Electronic structure Important question: Why certain materials are metals in addition to ot

Electronic structure Important question: Why certain materials are metals in addition to ot

Rosen, Peter, Founding Editor has reference to this Academic Journal, PHwiki organized this Journal Electronic structure Important question: Why certain materials are metals in addition to 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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(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.), in addition to 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 b in addition to gap occur in amorphous materials Yes What is the definition of semiconductors 1. Conductivity Conductivity is between metals in addition to 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 in addition to on lifting some electrons into an energy states of the conduction b in addition to . Electrons excited to the conduction b in addition to leave behind electron holes in the valence b in addition to . Both the conduction b in addition to electrons in addition to the valence b in addition to holes contribute to electrical conductivity. + – Most common definition The temperature dependence of resistivity at low temperature: = 0 exp(0/kB T ) T increasing, decreasing (In metal case: T increasing, 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 as long as bonding in CH4. While exciting 2s electrons into a 2p orbitals would, in theory, allow as long as 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 in addition to p wave functions, known as a hybridized orbital. In the case of carbon attempting to bond with four hydrogens, four orbitals are required. There as long as e, the 2s orbital “mixes” with the three 2p orbitals to as long as m four sp3 hybrids becomes 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 sp3

In CH4, four sp3 hybridised orbitals are overlapped by hydrogen’s 1s orbital, yielding four (sigma) bonds (that is, four single covalent bonds). The four bonds are of the same length in addition to strength. This theory fits the requirements. CH4 sp2 hybrids For example, ethene (C2H4). Ethene has a double bond between the carbons. For this molecule, carbon will sp2 hybridise, because one (pi) bond is required as long as the double bond between the carbons, in addition to only three bonds are as long as med per carbon atom. In sp2 hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals.

sp2 hybrids In ethylene (ethene) the two carbon atoms as long as m a bond by overlapping two sp2 orbitals in addition to each carbon atom as long as ms two covalent bonds with hydrogen by ssp2 overlap all with 120° angles. The bond between the carbon atoms perpendicular to the molecular plane is as long as med by 2p2p overlap. The hydrogen-carbon bonds are all of equal strength in addition to length, which agrees with experimental data. 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 sp2

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

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 b in addition to in addition to the top of the valence b in addition to . k selection rule must be satisfied: kphoton = ki  kf . (kphoton, ki in addition to , kf are the wave numbers of photons, electron of initial in addition to final states. Since kphoton is much smaller than ki in addition to 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. c-Si

Rosen, Peter Founding Editor

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