Electronic structure

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Electronic structure
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• 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

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• f(k,r) u(k,r) exp (ikr)
• where u(k,r) has the periodicity of the
  crystal lattice
• u(k,r)u(k,rR)
• (R is lattice translation vector.),
• and term exp(ikr) represents a plane wave.

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• 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.

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First Brillouin zone of FCC lattice showing
symmetry labels
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Electron density of states of c-Si Indirect
semiconductor
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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

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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?

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• 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.

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Most common definition

• The temperature dependence of resistivity at
  low temperature
• ? ?0 exp(e0/kB T )
• T increasing, ? decreasing
• (In metal case
• T increasing, ? increasing!)

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Electronic structure
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Covalent bonding

• Amorphous semiconductors are typically
  covalently bonded materials.

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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

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• 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.

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• 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

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sp3 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

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sp3
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• In CH4, four sp3 hybridised orbitals are
  overlapped by hydrogen's 1s orbital, yielding
  four s (sigma) bonds (that is, four single
  covalent bonds). The four bonds are of the same
  length and strength. This theory fits the
  requirements.

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CH4
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sp2 hybrids

• For example, ethene (C2H4). Ethene has a
  double bond between the carbons.
• For this molecule, carbon will sp2 hybridise,
  because one p (pi) bond is required for the
  double bond between the carbons, and only three s
  bonds are formed per carbon atom. In sp2
  hybridisation the 2s orbital is mixed with only
  two of the three available 2p orbitals.

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sp2 hybrids

• In ethylene (ethene) the two carbon atoms form a
  s bond by overlapping two sp2 orbitals and each
  carbon atom forms two covalent bonds with
  hydrogen by ssp2 overlap all with 120 angles.
  The p bond between the carbon atoms perpendicular
  to the molecular plane is formed by 2p2p
  overlap. The hydrogen-carbon bonds are all of
  equal strength and length, which agrees with
  experimental data.

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sp2 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

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sp2
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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

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IV. Column materials
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VI. Column materials(2s4p electrons gt 2s2
sigma bond 2 lone pair )
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Atomic charges
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• 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.

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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

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• Charge accumulation has an important influence
  on electron energy distribution and it plays an
  important role for the chemical shift in NMR
  measurements.

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• Electronic density of states
• (EDOS)

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a-Si RMC I.
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Measured structure factor (solid line), RMC
model (dashed line)
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Unconstrained model
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Constrains for a-Si

• Is it really possible?

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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)
• .

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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.

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• 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.

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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).

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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.

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Journal of Physics Conference Series 253 (2010)
012013
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The end
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Optical properties
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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.

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• Absorption coefficient a is defined by
  I(z) Io exp - a z
• where I(z) is the flux density if incident
  light is Io, z is the distance measured from the
  incident surface. Hence
• a - (1/I(z)) dI(z)/dz

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Absorption
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Tauc law (Tauc plot, A region)

• The absorption coefficient, a, due to
  interband transition near the band-gap is well
  described
• ah? B (h ? Eg)2
• h? is photon energy, Eg is optical gap.
• This Tauc plot defines the optical gap in
  amorphous semiconductors.

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Urbach tail (B region)

• The absorption coefficient at the photon
  energy below the optical gap (tail absorption)
  depends exponentially on the photon energy
• a(h ?) exp (h ?/Eu)
• where Eu is called Urbach energy.

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C region

• In addition, optical absorption by defects
  also appears at energy lower than optical gap.
  Likewise a is written as another exponential
  function of photon energy
• a(h?) exp (h?/Ed),
• Ed belongs to the width of the defect states.
  C region is rather sensitive to the structural
  properties of materials.

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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.

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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.

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• 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 k-selection rule (indirect-gap
  semiconductor). Transition is allowed by either
  absorption of phonons or their emission.

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c-Si