Rev.

Letters,

1969, 22, 705-708.

[lo]

SRINIVASAN (G.),

Phys.

Rev.,

1969, 178, 1244-1251.

[ l l ]

SHAM

(L. J.)

and ZIMAN

(J. M.), Sol. Stat. Phys., 1963, 15, 221-298.

[12]

VAN

VECHTEN

(J.

A.), Phys.

Rev.,

1969, 182, 891-905.