6.3 Quantum particles in periodic potentials

Quantum particles in periodic potentials behave differently than in free space. This topic explores how electrons move through crystal lattices, forming energy bands and gaps that determine a material's properties.

Understanding periodic potentials is crucial for grasping solid-state physics. We'll dive into Bloch's theorem, crystal momentum, and energy band structures, which are key to explaining conductivity in metals, semiconductors, and insulators.

Bloch Theorem and Energy Bands

Bloch's Theorem and Wave Functions

• Bloch's theorem describes electron behavior in periodic potentials
• States wave functions for electrons in periodic lattices take the form $\psi_k(r) = e^{ikr}u_k(r)$
• $e^{ikr}$ represents plane wave, while $u_k(r)$ has periodicity of crystal lattice
• Theorem applies to any particle in a periodic potential (electrons, phonons)
• Provides foundation for understanding electronic properties of crystalline solids

Energy Band Formation and Structure

• Energy bands form from overlapping atomic orbitals in crystal structures
• Continuous ranges of allowed electron energies within solids
• Band structure determines electrical and optical properties of materials
• Valence band contains highest occupied electron states at absolute zero
• Conduction band comprises lowest unoccupied states above valence band
• Band gaps separate allowed energy regions, crucial for semiconductor behavior
• Materials classified based on band structure (metals, semiconductors, insulators)

Allowed and Forbidden Energy Regions

• Allowed energy regions correspond to energy bands where electrons can exist
• Forbidden energy regions align with band gaps where electron states are not permitted
• Band structure determines material's electrical conductivity
• Metals have overlapping valence and conduction bands, allowing easy electron flow
• Semiconductors feature small band gaps, enabling controlled conductivity
• Insulators possess large band gaps, inhibiting electron movement between bands
• Energy band engineering enables creation of advanced electronic devices (lasers, LEDs)

Crystal Momentum and Brillouin Zones

Crystal Momentum Concept

• Crystal momentum represents electron's wave vector in periodic potential
• Differs from classical momentum due to lattice interactions
• Defined as $p = \hbar k$, where k is the wave vector
• Conserved in perfect crystals during electron scattering processes
• Plays crucial role in understanding electron transport in solids
• Determines electron's behavior in applied electric and magnetic fields

Brillouin Zones and Reciprocal Space

• Brillouin zones represent regions in reciprocal space of crystal lattice
• First Brillouin zone contains all unique wave vectors describing electron states
• Higher-order Brillouin zones obtained by translating first zone by reciprocal lattice vectors
• Zone boundaries correspond to electron diffraction conditions in crystals
• Essential for analyzing electronic and vibrational properties of solids
• Brillouin zone shape depends on crystal structure (cubic, hexagonal, tetragonal)
• Used to visualize and calculate band structures, phonon dispersions, and other properties

Models and Density of States

Kronig-Penney Model

• Simplified model for electron behavior in one-dimensional periodic potential
• Assumes potential as series of rectangular barriers separated by regions of constant potential
• Allows analytical solution of Schrödinger equation for periodic systems
• Demonstrates formation of energy bands and band gaps
• Provides insights into electron wave functions and dispersion relations
• Model parameters can be adjusted to approximate real material properties
• Serves as foundation for more complex band structure calculations

Density of States and Material Properties

• Density of states (DOS) describes number of available electron states per energy interval
• Crucial for understanding thermal and electrical properties of materials
• Varies with dimensionality of system (1D, 2D, 3D)
• In 3D, DOS typically increases with square root of energy within a band
• Features discontinuities or singularities at band edges (Van Hove singularities)
• Determines electron distribution in thermal equilibrium (Fermi-Dirac statistics)
• Influences material's specific heat, electrical conductivity, and optical absorption
• Can be measured experimentally through techniques like photoemission spectroscopy
• Plays key role in designing electronic and optoelectronic devices