CHEM 1066: Band Theory and Bonding in Solids

CHEM 1066: Band Theory and Bonding in Solids

Instructor Information

  • Name: Michael Forde, PhD
  • Email: Michael.forde@uwi.edu

Course Objectives

  • Band Theory
    • Describe and explain the bonding and properties of:
    • Semiconductors
    • Metals
    • Insulators

Recap from Molecular Orbital Theory (MOT) / Valence Bond Theory (VBT)

  • Chemical Bond Definition: A chemical bond is a redistribution of electronic charge when atoms are brought close together.
  • Valence Bond Theory (VBT): Focuses on a central atom bonding to others.
  • Molecular Orbital Theory (MOT): Describes interactions over the whole molecule for all bonds.

Types of Solids

Ionic Solids
  • Characteristics:
    • Extended array of interacting electrostatic charges due to atoms with large electronegativity differences.
    • Electrical insulating nature due to strongly bound valence electrons, wherein the energy gap E_g > 3 ext{eV} leads to charges remaining on the ions.
  • Common Groups: Typically formed from Group VI/VII elements in combination with Group I-III elements.
  • Limitations: The ionic solid concept fails for easily polarizable ions (e.g., I-).
  • Crystal Field Effect: A sub-class of ionic bonding in solids.
Covalent Solids
  • Characteristics:
    • Directional bonds between atoms that are either too polarizable to be ionic or insufficiently electropositive to be metallic.
    • Typically involve Group III-VII elements.
    • Energy minimization occurs through sharing electronic charge across molecules, forming bands in semi-infinite solids.
  • 8-N Rule:
    • The coordination number (N) is defined as 8−N8-N, where N is the number of outer shell electrons.
    • This creates valence and conduction bands in the solid.
Metallic Solids
  • Characteristics:
    • Atoms with significant electropositivity engage in metallic bonding.
    • Band Theory:
    • Central concept is a band of free electrons held by a core of positive charge.
    • Electrons are generally non-directional.
  • Properties:
    • Good electrical conductivity at room temperature (conductivity decreases with increasing temperature).
    • Exhibits metallic luster and high heat conductivity.
    • Malleable and ductile.
  • Types: Includes pure metals, alloys, inter-metallics, and solid solutions.
  • Metallic nanoparticles: Refers to metallic elements in zero oxidation state; insufficient atoms to form semi-infinite arrays.

Bonding in Metallic Solids - Models

Classical Free Electron Model
  • Applicable to conductors, semiconductors, and insulators (not limited to real metals).
  • Drude & Lorentz Theory:
    • Metals consist of immobile positive ion cores with detached valence electrons that move randomly among these cores.
    • Electrons adhere to classical mechanics (multiple electrons can possess the same velocity or energy), following a Boltzmann distribution, facilitating metal conductivity.
Quantum Free Electron Model (Sommerfeld 1928)
  • Free electrons obey Fermi-Dirac statistics exhibiting quantum behavior.
  • Electrons move in a periodic potential, thus resulting in band theory properties.
Band Theory (Bloch 1928)
  • Also supports the notion of free electrons as waves moving in a periodic potential provided by the lattice.
  • It elucidates behaviors of conductors, semiconductors, and insulators.

Origin of Bonding in Metallic Solids

  • As atoms approach each other, their energy levels interact leading to energy level splitting.
  • This can create bands of closely spaced energy levels with partial occupancy, relevant in conductivity classification.

Classification of Materials Based on Band Gap

  • Conduction Band: The energy level band above the valence band facilitated electron conductivity.
  • Valence Band: The energy level band occupied by electrons engaged in covalent bonding.
  • Fermi Level: The highest filled energy level in a conductor at absolute zero (0 K).
  • Band Gap (EgE_g) Determination:
    • Eg=0<br/>ightarrowextConductorE_g = 0 <br /> ightarrow ext{Conductor}
    • E_g < 2 ext{ eV}
      ightarrow ext{Semiconductor}
    • E_g > 3 ext{ eV}
      ightarrow ext{Insulator}

New Terms and Definitions

  • Band Gap: The difference between the top of the valence band and the bottom of the conduction band.
  • Conductance Band: The band characterized by free electrons contributing to electrical conduction.
  • Valence Band: The occupied band of electrons involved in covalent bonding.
  • Fermi Level: The maximal energy level filled with electrons at 0 K.

Shape and Energy of the Band

  • Influenced by crystal structure (atomic packing) and temperature dependence.
  • Fermi Level: Determines electron distribution at various temperatures, with defined statistical equations for Fermi-Dirac distribution.

Semiconductors

  • Characterization:
    • Full valence band with a small band gap (typically between 0 eV to 2 eV).
    • Limited conductivity at room temperature due to thermally excited charge carriers.
  • Types of Semiconductors:
    • Intrinsic: Chemically pure, electrical conduction solely arises from thermally excited electrons and holes.
    • Extrinsic: Doped materials where conductance is influenced by additional charge states from the dopants.
    • n-type: Doped with elements that provide excess electrons, making electrons the majority carriers.
    • p-type: Doped with elements that create holes, thus holes become the majority carriers.

Hole Mechanism Example in Semiconductors

  • The movement of holes is stimulated by energy (heat/light):
    1. Energy excites an electron, creating a positive hole.
    2. Adjacent electrons jump to fill the holes, creating new holes in the process.
    3. The hole ‘moves’ as neighboring electrons shift to fill the gaps.

Extrinsic Semiconductor Diagrams

  • P-type Doped Semiconductor: Conventional doping with trivalent impurities (e.g., Boron).
  • N-type Doped Semiconductor: Doping with pentavalent impurities (e.g., Phosphorus).
Applications of Semiconductor Technology
Diodes
  • Understand how n-type and p-type semiconductors function collectively to form diodes.
Photovoltaics
  • Photon excitation promotes electron movement to the conduction band, generating electrical current within semiconductor junctions.
Superconductivity
  • Properties: Materials exhibit zero resistance at temperatures below a critical threshold.
  • Meissner Effect: Exclusion of magnetic flux lines from a superconductor cooled below its critical temperature.

Van der Waals Solids

  • Comprised of inert gas/molecular components held by van der Waals interactions.
  • Types of Forces:
    • Dipole-Dipole Interactions
    • Induced Dipole Interactions
    • Spontaneous Dipole-Induced Dipole Interactions

London (Dispersion) Forces

  • Results from temporary dipoles due to asymmetric electron distributions around a nucleus.
  • Magnitude increases with atomic size and polarizability of molecules.

Structural Types of Hydrogen-Bonded Solids

  • Commonly include hydrides of F, O, N.
  • Held together by hydrogen bonds with specified energy ranges and bond lengths.

Ionic Solids - Structure and Bonding

Objectives for Ionic Bonding studies
  • Define various critical ionic-related concepts such as ionic radius, electronegativity, enthalpy, and thermodynamic principles.
Evidence of Crystal Structures Using XRD
  • Techniques illustrate close packed structures necessary for stability in ionic solids.
Packing of Atoms/ Ions/Molecules
  • Achieved through energy minimization techniques; definitions of unit cells and packing structures including HCP, FCC, BCC formations.

Ionic Solids Specific Structures and Types

  • Detailed review of NaCl, CsCl, ZnS structures focusing on characteristics, coordination numbers, and examples with underpinning principles for stability in those crystals.
Fajan's Rules and Ionic Bond Covalency
  • Fajan's Rules demonstrate the influences of ionic nature and covalent characteristics in ionic bonding:
    1. Small cations with high charge promote covalency.
    2. Large anions with low charge also influence semi-ionic character.

End of Notes