2.1 Free electron model

The free electron model simplifies how electrons behave in metals, treating them as a gas of non-interacting particles. This foundational concept helps explain electrical conductivity, thermal properties, and basic electronic behavior in materials.

While useful for alkali metals, the model has limitations. It ignores electron-electron interactions and crystal structure effects. More advanced models, like the Sommerfeld model, incorporate quantum mechanics to address these shortcomings and explain additional phenomena.

Free electron model basics

• Describes conduction electrons in metals as a gas of non-interacting particles moving freely within the material
• Provides a simplified framework for understanding electronic properties of metals in condensed matter physics
• Serves as a foundation for more advanced models in solid-state physics

Assumptions and limitations

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• Assumes electrons move freely without interaction with the ionic lattice
• Neglects electron-electron interactions and treats electrons as independent particles
• Ignores the periodic potential of the crystal structure
• Works well for alkali metals (sodium, potassium) but fails for transition metals
• Cannot explain certain phenomena like band gaps in semiconductors

Drude model vs Sommerfeld model

• Drude model applies classical mechanics to free electrons
  • Treats electrons as a classical gas following Maxwell-Boltzmann statistics
  • Predicts electrical and thermal conductivity but fails to explain specific heat
• Sommerfeld model improves upon Drude model by incorporating quantum mechanics
  • Uses Fermi-Dirac statistics to describe electron distribution
  • Accurately predicts electronic specific heat and explains Wiedemann-Franz law
• Both models assume constant electron density and isotropic electron mass

Quantum mechanical approach

• Incorporates wave-like nature of electrons using quantum mechanics principles
• Explains phenomena unexplained by classical models (specific heat, paramagnetism)
• Forms the basis for understanding more complex electronic structures in solids

Fermi-Dirac distribution

• Describes the probability of electron occupancy in energy states at thermal equilibrium
• Accounts for Pauli exclusion principle and indistinguishability of electrons
• Represented by the equation: $f(E) = \frac{1}{e^{(E-E_F)/k_BT} + 1}$
• Determines electron distribution in metals and semiconductors
• Explains temperature dependence of electronic properties

Density of states

• Represents the number of available electron states per unit energy interval
• Crucial for calculating electronic properties and carrier concentrations
• For free electrons in 3D: $g(E) = \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$
• Varies with dimensionality of the system (1D, 2D, 3D)
• Influences optical and transport properties of materials

Electronic properties

• Describes how free electrons contribute to various measurable properties of materials
• Provides insights into the behavior of metals and semiconductors under different conditions
• Forms the basis for designing and optimizing electronic devices

Electrical conductivity

• Measures a material's ability to conduct electric current
• Derived from Drude model: $\sigma = \frac{ne^2\tau}{m}$
• Depends on electron density, relaxation time, and effective mass
• Explains temperature dependence of conductivity in metals
• Affected by scattering mechanisms (phonons, impurities, defects)

Thermal conductivity

• Quantifies a material's ability to conduct heat
• Electronic contribution given by Wiedemann-Franz law: $\kappa_e = L\sigma T$
• L is the Lorenz number, approximately $\frac{\pi^2}{3}(\frac{k_B}{e})^2$ for free electrons
• Explains why good electrical conductors are also good thermal conductors
• Deviations from Wiedemann-Franz law indicate strong electron-electron interactions

Hall effect

• Occurs when a magnetic field is applied perpendicular to current flow
• Results in a transverse voltage (Hall voltage) across the sample
• Hall coefficient: $R_H = -\frac{1}{ne}$ for free electrons
• Allows determination of carrier type and concentration
• Sign of Hall coefficient indicates whether carriers are electrons or holes

Band structure

• Describes the range of energies electrons can have within a solid
• Crucial for understanding electronic and optical properties of materials
• Provides insights into the distinction between metals, semiconductors, and insulators

Nearly free electron model

• Treats electrons as almost free but perturbed by a weak periodic potential
• Introduces the concept of energy bands and band gaps
• Explains the formation of allowed and forbidden energy regions
• Accounts for the periodic nature of the crystal lattice
• Predicts the existence of energy gaps at Brillouin zone boundaries

Brillouin zones

• Represent the primitive cell of the reciprocal lattice in k-space
• Define the range of allowed electron wavevectors in a crystal
• First Brillouin zone contains all unique electronic states
• Determine the periodicity of electronic band structure
• Essential for understanding electron dynamics and scattering processes

Fermi surface

• Represents the surface of constant energy in k-space at the Fermi level
• Shape and topology determine many electronic properties of materials
• Crucial for understanding electron transport and optical properties
• Provides insights into the anisotropy of electronic properties

Fermi energy and wavevector

• Fermi energy (E_F) is the highest occupied energy level at absolute zero
• For free electrons: $E_F = \frac{\hbar^2}{2m}(3\pi^2n)^{2/3}$
• Fermi wavevector (k_F) relates to Fermi energy: $k_F = \sqrt{\frac{2mE_F}{\hbar^2}}$
• Determines the size of the Fermi sphere in k-space
• Influences electronic specific heat and density of states at the Fermi level

Fermi surface measurements

• Employ various experimental techniques to map Fermi surface topology
• de Haas-van Alphen effect measures oscillations in magnetic susceptibility
• Angle-resolved photoemission spectroscopy (ARPES) directly probes occupied electronic states
• Positron annihilation provides information on momentum distribution of electrons
• Compton scattering reveals electron momentum density

Optical properties

• Describe how materials interact with electromagnetic radiation
• Provide information about electronic structure and excitations
• Crucial for designing optoelectronic devices and understanding light-matter interactions

Plasma frequency

• Characteristic frequency at which free electrons oscillate collectively
• Given by: $\omega_p = \sqrt{\frac{ne^2}{\epsilon_0 m}}$
• Determines the optical response of metals at different frequencies
• Separates regions of high reflectivity (below ω_p) and transparency (above ω_p)
• Influences plasmon excitations and electromagnetic wave propagation in metals

Reflectivity and absorption

• Reflectivity describes the fraction of incident light reflected from a material surface
• For frequencies below plasma frequency, metals are highly reflective
• Absorption occurs when photons excite electrons to higher energy states
• Interband transitions contribute to absorption in visible and UV regions
• Free electron absorption dominates in the infrared for metals

Limitations and extensions

• Recognizes the shortcomings of the free electron model in explaining certain phenomena
• Introduces more sophisticated models to address these limitations
• Provides a bridge to more advanced topics in condensed matter physics

Failure cases

• Cannot explain the existence of band gaps in semiconductors and insulators
• Fails to account for the periodic potential of the crystal lattice
• Inadequate for describing strongly correlated electron systems
• Does not explain magnetic properties of materials
• Inaccurate for materials with complex band structures (transition metals)

Beyond free electron model

• Tight-binding model considers localized atomic orbitals and their interactions
• k·p theory provides a more accurate description of band structure near extrema
• Density functional theory (DFT) incorporates electron-electron interactions
• Many-body perturbation theory accounts for complex electron correlations
• Dynamical mean-field theory (DMFT) addresses strongly correlated electron systems

Experimental evidence

• Provides empirical support for the free electron model and its predictions
• Highlights the successes and limitations of the model in explaining observed phenomena
• Guides the development of more advanced theoretical frameworks

Specific heat measurements

• Electronic specific heat in metals follows linear temperature dependence
• Sommerfeld model accurately predicts the coefficient of electronic specific heat
• Deviations from free electron behavior indicate strong electron correlations
• Low-temperature measurements reveal contributions from lattice vibrations (phonons)
• Superconducting transition appears as a jump in specific heat

Magnetoresistance

• Describes the change in electrical resistance when a magnetic field is applied
• Positive magnetoresistance in simple metals follows B² dependence at low fields
• Quantum oscillations (Shubnikov-de Haas effect) reveal Fermi surface topology
• Negative magnetoresistance can occur due to weak localization effects
• Giant and colossal magnetoresistance observed in certain materials (multilayers, manganites)

Applications in materials

• Demonstrates the practical relevance of free electron model in understanding real materials
• Highlights the importance of electronic structure in determining material properties
• Provides insights for designing and optimizing materials for specific applications

Metals vs semiconductors

• Metals have partially filled bands with Fermi level within a band
• Semiconductors have fully occupied valence band and empty conduction band
• Band gap in semiconductors can be tuned by doping or alloying
• Effective mass of charge carriers differs significantly between metals and semiconductors
• Carrier concentration and mobility determine electrical properties

Nanostructures and quantum confinement

• Reduced dimensionality alters electronic density of states
• Quantum wells, wires, and dots exhibit discrete energy levels
• Confinement effects become significant when dimensions approach de Broglie wavelength
• Enables tailoring of electronic and optical properties through size control
• Forms the basis for various nanoelectronic and optoelectronic devices (quantum dot lasers, single-electron transistors)