electronic density of states.docx

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Electronic density of states

It has been a great triumph of the application of quantum mechanics to solid-

state physics that an understanding of why certain crystals are metals and others

are insulators has been achieved. The presence of perfect periodicity greatlysimplifies the mathematical treatment of the behaviour of electrons in a solid.

The electron states in this case can be written as 'Bloch waves' extending

throughout the crystal:

(5.1)

where the function u(k, r) has the periodicity of the crystal lattice (in which a

lattice translation vector R1 connects lattice points):

(5.2)

and this modulates the term exp (ik r) representing a plane wave. The allowed

wavevectors k of the electrons are intimately related to the symmetry of the

underlying crystal lattice since a reciprocal lattice (related to the unit cell

parameters) can be established in reciprocal or A-space.

The allowed energies of the electrons can thus be represented by means of

a 'band structure' in k-space. A free electron has an energy E(k)=2

k2

/2m, but thisparabolic dependence is distorted considerably if the electron experiences a

scattering potential.

Of course for a crystalline solid, these potentials arise from the periodic array of

atom centres, and the electron waves can Bragg reflect from the lattice planes.

This results in energy gaps opening up at values of k corresponding to certain

values of reciprocal lattice vector Bn/a for a linear array of atoms of separation a),

i.e. at the edge of the 'Brillouin zone'. The occurrence of energy ranges for which

there are no allowed electron states can be thought of as being due to destructive

interference of the Bragg reflected electron waves; in the 'nearly-free electron'

model, the size of the energy gap is determined by the Fourier component of the

potential corresponding to the Bragg condition (see e.g. Madelung 1978). A

simple one-dimensional band structure in the extended zone scheme is shown in

rikrkurk .exp,,

rkuRrku ,, 1


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Fig. 5.1. The difference between metals and insulators then simply amounts to

whether there are sufficient electrons available to fill all the states in a Brillouin

zone; if the band is only partly filled the solid is a metal, whereas if the band is

completely filled, there is a gap between occupied and unoccupied electron

states, and the material is an insulator at T=0.

Let us now consider the case of amorphous materials. The occurrence of

band gaps can be viewed as arising from Bragg reflection of electron waves and

hence is a direct consequence of periodicity, the question therefore arises: Should

band gaps occur in amorphous materials too ? The fact that window-glass (silica)

is transparent to visible light is direct experimental proof that a band gap ~2eV

must exist for this material (in fact the gap is nearer 10 eV).

Conventional (crystalline) solid-state physics theory is incapable of accounting for

this behaviour, and we shall see that concepts more akin to those of chemistry

can resolve the dilemma. Before we consider this matter further, it is perhaps

pertinent to discuss another consequence of the lack of long-range order on the

description of electron states in an amorphous solid.

Fig. 5.1 One-dimensional band structure in the extended zone scheme.

We have seen that the absence of periodicity in an amorphous solid dictates that

there can be no reciprocal space. In this case, electron states cannot be

represented by a band structure in the form E(k). The quantity that is equally valid

as a description of electron states for both crystalline and amorphous solids is, as

in the phonon case, the density of states. This can be written in the form:


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(5.3)

where g(E) is the density of states per unit volume per unit energy interval, and V

is the volume of the system. The quantity more often used is the integrated

density of states:

(5.4)

We can now address the problem of whether a gap can exist in the density of

states of an amorphous covalently bonded material. Weaire and Thorpe A971)

first showed that if short-range interactions between electrons are dominant,

then it is the short-range order which mainly determines the electronic density of

states.

In particular they showed using a simple model Hamiltonian for the electron

interactions that a gap is expected for an ideal tetrahedrally coordinated

amorphous solid, providing that the interactions are of a certain magnitude. The

amorphous structure is taken to be that in which each atom is in a perfect

tetrahedral environment, with presumably a wide distribution of dihedral angles

necessary to generate a random network. (Whether in practice a CRN could be

constructed without bond-angle distortions remains unclear, but this assumption

simplifies the treatment.) The alternative approach to the nearly-free electronapproximation, namely the tight-binding LCAO approach, is adopted, in which the

basis functions are localized at each atomic site rather than being extended plane

(Bloch) waves. The two interactions considered in the model are an intrasite

'banding' interaction V1 responsible for the width of the bands, and an intersite

'bonding' interaction V2, responsible for the separation of bonding and

antibonding bands; they are shown

schematically in Fig. 5.2, where the basis functions are taken to be sp3 hybridized

orbitals localized at each site. The Hamiltonian is thus written as:

(5.5)

All site orbitals are assumed to be orthogonal, and V1 and V2 are assumed to be

the same for all atoms of the network.

n

nEEV

Eg 1

dEEgEN

0

iiVijkiVHiij

21


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Fig. 5.2 Interactions and basis functions in the Weaire-Thorpe Hamiltonian.

By consideration of an isolated atom, it can be shown that V1 must be negative

(so that the s-states are lower in energy than the p- states), and V2 must also be

negative (so that bonding orbitals are lower in energy than antibonding orbitals).

Instead of using the states |i> as a basis, we can equally use bonding (B) and

antibonding (A) orbitals associated with pairs of neighbouring atoms:

(5.6)

If the assumption is made that the valence band is solely constructed from

bonding orbitals, i.e. , then the expectation value of the energy for

this state is:

(5.7)

where the sum is over all pairs of bonds. Contributions of V2 are obtained from

the terms for which i = I, and V1/2 from terms for which = ', i I giving:

(5.8)

It can be shown (see problem 5.1) that the valence band limits, i.e. E, must lie

between (V2 V1) (when each ibi=0) and (V2 + 3V1) (when each bi = bi ).

iiiA

iiiB

2

1,

2

1,

iBHiBbbHE

ii ,'',*

''

iBbbonds i

,


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Similarly, by assuming the conduction band is constructed solely from antibonding

orbitals, the band limits are found to be (- V2 + 3V1) and ( V2 + V1).

Thus, if |V2| > 2|V1|, there is no overlap between the bands, i.e. a true band gap

of magnitude

(5.9)

must exist. Note that this model is concerned only with the short-range structure

and says nothing at all about longer range structure, i.e. the results obtained hold

for amorphous and crystalline tetrahedral systems alike, irrespective of the

presence of periodicity. This is in accord with the chemist's view of covalent

bonding; there are four states per atom in the valence (and conduction) band,

and hence all bonds are satisfied in the valence band.

Although providing an 'existence theorem' for a gap in the density of states of an

amorphous semiconductor, the model is too simple to be expected to give

quantitative results.

As an example, the band structure calculated for diamond cubic Ge using the

Weaire-Thorpe Hamiltonian is compared in Fig. 5.3 with a more sophisticated

pseudopotential calculation; also shown is the density of states obtained from

such band structures. Agreement is seen to be qualitative for the valence band,

but very poor for the conduction band. Note that the Weaire-Thorpe model gives

a delta function in the density of states at the top of the valence band (resulting,

in the crystalline case, from the flat band in the band structure at the same

position), which contains pure p-like bonding states. This region is relatively

insensitive to the detailed structure (although the presence of like-atom bonds in

an alloy does affect it); in contrast, as we shall see later, the deeper-lying states in

the valence band density of states (mainly s-like for the deepest band and mixed

s- and p- like for the intermediate band) are very sensitive to structural variations.

In order to improve on these calculations for amorphous solids, a more realistic

Hamiltonian is required, which remedies at least two deficiencies, namely the lack

of the inclusion of longer range interactions and the absence of variations in the

12 22 VVEg


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interactions arising from fluctuations in the structure, e.g. bond-angle distortions

(see Yonezawa and Cohen 1981).

Fig. 5.3 (a) Band structure of crystalline Ge (diamond cubic structure) calculated using

pseudopotential theory. (b) (top) The density of states for this band structure (the zero of

energy marks the Fermi level). The bottom figure in (b) is the density of states calculated

using the Weaire-Thorpe Hamiltonian (Weaire et al. 1972).

1 Theoretical calculations

We have already seen that the allowed electron states for a

tetrahedrally bonded amorphous solid can be determined by using the Weaire-

Thorpe tight-binding Hamiltonian [5.5]. This is a very simple model, however, and

considerable effort has been expended in attempts to improve on this method

and to obtain more realistic estimates for the density of states of amorphous

solids.

Improvements can be made in two areas:

(a) A more realistic Hamiltonian can be used, involving more interactions than just

the two inter- and intrasite terms considered by Weaire and Thorpe (Fig. 5.2), inparticular including interactions involving more distant neighbours (see e.g.

Bullett and Kelly 1979).

(b) Topological disorder can be introduced quantitatively by considering a more

realistic structural model than that used by Weaire and Thorpe, which assumed


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perfect nearest-neighbour tetrahedral order, but did not address the problem of

longer range structure (since only short-range interactions were included).

Although this generality of structure is a feature of the Weaire- Thorpe model,

giving rise to an energy gap whatever the structure (if |V2|>2|V1|), it is

intuitively obvious that details of the structure, such as ring statistics for a

covalent solid, would be expected to influence the detailed shape of the density

of states. Thus the atomic coordinates from a structural model which fits well

experimental scattering data, such as a continuous random network (for a

covalent solid) or a dense random packed model, are a better starting point.

In this manner, quantitative disorder in the form of variations of the value of

interactions (e.g. V1 and V2) are naturally included as a result of the presence of

bond-angle and dihedral-angle variations for a CRN, or packing variations for aDRP model.

The use of large structural models in density of states calculations,

however, necessitates the employment of sophisticated numerical techniques in

order to cope with the diagonalization of the large matrices involved; a general

review of these techniques can be found in Kramer and Weaire (1979). Essentially

the same methods can be employed as are used in the vibrational density of

states calculations. The Lanczos method at the heart of the negative eigenvaluemethod and the equation-of-motion method for example, have both been used

in this regard. More widely used, however, are two equally successful techniques

which calculate the local density of states of a cluster , rather than effect the

diagonalization of large matrices corresponding to the whole cluster. These are

the 'recursion' method and the 'cluster-Bethe lattice' method.

The recursion scheme to calculate the local density of states

commences with a choice of |u0>, the orbital of interest (e.g. an sp3 hybrid for a

tetrahedral solid), together with an appropriate Hamiltonian. If the starting vector

is constructed from an equal contribution from each orbital in the system, but

with a random phase factor exp (i) for 0 < i < 2, a good approximation to the

total density of states is obtained since the starting orbital picks up an equal

contribution to the spectrum from each distinct energy.


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The local density of states is related to the diagonal Green function matrix

element n(E,m)= - 1/ Im

where |m> = |u0> in the recursion scheme. The average over a unit cell of a

crystalline local density of states is proportional to the total density of states; foran amorphous solid, on the other hand, an average should be performed over as

many atomic sites as possible since each n(E) reflects its own particular

environment, although 20 sites seem to be sufficient in practice. Boundary

effects are minimized by choosing sites near the centre of the cluster since,

although interactions with distant sites are included in the continued fraction,

their contribution becomes negligibly small if the cluster is large enough (say, a

few hundred atoms). Thus, there is no need for structural models with periodic

boundary conditions (e.g. the Henderson model) as required by those methodsmore commonly used for crystalline systems (e.g. using pseudopotentials).

As an example of the recursion method, we show in Fig. 5.4 calculations of

the valence-band densities of states for silicon in various structural forms. The

two crystalline forms are the diamond lattice and the ST-12 lattice (so named

because it has a simple tetragonal unit cell containing 12 atoms), which differ in

two respects; the diamond lattice is formed from sixfold rings with each atom

having the same tetrahedral bond angle, whereas the higher density ST-12structure has a variety of ring sizes, even and odd (the smallest being five-

membered) with a spectrum of bond angles ~25% about the tetrahedral value,

109 28'. The model amorphous structures employed are the Polk-Boudreaux

(even-odd ring) model and the Connell-Temkin (even ring) model.


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Fig. 5.4 Densities of states calculated using the recursion method for various

crystalline modifications and amorphous models of Si (Kelly 1980): (a) diamondcubic; (b) ST12 crystalline modification; (c) Polk-Boudreaux CRN; (d) Connell-

Temkin even-ring CRN. The zero of energy is self-energy of an isolated bond.

Fig. 5.5 Schematic density of states for tetrahedrally bonded semiconductors, (a)

homopolar and (b) heteropolar, in both the amorphous (dashed line) and

crystalline (solid line) phases (after Joannopoulos and Cohen 1976).


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The first point to note is that the local densities of states of all these structural

forms are qualitatively similar, supporting Weaire and Thorpe's contention that it

is the short-range order, i.e. the tetrahedral coordination, which determines the

gross features. However, there are distinct differences, despite the fact that the

same model Hamiltonian was used in each case, which must arise therefore from

topological differences between the various structural forms. In particular, the

amorphous forms are differentiated from the diamond cubic form by a filling-in of

the dip between s-states (at Xx) and a distinct skewing of the p-state distributions

at the top of the valence band towards the gap. These trends are also observed

experimentally in photoemission studies of Ge and Si (see later) and are shown

schematically in Fig. 5.5, lending credence to the validity of the theoretical

calculations and suggesting that odd-membered rings must be present in the

amorphous phase. Note however from Fig. 5.5(b) that these trends are different

for the case of heteropolar systems (e.g. GaAs), principally because of the

difference in ionicity of the anion and cation and the consequent avoidance of

like-atom bonds and hence odd-membered rings.

The other method commonly used to obtain the local density of states is

the 'cluster-Bethe-lattice' method - for a review of this approach see

Joannopoulos and Cohen A976) and Joannopoulos A979). In this, a small

symmetrical cluster (containing a few tens of atoms) has Bethe lattices (or Cayley

trees) attached to the surface dangling bonds; the Bethe lattice is characterized

by having the same connectivity as the host network, but contains no closed rings

of atoms.

The method is shown schematically in Fig. 4.4, and has the following advantages:

(a)The density of states of a Bethe lattice can be calculated exactly for avariety of model Hamiltonians, and furthermore generally yields a smooth

and feature- featureless spectrum.

(b) The local density of states of the atom at the centre of the cluster in a given

environment (say in a fivefold ring) can be calculated analytically, without

attendant problems of boundary effects, because of the attached Bethe lattices

which simulate the rest of the network.


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Experimental determination

Photoemission

The technique most widely used to explore the valence and conduction

band densities of states is photoemission; as its name implies, this measures the

energy distribution of photo-emitted electrons as a function of incident photon

energies. The commonest photon sources are the UV line spectra of rare gas

discharge lamps, or X-ray anodes; these give rise, respectively, to two

conventional branches of photoemission, namely 'ultra-violet photoemission

spectroscopy' (UPS) and 'X-ray photoemission spectroscopy' (XPS). The UV photon

energy range is 10-50 eV, whereas the K X-ray lines commonly employed are

1486.6 eV (Al) and 1253.6 eV (Mg), although this gap can be bridged by the use of

synchrotron radiation, rendering the difference between the two techniques less

distinct. The photo-emitted electrons can be energy analysed either by a

retarding grid analyser or a dispersive electrostatic analyser. The experiments

have to be conducted in ultra-high vacuum to minimize surface contamination,

since the escape depth of the photo-electrons is very small (-5 for UPS and -50

for XPS).

The photoemission process can be understood in terms of the 'three-step

model', namely: A) optical excitation of an electron; B) its transport through the

solid (including the possibility of inelastic scattering by other electrons; and C) the

escape through the sample surface into the vacuum, although this approach

drastically approximates the many-body processes that take place. The 'energy

distribution curves' (EDCs) of the photo-emitted electrons are given by :

where P(E, ) represents the distribution of photo-electrons of energy E

excited by a photon of energy , T(E) is a transmission function (weakly and

smoothly varying with E), and D(E) is an escape function also a smooth function of

E. Thus it is only P(E,) which contributes structure to the EDCs, and this may be

written, when k-conservation is not important, as is the case for amorphous

materials, as:

EDETEpEI ,,


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(5.15)

where Nc andNv are the conduction and valence band densities of states,

respectively, and M is a matrix element which may be taken to be constant for

limited photon energy ranges. Thus we see that the photoemission EDCs aredetermined by a joint density of states involving a convolution of occupied and

unoccupied electron states.

The origin of P(E,) is illustrated schematically in Fig. 5.7, where the features in

the EDCs can be traced back to structure in the densities of states of the bands.

The valence band density of states has been raised by hco with respect to the

conduction band to account for the term Nv (E ) and this is then convoluted

with NC(E) and T(E) to give P(E,) (assuming D(E) and M(E, E-) to be constant).

It can be seen from Fig. 5.7 that features in the EDCs obtained from UPS which do

shift in position when is varied are to be associated with peaks in the valence

band density of states, whereas conversely those features which do not shift in

position are associated with maxima in the conduction band density of states,

although conduction band states below the vacuum level of the semiconductor

are inaccessible to the technique and can only be rendered accessible by means

of a layer of caesium evaporated on to the surface to lower the work function. In

this way the valence band and conduction band densities of states can be

extracted independently from the EDCs. An illustration of this is given in Fig.

5.8(a) for the case of amorphous and crystalline Ge (Spicer 1974), where it is seen

that the EDCs for both materials alter dramatically (in different ways) upon

varying the UV photon energy. The densities of states deduced from these spectra

are shown in Fig. 5.8(b) using [5.15] for the amorphous case (although not for c-

Ge, where k is conserved and a different equation must be employed). If photon

energies greater than ~ 20 eV are employed, then the intensity modulation by the

final (conduction band) density of states rapidly becomes unimportant as the final

state N(E) approaches its featureless free-electron dependence E1/2. Thus soft X-

ray photoemission probes only the valence band density of states, and the EDC is

a direct reflection of this, modulated by slowly varying photo-ionization cross-

sections.

EEMENENEp vc ,,2


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Fig. 5.6 Comparison of densities of states for amorphous (solid line) and

crystalline thombohedral (dashed line) As. Experimental data are from XPS (Ley

et al 1973), and the theoretical curves are calculated using the recursion method

(see Kelly 1980) and the cluster-Bethe-lattice method (CBLM) (Pollard and

Joannopoulos 1978a).

Fig. 5.7 Schematic illustration of the origin of the features which are observed in

photoemission EDCs, P(E, ) (after Mott and Davis 1979).


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An example of the valence band density of states obtained from XPS data has

already been given in Fig. 5.6 for the case of amorphous and crystalline As. Higher

energy X-rays cause excitation from deep- lying core states, rather than the

valence band, and produce narrow peaks in the EDCs at kinetic energies below

the valence band spectra. The number of core levels, and their binding energies,

are characteristic of a given element, and moreover the exact value of binding

energy measured depends on the chemical environment of the element. This

phenomenon forms the basis of the technique 'electron spectroscopy for

chemical analysis' (ESCA) which is really only a surface probe, however, because

of the short sampling depth 10-60 ) of the XPS method. ESCA has been used to

determine whether 'wrong', i.e. homopolar, bonds exist in amorphous semi-

semiconductors, e.g. in the III-V compounds, InP, GaAs, etc., but the results are

somewhat inconclusive.

Ultra-violet and X-ray absorption

These techniques are analogous, and in a sense complementary, to

photoemission. The absorption (or reflectivity) associated with electronic

transitions from filled valence states to empty conduction states produced by

photons having energies greater than the band-gap energy will be discussed here;

excitation involving photon energies less than or equal to the band-gap energy,which probe states near the band edges of amorphous semiconductors, will be

left until later. The optical properties (for both UV and X-ray excitation) of

amorphous and crystalline semiconductors are almost entirely determined by the

imaginary part of the dielectric constant, where:

(5.16)

and 2() is given in a one-electron expression (see, e.g. Connell 1979) as:

(5.17)

where V is the sample volume, P is the momentum operator, and the summations

are over all initial valence states |i> and final conduction states |f>.

21 i

if

i f

EEipfm

ev

2

2

222


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The wavefunctions |i> and |f> of the valence and conduction bands respectively,

can be expanded in terms of a set of orthonormal, localized wavefunctions |nv>

and |nc> centred on different atoms n. Thus

(5.18a)

(5.18b)

For a crystal, the a's are plane waves and |nv> and |nc> are Wannier functions.

For an amorphous solid however, writing aAQlS, the phases 5 vary randomly

from site to site if the uncertainty in k is of the order of the wavevector itself, 8k ~

k (for strong scattering); the amplitude A may also vary randomly from atom to

atom, although being of the same magnitude for extended states. This 'random-

phase approximation' (RPA) is an embodiment of the rule introduced by Ioffe andRegel that for an electron mean free path l, kl < 1 is impossible, i.e. when the

wavefunction loses phase memory from atom to atom, the mean free path

cannot be less than the interatomic spacing. The RPA is not in general a realistic

model since it does not take account of the considerable degree of short-range

order exhibited by amorphous semiconductors, although it is a useful starting

point. The momentum matrix element, averaged over an ensemble of random

systems, can be evaluated for transitions between extended (delocalized) states:

Fig. 5.8 Photoemission results for amorphous and crystalline Ge (Spicer 1974): (a) EDCs for c-

Ge and a-Ge for different photon energies; (b) density of states for valence and conduction

bands derived from the EDCs in (a). The profile for the conduction band for a-Ge is only

approximate.

n

ninv

ncaf

nvai

inf


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