lectures 21 22 solid state materials electronic structure and conductivity 1 band theory n.
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Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory

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Lectures 21-22 Solid state materials. Electronic structure and conductivity 1) Band theory. The electronic structure of solids can also be described by MO theory. A solid can be considered as a supermolecule .

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lectures 21 22 solid state materials electronic structure and conductivity 1 band theory
Lectures 21-22Solid state materials. Electronic structure and conductivity1) Band theory
  • The electronic structure of solids can also be described by MO theory.
  • A solid can be considered as a supermolecule.
  • One mole of atoms (NA), each with X orbitals in the valence shell contributes X moles of atomic orbitals producing X moles of MO’s.

Consider qualitatively bonding between Nmetal atoms of ns1 configuration (Li, Na etc) arranged in a chain; N = 2, 4, NA. Assume that X=1 for simplicity.

  • In the case of N~NA atoms they form not bonds but bands.
  • The band appearing in the bonding region is called valenceband. The antibonding region is called conductionband.
  • In the case of metals the valence and conduction bands are immediately adjacent.
2 band theory insulators semiconductors conductors
2) Band theory. Insulators, semiconductors, conductors
  • If we apply now an electrostatic potential to a conductor, the population of the energy levels will tend to change and electrons will be able to flow using empty adjacent conductionband.
  • In the case of insulators and semiconductors, the energy gap between the valence and conduction bands is more or less significant; electrons cannot easily get into the conduction band and cannot move along the sample; thermal or photo-energy is needed to bring some electrons to the conduction band.
3 crystal orbital theory
3) Crystal Orbital theory
  • The band structure of a crystalline material of virtually any complexity can be found through the application of the MO theory for solid state materials (Crystal Orbital theory).
  • One of the ways to model a real (finite size) crystal is by using cyclic boundary conditions assuming that a chain of bound atoms forms a very large ring.
  • It turns out that the energy levels in a cyclic molecule composed of N hydrogen atoms look as shown below.
4 crystal orbitals bloch functions
4) Crystal orbitals (Bloch functions)
  • If we have N hydrogen atoms with atomic wave functions fm (m = 1 … N) related by symmetry and spaced at distance a, we can get N MO’s yn (n = -N/2, …, 0, …, N/2) which are called Bloch functions.
  • For the n-th crystal orbital, yn, we will have:
  • When n changes from 0 to N/2, variable k = 2pn/(aN)) (wave vector) changes from 0 to p/a and the type of the MO changes from the completely bondingy0 to the completely antibondingyN/2:
  • Energy levels of the resulting set of MO’s (band structure) can be described with help of continuous functions E and density of states dn/dE (DOS)

5) Bonding in solids: Crystal Orbital Overlap Population

  • A common way to analyze bonding in solids is by calculating and analyzing the crystal orbital overlap population (COOP).
  • COOP is defined in the same way as the bond order is defined in MO theory of molecules.
  • For any two atoms i and j COOP(i-j) = S2cicjSij (Sij is the overlap integral for two atomic wavefunctions; summation should be performed for all pairs of overlapping orbitals of atoms i and j). A negative value of COOP means antibonding situation while a positive value is characteristic for bonding.
  • For the chain of hydrogen atoms the lower half of the band is bonding while the upper half is antibonding (see diagram on the right).
6 simplified picture of bonding in crystalline metals
6) Simplified picture of bonding in crystalline metals
  • Using crystal orbital theory we can rationalize the well-known fact that the metals with highest melting points are those belonging to 6th and 7th groups (see diagram below).

7) The Peierls distortion

  • When working with highly symmetrical structures one has to be cautious.
  • Highly symmetrical structures with not completely filled degenerate or near-degenerate levels are a subject to distortions which lower the symmetry and the energy of the system (Peierls distortion).
  • Diagrams on the left and in the center show how we can form bands for polymeric dihydrogen (s-MO) with twice larger four-atomic unit 2a and then distort the polymer to produce an array of dihydrogen molecules (the diagram on the right).
  • Similarly an infinite polyene -HC=HC-HC=HC-… polyacetylene will have alternating HC-HC and HC=HC bonds due to the Peierls distortion. Because of the large band gap it will behave not as a conductor but as an semiconductor.

8)Band structure of one dimensional polymers: a stack of PtII square planar complexes

  • In some cases one dimensional consideration is sufficient for a satisfactory analysis of band structure of solids – one dimensional polymers. For example, we can get a satisfactory description of bonding and conductivity of K2[Pt(CN)4Clx] (x = 0 … 0.3) using just one-dimensional model of crystal.
  • The complexes K2[Pt(CN)4Clx] (x = 0 … 0.3) have Pt(CN)4 – squares stacked one above another with Pt-Pt separation of 3.3 (x = 0) or 2.7-3.3 Ǻ (0 < x < 0.3).
  • Purely PtII complex (x = 0 in the formula above) is an insulator while oxidized cyanoplatinates are low-dimensional conductors.

9) Forming bands: Principles

  • To predict a qualitative band structure of stacked [Pt(CN)4]2-, we will consider [PtH4]2- as a model.
  • We will need for this analysis a MO diagram of PtL4.
  • Each of the monomer’s MOs generates a band when we form a polymer. We can analyze all MO’s one by one and then combine all bands together.
  • To get an idea about bands width use the rule which states that better orbital overlap will produce a wider band (s>p>d):

10) How bands behave

  • To learn, how the “frontier” bands will run (“up” or “down”) let’s write corresponding Bloch functions for frontier orbitals, pz, and all d-orbitals, for k = 0 and k = p/a.

11) Band structure of a stacked [PtH4]2-

  • The predicted band structure of a stacked [PtH4]2- in the center match well a calculated diagram on the right.
  • With band structure or DOS diagram in hands we can answer the questions: 1) why oxidized K2[Pt(CN)4Clx] (x>0) is a conductor and 2) why Pt-Pt distance shortens as x increases.

12) Bonding and conductivity in stacked [PtH4]2-

  • Conductivity. The Fermi level of stacked [PtH4]2- is on the top of the z2-band since the monomer HOMO is dz2 orbital. The conduction band is pz-band which is almost 3 eV higher in energy.
  • When the z2-band is completely filled (case of PtIIL4), no conductivity is expected / observed. For partially oxidized materials z2–band is filled only partially and we expect and observe conductivity.
  • Bonding. In solids like in molecules if bonding and antibonding MO’s are completely filled, the net bonding is zero.
  • For partially oxidized materials K2[Pt(CN)4Clx] (x = 0 … 0.3) z2–band is partially empty and we observe s(dz2-dz2) bonding between Pt atoms.