Trends in the band structure of defect tetrahedral compound semiconductors : oxides and other systems with 4-2 local coordination

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Trends in the band structure of defect tetrahedral compound semiconductors : oxides and other systems

with 4-2 local coordination

M. Bensoussan, M. Lannoo

To cite this version:

M. Bensoussan, M. Lannoo. Trends in the band structure of defect tetrahedral compound semicon-

ductors : oxides and other systems with 4-2 local coordination. Journal de Physique, 1979, 40 (8),

pp.749-761. �10.1051/jphys:01979004008074900�. �jpa-00209159�

Trends in the band structure of defect tetrahedral compound semiconductors : oxides and other systems ^{with 4-2} local coordination

M. Bensoussan

Centre National d’Etudes des Telecommunications, 196, ^{rue de} Paris, ⁹²²²⁰ Bagneux, ^France and M. Lannoo

Equipe ^de Physique des Solides (*), ISEN, 3, ^rue François-Baës, 59046 Lille Cedex, ^France (Reçu ^{le 15} janvier 1979, accepté ^{le 2} avril 1979)

Résumé.

Un Hamiltonien de liaisons fortes est appliqué ^à l’étude de la structure électronique ^et du caractère local des fonctions d’onde dans une famille de composés ^à coordinance locale 4-2. Des théorèmes généraux ^sont

établis pour les composés AX2 ^{et AB} X4 reliant l’existence de bandes interdites à l’environnement local. Des

expressions générales du caractère « s » ou « p » ^sur chaque ^{atome sont obtenues. Dans tous les cas, la bande de}

valence la plus ^{haute est «} p ^» pure à caractère ^non liant sur l’atome X et correspond ^{à un « doublet libre ». De} façon générale ^{la bande de valence des} systèmes AX2 ^{se scinde en trois} sous-bandes _pour lesquelles il n’existe que deux possibilités distinctes. Des bandes interdites additionnelles existent pour les composés ^AB X4. ^Une applica-

tion numérique ^{détaillée est} faite dans le cas important ^des oxydes.

Abstract.

A tight binding Hamiltonian is used to investigate ^the electronic structure and the average local character of eigenstates ^{in a} large ^{class of} compounds ^{with 4-2} local coordination. General theorems are established for AX2 ^{and AB} X4 compounds relating the existence of _gaps to the local environment. General expressions ^are

derived concerning ^the « s » or « p ^» character of the eigenstates ^on any ^atom. The highest valence band is shown

to be in all cases a pure p ^non bonding ^{or lone} pair ^{band on the X atom. One finds} quite generally ^{that the valence}

band of AX2 systems splits ^into three subbands for which only ^{two distinct} possibilities ^{can exist.} Additional _gaps

are obtained for AB X4 systems. ^A detailed numerical investigation ^{is made} especially ^{in the} important ^{case of}

oxides.

Classification Physics ^Abstracts 71.25T - 71.20

Introduction.

An interesting ^{area in the} study of

semiconductors is now offered by binary ^or temary defect tetrahedral compounds. Many ^{of them are} amorphous ^{over most of the} composition range and when they crystallize, ^{their structure is often} extremely complicated. ^A good example ^{of such a} behaviour is

GexSe 1 _ x [1-4]. ^The electronic structure of such systems will become increasingly ^{difficult to obtain} by standard numerical techniques. ^{A few} attempts have been made, ⁱⁿ Si02 [5-7], SixGel-x02 [8], AIP04 [9] for instance. However these are still simple crystalline systems ^with relatively small unit cells.

We then believe that it is useful to extend to the more

complicated ^cases simplifying techniques ^{such as the}

Hückel theory ^or tight binding approximation ^which

(*) ^{L.A. No 253 au C.N.R.S.}

has provided ^the simplest meaningful analysis ^of covalent, III-V and II-VI semiconductors [10-16].

It also applies ^{to lone} pair semiconductors such as Se and Te, leading ^{to an} essentially ^correct description

of their valence bands [17-20]. ^{One can thus} hope ^to

obtain a similar success for defect tetrahedral ^com-

pounds.

Our aim in this work is to use such a tight binding

Hamiltonian and derive general properties ^which only depend ^{on the local} coordination. This is reminis- cent of the work done in normal tetrahedral semi- conductors [10-16]. ^We shall consider here the general

class of solids having local 4-2 coordination. Some of the sites have four nearest neighbours ^{at the corners}

of a practically regular tetrahedron, ^{the other atoms} having two nearest neighbours.

The most important group of this family ^corres- ponds ^to compounds having only ^two types ^{of atoms A}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01979004008074900

and X, ^{the A atoms} being tetrahedrally coordinated with X atoms as nearest neighbours. The stoichio- metric formula for these compounds ^is AX2. ^Such

structures are obtained essentially ^for II-VII2 ^and IV-VI2 compounds [21]. ^{To these one must add} Be(CH3)2 ^and Mg(CH3)2 [22]. ^These systems crys- tallize either in chains, layers ^or tridimensional

structures.

A second group corresponds ^{to AB} X4 compounds

derived from the ’V-V’2 ^{and which are either} III-V-VI4

or II-VI-VI4 systems. They ^{are obtained} by replacing

the A sublattice of the first _group by ^{an ordered AB} compound. Again ^{one must add to these} FeP04, MnP04 ^and LiAI(C2H5)4 [22] ^{and a} large group of

crystals ^{of which} ZrSi04 (Zircon) ^{is a} typical example [23].

The third group includes all systems ^{of formula} A X2n ^{and AB} X4n (n being ^an integer greater ^than unity), the tetrahedra being ^{centred on A and B atoms.}

We begin by giving ^{a formal treatment} of the band structure without any specification ^{for the} bonding

units. This is done in sections 1 and 3 for homopolar

and heteropolar tetrahedral sublattices respectively.

We also calculate formally ^{the « s » or «} p » character of the wave functions on the tetrahedrally coordinated atoms in sections 2 and 4. This is then applied ⁱⁿ

section 5 to the specific ^case of AX2 ^{and AB} X4 ^com- pounds for which the general trends of the band struc- ture with ionicity ^{and bond} angle ^{are discussed.}

Finally section 6 describes the numerical results obtained for these systems and discusses the trends in

more details.

1. Homopolar tetrahedral sublattice : Formal treat- ment of the band structure.

We consider first the

case of identical atoms on the tetrahedrally ^coordinat-

ed sublattice (figure ¹ where A and B atoms are

identical). ^{On these atoms we build} sp3 hybrid ^orbitals pointing towards their four nearest neighbours. ^We

assume that _any sp3 ^{orbital is} coupled ^{to one and} only ^one bonding unit, ^each bonding ^unit being coupled ^{to its two} neighbouring sp3 orbitals. We also consider the interaction between sp3 hybrids ^{of the}

same atom which we take to be d (this ^is equal ^to 1/4

of the atomic _s-p promotion energy [11]). ^{We shall}

demonstrate in the following that the band structure of the whole system ^{can be} completely ^{deduced from}

Fig. ^1.

Coupling ^{of one} bonding ^{unit to the} sp’ orbitals 1 0 >

and 11 > ^{of two} tetrahedrally coordinated atoms A and B.

the eigenvalues ^{of the} connectivity ^matrix correspond- ing ^to the tetrahedral A sublattice. This property ^is

independent ^{of the structure of the} bonding ^units

which we do not specify ^{for the moment.}

For this we use a Green’s function approach. ^The

total Hamiltonian matrix H can be written under the form

Here the index A refers to basis states of the A sublat-

tice, ^{R to the} remaining ^basis states, corresponding ^to

the bonding ^{units. To H one} may associate the resol- vent matrix G defined as

One can also define an unperturbed Hamiltonian matrix h as

with its corresponding resolvent matrix _g. With our

assumptions ^the bonding ^{units are} disconnected in

HRRI ^so that g ^{will have} poles ^at energies equal ^{to the} eigenvalues of the Hamiltonian of _any isolated bond-

ing ^{unit, or also} equal ^{to the s and} p energies ^{of the}

isolated A atoms.

The Dyson equation ^{allows us to write :}

where V describes the interaction between the A atoms and the bonding ^{units. The} eigenvalues ^{of H are}

related to the poles of G. The resolvent matrix ^can also be written

and for the coupled system it is in general enough

to know the poles ^of GAA. ^{Let us} then write (4) ^{for the}

submatrix G AA ^{and then} GRA :

Combining ^{these two relations one obtains}

Equations (7) represent ^a completely general ^result showing ^{that one can write} Dyson’s equation ^{for the A}

sublattice alone, adding ^to the direct interactions between its basis states indirect interactions U which

occur via the bonding units. An important ^charac-

teristic is that UAA ^is energy dependent through ^the

resolvent matrix _9RR.

In practice, ^{for the situation of} figure ¹ there will be

only ^{two différent terms in U, the first one} correspond- ing ^to Uoo ⁼ Ui 1, ^{the second one to} VOl (one ^must

recall that there is also one direct interaction A).

Let us call 1 a > and Ea ^the eigenvectors ^and eigenvalues

of the bonding ^{unit of} figure 1 ; ^{then one can write} quite generally

We can now go further and obtain general ^informa-

tion about the eigenvalues ^of the Hamiltonian (1) by considering ^{the A} sublattice alone. The whole problem

of the electronic structure is reduced to finding ^the

solutions of ^an effective Hamiltonian which only

concerns the tetrahedral sublattice and includes the direct interactions between the corresponding sp3

orbitals together ^{with the} effective interactions Uij

defined by (8).

An eigenstate l ’P A > of ^{the A} sublattice can always

be written as

where 1 ii’ > ^{is the} spl hybrid pointing ^{from atom i}

towards the bonding ^{unit common to atoms i and i’.}

The coefficients _{aü, and al’i} relative to the Sp3 ^orbitals

connected to the same bonding ^{unit must} satisfy ^the following coupled equations :

where the origin ^{of the} energies ^{is taken at the A atom} sp3 energy.

To derive (10) ^we have considered an homogeneous

system, ^all matrix elements of U being equal ^{either to} Uoo ^{or to} U 01 ^{in the notations of} figure ^{1. We have also} used Si = E ^au,.

r

Injecting the second of equations (10) ^{into the first}

one, ^one obtains :

valid for _any couple ^{ii’. A first non trivial solution of} (11) corresponds ^{to all} Si ^{= 0. For non zero} aii, ^one must then have

This is the equation ^of the characteristic flat bands

occurring ^{in the Leman,} Thorpe and Weaire Hamilto- nian [10-16]. Their number depends ^{on the structure}

of the indirect interaction U. To obtain the other bands

one must sum (11) ^{over i’ which leads to}

The allowed values for _cp are the eigenvalues ^{of the} connectivity ^matrix corresponding ^{to the A sublattice}

alone. These depend upon the detailed structure of this sublattice. However, the Gershgorin ^theorem applied ^{to fourfold} coordination gives 1 cp 1 4,

which allows us to write that the allowed energies

must always satisfy ^the inequality :

This is a generalization ^{of the} Leman, Friedel, Thorpe ^{and Weaire theorem. It} gives directly ^the

limits of the allowed bands without need for ^a band structure calculation. We can notice that - L1 and + 3 L1 are respectively ^{the « s » atomic energy} Et-

and the « p » atomic energy Ep ^{relative to the average} sp3 energy. The band limits correspond ^{to the values}

of E such that the first member in (14) vanishes. This

gives ^{the two sets} of solutions

The second of these equations ^is identical with

equation (12). ^{This means that the flat bands are} necessarily ^at the limits of broad bands, ^{as was the}

case in normal tetrahedral lattices.

2. Homopolar tetrahedral sublattice : ^« s » and ^« p » character of the subbands.

Precise statements can

be made about the average loca fharacter ^{of the eigen-}

states. Such statements are /useful in interpreting experiments ^{where the effect of selection rules on the}

matrix elements are important. ^For example, experi-

mental X-ray absorption spectra ^often give ^evidence

of the « s » and ^« _p ^» character of the different valence band states on a given type ^{of atoms.} This has been

analysed ^{in detail for} Si02 [5-7,24]. It is thus interesting

to derive theoretically such information. We shall do this for the atoms of the tetrahedral sublattice in the homopolar ^case, deriving analytical expressions

in terms of the indirect interactions, ^{which in turn} depend ^{on the structure of the} bonding ^units.

For this let us first notice that _any eigenstate [ # )

of the whole Hamiltonian can be written as

1 kA > and 1 t/lR > being ^the projections ^{on the basis}

states of the A sublattice and the bonding ^units

sublattice respectively. ^{In our} tight binding approxi-

mation all atomic basis states are orthonormal so

that 1 wA > and II/IR > ^are orthogonal.

To express, ^{as was} done before, ^all properties ⁱⁿ

terms of the A sublattice alone, ^the first task is to obtain the normalization condition ( # [ gi )

¹

in terms of 1 w’A >. This first gives

However for the eigenstates ^{of the} coupled systems ^we have the equation

which for 1 t/lR > ^can be written as

Combining equations (16) ^and (18) ^{leads to the} following expression

Noticing ^that gRR gRR is equal ^{to -} dEgRR ^(the

eigenstates ^{of H are} distinct from those of HRR ^{so that}

at these energies gRR is real), ^{this can} be rewritten

using (7), ^as

We are now in position ^to derive the ^« s » character of the wave function on the atoms of the A sublattice.

This is given by

where 1 st > ^{is the s state} corresponding ^{to the ith atom}

of the A sublattice. This can be expressed ^{in terms of} 1 wA > ^{alone as}

One can use the expansion (9) of 1 wA > ^{to obtain cs}

in a more detailed form. Using the notations of sec-

tion 1, ^{one has}

In (23) ^the prime ^{on the sum} indicates that only couples ^of values i, i’ ^can occur, corresponding ^to nearest-neighbours ^{A atoms. To} go further, ^{one must}

evaluate the quantity

For this one takes the _square modulus of (10), ^then replaces ^{in it} ai’i by its value deduced from (10), ^and finally ^uses (13) ^{to obtain}

with

The other term to be determined is

This one can be expressed ^{in terms of} c’, by ^multi-

plying ^{the first of} equations (10) by ai,i and summing.

One thus finally obtains for _cs the expression

This expression ^is only valid in the wide bands. In the

case where Uoo ^and U01 ^become energy independent

one finds _Cg equal ^to co ^{which must then be the} expres- sion for the ^{« s »} character in a pure covalent system such as C, Si, Ge, ^{Sn. In fact one recovers} the formula first derived by ^Leman and Friedel [11] ^{in the} crystal-

line case.

The « p » character in the wide bands can be obtained exactly by ^{the same} arguments ^{and it is not} difficult to show that ^one obtains

As to the flat bands they ^are pure « p » ^on the

A atoms, because all Si vanish in this case. In this

limit expression (26) thus remains valid, ^with c’ ^{= 0.}

We have then obtained general formulae for the

« s » and « p » character of the ^wave function on the A atoms. These formulae still simplify ^{further near the}

limits of the bands (where rp ⁼ ± 4, ^{s =} ± U 01

as exhibited by equation (14)). ^{In this case} co ^{is either} equal ^{to zero or} unity, ^{and the states are} pure ^{« s »}

or pure ^« p ^». One can easily show that for the _pure

« p » ^{states :}

and for the _pure ^« s » states :

These expressions ^{are the most} simple ^to calculate and

give very accessible and useful information as will be demonstrated in the following.

The calculation of the atomic ^« s », ^« p ^»... charac- ters on the other atoms depends ^{on the structure of the} bonding units which has not been specified ^{until now.}

3. Extension to a heteropolar tetrahedral sublattice.

-

We want now to generalize ^our previous ^{results to}

the case where the tetrahedral sublattice is built from two types ^{of atoms} forming ^{an A-B} compound.

Such a situation occurs, for instance when one goes from Si02 ^to AlP04.

The treatment of section 1 leading ^{to the} concept of effective interactions U still remains valid. However, using ^{the same} assumptions and the notations of

figure ^{1, U has now} three distinct matrix elements

It is also clear that the intraatomic interorbital interaction A now takes two values AA ^or AB. ^Moreover

the sp’ ^{atomic energy} presents ^two different values _aA and _aB. With this in mind one can try ^to generalize

the theorems concerning the existence of gaps.

For this, equations (10) ^{have to be} replaced by

which can be rewritten as

with

Here again the summation over i and i’ respectively

leads to :

with

A first non trivial solution to (32) corresponds ^{to all} Si ^and Si, equal ^{to zero. From} (31) ^this gives f(E)

⁰

which again ^{is the} equation ^{of pure p} flat bands.

For the wide bands one must combine the first two

equations (32) which lead to :

where the sum over j corresponds ^{to the nearest-} neighbours of i in the A sublattice. Their number is 12,

so that a direct application ^{of the} Gershgorin ^theorem

leads to qJA (OB - ⁴ 12, ^i.e. (PA (PB 1 ^{16. This} gives

the new band limits.

However another inequality ^{can be} derived for alternate compounds. ^{In this case} lfJA qJB - 4 reduces to qJ2 - ^{4 when A =} B. Thus the connectivity ^matrix occurring ⁱⁿ (33) ^must satisfy ^all inequalities imposed by the known bounds on ço 2which ^{are 0 and 16.}

This means that _{ÇOA qJB must} also lie within this interval.

The _upper bound is the same as before but the lower bound gives ^another inequality ^{to be satisfied and this} inequality ^{creates new} gaps with respect ^{to the homo-} polar ^situation.

The first inequality, ^which generalizes (14), ^{can be}

written

where we have replaced aA - L1A by Ep , ^{aA + 3 d A}

by EsA ^{which are} respectively ^the p and s energies ^{for A}

atoms, ^{the same} being ^{true for B atoms. The second}

inequality which introduces new gaps in the band structure gives :

The determination of the s and _p character of the different states on the A and B atoms is much more

complicated ^{than in the} homopolar ^{case. However}

this ^can again ^{be done} simply ^for the band limits.

Let us first consider the ^« s » character c.1 ^{on the A}

sublattice (the situation is completely symmetric ^for

the B one). ^Trivial considerations analogous ^{to those}

of section 2 allow us to write generally :

Let us consider the « s » band limits given by ^the equations :

This equation ^{is satisfied} only ^{when all} SA ^are equal

on the one hand, and all SH ^{on the other hand} (this effectively gives ÇOA 9B ⁼ 12 in equation (33)). ^With

this, equation (31) ^{leads to the fact} that all coefficients au, ^are equal ^{to a common value} a, and all bii, ^{to a}

common value b. Then equation (30) ^{leads to :}

One can thus détermine b/a from the first of these

equations ^{for instance.} Putting ^{this in} (35) ^and noticing ^that st ^{= 4 a one} finally ^{obtains for} csA

the expression

with a symmetrical expression ^for cB. Evidently ^these expressions reproduce ^the homopolar limit, ^when A = B and Uoo = Ull.

The case of the _p character is still simpler ^{to derive}

at the pure « p » band limits. There all SA ^and i.

vanish in (30) leading ^{to an} equation ^which directly corresponds ^{to (37). The final} result thus ^comes out to be

(38) ^and (39) ^are completely equivalent. They ^allow CsA, cpA ^{(and also c.,} c§) ^{to be} calculated directly ^once

the energies ^{of the} corresponding band limits are

known. These expressions evidently ^{reduce to} (27)

and (28) ^{in the} homopolar ^case.

If we now consider the ^new band limits introduced

by ^{the new} inequality (34bis) corresponding ^to gaps which open in the band ^structure deduced from the

homopolar ^{case, it is} possible ^{to derive} c. and cp along

the same lines as before. These limits correspond ^to

(pA ⁼ 0 or (pB ⁼ 0. For _ÇOA ⁼ 0, ^all SA ^and aii, ^are equal

while all SiB ^{= 0. The} symmetrical situation holds when _çoB ⁼ 0. One then obtains equations ^similar

to (37) and finds that csA ^and cl’ ^are given by (38) ^and (39) ^{in that case too.}

We are now in a position ^to analyse ^concrete examples. ^{For this we have to} specify ^{the structure}

of the bonding units in order to obtain Uoo, U11’ U01

in explicit ^form.

4. Application ^to AX2 ^or AB X4 compounds.

Here we shall apply ^the general ^{results obtained in the} preceding ^{sections to} simple ^{cases such as} AX2 ^or

AB X4 compounds. ^{In these} compounds ^{the structure}

of the bonding ^{unit is reduced to its} simplest ^{form and}

is made up with only one ^atom named X. We shall consider that a sp basis is accurate enough ^{to describe}

Fig. ^2.

^{Case of a} bonding ^unit containing ^{one X atom.} px, p.,

and Pz (perpendicular ^{to the AXB} plane) ^are the three p atomic

orbitals on the X atom.

the valence electrons of thèse atoms. For each of them the operator gRR ^can thus be written

Ex ^and Ep ^{being the s and p levels of the X atom.}

The next step ^{is to determine} the effective interac- tions U, ^{i.e. the different terms} VOIX ^and Yla ^between Sp3

orbitals of the AB sublattice and the atomic states of their nearest X neighbour. ^{There are in} principal eight possible ^values Vos, Vis, VO., Viz, VOx, V1x, VOY5 Viy

in the notations of figure ^{2. From} purely geometrical

considerations all these interactions can be expressed

in terms of four distinct parameters, ^{which are}

The Pz ^states of the X atoms remain uncoupled ⁱⁿ

our model and will give ^{rise to an} atomic like flat band of non bonding character. The _px, py ^{states will} be coupled ^to the sp3 orbitals and will produce ^bond- ing ^and antibonding bands. To obtain these one must

inject (41) ^into (29) ^which gives for the effective inter- actions :

These expressions ^can be used in (34) ^{to calculate}

the band limits for which the first member of (34) ^is equal ^{to zero.} Then the allowed energy ranges ^are obtained from the inequality. After that for each of these limits one can compute ^the individual atomic s or

p characters when required, using (38) ^and (39).

To illustrate the method we shall make the further

assumption ^that Ex ^is sufficiently ^far apart ^from

E;, ^{Es ,} EA, E.B ^and Ep ^{to have a} negligible ^effect,

in which case the X « s » band will remain atomic- like. Such ^a simplification ^{seems to} be valid for all the

compounds described in the introduction and _espe-

cially ^for II-(VII)2 compounds ^like BeF2. ^{In what}

follows we shall then only retain the ^« _p ^» part ^of (42), keeping in mind that this is not an essential approxi-

mation (other ^{atomic states like «} d » orbitals could also be incorporated ⁱⁿ go (40) ^without any funda- mental difficulty). ^Its advantage ^{is that it leads to} analytical results which can be discussed much ^more

easily.

In the homopolar ^case Uoo ^becomes equal ^to Ui i

and the effective interactions are obtained from (42)

by replacing PA and by ^{their common value} fi, letting yA and _yB be zero. ’The _energy of the flat ^« p » bands given by equation (12) ^or the second _equa- tion (l4bis) ^{can now be written}

The equations for the broad band limits which derive from both equations (14bis) have solutions

given by (43a) ^{for the «} p ^» band limits and lead for the ^« s » band limits to :

Let us first look at the general ordering ^{of the band}

limits given by (43). They fall into two groups, all

bonding ^states being lower than E;, ^all antibonding

states having energies higher ^than EpX . ^Moreover,

from (43a ^and b), ^{one can} easily show that within each group the lower band limit is necessarily ^{a « s »}

one, the higher ^limit being ^a « p » ^one. Only ^the

intermediate _p and s limits can be inverted when the two fundamental parameters p ^{and 0 are varied.}

The density ^{of states} per AX2 ^{unit can} only present

two different shapes pictured ⁱⁿ figure 3 within the

bonding ^{bands as well as} within the antibonding ^bands.

On this figure ^{there is a} pure p flat band at all the _p band limits given by (43a) while all other band limits obtained from (43b) ^have pure ^{« s »} character on the A atoms. In all cases the total number of states per

AX2 ^{unit is} 10, ^the bonding group of bands containing

4 states, ^the antibonding ^group also. There remain 2 states per unit which correspond ^to pz orbitals perpen- dicular to the A X A plane ^{and which are} decoupled

Fig. ^3.

^{A schematic} representation ^{of the two} possibilities ^which

can occur for the bonding ^or antibonding ^states of AX2 systems.

Each broad or flat band contains 1 state per AX2 ^unit.

in our model. The filling of the bands can now be obtained from the fact that there are 12 electrons available. The conclusion is that, ^{in our} model, ^the AX2 compounds ^are always semiconductors or insu- lators, ^the highest valence band being ^a «Pz» ^non

bonding ^{band or lone} pair ^{band on the X atom} exactly

as in selenium or tellurium [17-20]. This conclusion

might be altered in a more refined model where the

« Pz» band could possibly be broadened sufficiently

to overlap ^the antibonding ^states. However this is very unlikely for all the cases we consider later, ^all the fundamental gaps being greater ^{than 2 eV.}

The shape of the bands when the parameter f ^is

varied is pictured ^on figure ^{4 in one} typical ^{case. The}

same is done versus 0 in figure ^{5. The bands are} very sensitive to this angle ^{and band} crossing ^{can occur}

for critical values of 0.

We now briefly discuss the ionic character of these

compounds. For this it is convenient to consider two

possibilities :

- E; EsA EA : let ^{us start} from the atomic limit p 0 (or d --+ oo) where the bands condense into the atomic levels. In this case all 12 electrons of the

AX2 unit will _occupy the _p states of the X atom.

This is thus the completely ionic limit where all the

Fig. ^4.

Broadening ^{of the bands versus} the interatomic distance for AX2 systems, ^{for a different} ordering of the atomic levels.

Fig. ^5.

Evolution of the bands versus the bond angle ^{2 0 for} AX2 systems.

electrons are on the more electronegative ^{X atom.}

The corresponding chemical formula would be

A4+(X2 -)2 ^for IV-(VI)2 compounds ^and A 2 ’(X-)2

for II-(VII)2 compounds. ^{In contrast the} strongly

covalent situation 1 f 1 --> ^{oo leads to} bonding ^states

whose population ^is equally shared between the A and X atoms. The formulae would then become

A°(X°)2 ^and A2-(X+)2 respectively.

E s A Ex Ep : the situation in the atomic limit now becomes less ionic than before. One has 2 s

electrons on the A atom. One can refer to this as the

partially ^ionic limit, ^{with formulae} A 2 ’ (X -)2 ^for IV-VI2 compounds ^and A 0(XO)2 ^for II-VII2 ^com- pounds. The covalent limit gives ^{the same result as}

before.

To end this section we briefly ^consider AB X4 compounds treated within the same model as AX2 systems. The second order equations giving ^{the band}

limits for AX2 compounds ^are changed into fourth order equations. The number of band limits is thus doubled which means that new gaps related ^to the AB

ionicity ^{are now} opened ^{as is} intuitively ^{evident. A}

typical example of this is given ^on figure ⁶ (which

corresponds ^to figure ^{4 for} AX2).

Fig. ^6.

Broadening of the bands for AB X4 systems.

5. Discussion.

We must now devise a systematic

method of choosing ^the parameters ^to be used in our

tight binding Hamiltonian. The intraatomic _para- meters are taken to be equal ^{to the free atom term}

values [25, 26]. ^This approach has been used with fair

success for AB tetrahedral compounds [27, 28] ^{as well}

as for Se and Te [19]. For the interatomic terms we

only keep nearest-neighbour interactions, ^i.e. Pss, Psa, Paa ^and P1C1C with standard notations. Their nume-

rical values are determined from the following simple

considerations :

-

The four fi parameters ^{must lead to the model} Hamiltonian of Leman, Friedel, Thorpe ^and

Weaire [11, 12] in covalent systems. ^{This is} achieved,

as shown in the Appendix, when these interactions ^can be expressed ^{in terms of one} parameter fl :

where fi ^{is the} interaction between two nearest-neigh-

bours sp3 hybrids pointing towards each other.

-

Assuming that relations (44) ^{hold true in all}

our systems, the p ^{values are} chosen for C, Si, Ge, ^Sn

as well as for Se and Te to give ^{the best} agreement with known band structures. Details of this are given

in the Appendix.

-

When plotting p ^versus (l/d)2 where d is the interatomic distance, ^{one finds} roughly ^a straight

line pictured ^on figure ^{7. We assume} that this is a

universal law. Such a procedure ^has been used in [29]

for tetrahedral systems.

Fig. ^7.

^Plot of 1/31 versus l/d2.

The values of the intraatomic parameters ^are given

in table I for all cases which we have considered. They

have been taken from refs. [25] ^or [26] (see 127, 28] ^for

more details). The values of the interatomic distance d,

bond angle ² 0, and interatomic resonance intégrais

are also given in table I.

Table 1.

Numerical values of ^the tight binding parameters.

To show the sort of agreement ^{we can obtain we} _give

on figure ⁸ the calculated values of the densities of valence band states for Si02 compared ^to photo-

emission results [30] (the calculation has been done for a Bethe lattice and the density ^{of states has been}

convoluted with a Gaussian of width 0.5 eV as is often done to compare with photoemission results).

It is clear that our results are essentially ^correct.

Furthermore figure ^{8 shows that the} partial ^{« s » and}

« p » densities of ^{states are} in good agreement ^with X-ray ^emission measurements [31]. ^The major ^dis-

crepancy ^comes from the fact that our valence bands

are too narrow, leading ^{to an} important gap in the valence band. However this is a well known deficiency

of our simple ^model [11] ^{which can be} easily ^remedied

for when adding further interactions. One must also notice that our model gives only ^{a crude} estimate of the fundamental _gap and a poor description of the conduc- tion band as in normal tetrahedral systems [11].

Fig. ^8.

^Predicted partial ^{and total} densities of states for Si02 compound compared ^to experimental X-ray emission and UPS spectra.

Let us now discuss the trends which we predict along the families of AX2 systems. ^{To allow} for some

classification we shall distinguish ^between moderately

ionic compounds (EA EpX EpA ) ^{and strongly}

ionic compounds (Ep ^Es E A with ^{one class of}

intermediate systems (Ex , p E s EpA ).

5.1 MODERATELY IONIC SYSTEMS EsA Ex E A

-

Here X atoms are S or Se and A atoms Si or Ge.

These chalcogenide compounds ^{exist in different} crystalline ^{forms and also in the} amorphous ^state [21].

One can distinguish ^{between two} différent situations :

la) ^Chain structure : this is the normal form of

SiS2 ^and SiSe2. Here tetrahedra are sharing edges.

Thus the bonding angle is about 80°. Results for SiS2

are given ^on figure ^9.

lb) Three-dimensional network : this concems GeS2

and GeSe2 ⁱⁿ crystalline ^{as well as in} amorphous ^form.

The difficulty here is that there is a distribution of bond

angles around 90°. We give ^results only ^for GeSe2 (GeS2 ^is similar) ^{and for 2 0 = 90°. Variations}

in bond angles ² 0 will result in a broadening ^{of these}

bands but will not essentially alter the results.

Both SiS2 ^and GeS2 ^{have the same} structure at high

pressures, ^a compressed cristobalite like structure [21]

in which the 0 bond angle ^{is close to the} regular ^tetra-

hedral value ( = 109°). This will in fact broaden the bands with respect ^to the situation where 2 0 = 80°

but will not change ^their qualitative ^nature.

5.2 STRONGLY IONIC SYSTEM Ep ^EsA.

^This

class essentially corresponds ^to II-VII2 compounds.

Here too there exist different structures which fall into three _{groups :}

2a) ^Chain structure : BeC’21 BeBr2 ^and BeI2 belong

to this _group [22]. ^{As for} SiS2 ^{the bond} angle ^{is near}

80°. We have explicitely considered BeC’2. Figure ⁹

shows that we are now in a limiting ^case of nearly ^band crossing for the valence band states.

2b) Layer structure : the most representative ^com- pound ^{of this} group is red HgI2 [21]. It also contains

ZnC’21 ZnBr2 ^and ZnI2. ^{Here 2 0 is} of the order of 103°. We have chosen ^as representative example ZnCl2. Figure ^{9 shows} that for this _group the results are

essentially equivalent ^to those found for the chain structure 2a).

2c) Three-dimensional network : BeF2 ^{is the most}

well known compound ^{of this} group. It is formed from

BeF4 tetrahedra linked through vertices and _may have cristobalite, _quartz ^or silica-like structure, ^{with a bond}

angle ^{20 = 155°} [22]. ^{Here, as shown on} figure 9, there is no ambiguity in the band structure which is,

as we shall see, qualitatively ^{similar to what is obtained}

for quartz.

5.3 Si02 ^AND Ge02. - Although Si02 is ^the

more representative of AX2 compounds with 4-2 local

coordination, ^{it can with} Ge02 ^be thought ^{as a}

bordeline material between the two previous ^classes.

Here the _{oxygen p} level Ep ^is practically equal ^to Es’

or EsGe.

The chain form of Si02 (2 0 = 75°) gives ^results

similar to SiS2 (Fig. 9). All other forms are three- dimensional with bond angles ^{2 0} varying ^{from 140°}

to 160° for crystals ^{and within a} range of 40° around an

Fig. ^9.

Predicted band limits for families of compounds. The shaded regions correspond ^{to filled} bands. Pure p flat bands ^are drawn.

The arrows indicate the fundamental gap. The numbers give ^{the s or} p character ^on the A atom at the corresponding ^band limits, ^for AX2 compounds.

average value of 150° for amorphous materials. In all

cases the results will be qualitatively ^{similar to those}

obtained for quartz (Fig. 9) ^with hybridized ^bands.

As demonstrated on figure 8 this band structure is in very good agreement ^with experiment. ^{The same is}

true for Ge02 ^{and our model} predicts correctly ^the

essential features of the evolution in band structure when going ^from S’02 ^to Ge02 [8].

AB X4 compounds : ^{Here too one can} distinguish

among différent cases according ^{to the} ionicity.

However the detailed ^structure of these compounds

is still quite badly ^known. Among ^them AIP04 ^has

been the most studied. It may have cristobalite, quartz ^or silica like structures. The bond angle ^is always of the order of 150°. The corresponding ^band

structure is given ^on figure ^{9. With} respect ^to Si02

new gaps open within the valence bands.

We shall end this section by ^an overall discussion of the ionic character of thèse compounds. ^On figure ⁹

we give, ^{for all} AX2 compounds, ^the numerical value

of cs and cp ^{for the} corresponding ^{valence band limits.}

This directly reflects the corresponding contribution

to the population ^{on the A atom. It} appears clearly

that when going from class 1) ^{to class} 3) ^and finally ^to

class 2) the electron population ^{on the A atom tends to}

decrease which reflects the increase in ionic character of the bonds. From this one can have directly ^a rough

idea of this population (an ^exact value would require

numerical integration). For instance cg and cp ⁱⁿ

BeCl2 ^{have an} average value of about 0.23. The elec- tron population ^{on the Be atom} will thus be equal ^to

0.23 x 8 ⁼ 1.84 (which ^{is much} less than 4, ^the

number one would obtain in the purely ^covalent limit) ^{but leads to a} practically neutral Be atom in that system. ^A similar calculation for Si02 ^quartz gives

about 2.08 electrons _per Si atom. This is slightly

more covalent than BeCl2 ^{but leads to} the formulae

Si2+(O-)2 ^{which now indicates a} strong ^electron transfer. Finally ⁱⁿ category 1) ^i.e. SiS2 ^{one now}

finds that the Si atom has 3.2 electrons a still more

covalent situation leading ^to the formula

Thus, ^as expected, ^our classification is related to the

ionicity of the bond ^or the difference in electron

population, ^{but not to the} charge ^transfer.

6. Conclusion.

We have applied ^a tight binding

Hamiltonian to study the trends of the electronic structure along families of 4-2 coordinated defect tetrahedral compounds. This Hamiltonian is similar.

in nature to the one used by Leman, Friedel, Thorpe

and Weaire for tetrahedral systems [10-16]. ^{It has}

allowed us to derive general ^theorems relating energy

gaps occurring ^{in these} compounds ^{to the short} range atomic configurations. ^Another interesting ^{feature of}

this description ^{is that we} have been able to derive closed analytic expressions ^{for the « s » or} «p»

character of the eigenstates ^{on the} tetrahedrally

coordinated atoms. All these quantities ^{have been}

calculated numerically using ^a systematic ^{method to}

determine the parameters.

We find that ^a first important parameter ^{for the} electronic properties of AX2 compounds ^{is the} posi-

tion of the _p atomic level of the X atom compared

to the s level of the A atom. This largely determines the ionic character of the bands. Another fundamental parameter is furnished by ^{the bond} angle ^{2 0 to which}

the band structure is extremely sensitive, especially

the ordering ^{and nature} of the subbands.

We consider this work as a preliminary step ^{to a} classification of the electronic properties ^{of this} family

of compounds. ^Our predicted ^{valence band structures}

are in agreement ^with experimental information for

compounds where these exist. A natural extension is thus to study other electronic properties, ^{such as}

dielectric susceptibilities, ^and classify ^{them with an} ionicity parameter ^{which can} be defined from ^our Hamiltonian. Another _very important ^{field of} appli-

cation of this study corresponds ^to SiO,, compounds

or the Si-SiO2 interface where preliminary calculations

along ^the lines of this model show interesting similarity

with experimental spectra. Refinements of the model

are also being currently investigated ^{in order to} remedy

to some of its deficiencies while keeping its attractive

simplicity.

Appendix.

^We apply ^{here our} simple ^model

Hamiltonian to the two following classes of semi- conductors : the _group IV elements and the _group VI elements Se and Te.

1. COVALENT SEMICONDUCTORS C, Si, Ge, ^Sn.

Here our model Hamiltonian reduces to the Leman, Friedel, Thorpe and Weaire Hamiltonian [10-16],

for which, apart from the atomic levels Es ^and Ep ^there

exists only ^one interatomic term fl connecting ^two nearest-neighbours sp’ hybrids belonging ^{to the}

Table II.

Parameters for ^covalent systems.

same bond. This term can be written in terms of Pss, P sa’ P aa ^and P xx

If we impose that all other interatomic terms vanish, then all other independent combinations of the Pa./J

must also vanish which leads to the following ^rela-

tions :

In the following ^{we assume these relations to be}

fulfilled. Then the only parameter ^to ohe determined

is fi which will give ^{us all the} Pij ^{with the help of} (A. 2). ^{As was done before we take for ES and} Ep

the atomic values and determine fi through ^{a fit of the}

direct energy gap Ed ^{which in our model} tums out to be

given by 2 1 P 1 - (Ep - Es). Corresponding ^{values of} Ed, Ep - ^{ES and fi are given} in table II.

2. LONE PAIR SEMICONDUCTORS : Se, ^Te.

^{In this}

case the band structure can also be quite easily

described. One starts from the s and _p atomic levels

separated by about 10 eV. This promotion energy

being quite substantial there is practically ^no hybri-

dization between s and _p states. Further, ^{in a nearest-} neighbour approximation, ^these systems ^{consist of} linear chains. In this ^case the s band simply ^broadens by 4 1 Pss 1. ^The problem ^{of the p band can be consi-}

dered as follows. On each atom one can again, ^{as in} AX2 systems, ^{build a} p, orbital perpendicular ^{to the} plane containing ^{this atom and its two nearest} neigh-

bours. This orbital will be practically uncoupled (this is detailed in ref. [19]) ^and give ^a practically ^flat

lone pair ^{band. As} regards ^{the two other} p orbitals,

one can build from them two equivalent ^orbitals approximately pointing towards the nearest-neigh-

bours in the chain. One can then take bonding ^and antibonding combinations of these orbitals. It is not difficult to show (ref. [19]) that this leads to p bonding

and antibonding bands at average energies - + 1 Ptla I.

Now in Se and Te the average energy separating

the bonding ^and antibonding ^{bands is} respectively ⁸

and 6 eV. This leads to 1 Paa 1 ^{4 and 3 eV. If one}

assumes relations (A. 2) ^{to be} fulfilled, this gives p - 5.33 eV and 4 eV. A test of the validity ^{of our} procedure ^is provided by the width of the s band.

This is equal ^{to 4} 1 Pss 1 = 1 P 1 ^from (A. 2). Experi-

mental results show that this is indeed a correct order

of magnitude.

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