11.4 Free Electron Model and Band Theory

The free electron model simplifies how electrons move in metals, treating them like a gas. It explains some properties well but falls short on others. This model helps us understand conductivity but can't tell metals from insulators.

Energy bands form when atoms come together in crystals. These bands determine if a material conducts electricity, insulates, or acts as a semiconductor. Understanding band theory is key to grasping how materials behave electrically.

Assumptions and Limitations of the Free Electron Model

Key Assumptions of the Free Electron Model

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• Treats conduction electrons in metals as a gas of non-interacting particles
• Ignores electron interactions with the ionic lattice and each other
• Assumes constant potential energy of electrons throughout the metal creates a "potential well" with infinite barriers at the surface
• Considers electrons move freely within the metal subject only to collisions with sample boundaries
• Successfully explains electrical and thermal conductivity in metals
• Accounts for the linear term in the heat capacity of metals

Limitations and Breakdown of the Model

• Unable to explain the periodic table's structure
• Fails to account for some magnetic properties of materials
• Cannot differentiate between metals and insulators
• Breaks down when considering tightly bound electrons
• Becomes inaccurate when electron-electron interactions become significant
• Oversimplifies the complex quantum mechanical nature of electrons in solids

Energy Levels and Density of States for Free Electrons

Quantum Mechanical Description of Free Electrons

• Derives energy levels using the Schrödinger equation with boundary conditions for a three-dimensional box
• Quantizes resulting energy eigenvalues depending on three quantum numbers (nx, ny, nz) corresponding to spatial dimensions
• Utilizes the concept of k-space to represent electron states in momentum space
• Applies periodic boundary conditions to account for the large number of electrons in a macroscopic solid

Density of States and Fermi Energy

• Defines density of states g(E) as the number of available electron states per unit energy interval
• Derives g(E) by counting states within a spherical shell in k-space and relating to energy through dispersion relation
• Demonstrates density of states for a three-dimensional system proportional to square root of energy: g(E) ∝ √E
• Introduces Fermi energy (EF) representing highest occupied energy level at absolute zero temperature
• Describes Fermi-Dirac distribution function for electron occupancy probability at finite temperatures
• Modifies sharp cutoff at EF due to thermal excitation of electrons

Energy Bands in Crystalline Materials

Formation of Energy Bands

• Arises from overlap and splitting of atomic energy levels when atoms form crystal lattice
• Results from periodic potential of crystal lattice leading to allowed and forbidden energy ranges
• Applies Bloch's theorem to describe wave functions of electrons in periodic potential
• Introduces concepts of crystal momentum and Brillouin zone
• Utilizes tight-binding approximation and nearly-free electron model as complementary approaches
• Represents band structure through energy vs. crystal momentum diagrams
• Plots along high-symmetry directions in Brillouin zone (Γ, X, L points)

Characteristics of Energy Bands

• Determines width and shape of energy bands based on strength of interatomic interactions
• Influences band structure by crystal structure (face-centered cubic, body-centered cubic, etc.)
• Defines band gaps as energy ranges where no electron states exist
• Plays crucial role in determining material's electrical properties through band gap size
• Exhibits different band structures for various materials (metals, semiconductors, insulators)

Conductors, Insulators, and Semiconductors: Band Theory

Band Structure and Electrical Properties

• Classifies conductors with partially filled bands or overlapping valence and conduction bands
• Allows easy electron movement and high electrical conductivity in conductors
• Defines insulators with large band gap (typically > 4 eV) between fully occupied valence band and empty conduction band
• Prevents significant electron excitation at room temperature in insulators
• Characterizes semiconductors with smaller band gap (typically < 4 eV)
• Enables thermal or optical excitation of electrons from valence to conduction band in semiconductors

Fermi Level and Material Behavior

• Positions Fermi level within a band for conductors
• Locates Fermi level in the band gap for insulators and semiconductors
• Modifies semiconductor properties through doping by introducing additional energy levels within band gap
• Creates n-type (electron-rich) or p-type (hole-rich) semiconductors through doping
• Exhibits different temperature dependence of conductivity among materials
• Increases resistance with temperature in conductors
• Demonstrates increased conductivity with temperature in semiconductors