Solid State Chemistry

• The correct unit cell is the smallest that displays the full symmetry
• How many lattice points are in primitive, body-centred and face-centred?

  1, 2 and 4
• What is the order of stability for vertexes, edges and face sharing?

  Vertex sharing > Edge sharing > Face sharing
• Why in band gap energies do the energy levels appear as a continuous band?

  Because N is so large (so many energy levels)and there is a small energy of separation
• What type of bonding does a partially filled band suggest?

  Metallic bonding
• What does a broader band in DoS diagrams suggest?

  Broader band = stronger bond = better overlap, hence easier for electrons to move through the lattice
• How does an electric field affect electrons in crystal orbitals?


  Electrons travelling in the field direction are accelerated and their crystal orbital potential energy is lowered. If against the direction they are decelerated and their crystal orbital potential energy is increased.
• How do electrons move within partially filled bands when an electric field is applied and why is this only for partially filled bands?

  Once the crystal orbital energy is lowered, higher energy electrons drop down into empty levels causing a net movement of electrons. This is only for partially filled bands as empty states just above the Ef are required.
• How does temperature affect resistance (in terms of scattering frequency)?

  Higher temperature = Higher amplitude vibrations = Increased scattering frequency = increased resistance
• How does a band gap size relate to conductivity?

  Larger band gap = Better overlap = Stronger bonding = Poorer Conductor
• How does ionic bonding affect conductivity?

  Ionic bonding leads to a large band gap, some conduction at very high temperatures or very high electric fields
• What are the two types of charge carriers?

  • Electrons
  • Holes (h+)
  • These limit the conductivity in semi-conductors
• What are intrinsic semiconductors?

  Pure semiconductors where promotion generates equal amounts of electrons and holes
• What are extrinsic semiconductors?

  Caused by doping with small amounts of non-native atoms (which are more conductive than the undoped material)
• How does doping affect the Fermi level?

  It shifts the Fermi level to produce a dominant charge carrier
• What are the two types of doping?

  • n-type: the dopant has more electrons than native electrons hence conduction occurs through additional electrons causing the Fermi level to shift upwards (Electrons are promoted to the conduction band)
  • p-type: the dopant has less electrons than native electrons hence conduction occurs through positive holes causing the Fermi level to shift downwards (Electrons are promoted to the Fermi level)
• Explain the Temperature dependence graph shown below (3 stages)

  1. Extrinsic: Localised dopant charge carriers are ionised at low temperatures due to the small gap between dopant energy levels and the band edge
  2. Saturation: All donor sites are ionised so the number of dopant atoms = the number of charge carriers however there is insufficient energy to promote electrons across the band
  3. Intrinsic: There is sufficient energy to promote electrons across the band and the charge carrier concentrations increase with temperature

  
• What are the differences between electron-rich and poor dopants

  • Electron-rich dopants provide donor sites and n-type conductivity (contain more electrons)
  • Electron-poor dopants provide acceptor sites and p-type conductivity (contain more holes)
  • Hence n-type conductivity is higher than p-type
• Define Ionisation energy and the requirement for an atom to act as a dopant

  Ionisation energy = the energy required to release an electron to the conduction band

  For an atom to act as a dopant it must have the correct number of electrons
• How is covalent and ionic bonding inferred?

  • Covalent compounds depend on orbital overlap (which is inferred from bond length)
  • The ionic component is inferred by the electronegativity difference (ionic interactions increase bond strength without affecting bond length)
  • Longer bonds have more diffuse orbitals hence poorer overlap and a smaller bandgap
• How does bond length affect bandgap?

  Longer bonds have more diffuse orbitals hence poorer overlap (weaker bonding) and so have a smaller bandgap
• Explain the anomalies

  IN, ZnO and CdO have mismatched orbitals hence poor overlap
• Different types of doping

  • n-type: assume substitutional doping, any more electron rich atom can act as the n-dopant. Replacing with dopant atoms causes an imbalance leading to 'cations' and 'anions'. Cation vacancies result in excess anions and vice versa
  • p-type: Requires an electron poor dopant to accept electrons leaving holes in the valence band, holes can also be induced through anion vacancies. P-type doping requires vacancy on the more electronegative atom.
• How does X-ray diffraction work?

  • A hot tungsten cathode is used and thermionic emission produces electrons which are accelerated through a high voltage into a metal anode target (water cooling keeps the cathode cool). High-energy electron bombardment ejects core electrons so valence electrons drop down to fill the gaps which emits X-rays
  • Background is caused by Bremsstrahlung
• Metal Foils in X-ray Diffraction

  • Metal foils absorb X-rays up to their absorption edge
  • X-ray energy > Required transition energy = Strong absorption
  • X-ray energy < Required transition energy = no absorption
  • Foil filtered don't create perfect monochromatic beams they allow K-alpha through
  • When waves combine they create interference which can be constructive or deconstructive
• Thomson Scattering

  • Electrons absorb and re-emit X-rays in all directions (but are more likely to emit at low angles relative to the incoming beam)
  • Powder crystalline samples only irradiate X-ray scatters of X-rays at specific angles
  • X-rays are reflected off of miller planes
  • For 2 beams to be in phase the path difference must be a whole number of wavelengths
  • Scattering is weaker at high angles and for lighter elements
• Debye-Scherrer Cones

  • Cannot be used to distinguish between equivalent planes
  • Heavy elements have more electrons hence more scattering
  • Higher angles have a larger path difference and so increased destructive interference
• Crystal Monochromation

  • A single crystal is cut along a specific miller plane and the beam is aligned so only CuK-alpha 1 radiation satisfies the Bragg condition
  • The Bragg Condition is that scattered radiation undergoes constructive interference such that the crystal appears to have reflected X-rays
• Modelling wavefunctions and energies for a 1D crystal

  • Condition 1: Periodic Boundary
  • Condition 2: Electron Density
  • Each crystal orbital in the band has a unique K value
  • In a 1D chain electrons in equal number of states move in either a positive or negative direction
• E vs K diagrams

  • The beta value is derived by allowing electrons to exchange places (the interaction beyond nearest neighbours is 0)
  • The p band is inverted as p-p interactions are antibonding
  • The band-width is equal to |4β| hence strong orbital interactions = wider bands
  • DoS diagrams are inversely proportional to the slope of the E-K plot, a shallow E-K curve = higher DoS slope
  • The area of the DoS plot is proportional to the number of states (this tends to be the reverse for 3D)
  • Indirect semiconductors must be coupled with a phonon absorption due to having indirect band gaps
• Indexing

  • Missing peaks are called systematic absences (hkl peaks not seen due to inherent destructive interference)
  • P centring; no restrictions
  • F centring; individual hkl are all odd or all even
  • I centring; the sum of hkl must be even
  • C centring; h+k must be even
  • Integers without 7 = Primitive
  • Non-integers or integers with 7 = Body centred
  • Thirds = Face centred
• 2D Band Theory

  • Kx and Ky of the wavevector take values between -π/a to +π/a
  • The minimum point on the plot is known as the gamma point
  • The range of Kx and Ky is known as the first Brillouin zone
• Modelling Graphite with 2D Band Theory

  • Modelled as having localised sp2 bonding within its planes and weak dispersion forces between the layers
  • The remaining 1 electron per carbon is delocalised in overlapping Pz orbitals
  • Graphite is a heavy metal and hence a better conductor than a semiconductor but not better than a metal as it still has a bandgap
• 3D Band Theory

  • Curvature is related to orbital overlap and directions where charge carriers are mobile
  • Allows the determination of direct or indirect bandgaps
• The Free Electron Model (3D Band Theory)

  • Ignores interactions between electrons and interactions between electrons and the periodic lattice of cations
  • Predicts the same band structure for all materials
  • Effective electron mass relates curvature based on how easy it is for an electron to move through the material
  • Good orbital overlap means a lighter electron mass hence a broader DoS diagram and so more curved E-K bands
• The Free Electron Theory (3D Band Theory) CONTINUED

  • Holes have different mobilities depending on what type of band they are in
  • Shallower = Heavier electron
  • Doping semiconductors with charge carriers at the Mott Criteria (concentration) allows them to become metallic
• X-Ray Diffraction

  • Peak positions depend on d-spacing and lattice parameters
  • Peak intensity is determined by atom types and positions within the unit cell
  • In powder diffraction, a peak corresponding to hkl will include reflection from all miller planes sharing the same d-spacing value
  • Lower multiplicities occur if the combinations are not symmetrically equivalent
  • Lorentz and Polarisation lead to a decrease in intensity with increasing angle
  • The general case where we replace cos(x) with e^ix for the atomic form factor assumes centro-symmetric structures
• Electronic transitions within solids in a metal can be within a band (intraband) or in semiconductors can be between the bands (interband)

  Electrons can be excited at any point in the band to any empty portion of the band
• All flat surfaces reflect some light if k is greater than 0 and n does not equal 1
• Refraction is the slowing of light as it passes through a medium where n is greater than 1 (the light's frequency is constant but its wavelength changes).