Transcript Electronic structure

Electronic
structure

Important question: Why certain materials
are metals and others are insulators?

The presence of perfect periodicity greatly
simplifies the mathematical treatment of
the behaviour of electrons in a solid. The
electron states can be written as Block
waves extending throughout the crystal:
φ(k,r) = u(k,r) exp (ikr)
where u(k,r) has the periodicity of the
crystal lattice
u(k,r)=u(k,r+R)
(R is lattice translation vector.),
and term exp(ikr) represents a plane wave.
The allowed wavevectors k of
the electrons are related to the
symmetry of lattice.
Since that a reciprocal lattice
related to the unit cell parameters
can be established in k-space.
First Brillouin zone of FCC lattice
showing symmetry labels
Electron density of states of c-Si
Indirect semiconductor
Amorphous materials?
There is no periodicity!
 Hence there can be no reciprocal
k-space. No k vector.
 The electrons can not be represented as
Block states.

Should band gap occur in amorphous
materials?
Yes
What is the definition of
semiconductors?
1. Conductivity?
Conductivity is between metals and
insulators?
2. Gap size?
It has a gap of 1 – 2 eV?
3. Or?
As the temperature of a
semiconductor rises above
absolute zero, there is more
energy to spend on lattice
vibration and on lifting some
electrons into an energy
states of the conduction
band.
 Electrons excited to the
conduction band leave
behind electron holes in the
valence band.
 Both the conduction band
electrons and the valence
band holes contribute to
electrical conductivity.

-
+
Most common definition
The temperature dependence of resistivity at low
temperature:
ρ = ρ0 exp(ε0/kB T )
T increasing, ρ decreasing
(In metal case:
T increasing, ρ increasing!)
Electronic
structure
Covalent bonding
Amorphous semiconductors are
typically covalently bonded materials.
sp3 hybrids
 Hybridisation
describes the bonding atoms
from an atom's point of view. A tetrahedrally
coordinated carbon (e.g., methane, CH4), the
carbon should have 4 orbitals with the correct
symmetry to bond to the 4 hydrogen atoms.
 The problem with the existence of methane is
now this: carbon's ground-state configuration
is 1s2, 2s2, 2px1, 2py1
Ground state orbitals cannot be used
for bonding in CH4. While exciting 2s
electrons into a 2p orbitals would, in
theory, allow for four bonds according
to the valence bond theory, this
would imply that the various bonds of
CH4 would have differing energies
due to differing levels of orbital
overlap. This has been experimentally
disproved.

The solution is a linear combination of the
s and p wave functions, known as a
hybridized orbital. In the case of carbon
attempting to bond with four hydrogens,
four orbitals are required. Therefore, the
2s orbital "mixes" with the three 2p
orbitals to form four sp3 hybrids
becomes
3
sp
orbitals
 1.
sp3 = ½ s - ½ px - ½ py + ½ pz
 2. sp3 = ½ s - ½ px + ½ py - ½ pz
 3. sp3 = ½ s + ½ px - ½ py - ½ pz
 4. sp3 = ½ s + ½ px + ½ py + ½ pz
Linear Combination of Atomic Orbitals
Scalar product:
(n.sp3; m.sp3) = 0
3
sp
In CH4, four sp3 hybridised
orbitals
are
overlapped
by
hydrogen's 1s orbital, yielding four
σ (sigma) bonds (that is, four
single covalent bonds). The four
bonds are of the same length and
strength. This theory fits the
requirements.
CH4
2
sp
hybrids
For example, ethene (C2H4). Ethene has
a double bond between the carbons.
 For this molecule, carbon will sp2
hybridise, because one π (pi) bond is
required for the double bond between the
carbons, and only three σ bonds are
formed per carbon atom. In sp2
hybridisation the 2s orbital is mixed with
only two of the three available 2p orbitals.
2
sp

hybrids
In ethylene (ethene) the two carbon atoms form
a σ bond by overlapping two sp2 orbitals and
each carbon atom forms two covalent bonds
with hydrogen by s–sp2 overlap all with 120°
angles. The π bond between the carbon atoms
perpendicular to the molecular plane is formed
by 2p–2p overlap. The hydrogen-carbon bonds
are all of equal strength and length, which
agrees with experimental data.
2
sp
orbitals
1. sp2 = (1/3)½ s + (2/3)½ px
 2. sp2 = (1/3)½ s - (1/6)½ px + (1/2)½ py
 3. sp2 = (1/3)½ s - (1/6)½ px - (1/2)½ py

Linear Combination of Atomic Orbitals
Scalar product:
(n.sp2; m.sp2) = 0
2
sp
sp hybrid
In C2H2 molecule. Only two sigma bonds:
1. sp3 = (1/2)½ s - (1/2)½ px
2. sp3 = (1/2)½ s + (1/2)½px
IV. Column materials
VI. Column materials
(2s4p electrons =>
2s+2 sigma bond +2 lone pair )
Atomic charges
In crystalline case on monoatomic
semiconductors there is no charge
transfer among the same atoms because
of translation symmetry.
 In non-crystalline case there is charge
transfer because of distorted sp3
hybridization.

distorted sp3 hybridization
 1.
sp3 = ? s - ? px - ? py + ? pz
 2. sp3 = ? s - ? px + ? py - ? pz
 3. sp3 = ? s + ? px - ? py - ? pz
 4. sp3 = ? s + ? px + ? py + ? Pz
Charge accumulation has an
important influence on
electron energy distribution
and it plays an important
role for the chemical shift in
NMR measurements.
Electronic density
of states
(EDOS)
a-Si RMC I.
Measured structure factor (solid line),
RMC model (dashed line)
Unconstrained model
Is it really
possible?
Constrains for a-Si
Tight Binding Molecular Dynamics
Simulations
We have developed
a tight binding
molecular dynamics (TB-MD) computer code
to simulate the real preparation procedure of
an amorphous structure, which is grown by
atom-by-atom deposition on a substrate. This
method differs from most other molecular
dynamics (MD) studies where the amorphous
networks are formed by rapid cooling from the
liquid state. Our MD method was successfully
used for the description of the amorphous
carbon growth.
(K. Kohary and S. Kugler, Phys. Rev. B 63 (2001) 193404; and K.
Kohary, PhD thesis, Budapest-Marburg (2001), cond-mat/0201312)
Density of States calculations
Quantum chemical cluster calculations at
the AM1 level were also carried out in
order to find out whether the presence
of triangles and/or
squares
cause
variations in terms of the electronic
properties.
The electronic density of
states (EDOS) of the WWW model and
the modified WWW models containing
triangles and squares were calculated.
The reference cluster (a part of the
WWW
model) contained about 100
fourfold coordinated Si atoms and a
sufficient
number
of
hydrogens
saturating the dangling bonds on the
boundary of the cluster. It contains no
significant deviation from a locally nearly
perfect tetrahedral order.
First calculation
Based on reference network,
we
constructed other clusters adding silicon
(and hydrogen) atoms which formed one,
two and three fused
or individual
triangles
and
squares. Significant
differences were observed in terms of
the
EDOS:
additional higher energy
states appeared in the mobility gap, which
are localized on the triangle(s)
and
square(s).
Second calculation
Next figure shows the EDOS computed
for the central part of the RMC structural
model obtained at the 10th stage, as
compared to the EDOS of the reference
(WWW) cluster. The new states in the
gap correspond to a bond angle of
about 74 deg. in the RMC model. Here, it
is demonstrated that these states are due
exclusively to bond angles that
are
smaller than the tetrahedral ones.
Journal of Physics: Conference Series 253 (2010) 012013
The end
Optical
properties
General aspects
Optical absorption and luminescence
occur by transition of electrons and holes
between electronic states (bands, tail
states, gap states). If electron-phonon
coupling is strong enough self-trapping
occurs.

Absorption coefficient α is defined by
I(z) = Io exp {- α z}
where I(z) is the flux density if incident
light is Io, z is the distance measured from
the incident surface. Hence
α = - (1/I(z)) dI(z)/dz
Absorption
Tauc law (Tauc plot, A region)
The absorption coefficient, α, due to
interband transition near the band-gap is
well described:
αħω = B (ħ ω – Eg)2
ħω is photon energy, Eg is optical gap.
This Tauc plot defines the optical gap in
amorphous semiconductors.
Urbach tail (B region)
The absorption coefficient at the photon
energy below the optical gap (tail
absorption) depends exponentially on the
photon energy:
α(ħ ω) ~ exp (ħ ω/Eu)
where Eu is called Urbach energy.
C region
In addition, optical absorption by defects
also appears at energy lower than optical
gap. Likewise α is written as another
exponential function of photon energy:
α(ħω) ~ exp (ħω/Ed),
Ed belongs to the width of the defect
states. C region is rather sensitive to the
structural properties of materials.
Photoluminescence
Photoluminescence occurs as a result of
the transition of electrons and holes from
excited states to ground state.
 After interband excitation, electrons
(holes) relax to the bottom (top) of the
conduction (valence) band by emitting
phonons much more quickly than the
radiative transition.

Direct/indirect transition

In the case of crystalline semiconductors
(without defects, there is no localized
state) photoluminescence occurs by
transition between the bottom of the
conduction band and the top of the
valence band. k selection rule must be
satisfied: kphoton = ki – kf . (kphoton, ki and,
kf are the wave numbers of photons,
electron of initial and final states.
Since kphoton is much smaller than ki and
kf, we can rewrite the selection rule:
ki = kf.
The semiconductors satisfying this
condition is called direct-gap
semiconductors. c-Si is not satisfying kselection rule (indirect-gap
semiconductor). Transition is allowed by
either absorption of phonons or their
emission.
c-Si