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Electronic structure. Important question: Why certain materials are metals and others are insulators?
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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:
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 oflattice. Since that a reciprocal lattice related to the unit cell parameters can be established in k-space.
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!)
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
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
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.
sp2 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.
sp2 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.
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
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
VI. Column materials(2s4p electrons => 2s+2 sigma bond +2 lone pair )
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)
Measured structure factor (solid line), RMC model (dashed line)
Constrains for a-Si Is it really possible?
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.
