2.2 Nearly free electron model

The nearly free electron model builds upon the free electron model by introducing weak periodic potentials from ion cores. It's a fundamental approach in condensed matter physics for understanding electronic properties of metals and semiconductors, bridging the gap between simple free electron theory and complex band structures.

This model applies perturbation theory to solve the Schrödinger equation with a weak periodic potential. It predicts the formation of energy gaps at Brillouin zone boundaries and introduces concepts like Bloch's theorem, energy band structures, and effective mass, crucial for understanding solid-state physics.

Nearly free electron approximation

• Builds upon the free electron model by introducing weak periodic potential from ion cores
• Serves as a fundamental approach in condensed matter physics to understand electronic properties of metals and semiconductors
• Bridges the gap between simple free electron theory and more complex band structure calculations

Assumptions and limitations

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• Assumes electrons in a solid experience a weak periodic potential from ion cores
• Treats the periodic potential as a small perturbation to the free electron Hamiltonian
• Limited to materials with delocalized electrons (metals, some semiconductors)
• Breaks down for materials with strongly localized electrons or strong electron-electron interactions

Perturbation theory approach

• Applies first-order perturbation theory to solve the Schrödinger equation with a weak periodic potential
• Utilizes Fourier series expansion of the periodic potential: $V(\mathbf{r}) = \sum_{\mathbf{G}} V_{\mathbf{G}} e^{i\mathbf{G} \cdot \mathbf{r}}$
• Calculates energy corrections and modified wavefunctions due to the perturbation
• Predicts the formation of energy gaps at Brillouin zone boundaries

Bloch's theorem

• Fundamental theorem in solid-state physics describing electron behavior in periodic potentials
• Provides a mathematical framework for understanding electronic states in crystalline solids
• Forms the basis for band structure calculations and understanding of electronic properties

Periodic potential effects

• Introduces periodicity in the electron wavefunction matching the crystal lattice
• Modifies the free electron dispersion relation, leading to the formation of energy bands
• Creates allowed and forbidden energy regions (band gaps) in the electronic structure
• Influences electron scattering processes and transport properties in solids

Wavefunction in periodic lattice

• Bloch's theorem states that the wavefunction takes the form: $\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} u_{\mathbf{k}}(\mathbf{r})$
• $e^{i\mathbf{k} \cdot \mathbf{r}}$ represents a plane wave, while $u_{\mathbf{k}}(\mathbf{r})$ has the periodicity of the lattice
• Introduces the concept of crystal momentum $\hbar\mathbf{k}$, distinct from the electron's true momentum
• Allows for the description of electronic states using quantum numbers $\mathbf{k}$ and band index $n$

Energy band structure

• Describes the relationship between electron energy and momentum in crystalline solids
• Arises from the interaction of electron waves with the periodic potential of the crystal lattice
• Plays a crucial role in determining electronic, optical, and thermal properties of materials

Formation of energy gaps

• Occurs at the Brillouin zone boundaries due to Bragg reflection of electron waves
• Results from the lifting of degeneracy in free electron states by the periodic potential
• Gap size depends on the strength of the periodic potential and the specific crystal structure
• Determines whether a material behaves as a metal, semiconductor, or insulator

Brillouin zones

• Represent the primitive cell of the reciprocal lattice in momentum space
• First Brillouin zone contains all unique $\mathbf{k}$ values needed to describe electronic states
• Higher-order Brillouin zones relate to the periodicity of the energy bands in reciprocal space
• Provide a convenient framework for analyzing and visualizing band structures

Effective mass

• Describes the response of electrons to external forces in a crystal lattice
• Accounts for the effects of the periodic potential on electron motion
• Can be positive, negative, or even anisotropic depending on the band structure

Concept and significance

• Relates the acceleration of electrons in a crystal to that of free electrons
• Defined as: $m^* = \hbar^2 \left(\frac{d^2E}{dk^2}\right)^{-1}$
• Determines carrier mobility and transport properties in semiconductors
• Influences the density of states and optical properties of materials

Calculation methods

• Derived from the curvature of the energy bands in k-space
• Can be obtained experimentally through cyclotron resonance measurements
• Tensor quantity for anisotropic materials, requiring directional considerations
• Often calculated using computational methods (density functional theory)

Density of states

• Describes the number of available electronic states per unit energy interval
• Crucial for understanding thermal and electrical properties of materials
• Influences the behavior of electrons in various physical phenomena (heat capacity, electrical conductivity)

Free electron vs nearly free

• Free electron model: DOS proportional to $\sqrt{E}$ in 3D systems
• Nearly free electron model: DOS modified by the presence of energy gaps and band structure
• Introduces van Hove singularities at critical points in the Brillouin zone
• Affects the temperature dependence of various material properties

Energy dependence

• Varies with dimensionality of the system (3D, 2D, 1D, 0D)
• In 3D: $g(E) \propto \sqrt{E}$ for free electrons, modified by band structure effects
• In 2D: Constant DOS within each subband (quantum wells, graphene)
• In 1D: Inverse square root dependence (quantum wires, carbon nanotubes)

Fermi surface

• Represents the surface of constant energy in k-space at the Fermi level
• Separates occupied from unoccupied electronic states at absolute zero temperature
• Crucial for understanding electronic and thermal properties of metals and semimetals

Shape and properties

• Determined by the band structure and electron filling of the material
• Can be spherical (free electron-like metals), or have complex shapes (transition metals)
• Topology influences various phenomena (quantum oscillations, superconductivity)
• Nesting features can lead to charge density waves or spin density waves

Experimental observations

• Measured using techniques such as de Haas-van Alphen effect
• Angle-resolved photoemission spectroscopy (ARPES) provides direct visualization
• Compton scattering and positron annihilation also probe Fermi surface features
• Comparison with theoretical calculations validates band structure models

Electron dynamics

• Describes the motion of electrons in response to external fields and scattering processes
• Crucial for understanding transport phenomena in solids (electrical conductivity, Hall effect)
• Influenced by the band structure and scattering mechanisms in the material

Group velocity vs phase velocity

• Group velocity: $\mathbf{v}_g = \frac{1}{\hbar} \nabla_{\mathbf{k}} E(\mathbf{k})$
• Represents the velocity of electron wave packets in the crystal
• Phase velocity: $\mathbf{v}_p = \frac{E(\mathbf{k})}{\hbar k}$
• Describes the propagation of individual plane wave components

Acceleration in electric field

• Classical acceleration modified by the effective mass: $\mathbf{a} = -\frac{e\mathbf{E}}{m^*}$
• Electrons can exhibit negative effective mass, leading to counterintuitive behavior
• Bloch oscillations occur in very strong fields or superlattices
• Scattering processes limit the acceleration and lead to drift velocity

Optical properties

• Describe how materials interact with electromagnetic radiation
• Determined by the electronic band structure and allowed transitions
• Crucial for understanding and designing optoelectronic devices

Interband vs intraband transitions

• Interband: Transitions between different energy bands (valence to conduction)
• Responsible for absorption and emission of photons in semiconductors
• Intraband: Transitions within the same band (free carrier absorption)
• Dominate in metals and heavily doped semiconductors

Absorption and reflection

• Absorption coefficient related to the joint density of states and transition matrix elements
• Reflection determined by the complex dielectric function of the material
• Band gap influences the onset of strong absorption in semiconductors
• Plasma frequency separates reflecting and transmitting regions in metals

Applications in solids

• Nearly free electron model provides insights into various solid-state phenomena
• Useful for understanding and predicting material properties for technological applications
• Serves as a starting point for more sophisticated electronic structure calculations

Metals vs semiconductors

• Metals: Partially filled bands with Fermi level within a band
• Exhibit high electrical and thermal conductivity, free-electron-like behavior
• Semiconductors: Fully occupied valence band separated from empty conduction band by a gap
• Properties highly tunable through doping, crucial for electronic devices

Alloys and impurities

• Alloying modifies the band structure and Fermi surface of materials
• Can be used to engineer desired electronic and optical properties
• Impurities introduce localized states within the band gap of semiconductors
• Dopants control carrier concentration and type (n-type or p-type)

Limitations and extensions

• Nearly free electron model provides valuable insights but has limitations
• More sophisticated approaches needed for strongly correlated or complex materials
• Ongoing research in condensed matter physics addresses these limitations

Beyond nearly free electron model

• Tight-binding model: Suitable for materials with more localized electrons
• Pseudopotential methods: Improve the description of electron-ion interactions
• Density functional theory: Provides accurate electronic structure calculations
• $\mathbf{k} \cdot \mathbf{p}$ theory: Describes band structure near specific points in k-space

Strongly correlated systems

• Materials where electron-electron interactions dominate (transition metal oxides)
• Exhibit phenomena not captured by single-particle models (Mott insulators, high-Tc superconductors)
• Require many-body approaches (Hubbard model, dynamical mean-field theory)
• Active area of research in condensed matter physics and materials science