Post on 03-Apr-2018
7/29/2019 Electronic density of states.docx
1/16
1
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
7/29/2019 Electronic density of states.docx
2/16
2
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:
7/29/2019 Electronic density of states.docx
3/16
3
(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
7/29/2019 Electronic density of states.docx
4/16
4
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
,
7/29/2019 Electronic density of states.docx
5/16
5
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
7/29/2019 Electronic density of states.docx
6/16
6
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
7/29/2019 Electronic density of states.docx
7/16
7
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.
7/29/2019 Electronic density of states.docx
8/16
8
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.
7/29/2019 Electronic density of states.docx
9/16
9
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).
7/29/2019 Electronic density of states.docx
10/16
10
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.
7/29/2019 Electronic density of states.docx
11/16
11
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 ,,
7/29/2019 Electronic density of states.docx
12/16
12
(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
7/29/2019 Electronic density of states.docx
13/16
13
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).
7/29/2019 Electronic density of states.docx
14/16
14
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
7/29/2019 Electronic density of states.docx
15/16
15
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
7/29/2019 Electronic density of states.docx
16/16
16